Papers Repository of the International Centre for Numerical Methods in Engineering (CIMNE)

The Papers Repository of the International Centre for Numerical Methods in Engineering (CIMNE) is an online archive aimed at collecting, preserving, and disseminating digital copies of scientific papers published by researchers of the International Center for Numerical Methods in Engineering (CIMNE).

The goal of the repository is to widely disseminate the results of the research performed at CIMNE, by sharing the preprints or postprints -depending on the article sharing policies of the original ... show more

DOCUMENTS705
VIEWS5625
SCORE5

SCORE PERCENTILE100

Desarrollos y aplicaciones de modelos de fractura en la escuela de ingenieros de caminos de Barcelona
E. Oñate

Anales de Mecánica de Fractura (2000). Vol. 17, pp. 127-154

Abstract
El artículo es una panorámica de los aspectos teóricos y algunas aplicaciones prácticas de los modelos de fractura desarrollados por diversos grupos en la Escuela de Ingenieros de Caminos de Barcelona (EICB) durante los últimos quince años para el análisis no lineal de estructuras. La motivación fundamental para el desarrollo de estos modelos se centra en el análisis de la seguridad de estructuras de hormigón en masa y armado. La mayor parte de los modelos se basan en la teoría de daño continuo y utilizan el método de los elementos finitos para la solución numérica. Los modelos de daño se han extendido y aplicado también con éxito al análisis de diversas estructuras de edificios históricos. Los desarrollos más recientes de estos modelos en la EICB incluyen la predicción de fenómenos de localización en estructuras de hormigón y el análisis del comportamiento no lineal de estructuras con materiales compuestos. De todos estos modelos se presentan en el artículo unas breves pinceladas, las aplicaciones más relevantes y las referencias donde pueden encontrarse los detalles sobre cada caso.
Abstract
El artículo es una panorámica de los aspectos teóricos y algunas aplicaciones prácticas de los modelos de fractura desarrollados por diversos grupos [...]
- 0
- read
Análisis CFD del movimiento de balance de un portacontenedores
J. García-Espinosa, E. Oñate, J. Helmers, M. Chakkor

Revista de Ingeniería Naval (2004). Vol. 814, pp. 104-108

Abstract
In the present work a novel approach has been developed for the resolution of this problem of analysis of the movement of roll of a ship. The methodology used for this is based on the modification of the differential equations of the fluid dynamics (RANSE equations), including the movement of the free surface, by applying the finite calculus method. The modified equations are solved using an implicit predictor-corrector scheme and the finite element method (FEM). This resolution scheme is considered optimal for these types of problems, both in accuracy and in calculation time. As an example, it should be noted that for typical problems (more than 1,000,000 finite elements) less than 4 hours of CPU are required to solve several cycles of ship movement (Pentium IV). This allows the analysis of an array of tests in a few days or even hours, having a computer network or sufficient computing power.
Abstract
In the present work a novel approach has been developed for the resolution of this problem of analysis of the movement of roll of a ship. The methodology used for this is [...]
- 0
- read
A finite element model for fluid–structure interaction problems involving closed membranes, internal and external fluids
P. Ryzhakov, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 326, pp. 422-445

Abstract
In this paper we propose a Finite Element model for analyzing closed membranes (“bags”) interacting with internal and external (surrounding) fluids. The approach is based on embedding a Lagrangian monolithic model describing the membrane containing an internal fluid into an Eulerian external fluid model. The combination of kinematic frameworks allows us to accurately track the location of the membrane and naturally represent flow variables discontinuities across it. In order to obtain stable coupling for membrane materials with low density, a slight fluid compressibility is assumed. The coupling between the membrane and the internal fluid is automatically accounted for by a monolithic set-up. The filled membrane and the external fluid are coupled in a Dirichlet–Neumann fashion. The model is validated in several numerical examples and its potential application to a civil engineering problem of coast protection via water-filled bag reefs is shown.
Abstract
In this paper we propose a Finite Element model for analyzing closed membranes (“bags”) interacting with internal and external (surrounding) fluids. The approach [...]
- 0
- read
Cool steam method for desalinating seawater
P. Arnau, N. Navarro, J. Soraluce, J. Martínez-Iglesias, J. Illas, E. Oñate

Water (2019). Vol. 11 (11), 16 pp.

Abstract
Cool steam is an innovative distillation technology based on low-temperature thermal distillation (LTTD), which allows obtaining fresh water from non-safe water sources with substantially low energy consumption. LTTD consists of distilling at low temperatures by lowering the working pressure and making the most of low-grade heat sources (either natural or artificial) to evaporate water and then condensate it at a cooler heat sink. To perform the process, an external heat source is needed that provides the latent heat of evaporation and a temperature gradient to maintain the distillation cycle. Depending on the available temperature gradient, several stages can be implemented, leading to a multi-stage device. The cool steam device can thus be single or multi-stage, being raw water fed to every stage from the top and evaporated in contact with the warmer surface within the said stage. Acting as a heat carrier, the water vapor travels to the cooler surface and condensates in contact with it. The latent heat of condensation is then conducted through the conductive wall to the next stage. Net heat flux is then established from the heat source until the heat sink, allowing distilling water inside every parallel stage.
Abstract
Cool steam is an innovative distillation technology based on low-temperature thermal distillation (LTTD), which allows obtaining fresh water from non-safe water sources with [...]
- 0
- read
PFEM formulation for thermo-coupled FSI analysis. Application to nuclear core melt accident
A. Franci, E. Oñate, J. Carbonell, M. Chiumenti

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 325, pp. 711-732

Abstract
The aim of this paper is to present a Lagrangian formulation for thermo-coupled fluid–structure interaction (FSI) problems and to show its applicability to the simulation of hypothetical scenarios of a nuclear core melt accident. During this emergency situation, an extremely hot and radioactive lava-like material, the corium, is generated by the melting of the fuel assembly. The corium may induce collapse of the nuclear reactor devices and, in the worst case, breach the reactor containment and escape into the environment. This work shows the capabilities of the proposed formulation to reproduce the structural failure mechanisms induced by the corium that may occur during a meltdown scenario. For this purpose, a monolithic method for FSI problems, the so-called Unified formulation, is here enhanced in order to account for the thermal field and to model phase change phenomena with the Particle Finite Element Method (PFEM). Several numerical examples are presented. First, the convergence of the thermo-coupled method and phase change algorithm is shown for two academic problems. Then, two complex simulations of hypothetical nuclear meltdown situations are studied in 2D as in 3D.
Abstract
The aim of this paper is to present a Lagrangian formulation for thermo-coupled fluid–structure interaction (FSI) problems and to show its applicability [...]
- 0
- read
A FIC-based stabilized finite element formulation for turbulent flows
J. Cotela-Dalmau, R. Rossi, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2017). (preprint) Vol. 315, pp. 607-631

Abstract
We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework (Oñate, 1998). In comparison to existing FIC approaches for fluids, this formulation involves a new term in the momentum equation, which introduces non-isotropic dissipation in the direction of velocity gradients. We also follow a new approach to the derivation of the stabilized mass equation, inspired by recent developments for quasi-incompressible flows (Oñate et al., 2014). The presented FIC-FEM formulation is used to simulate turbulent flows, using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation.
Abstract
We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework (Oñate, [...]
- 0
- read
A shallow water model by finite point method
C. Bouchart, W. Kanok-Nukulschai, E. Ortega, E. Oñate

International Journal of Computational Methods (2014). Vol. 11 (1), 27 pp., Paper N: 1350047

Abstract
In this paper, "finite point method" (FPM) is presented for modeling 2D shallow water flow problem. The method is based on the use of a weighted least-square approximation procedure, incorporating QR factorization and an iterative adjustment of local approximation parameters. The stabilization of the convective term in this present work is derived from the approximate Riemann solver proposed by Roe. The present method is shown to produce competitive accuracy in the comparisons with the analytical solutions and the well-known Galerkin characteristic-based split (CBS) algorithm.
Abstract
In this paper, "finite point method" (FPM) is presented for modeling 2D shallow water flow problem. The method is based on the use of a weighted least-square approximation [...]
- 0
- read
Advances in the development of a FEM model for evaluation of a Surface-Effect Ship (SES) including skirt dynamics
J. García-Espinosa, B. Camas, D. Di Capua, P. Ubach de Fuentes, E. Oñate

Ingeniería Naval (2013). Vol. 1 (920), pp. 68-91

Abstract
El “Innovative Naval Prototype Transformable Craft” (T-Craft) es un nuevo concepto de buque desarrollado por la marina de los Estados Unidos de América para la conexión de una base en alta mar con tierra. El TCraft puede operar en múltiple modos, usándose para conectar una base intermedia de soporte con la principal, y luego ser utilizado como conexión de alta velocidad con la costa, transportando vehículos y otros equipos y carga pesada hasta la playa. El T-Craft se ha concebido como un buque de efecto superficie (Surface-Effect Ship, SES) con un colchón de aire activo entre dos cascos rígidos, que permite al buque operar a desplazamiento completo, con soporte parcial del colchón o con el colchón plenamente activo. Este artículo presenta el trabajo reciente de los autores en el desarrollo de un modelo de evaluación del comportamiento en la mar y la maniobrabilidad del T-Craft, incluyendo la interacción del faldón. Este modelo está basado en la resolución de las ecuaciones de flujo potencial en el dominio del tiempo, mediante el método de los elementos finitos, integrando la superficie libre a lo largo de las líneas de corriente. Este artículo revisa el algoritmo de interacción fluido estructura desarrollado para la simulación de la compleja dinámica de los sellos, en la interfaz entre el colchón de aire y el agua. El algoritmo se basa, por una parte, en un algoritmo explícito alternado que permite la comunicación entre los solvers fluido-dinámico y estructural mediante TCP-IP sockets, intercambiando fuerzas de presión y desplazamientos en los sellos durante la ejecución, y por otra parte, en un innovador esquema de secado-mojado que permite predecir la interacción del agua con los sellos. Se incluyen varios casos de aplicación de la metodología desarrollada sobre el modelo l XR-1B.
Abstract
El “Innovative Naval Prototype Transformable Craft” (T-Craft) es un nuevo concepto de buque desarrollado por la marina de los Estados Unidos de América [...]
- 0
- read
Analysis of the melting, burning and flame spread of polymers with the particle finite element method
E. Oñate, J. Marti, P. Ryzhakov, R. Rossi, S. Idelsohn

Computer Assisted Methods in Engineering and Sciences (2013). Vol. 20 (3), pp. 165-184

Abstract
A computational procedure for analysis of the melting, burning and flame spread of polymers under fire conditions is presented. The method, termed particle finite element method (PFEM), combines concepts from particle-based techniques with those of the standard finite element method (FEM). The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes (particles) in the thermoplastic material. Nodes are viewed as material points which can freely move and even separate from the main analysis domain representing, for instance, the effect of melting and dripping of polymer particles. A mesh connects the nodes defining the discretized domain where the governing equations are solved using the FEM. An incremental iterative scheme for the solution of the nonlinear transient coupled thermal-flow problem, including radiation, loss of mass by gasification and combustion is used. Examples of the possibilities of the PFEM for the modelling and simulation of the melting, burning and flame spread of polymers under different fire conditions are described.
Abstract
A computational procedure for analysis of the melting, burning and flame spread of polymers under fire conditions is presented. The method, termed particle finite element [...]
- 0
- read
Analysis of the discharge capacity of radial-gated spillways using CFD and ANN – Oliana Dam case study
F. Salazar, R. Morán Moya, R. Rossi, E. Oñate

Journal of Hydraulic Research (2013). (preprint) Vol. 51 (3), pp. 244-252

Abstract
The paper focuses on the analysis of radial-gated spillways, which is carried out by the solution of a numerical model based on the finite element method (FEM). The Oliana Dam is considered as a case study and the discharge capacity is predicted both by the application of a level-set-based free-surface solver and by the use of traditional empirical formulations. The results of the analysis are then used for training an artificial neural network to allow real-time predictions of the discharge in any situation of energy head and gate opening within the operation range of the reservoir. The comparison of the results obtained with the different methods shows that numerical models such as the FEM can be useful as a predictive tool for the analysis of the hydraulic performance of radial-gated spillways.
Abstract
The paper focuses on the analysis of radial-gated spillways, which is carried out by the solution of a numerical model based on the finite element method (FEM). The Oliana [...]
- 0
- read
Olgierd Cecil Zienkiewicz obituary
M. Kleiber, E. Oñate

Archives of Comp. Meths. Engng. (2009). Vol. 16, pp. 465
- 0
- read
Control of the finite element discretization error during the convergence of structural shape optimization algorithms
G. Bugeda, J. Ródenas, J. Albelda, E. Oñate

Int. J. Simul. Multidisci. Des. Optim. (2009). Vol. 3 (3), pp. 363-369

Abstract
This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by structural shape optimization algorithms over the behaviour of the algorithm. If the FE analyses are not accurate enough, the final solution will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper also proposes the use of the algorithm described in [1] in order to reduce the computational cost associated to the adaptive FE analysis of each geometrical configuration when evolutive optimization algorithms are used.
Abstract
This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by structural shape optimization [...]
- 0
- read
Advances in the particle finite element method for fluid-structure interaction problems
E. Oñate, S. Idelsohn, M. Celigueta, R. Rossi

Computational Mechanics (2006). Vol. 6, pp. 41-62

Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.
Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use [...]
- 0
- read
The rotation-free BST shell element for linearized buckling analysis of steel structures
A. Agüero, F. Pallarés, E. Oñate

Int. J. Computational Methods in Engineering Science and Mechanics (2006). Vol. 7 (3), pp. 209-215

Abstract
In this paper the ability of the basic shell triangular (BST) element to perform linearized buckling analysis is evaluated. The results have been compared with analytical solutions and other finite elements in the literature, such as ANDES3, ANDES4, QSEL, FFQC and BCIZ. This type of analysis is applied to the design of steel structures to obtain the collapse load of panels using the effective width of the plate. In this approach, which is currently recommended by most of the design standards, a semi-empirical/analytical method is used to take into account the nonlinear geometric and material behavior, geometric imperfections and residual stresses. To obtain the critical loads two methods have been mainly applied: the finite strip method and the finite element method, which is more appropriate to deal with any boundary condition and loading pattern. In practice, to obtain the local and distortional buckling, some assumptions related to the interaction between plates are made in order to obtain practical formulae that can be applied; these simplifications can lead to unsafe results, so the linearized buckling analysis must be carried out in a proper way, taking into account the interactions. It is concluded that the BST element presents an excellent behavior to predict the critical loads in compression and shear, and therefore this element must be utilized in future codes when this type of analysis turn out to be mandatory.
Abstract
In this paper the ability of the basic shell triangular (BST) element to perform linearized buckling analysis is evaluated. The results have been compared with analytical [...]
- 0
- read
Simulación de problemas de desgaste, en la interacción herramienta de corte terreno, empleando el método de los elementos discretos
C. Recarey Morfa, E. Oñate, J. Canet, J. Rojek, F. Zarate, S. Burrel

Ingeniería Civil (2006). Vol. 141, pp. 19-34

Abstract
Se presenta un modelo numérico que emplea elementos discretos esféricos o también denominados elementos distintos. Este modelo se aplica en la simulación de problemas de desgaste. El movimiento de elementos esféricos se describe por medio de las ecuaciones de Newton-Euler. Se emplea en la formulación una integración explícita, la cual, brinda una buena eficiencia computacional. Los elementos esféricos interactúan recíprocamente entre sí a través de las fuerzas de contacto. El esquema de búsqueda de contacto es muy eficaz, y está basado en las estructuras de oct-tree. El modelo constitutivo de contacto presenta una singularidad muy especial, ya que puede simular las fuerzas de cohesión que permiten modelar la fractura y descohesión de los materiales. El estudio de desgastes se realiza a partir de una formulación termoacoplada que se aplica a nivel experimental, o escala reducida, y además in-situ o escala real.
Abstract
Se presenta un modelo numérico que emplea elementos discretos esféricos o también denominados elementos distintos. Este modelo se aplica en la simulación [...]
- 0
- read
Posibilidades de los métodos numéricos en el mundo industrial
E. Oñate

Col·lecció Aula de Ciència i Cultura (2002). Nº 13, pp. 9-33

Abstract
Se presentan en el artículo las ideas básicas de qué son los métodos numéricos, cuáles son los métodos numéricos más populares, cómo se aplican para resolver ecuaciones diferenciales de interés práctico en ingeniería y qué posibilidades y limitaciones tienen para ayudarnos a entender mejor el mundo que nos rodea. El contenido del artículo se completa con diversas aplicaciones prácticas del método de elementos finitos a diversos problemas de ingeniería.
Abstract
Se presentan en el artículo las ideas básicas de qué son los métodos numéricos, cuáles son los métodos numéricos más [...]
- 0
- read
A large strain plasticity model for anisotropic materials - composite material application
E. Car, S. Oller, E. Oñate

International Journal of Plasticity (2001). Vol. 17 (11), pp. 1437-1463

Abstract
In this work a generalized anisotropic model in large strains based on the classical isotropic plasticity theory is presented. The anisotropic theory is based on the concept of mapped tensors from the anisotropic real space to the isotropic fictitious one. In classical orthotropy theories it is necessary to use a special constitutive law for each material. The proposed theory is a generalization of classical theories and allows the use of models and algorithms developed for isotropic materials. It is based on establishing a one-to-one relationship between the behavior of an anisotropic real material and that of an isotropic fictitious one. Therefore, the problem is solved in the isotropic fictious space and the results are transported to the real field. This theory is applied to simulate the behavior of each material in the composite. The whole behavior of the composite is modeled by incorporating the anisotropic model within a model based on a modified mixing theory.
Abstract
In this work a generalized anisotropic model in large strains based on the classical isotropic plasticity theory is presented. The anisotropic theory is based on the concept [...]
- 0
- read
Optimización de embarcaciones de recreo mediante la utilización de un código CFD
M. López-Rodríguez, J. García-Espinosa, E. Oñate

Ingeniería Naval (2001). Vol. 784, pp. 122-130

Abstract
En la actualidad las más avanzadas técnicas para diseño naval ya no se restringen sólo a proyectos de alto coste; si no que también ahora son de aplicación en el campo de las embarcaciones de recreo y turísticas, dando con ello un valor añadido al proyecto final. La utilización de un código CFD para el mayor conocimiento del comportamiento hidrodinámico de las embarcaciones de recreo y turísticas es de una gran utilidad debido a las complejidades que en el diseño de estas se presentan. Estas herramientas nos van a proporcionar una serie de datos como sustentación lateral en las embarcaciones de vela o interacción entre cascos en catamaranes con deriva lateral, que de otra forma nunca pudieran conocerse. Los resultados hidrodinámicos obtenidos nos sirven para comparar las diferentes alternativas, y para además obtener unos valores que directamente son usados para la predicción de velocidad de la embarcación, que más tarde se verán refrendadas por las pruebas de mar. La utilización de técnicas CFD, basadas en el método de los elementos finitos, para la resolución de las ecuaciones de Reynolds (RANSE), incluyendo efectos de superficie libre, hace que todo esto sea posible. En este trabajo se presentan diversos resultados de aplicación del código CFD SHYNE a varias embarcaciones deportivas y de recreo. SHYNE está basado en la técnica del Cálculo Finitesimal para la integración de las ecuaciones de Reynolds.
Abstract
En la actualidad las más avanzadas técnicas para diseño naval ya no se restringen sólo a proyectos de alto coste; si no que también ahora [...]
- 0
- read
ED-Elas2D: An Educational Program for Computer-aided Training in Structural Analysis Using the Finite Element Method
B. Suarez, E. Blanco, L. Gil, P. Zapata, E. Oñate

European Journal of Engineering Education (1998). Vol. 23 (2), pp. 243-254

Abstract
In this paper, a methodology for computer-aided training of structural problems by the finite element method is presented. The resulting educational program, named ED-Elas2D, has been developed by a group of researchers at the School of Civil Engineering at the Technical University of Catalonia (UPC) in Barcelona, UPC. The program combines the standard concept which appears in the traditional teaching of finite element methods for structural analysis, such as the introduction to the basic structural theory, data definition (pre-processor), description of the analysis procedure (processor) and interpretation of results (post-processor). The program is addressed to both students who enrol in the study on finite element methods and to academic and technical staff in universities and companies who wish to update their knowledge of finite element techniques. The educational program ED-Elas2D works in the Windows system and it is characterized by its interactivity and graphic utilities.
Abstract
In this paper, a methodology for computer-aided training of structural problems by the finite element method is presented. The resulting educational program, named ED-Elas2D, [...]
- 0
- read
A finite element model for thermomechanical analysis in casting processes
D. Celentano, S. Oller, E. Oñate

Journal of Physique IV (1993). Vol. 3 (C7), pp. 1171-1180

Abstract
This paper summarizes the recent work of the authors in the numerical simulation of casting processes. In particular, a coupled thermomechanical model to simulate the solidification problem in casting has been developed [7,8,9]. The model, based on a general isotropic thermoelasto-plasticity theory and formulated in a macroscopical point of view, includes generalized phase-change effects and considers the different thermomechanical behaviour of the solidifying material during its evolution from liquid to solid. For this purpose, a phase-change variable, plastic evolution equations and a temperature-dependent material constitutive law have been defined. Some relevant aspects of this model are presented here. Full thermomechanical coupling terms have been considered as well as variable thermal and mechanical boundary conditions : the first are due to air gap formation, while the second involve a contact formulation. Particular details concerning the numerical implementation of this model are also mentioned. An enhanced staggered scheme, used to solve the highly non-linear fully coupled finite element equations, is proposed. Furthermore, a proper convergence criterion to stop the iteration process is adopted and, although the quadratic convergence of Newton-Rapshon's method is not achieved, several numerical experiments demonstrate reasonable convergence rates [9]. Finally, an experimental cylindrical casting test problem, including phase-change phenomena, temperature-dependent constitutive properties and contact effects, is analyzed. Numerical results are compared with some laboratory measurements.
Abstract
This paper summarizes the recent work of the authors in the numerical simulation of casting processes. In particular, a coupled thermomechanical model to simulate the solidification [...]
- 0
- read
Análisis no lineal de tanques criogénicos bajo cargas térmicas
J. Oliver, E. Oñate, J. Peraire, R. Chueca

Hormigón y Acero (1985). Vol. 155, pp. 99-123
- 0
- read
La formulación del flujo viscoplástico y sus diversas aplicaciones prácticas por el método de los elementos finitos
E. Oñate

Revista de Obras Públicas (1980). Vol. 127 (3178), pp. 115-129
- 0
- read
Diseño óptimo de vasijas de presión usando el método de los elementos finitos y un optimizador no lineal
J. Middleton, E. Oñate

ETSIIT (1977). Vol. 11, pp. 309-325

Abstract
Se presenta un esquema de diseño para vasijas de presión metálicas de manera que se minimice la tensión tangencial en la vasija . La geometría final se establece mediante un procedimiento iterativo que optimiza un modelo matemático que expresa la variación de la tensión tangencial en la vasija. Los parámetros del modelo se calculan para cada interacción a partir de los valores de las tensiones tangenciales en la vasija, determinados por el método de los elementos finitos.
Abstract
Se presenta un esquema de diseño para vasijas de presión metálicas de manera que se minimice la tensión tangencial en la vasija . La geometría [...]
- 0
- read
A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties
H. Garikapati, S. Zlotnik, C. Verhoosel, E. Brummelen, P. Diez

Computational Mechanics (2019). (Preprint) pp. 1-23

Abstract
Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a postprocessing operation on the explicit solution provided by PGD.We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.
Abstract
Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random [...]
- 0
- read
A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates
N. Parés, P. Diez

Comput. Methods Appl. Mech. Engrg., (2019). (Preprint) Vol. 120, pp. 391-432

Abstract
The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations of the reaction‐diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bound property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise‐linear. The new approach is an extension of the flux‐free methodology introduced by Parés and Díez in the paper “A new equilibrated residual method improving accuracy and efficiency of flux‐free error estimates”, which introduces a guaranteed, low‐cost, and efficient flux‐free method substantially reducing the computational cost of obtaining guaranteed bounds using flux‐free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three‐dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions, which improves the quality of standard techniques while having a similar computational cost.
Abstract
The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations [...]
- 0
- read
Sampling-based stochastic analysis of the PKN model for hydraulic fracturing
H. Garikapati, C. Verhoosel, E. Brummelen, S. Zlotnik, P. Diez

Computational Geosciences (2019). (Preprint) Vol. 23 (1), pp. 81-105

Abstract
Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture.
Abstract
Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic [...]
- 0
- read
Algebraic and parametric solvers for the power flow problem: towards real-time and accuracy-guaranteed simulation of electric systems
R. García-Blanco, P. Diez, D. Borzacchiello, F. Chinesta

Archives of Comp. Meths. Engng. (2018). (Preprint) Vol. 25 (4), pp. 1003-1026

Abstract
The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers for the power flow equations, in both algebraic and parametric version. The application of the Alternating Search Direction method to the power flow problem is also detailed. This results in a family of iterative solvers that combined with Proper Generalized Decomposition technique allows to solve the parametric version of the equations. Once the solution is computed using this strategy, analyzing the network state or solving optimization problems, with inclusion of generation in real-time, becomes a straightforward procedure since the parametric solution is available. Complementing this approach, an error strategy is implemented at each step of the iterative solver. Thus, error indicators are used as an stopping criteria controlling the accuracy of the approximation during the construction process. The application of these methods to the model IEEE 57-bus network is taken as a numerical illustration.
Abstract
The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers [...]
- 0
- read
Algebraic PGD for tensor separation and compression: An algorithmic approach
P. Diez, S. Zlotnik, A. García-González, A. Huerta

Comptes Rendus Mécanique (2018). (Preprint) Vol. 346 (7), pp. 501-514

Abstract
Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated with the space of parameters defining the problem). The PGD philosophy consists in providing a separated representation of the multidimensional solution using a greedy approach combined with an alternated directions scheme to obtain the successive rank-one terms. This paper presents an algorithmic approach to high-dimensional tensor separation based on solving the Least Squares approximation in a separable format of multidimensional tensor using PGD. This strategy is usually embedded in a standard PGD code in order to compress the solution (reduce the number of terms and optimize the available storage capacity), but it stands also as an alternative and highly competitive method for tensor separation.
Abstract
Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated [...]
- 0
- read
Numerical estimation of the bearing capacity of resistance spot welds in martensitic boron steels using a J-integral fracture criterion
D. Dorribo, L. Greve, P. Diez, I. Arias, X. Larráyoz-Izcara

Theoretical and Applied Fracture Mechanics (2018). (Preprint) Vol. 96, pp. 497-508

Abstract
Predicting the bearing capacity of resistance spot welds (RSW) during vehicle crash tests has become a crucial task for the automotive industry, since the recent introduction of advanced high strength steels (AHSS) such as martensitic boron steels (e.g. 22MnB5). The spot weld joints of these steels exhibit relatively low bearing strengths, compared to those of more ductile high strength steels. Currently, the bearing capacity of spot weld joints is characterized through extensive experimental campaigns. In this article, a model for quantification of the bearing capacity of RSW using a finite-element J-integral fracture criterion is presented. The model takes into account geometric and mechanical features of the spot weld, namely the weld diameter and the mechanical properties distribution resulting from the welding process. An experimental loading test campaign is carried out for calibration and validation purposes, considering multiple sheet thickness combinations, loading angles and weld sizes. Experimental observations of the failed spot welds and preliminary simulations show that failure is caused mostly by stress concentration around the sharp weld notch. Consequently, the J-integral obtained from detailed finite element simulations is used to asses the stress/strain concentration along the first crack advance direction predicted by the acoustic tensor. The computed J-integral values are compared to the material toughness to obtain the joint’s maximum force. The resulting simulated and experimental bearing capacities show a good agreement for all tested configurations.
Abstract
Predicting the bearing capacity of resistance spot welds (RSW) during vehicle crash tests has become a crucial task for the automotive industry, since the recent introduction [...]
- 0
- read
A non-intrusive proper generalized decomposition scheme with application in biomechanics
X. Zou, M. Conti, P. Diez, F. Auricchio

Int. J. Numer. Meth. Engng. (2018). (Preprint) Vol. 113 (2), pp. 230-251

Abstract
Proper generalized decomposition (PGD) is often used for multi-query and fast-response simulations. It is a powerful tool alleviating the curse of dimensionality affecting multi-parametric partial differential equations. Most implementations of PGD are intrusive extensions based on in-house developed finite element (FE) solvers. In this work, we propose a non-intrusive PGD scheme using off-the-shelf FE codes (such as certified commercial software) as an external solver. The scheme is implemented and monitored by in-house flow-control codes. A typical implementation is provided with downloadable codes. Moreover, a novel parametric separation strategy for the PGD resolution is presented. The parametric space is split into two- or three-dimensional subspaces, to allow PGD technique solving problems with constrained parametric spaces, achieving higher convergence ratio. Numerical examples are provided. In particular, a practical example in biomechanics is included, with potential application to patient-specific simulation.
Abstract
Proper generalized decomposition (PGD) is often used for multi-query and fast-response simulations. It is a powerful tool alleviating the curse of dimensionality affecting [...]
- 0
- read
Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD)
A. Sibileau, A. García-González, F. Auricchio, S. Morganti, P. Diez

Computational Mechanics (2018). (Preprint) Vol. 62 (4), pp. 871-891

Abstract
Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical properties. One may obtain specific macro-scale (or bulk) properties in the resulting architectured material by properly designing the unit-cell. Typically, this is stated as an optimal design problem in which the parameters describing the shape and mechanical properties of the unit-cell are selected in order to produce the desired bulk characteristics. This is especially pertinent due to the ease manufacturing of these complex structures with 3D printers. The proper generalized decomposition provides explicit parametic solutions of parametric PDEs. Here, the same ideas are used to obtain parametric solutions of the algebraic equations arising from lattice structural models. Once the explicit parametric solution is available, the optimal design problem is a simple post-process. The same strategy is applied in the numerical illustrations, first to a unit-cell (and then homogenized with periodicity conditions), and in a second phase to the complete structure of a lattice material specimen.
Abstract
Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical [...]
- 0
- read
Numerical Modelling of Multi-Phase Multi-Component. Reactive Transport in the Earth’s interior
B. Oliveira, J. Afonso, S. Zlotnik, P. Diez

Geophysical Journal International (2018). (Preprint) Vol. 212 (1), pp. 345-388

Abstract
We present a conceptual and numerical approach to model processes in the Earth’s interior that involve multiple phases that simultaneously interact thermally, mechanically and chemically. The approach is truly multiphase in the sense that each dynamic phase is explicitly modelled with an individual set of mass, momentum, energy and chemical mass balance equations coupled via interfacial interaction terms. It is also truly multicomponent in the sense that the compositions of the system and its constituent phases are expressed by a full set of fundamental chemical components (e.g. SiO2, Al2O3, MgO, etc.) rather than proxies. These chemical components evolve, react with and partition into different phases according to an internally consistent thermodynamic model. We combine concepts from Ensemble Averaging and Classical Irreversible Thermodynamics to obtain sets of macroscopic balance equations that describe the evolution of systems governed by multiphase multicomponent reactive transport (MPMCRT). Equilibrium mineral assemblages, their compositions and physical properties, and closure relations for the balance equations are obtained via a ‘dynamic’ Gibbs free-energy minimization procedure (i.e. minimizations are performed on-the-fly as needed by the simulation). Surface tension and surface energy contributions to the dynamics and energetics of the system are taken into account. We show how complex rheologies, that is, visco-elasto-plastic, and/or different interfacial models can be incorporated into our MPMCRT ensemble-averaged formulation. The resulting model provides a reliable platform to study the dynamics and nonlinear feedbacks of MPMCRT systems of different nature and scales, as well as to make realistic comparisons with both geophysical and geochemical data sets. Several numerical examples are presented to illustrate the benefits and limitations of the model.
Abstract
We present a conceptual and numerical approach to model processes in the Earth’s interior that involve multiple phases that simultaneously interact thermally, mechanically [...]
- 0
- read
Generalized parametric solutions in Stokes flow
P. Diez, S. Zlotnik, A. Huerta

Comput. Methods Appl. Mech. Engrg. (2017). (Preprint) Vol. 326, pp. 223-240

Abstract
Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper Generalized Decomposition (PGD) provides a separable solution, a computational vademecum explicitly dependent on the parameters, efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored. This strategy has been successfully employed in many problems in computational mechanics. The application to problems with saddle point structure raises some difficulties requiring further attention. This article proposes a PGD formulation of the Stokes problem. Various possibilities of the separated forms of the PGD solutions are discussed and analyzed, selecting the more viable option. The efficacy of the proposed methodology is demonstrated in numerical examples for both Stokes and Brinkman models.
Abstract
Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper [...]
- 0
- read
A semi‐analytical scheme for highly oscillatory integrals over tetrahedra
R. Hospital-Bravo, J. Sarrate, P. Diez

Comput. Methods Appl. Mech. Engrg., (2017). (Preprint) Vol. 111 (8), pp. 703-723

Abstract
This paper details a semi‐analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the partition of unity method enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non‐oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared with standard Gauss–Legendre quadratures.
Abstract
This paper details a semi‐analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly [...]
- 0
- read
Monitoring a PGD solver for parametric power flow problems with goal-oriented error assessment
R. García-Blanco, D. Borzacchiello, F. Chinesta, P. Diez

Int. J. Numer. Meth. Engng. (2017). Vol. 111 (6), pp. 529-552

Abstract
The parametric analysis of electric grids requires carrying out a large number of Power Flow computations. The different parameters describe loading conditions and grid properties. In this framework, the Proper Generalized Decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to 1) iterating algebraic solver, 2) number of terms in the separable greedy expansion and 3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal-oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end- user. The paper discusses how to compute the goal-oriented error estimates. This requires linearizing the error equation and the Quantity of Interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems.
Abstract
The parametric analysis of electric grids requires carrying out a large number of Power Flow computations. The different parameters describe loading conditions and grid properties. [...]
- 0
- read
Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems
K. Serafin, B. Magnain, E. Florentin, N. Parés, P. Diez

Int. J. Numer. Meth. Engng. (2017). (Preprint) Vol. 110 (5), pp. 440-466

Abstract
This work focuses on providing accurate low-cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In [E. Florentin, P. Diez, Adaptive reduced basis strategy based on goal oriented error assessment for stochastic problems, Comput. Methods Appl. Mech. Engrg. 225-228 (2012) 116-127], an adaptive strategy has been introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU cost gain. Technical insight of these two topics are presented in details and numerical examples show the interest of these new developments.
Abstract
This work focuses on providing accurate low-cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In [E. Florentin, P. Diez, [...]
- 0
- read
Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems
M. Signorini, S. Zlotnik, P. Diez

Int. J. Numer. Meth. Engng. (2017). (Preprint) Vol. 109, (8), pp. 1085-1102

Abstract
The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples.
Abstract
The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are [...]
- 0
- read
A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates
N. Parés, P. Diez

Comput. Methods Appl. Mech. Engrg., (2017). (Preprint) Vol. 313 (1), pp. 785-816

Abstract
This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction-diffusion equation. The new approach revisits the ideas in [20, 18] with the goal of substantially reducing the com- putational cost of the flux-free method while retaining the good quality of the bounds. The new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies. The zeroth-order equilibration conditions are imposed using an alternative less restrictive form of the first-order equilibration conditions, along with a new efficient minimization criterion. This new equilibration strategy provides much more accurate up- per bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.
Abstract
This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction-diffusion [...]
- 0
- read
A reduced order modeling approach for optimal allocation of Distributed Generation in power distribution systems
R. García-Blanco, P. Diez, D. Borzacchiello, F. Chinesta

International Energy Conference (ENERGYCON) (2016). (preprint) pp, 1-6

Abstract
This paper presents an “offline-online” strategy for optimal allocation and sizing of Distributed Generation. In traditional optimization approaches, each function evaluation requires the solution of a power flow problem, which makes global optimality a computationally challenging goal. In the proposed strategy the power flow solver is invoked only once and a parametric solution is constructed with a monolithic solver. Despite the fact that the parametrized power flow equations result in a high-dimensional problem, the proposed algorithm is specifically designed to circumvent the curse of dimensionality. This is achieved through the application of Model Reduction, in particular the Proper Generalized Decomposition combined with a nonlinear solver. Numerical examples are carried out for showing the validity of the proposed method.
Abstract
This paper presents an “offline-online” strategy for optimal allocation and sizing of Distributed Generation. In traditional optimization approaches, each function [...]
- 0
- read
Unified formulation of a family of iterative solvers for power system analysis
F. Chinesta, P. Diez, D. Borzacchiello, M. Malik, R. García-Blanco

Electric Power Systems Research (2016). (Preprint) Vol. 140, pp. 201-208

Abstract
This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is fit to the particular algebraic structure of the power flow problem resulting from the combination of a globally linear set of equations and nonlinear local relations imposed by power conversion devices, such as loads and generators. The choice of the search directions is shown to be crucial for improving the overall robustness of the solver. A noteworthy advantage is that constant search directions yield stationary methods that, in contrast with Newton or Quasi-Newton methods, do not require the evaluation of the Jacobian matrix. Such directions can be elected to enforce the convergence to the high voltage operative solution. The method is explained through an intuitive example illustrating how the proposed generalized formulation is able to include other nonlinear solvers that are classically used for power flow analysis, thus offering a unified view on the topic. Numerical experiments are performed on publicly available benchmarks for large distribution and transmission systems.
Abstract
This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is [...]
- 0
- read
Numerical modeling of undersea acoustics using a partition of unity method with plane waves enrichment
R. Hospital-Bravo, J. Sarrate, P. Diez

Computational Mechanics (2016). (Preprint) Vol. 57 (5), pp. 717-732

Abstract
A new 2D numerical model to predict the underwater acoustic propagation is obtained by exploring the potential of the Partition of Unity Method (PUM) enriched with plane waves. The aim of the work is to obtain sound pressure level distributions when multiple operational noise sources are present, in order to assess the acoustic impact over the marine fauna. The model takes advantage of the suitability of the PUM for solving the Helmholtz equation, especially for the practical case of large domains and medium frequencies. The seawater acoustic absorption and the acoustic reflectance of the sea surface and sea bottom are explicitly considered, and perfectly matched layers (PML) are placed at the lateral artificial boundaries to avoid spurious reflexions. The model includes semi-analytical integration rules which are adapted to highly oscillatory integrands with the aim of reducing the computational cost of the integration step. In addition, we develop a novel strategy to mitigate the ill-conditioning of the elemental and global system matrices. Specifically, we compute a low-rank approximation of the local space of solutions, which in turn reduces the number of degrees of freedom, the CPU time and the memory footprint. Numerical examples are presented to illustrate the capabilities of the model and to assess its accuracy.
Abstract
A new 2D numerical model to predict the underwater acoustic propagation is obtained by exploring the potential of the Partition of Unity Method (PUM) enriched with plane waves. [...]
- 0
- read
Adaptividade e estimativas de erro orientadas por metas aplicadas a um benchmark test de propagação de onda
L. Steffens, P. Diez, M. Alves

Tendências em Matemática Aplicada e Computacional (2016). (preprint) Vol. 17 (1), pp. 35-53

Abstract
O objetivo deste artigo é estudar a eficiência e a robustez de técnicas adaptativas e estimativas de erro orientadas por metas para um benchmark test. As técnicas utilizadas aqui são baseadas em um simples pós-processo das aproximações de elementos finitos. As estimativas de erro orientadas por metas são obtidas por analisar o problema direto e um problema auxiliar, o qual está relacionado com a quantidade de interesse específico. O procedimento proposto é válido para quantidades lineares e não-lineares. Além disso, são discutidas diferentes representações para o erro e é analisada a influência do erro de dispersão. Os resultados numéricos mostram que as estimativas de erro fornecem boas aproximações ao erro real e que a técnica de refino adaptativo proposta conduz a uma redução mais rápida do erro.
Abstract
O objetivo deste artigo é estudar a eficiência e a robustez de técnicas adaptativas e estimativas de erro orientadas por metas para um benchmark test. [...]
- 0
- read
An error estimator for real-time simulators based on model order reduction
I. Alfaro, D. Gonzalez, S. Zlotnik, P. Diez, E. Cueto, F. Chinesta

Adv. Model. and Simul. in Eng. Sci. (2015). Vol. 2 (1), pp. 1-30

Abstract
Model order reduction (MOR) is one of the most appealing choices for real-time simulation of non-linear solids. In this work a method is presented in which real time performance is achieved by means of the o-line solution of a (high dimensional) parametric problem that provides a sort of response surface or computational vademecum. This solution is then evaluated in real-time at feedback rates compatible with haptic devices, for instance (i.e., more than 1kHz). This high dimensional problem can be solved without the limitations imposed by the curse of dimensionality by employing Proper Generalized Decomposition (PGD) methods. Essentially, PGD assumes a separated representation for the essential eld of the problem. Here, an error estimator is proposed for this type of solutions that takes into account the non-linear character of the studied problems. This error estimator allows to compute the necessary number of modes employed to obtain an approximation to the solution within a prescribed error tolerance in a given quantity of interest.
Abstract
Model order reduction (MOR) is one of the most appealing choices for real-time simulation of non-linear solids. In this work a method is presented in which real time performance [...]
- 0
- read
Effect of the separated approximation of input data in the accuracy of the resulting PGD solution
S. Zlotnik, P. Diez, D. Gonzalez, E. Cueto, A. Huerta

Adv. Model. and Simul. in Eng. Sci. (2015). Vol. 2, art. 28

Abstract
The Proper Generalized Decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation…) and control the nal accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice or the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.
Abstract
The Proper Generalized Decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases [...]
- 0
- read
Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition
E. Nadal Soriano, F. Chinesta, P. Diez, F. Fuenmayor, F. Denia

Comput. Methods Appl. Mech. Engrg., (2015). (preprint) Vol. 296 (1), pp. 113-128

Abstract
Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process requires obtaining a solution of the numerical model at the characteristic frequency of the process (real-time). This paper introduces a computational strategy allowing to efficiently exploit measurements of those industrial processes, providing the solution of the model at the required frequency. This is particularly interesting in the framework of control algorithms that rely on a model involving a set of parameters. For instance, the curing process of a composite material is modelled as a thermo-mechanical problem whose corresponding parameters describe the thermal and mechanical behaviours. In this context, the information available (measurements) is used to update the parameters of the model and to produce new values of the control variables (data assimilation). The methodology presented here is devised to ensure the possibility of having a response in real-time of the problem and therefore the capability of integrating it in the control scheme. The Proper Generalized Decomposition is used to describe the solution in the multi-parametric space. The real-time data assimilation requires a further simplification of the solution representation that better fits the data (reconstructed solution) and it provides an implicit parameter identification. Moreover, the analysis of the assimilated data sensibility with respect to the points where the measurements are taken suggests a criterion to locate the sensors.
Abstract
Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process [...]
- 0
- read
An efficient and general approach for implementing thermodynamic phase equilibria information in geophysical and geodynamic studies
J. Alfonso, S. Zlotnik, P. Diez

Geochmistry, geophysics, geosystems (2015). (preprint) Vol. 16 (10), pp. 3767-3777

Abstract
We present a flexible, general, and efficient approach for implementing thermodynamic phase equilibria information (in the form of sets of physical parameters) into geophysical and geodynamic studies. The approach is based on Tensor Rank Decomposition methods, which transform the original multidimensional discrete information into a separated representation that contains significantly fewer terms, thus drastically reducing the amount of information to be stored in memory during a numerical simulation or geophysical inversion. Accordingly, the amount and resolution of the thermodynamic information that can be used in a simulation or inversion increases substantially. In addition, the method is independent of the actual software used to obtain the primary thermodynamic information, and therefore, it can be used in conjunction with any thermodynamic modeling program and/or database. Also, the errors associated with the decomposition procedure are readily controlled by the user, depending on her/his actual needs (e.g., preliminary runs versus full resolution runs). We illustrate the benefits, generality, and applicability of our approach with several examples of practical interest for both geodynamic modeling and geophysical inversion/modeling. Our results demonstrate that the proposed method is a competitive and attractive candidate for implementing thermodynamic constraints into a broad range of geophysical and geodynamic studies. MATLAB implementations of the method and examples are provided as supporting information and can be downloaded from the journal's website.
Abstract
We present a flexible, general, and efficient approach for implementing thermodynamic phase equilibria information (in the form of sets of physical parameters) into geophysical [...]
- 0
- read
Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields
R. Cottereau, P. Diez

Computational Mechanics (2015). (preprint) Vol. 56 (4), pp. 601-612

Abstract
We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation methods. This approach tailors a specific mesh for each sample of the problem. It only requires the computation of the solution of a single deterministic problem with the same geometry and the average parameter, whose numerical cost becomes marginal for large number of samples. Starting from the mesh used to solve that deterministic problem, the nodes are moved depending on the particular sample of mechanical parameter field. The reduction in the error is small for each sample but sums up to reduce the overall bias on the statistics estimated through the Monte Carlo scheme. Several numerical examples in 2D are presented.
Abstract
We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation [...]
- 0
- read
Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications
S. Zlotnik, P. Diez, D. Modesto, A. Huerta

Int. J. Numer. Meth. Engng. (2015). (preprint) Vol. 103 (10), pp. 737-758

Abstract
The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The proper generalized decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., “Parametric solutions involving geometry: A step towards efficient shape optimization.” Comput. Methods Appl. Mech. Eng., 2014; 268:178–193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10‐dimensional space, but the PGD solution provides a fairly accurate approximation (error ≤1%) using less that 150 terms in the PGD expansion.
Abstract
The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the [...]
- 0
- read
A recovery-explicit error estimator in energy norm for linear elasticity
E. Nadal Soriano, P. Diez, J. Ródenas, M. Tur, F. Fuenmayor

Comput. Methods Appl. Mech. Engrg., (2015). (preprint) Vol. 287, pp. 172-190

Abstract
Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recovery-based estimates is proposed. The idea is (1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, (2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and (3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications.
Abstract
Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the [...]
- 0
- read
An error estimator for separated representations of highly multidimensional models
A. Ammar, F. Chinesta, P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (25-28), pp. 1872-1880

Abstract
Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, Fokker-Planck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.
Abstract
Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most [...]
- 0
- read
Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution
F. Verdugo, N. Parés, P. Diez

Computational Mechanics (2014). Vol. 54 (2), pp. 331-352

Abstract
This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed solution has an error in the quantity of interest below a user-defined tolerance. The methodology is based on a goal-oriented error estimate that requires accounting for an auxiliary adjoint problem. The major novelty of this paper is using modal analysis to obtain a proper approximation of the adjoint solution. The idea of using a modal-based description was introduced in a previous work for error estimation purposes. Here this approach is used for the first time in the context of adaptivity. With respect to the standard direct time-integration methods, the modal solution of the adjoint problem is highly competitive in terms of computational effort and memory requirements. The performance of the proposed strategy is tested in two numerical examples. The two examples are selected to be representative of different wave propagation phenomena, one being a 2D bulky continuum and the second a 2D domain representing a structural frame.
Abstract
This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed [...]
- 0
- read
Error assessment in structural transient dynamics
F. Verdugo, N. Parés, P. Diez

Archives of Comp. Meths. Engng. (2014). Vol. 21 (1), pp. 59-90

Abstract
This paper presents in a unified framework the most representative state-of-the-art techniques on a posteriori error assessment for second order hyperbolic problems, i.e., structural transient dynamics. For the sake of presentation, the error estimates are grouped in four types: recovery-based estimates, the dual weighted residual method, the constitutive relation error method and error estimates for timeline-dependent quantities of interest. All these methodologies give a comprehensive overview on the available error assessment techniques in structural dynamics, both for energy-like and goal-oriented estimates.
Abstract
This paper presents in a unified framework the most representative state-of-the-art techniques on a posteriori error assessment for second order hyperbolic problems, i.e., [...]
- 0
- read
Modal based goal-oriented error assessment for timeline-dependent quantities in transient dynamics
F. Verdugo, N. Parés, P. Diez

Int. J. Numer. Meth. Engng. (2013). (preprint) Vol. 95 (8), pp. 685-720

Abstract
This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline-dependent quantities) is proposed. These quantities are scalar time-dependent outputs of the transient solution which are better suited to time-dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline-dependent quantities of interest. The key ingredient is the modal-based approximation of the associated adjoint problems which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post-processed to produce an enhanced solution, requiring only one spatial post-process for each vibration mode and using the time-harmonic hypothesis to recover the time dependence. Thus the proposed goal-oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low.
Abstract
This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities [...]
- 0
- read
A stable XFEM formulation for multi-phase problems enforcing the accuracy of the fluxes through Lagrange multipliers
P. Diez, R. Cottereau, S. Zlotnik

Int. J. Numer. Meth. Engng. (2013). (Preprint) Vol. 96 (5), pp. 303-322

Abstract
When applied to diffusion problems in a multi‐phase setup, the popular extended finite element method (XFEM) strategy suffers from an inaccurate representation of the local fluxes in the vicinity of the interface. The XFEM enrichment improves the global quality of the solution but it is not enforcing any local feature to the fluxes. Thus, the resulting numerical fluxes in the vicinity of the interface are not realistic, in particular when conductivity ratios between the different phases are very high. This paper introduces an additional restriction to the XFEM formulation aiming at properly reproducing the features of the local fluxes in the transition zone. This restriction is implemented through Lagrange multipliers, and the stability of the resulting mixed formulation is tested satisfactorily through the Chapelle–Bathe numerical procedure. Several examples are presented, and the solutions obtained show a spectacular improvement with respect to the standard XFEM.
Abstract
When applied to diffusion problems in a multi‐phase setup, the popular extended finite element method (XFEM) strategy suffers from an inaccurate representation of the local [...]
- 0
- read
Meshless methods in dual analysis: theoretical and implementation issues
V. Ivannikov, C. Tiago, J. Almeida, P. Díez

Engineering Analysis with Boundary Elements (2013). Vol. 37 (12), pp. 1728-1744

Abstract
This paper presents a meshless implementation of dual analysis for 2D linear elasticity problems. The derivation of the governing systems of equations for the discretized compatible and equilibrated models is detailed and crucial implementation issues of the proposed algorithm are discussed: (i) arising of deficiencies associated with the independent approximation field used for the imposition of the essential boundary conditions (EBC) for the two parts of the boundary sharing a corner and (ii) determination of the Lagrange multipliers functional space used to impose EBC. An attempt to implement the latter resulted in an approximation which is nothing more than the trace on the essential boundary of the domain nodal functions. The difficulties posed by such approximation are explained using the inf-sup condition. Several examples of global (energy) and local (displacements) quantities of interest and theirs bounds determination are used to demonstrate the validity of the presented meshless approach to dual analysis. Numerical assessment of the convergence rates obtained for both models is made, for different polynomial basis degree.
Abstract
This paper presents a meshless implementation of dual analysis for 2D linear elasticity problems. The derivation of the governing systems of equations for the discretized [...]
- 0
- read
An algorithm for mesh refinement and un-refinement in fast transient dynamics
L. Casadei, P. Diez, F. Verdugo

Int. J. Computational Meths. (2013). (preprint) Vol. 10 (4), pp. 1-31

Abstract
A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen refinement strategy generates only elements of the same type as their parents, but also produces so-called hanging nodes along non-conforming element-to-element interfaces. Continuity of the solution across such interfaces is enforced strongly by Lagrange multipliers. The element split and un-split algorithm is entirely integer-based. It relies only upon element connectivity and makes no use of nodal coordinates or other real-number quantities. The chosen data structure and the continuous tracking of the nature of each node facilitate the treatment of natural and essential boundary conditions in adaptivity. A generalization of the concept of neighbor elements allows transport calculations in adaptive fluid calculations. The proposed procedure is tested in structure and fluid wave propagation problems in explicit transient dynamics.
Abstract
A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen [...]
- 0
- read
SUPG-based stabilization of Proper Generalized Decompositions for the high-dimensional Advection-Diffusion Equations
D. Gonzalez, E. Cueto, F. Chinesta, P. Diez, A. Huerta

Int. J. Numer. Meth. Engng. (2013). Vol. 94 (13), pp. 1216-1232

Abstract
This work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing multidimensional models, the use of mesh-based discretization fails because the exponential increase of the number of degrees of freedom related to a multidimensional mesh or grid, and alternative discretization strategies are needed. Separated representations involved in the so-called proper generalized decomposition method are an efficient alternative as proven in our former works; however, the issue related to efficient stabilizations of multidimensional advection–diffusion equations has never been addressed to our knowledge. Thus, this work is aimed at extending some well-experienced stabilization strategies widely used in the solution of 1D, 2D, or 3D advection–diffusion models to models defined in high-dimensional spaces, sometimes involving tens of coordinates.
Abstract
This work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing [...]
- 0
- read
Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation
N. Parés, P. Diez, A. Huerta

Int. J. Numer. Meth. Engng. (2013). Vol. 93 (5), pp. 483-509

Abstract
The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.
Abstract
The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. [...]
- 0
- read
Enforcing interface flux continuity in enhanced XFEM: stability analysis
P. Diez, S. Zlotnik, R. Cottereau

Oberwolfach Reports (2012). Vol. 9 (1), pp. 523-526

Abstract
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications.
Abstract
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical [...]
- 0
- read
Computable bounds of functional outputs in linear visco-elastodynamics
F. Verdugo, P. Diez

Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 245-246, pp. 313-330

Abstract
This work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco-elastodynamics. A novel expression for the error representation is introduced, alternative to the previous ones using the Cauchy–Schwarz inequality. The proposed formulation utilizes symmetrized forms of the error equations to derive error bounds in terms of energy error measures. The practical implementation of the method is based on constructing admissible fields for both the original problem and the adjoint problem associated with the quantity of interest. Here, the flux-free technique is considered to compute the admissible stress fields. The proposed methodology yields estimates with better quality than the ones based on the Cauchy–Schwarz inequality. In the studied examples the bound gaps obtained are approximately halved, that is the estimated intervals of confidence are reduced
Abstract
This work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco-elastodynamics. A novel expression for the error representation [...]
- 0
- read
Adaptive reduced basis strategy based on goal oriented error assessment for stochastic problems
E. Florentin, P. Diez

Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 225, pp. 116-127

Abstract
In the framework of stochastic non-intrusive finite element modeling, a common practice is using Monte Carlo simulation. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. The different Monte Carlo samplings correspond to realizations of the random variables characterizing the stochastic behavior of the model. Thus, this requires solving a set deterministic problems with the same structure, that is with variations concerning the material parameters and the loading data. Consequently, the different problems to be solved are in practice similar to each other. The reduced basis strategy is therefore a sensible option to reduce computational cost, provided that the quality of the numerical solution is guaranteed. The paper introduces a goal-oriented strategy allowing to successively enrich the reduced basis along the Monte Carlo process. The method is based on assessing the error of the reduced basis solution with a residual estimate for the prescribed quantity of interest. The efficiency of the proposed approach, which is particularly important if the number of independent random variables is large, is illustrated in 1D and 2D mechanical examples.
Abstract
In the framework of stochastic non-intrusive finite element modeling, a common practice is using Monte Carlo simulation. The main drawback of this approach is the computational [...]
- 0
- read
Modeling steel fiber reinforced concrete: numerical immersed boundary approach and a phenomenological mesomodel for concrete-fiber interaction
A. Pros, P. Diez, C. Molins

Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 90 (1), pp. 65-86

Abstract
Steel fiber reinforced concrete (SFRC) allows overcoming brittleness and weakness under tension, the main drawbacks of plain concrete. The influence of the fibers on the behavior of SFRC depends on their shape, length, slenderness, and also on their orientation and distribution into the plain concrete. The goal of this paper is to develop an ad hoc numerical strategy to account for the contribution of the fibers in the simulation of the mechanical response of SFRC. In the model presented, the individual fibers immersed in the concrete bulk are accounted for in their actual location and orientation. The selected approach is based on the ideas introduced in the immersed boundary (IB) methods. These methods were developed to account for 1D (or 2D) solids immersed in 2D (or 3D) fluids. Here, the concrete bulk is playing the role of the fluid and the cloud of steel fibers is acting as the immerse boundary (that is, a 1D structure in a 2D or 3D continuous). Thus, the philosophy of the IB methodology is used to couple the behavior of the two systems, the concrete bulk and fiber cloud, precluding the need of matching finite element meshes. Note that, considering the different size scales and the intricate geometry of the fiber cloud, the conformal matching of the meshes would be a restriction resulting in a practically unaffordable mesh. In the proposed approach, the meshes of the concrete bulk and fiber cloud are independent, and the models are coupled imposing displacement compatibility and equilibrium of the two systems. In the applications presented here, the concrete bulk is modeled using a standard nonlinear damage model. The constitutive model for the fibers is designed to account for the complex interaction between fibers and concrete. The fiber models are based on the previous investigations describing the concrete-fiber interaction and its dependence on the factors identified to be relevant: shape of the fiber (straight or hooked) and angle between the fiber and crack plane.
Abstract
Steel fiber reinforced concrete (SFRC) allows overcoming brittleness and weakness under tension, the main drawbacks of plain concrete. The influence of the fibers on the behavior [...]
- 0
- read
Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation
L. Steffens, N. Parés, P. Diez

Int. J. Numer. Meth. Engng. (2011). Vol. 86 (10), pp. 1197-1224

Abstract
An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in [28]. In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post-processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in [28] is based in a polynomial least squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem.
Abstract
An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension [...]
- 0
- read
Numerical modeling of the Double Punch Test for plain concrete
A. Pros, P. Diez, C. Molins

Int. J. Solids and Structures (2011). Vol. 48 (7-8), pp. 1229-1238

Abstract
Double punch test is used to indirectly assess the tensile strength of plain concrete, ft. For this normalized test, the tensile strength is obtained as a function of the failure load, P, which is expressed as ft = F(P). Different authors have proposed different expressions for the relation F(.), yielding scattered values of ft. None of these alternatives is universally recognized as being more suitable than the others. In fact, these expressions are mainly based on elastic models considering the maximum tensile stress under the load P and ft is obtained as an output of the linear model. A numerical simulation allows using models in which ft is an input of the material model and the corresponding failure load P is obtained associated with each value of ft. In the present work, double punch test is simulated numerically considering two alternatives for modeling plain concrete accounting for damage and cracking: (a) the nonlocal Mazars damage model and (b) an heuristic crack model including joint elements in an a priori defined crack pattern. Numerical results are validated with experimental data and compared with the analytical expressions available in the literature.
Abstract
Double punch test is used to indirectly assess the tensile strength of plain concrete, ft. For this normalized test, the tensile strength is obtained as a function of the [...]
- 0
- read
Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies
L. Steffens, N. Parés, P. Diez

Computational Mechanics (2011). Vol. 47 (6), pp. 681-699

Abstract
This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: 1) different error representations, 2) assessment of the dispersion error, and 3) different remeshing criteria.
Abstract
This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character [...]
- 0
- read
Numerical modeling of erosion using an improvement of the extended finite element method
R. Cottereau, P. Diez

European Journal of Environmental and Civil Engineering (2011). Vol. 15 (8), pp. 1187-1206

Abstract
We present in this paper a numerical model of the erosion of a soil that accounts for both the flow in the open fluid and the flow of fluid through the porous soil. The interface between the open fluid and the soil is represented using a level-set function, and the erosion is controlled by the shear stress vector. The evaluation of the approximate value of this gradient is particularly focused on, and an improved method, called XFE+ method, is presented. Numerical results in 2D and 3D illustrate the accuracy and the potentiality of this method.
Abstract
We present in this paper a numerical model of the erosion of a soil that accounts for both the flow in the open fluid and the flow of fluid through the porous soil. The interface [...]
- 0
- read
Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method
R. Cottereau, L. Chamoin, P. Diez

Mécanique & Industries (2010). Vol. 11 (3-4), pp. 249-254

Abstract
We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they refer to the exact solution of the continuous equations, rather than to some FE computation over a refined esh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements
Abstract
We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the [...]
- 0
- read
Error estimation and quality control
P. Diez, N. Parés, A. Huerta

Encyclopedia of Aerospace Engineering (2010). Vol. 3 (15), pp. 1725-1734

Abstract
This chapter provides a description of the error assessment tools available for general finite element analysis, in particular those for solid and structural mechanics. The text focuses on goal-oriented error estimation, in terms of general quantities of interest rather than on energy norm. However, the energy norm estimates are also discussed because they are seen as basic tools that must be used for assessing the error in arbitrary functional outputs of the solution. Attention is paid to the classification of the different methodologies and their main characteristics.
Abstract
This chapter provides a description of the error assessment tools available for general finite element analysis, in particular those for solid and structural mechanics. The [...]
- 0
- read
A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation
F. Larsson, P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (37-40), pp. 2383-2402

Abstract
In this contribution, we present an a posteriori error estimator for the incom- pressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators de- veloped for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity ”energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computa- tion of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor-Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds.
Abstract
In this contribution, we present an a posteriori error estimator for the incom- pressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point [...]
- 0
- read
Modeling, with a unified level-set representation, of the expansion of a hollow in the ground under different physical phenomena
R. Cottereau, P. Diez, A. Huerta

Computational Mechanics (2010). Vol. 46 (2), pp. 315-327

Abstract
This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. In particular, the dissolving action of a flow of water in a saturated soil, and that of a jet of particles of water in a non-saturated one, are represented in a common framework. In that manner, the complex geometrical evolutions of the hollow can be followed without the need for remeshing and this approach allows for a smooth transition between saturated and non-saturated models of the soil. Implementation and numerical difficulties are discussed and two applications of industrial interest are considered. The first one describes the modeling of the piping phenomenon, and the second one the evolution of an excavation created by a leaking duct.
Abstract
This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. [...]
- 0
- read
Accurate recovery-based upper error bounds for the extended finite element framework
J. Ródenas, A. Estrada, P. Diez, F. Fuenmayor

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199, pp. 2607-2621

Abstract
This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests.
Abstract
This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended [...]
- 0
- read
SUPG-based stabilization using a separated representations approach
D. Gonzalez, L. Debeugny, E. Cueto, F. Chinesta, P. Diez, A. Huerta

Int. J. of Material Forming (2010). Vol. 3 (1), pp. 883-886

Abstract
We have developed a new method for the construction of Streamline Upwind Petrov Galerkin (SUPG) stabilization techniques for the resolution of convection-diffusion equations based on the use of separated representations inside the Proper Generalized Decompositions (PGD) framework. The use of SUPG schemes produces a consistent stabilization adding a parameter to all the terms of the equation (not only the convective one). SUPG obtains an exact solution for problems in 1D, nevertheless, a generalization does not exist for elements of high order or for any system of convection-diffusion equations. We introduce in this paper a method that achieves stabilization in the context of Proper Generalzied Decomposition (PGD). This class of approximations use a representation of the solution by means of the sum of a finite number of terms of separable functions. Thus it is possible to use the technique of separation of variables in the context of problems of convection-diffusion that will lead to a sequence of problems in 1D where the parameter of stabilization is well known.
Abstract
We have developed a new method for the construction of Streamline Upwind Petrov Galerkin (SUPG) stabilization techniques for the resolution of convection-diffusion equations [...]
- 0
- read
Error assessment and mesh adaptivity for regularized continuous failure models
T. Pannachet, P. Diez, H. Askes, L. Sluys

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (17-20), pp. 961-978

Abstract
This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since this model is regularized in the post-peak regime, the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement. However, the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the postpeak loading stages. The accuracy of the nonlocal equivalent strain field (and the corresponding damage quantity) also greatly influences the accuracy of the quantity of interest. Two error measures have been proposed. The goal-oriented error estimates have provided similar error distributions, although some small differences have been found in the softening regime. Objective error estimates, together with adaptive criteria, have been used to perform automated h-adaptivity during computation.
Abstract
This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since [...]
- 0
- read
Assembling sparse matrices in MATLAB
S. Zlotnik, P. Diez

Commun. Numer. Meth. Engng (2010). Vol. 26 (6), pp. 760-769

Abstract
The assembly of sparse matrices is a key operation in finite element methods. In this study we analyze several factors that may have an influence on the efficiency of the assembly procedure. Different insertion strategies are compared using two metrics: a Cost function (the number of memory movements) and actual computing time. An improved algorithm implemented in MATLAB is proposed. It reduces both memory operations and computing time for all tested cases. The efficiency of the assembly process is found to be highly dependent on node and element numbering. The effect of the classic reverse Cuthill–McKee algorithm is, in most cases, positive and reduces computation costs. Finally, the case where a sparse matrix has to be re‐assembled at each time step is studied. The efficiency of the assembly is improved if the matrix pattern is entirely or partially inherited from previous steps.
Abstract
The assembly of sparse matrices is a key operation in finite element methods. In this study we analyze several factors that may have an influence on the efficiency of the [...]
- 0
- read
Guaranteed energy error bounds for the Poisson equation using a flux-free approach: solving the local problems in subdomains
N. Parés, H. Santos, P. Diez

Int. J. Numer. Meth. Engng. (2009). Vol. 79 (10), pp. 1203-1244

Abstract
A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain based a posteriori error estimate which yields self-equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux-free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds.
Abstract
A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed [...]
- 0
- read
Exact bounds for linear outputs of the advection-diffusion-reaction equation using flux-free error estimates
N. Parés, P. Diez, A. Huerta

Journal on Scientific Computing (SIAM) (2009). Vol. 31 (4), pp. 3064-3089

Abstract
The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.
Abstract
The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. [...]
- 0
- read
Hierarchical X-FEM for n-phase flow (n>2)
S. Zlotnik, P. Diez

Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (30-32), pp. 2329-2338

Abstract
The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more than two phases, the X-FEM has to be further eXtended. The proposed approach is presented in the case of a quasi-static Stokes n-phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases. A level set hierarchy allows describing triple junctions avoiding overlapping or “voids” between materials. Moreover, an enriched solution accounting for several simultaneous phases inside one element is proposed. The interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets. The computational implementation of this scheme involves calculating integrals in elements having several materials inside. An adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals. Examples of the hierarchical X-FEM approach are given for a n-phase Stokes problem in 2 and 3 dimensions.
Abstract
The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more [...]
- 0
- read
Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method
R. Cottereau, P. Diez, A. Huerta

Computational Mechanics (2009). Vol. 44 (4), pp. 533-547

Abstract
We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented
Abstract
We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are [...]
- 0
- read
A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation
L. Steffens, P. Diez

Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (15-16), pp. 1389-1400

Abstract
The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing the quantity of interest. For the assessment of the error in the approximation of the wave number for the Helmholtz problem (also referred to as dispersion or pollution error), this strategy cannot be applied. This is because there is no linear extractor producing the wave number from the solution of the acoustic problem. Moreover, in this context, the error assessment paradigm is reverted in the sense that the exact value of the wave number, κ, is known (it is part of the problem data) and the effort produced in the error assessment technique aims at obtaining the numerical wave number, κH, as a postprocess of the numerical solution, uH. The strategy introduced in this paper is based on the ideas used in the a priori analysis. A modified equation corresponding to a modified wave number κm is introduced. Then, the value of κm such that the modified problem better accommodates the numerical solution uH is taken as the estimate of the numerical wave number κH. Thus, both global and local versions of the error estimator are proposed. The obtained estimates of the dispersion error match the a priori predicted dispersion error and, in academical examples, the actual values of the error in the wave number.
Abstract
The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing [...]
- 0
- read
Bounds for quantities of interest and adaptivity in the Element Free Galerkin method
Y. Vidal, N. Parés, P. Diez, A. Huerta

Int. J. Numer. Meth. Engng. (2008). Vol. 76 (11), pp. 1782-1818

Abstract
A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision.
Abstract
A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower [...]
- 0
- read
Modelling gravitational instabilities: Slab break–off and Rayleigh–Taylor diapirism
S. Zlotnik, M. Fernández, P. Diez, J. Vergés

Pure and Applied Geophysics (2008). Vol. 165 (8), pp. 1491-1510

Abstract
A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are described: A level set technique to track the position of the materials and an enrichment of the solution to allow the strain rate to be discontinuous across the interface. These techniques have low computational cost and can be used in standard desktop PCs. Examples of phase tracking with level set are presented in two and three dimensions to study slab detachment in subduction processes and Rayleigh–Taylor instabilities, respectively. The modelling of slab detachment processes includes realistic rheology with viscosity depending on temperature, pressure and strain rate; shear and adiabatic heating mechanisms; density including mineral phase changes and varying thermal conductivity. Detachment models show a first prolonged period of thermal diffusion until a fast necking of the subducting slab results in the break–off. The influence of several numerical and physical parameters on the detachment process is analyzed: The shear heating exerts a major influence accelerating the detachment process, reducing the onset time to one half and lubricating the sinking of the detached slab. The adiabatic heating term acts as a thermal stabilizer. If the mantle temperature follows an adiabatic gradient, neglecting this heating term must be included, otherwise all temperature contrasts are overestimated. As expected, the phase change at 410 km depth (olivine–spinel transition) facilitates the detachment process due to the increase in negative buoyancy. Finally, simple plume simulations are used to show how the presented numerical methodologies can be extended to three dimensions.
Abstract
A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are [...]
- 0
- read
Small-scale gravitational instabilities under the oceans: implications for the evolution of oceanic lithosphere and its expression in geophysical observables
S. Zlotnik, J. Alfonso, P. Diez, M. Fernández

Philosophical Magazine (2008). Vol. 88 (28-29), pp. 3197-3217

Abstract
Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although the existence of SSC is generally accepted, its ability to effectively produce a constant lithospheric thickness (i.e. flattening of observables) is a matter of debate. Here we study the development and evolution of SSC with a 2D thermomechanical finite-element code. Emphasis is put on (i) the influence of various rheological and thermophysical parameters on SSC, and (ii) its ability to reproduce geophysical observables (i.e. seafloor depth, surface heat flow, and seismic velocities). We find that shear heating plays no significant role either in the onset of SSC or in reducing the lithospheric thickness. In contrast, radiogenic heat sources and adiabatic heating exert a major control on both the vigour of SSC and the thermal structure of the lithosphere. We find that either dislocation creep, diffusion creep, or a combination of these mechanisms, can generate SSC with rheological parameters given by laboratory experiments. However, vigorous SSC and significant lithospheric erosion are only possible for relatively low activation energies.
Abstract
Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although [...]
- 0
- read
Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the Discontinuous Galerkin time discretization
N. Parés, P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), pp. 1641-1660

Abstract
Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques.
Abstract
Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds [...]
- 0
- read
Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution
N. Parés, P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), 1661-1679

Abstract
The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds account for the error both in space and time. The assumption stating that the error introduced by the time marching scheme is negligible, used in the first part, is removed here. The bounds are computed starting from an approximation of the exact solution, associated with a spatial mesh and a time grid. Nevertheless, the bounds are guaranteed with respect to the exact solution, with no reference to any mesh or time discretization.
Abstract
The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds [...]
- 0
- read
Numerical modelling of tectonic plates subduction using X-FEM
S. Zlotnik, P. Diez, M. Fernández, J. Vergés

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (41-44), pp. 4283-4293

Abstract
The numerical modelling of plate subduction requires solving a coupled thermo-mechanical highly-nonlinear transient problem. The mechanical description of the phenomenon results in a multiphase quasi-static Stokes ﬂow, where the inertia terms are neglected. The transient thermal problem is dominated by the advection term. Here, the representation and evolution of the different phases are described using level sets. The phase tracking is carried out transporting the level set along with the material, using a pure advective model. The gradient discontinuities induced by the viscosity jump across the interface are resolved numerically by enriching the solution using a partition of unity method in a eXtended Finite Element Method (X-FEM) context. These numerical tools are used to simulate plate subduction with different parameters and to derive useful correlations between relevant geophysical factors.
Abstract
The numerical modelling of plate subduction requires solving a coupled thermo-mechanical highly-nonlinear transient problem. The mechanical description of the phenomenon results [...]
- 0
- read
Equilibrated Patch Recovery error estimates: simple and accurate upper bounds of the error
P. Diez, J. Ródenas, O. Zienkiewicz

Int. J. Numer. Meth. Engng. (2007). Vol. 69 (10), pp. 2075-2098

Abstract
This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ℒ︁2 norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ℒ︁2 norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ℒ︁2 norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests.
Abstract
This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires [...]
- 0
- read
Goal-oriented error estimation for transient parabolic problems
P. Diez, G. Calderón

Computational Mechanics (2007). Vol. 39 (5), pp. 631-646

Abstract
This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time domain is discretized using a Discontinuous Galerkin (DG) approach. This allows to formulate the time stepping procedure in a variational format. The error is measured in the basis of an output of interest of the solution, defined by a linear functional. A dual problem, associated with this linear output is introduced. The dual problem has to be solved backward in time. An error representation is introduced, based on the weak residual of the primal error applied to the dual solution. Two different alternatives are studied to estimate the error in the dual solution: 1) recovery based error estimators and 2) implicit residual type estimators. Once the error assessment is performed implicitly in the dual problem, the obtained estimate is plugged into the primal residual to obtain the error in the quantity of interest. The implementation of the estimator is drastically simplified by using the weak version of the residual instead of the strong version used in previous works. Thus, the output error is assessed using a mixed technique, explicit for the primal problem and implicit for the dual. In the framework of adaptive computations of transient problems, this approach is very attractive because it allows using first the implicit scheme for the dual problem and then integrating the primal problem, estimating the error explicitly and eventually adapting the space-time grid. Thus, at every time step of the time marching scheme, the estimate of the dual error is injected into the primal residual (explicit estimate for the primal problem).
Abstract
This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time [...]
- 0
- read
Remeshing criteria and proper error representations for goal oriented h-adaptivity
P. Diez, G. Calderón

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (4-6), pp. 719-733

Abstract
A key ingredient in h-adaptivity pertains to the transformation of output data from a given error estimator into input data, usually in the form of an element-size distribution, that needs to be supplied to a mesh generator. This paper analyzes the different possibilities of de¯ning remeshing criteria in the framework of goal oriented adaptivity. In standard energy norm driven adaptivity, the optimal mesh is clearly obtained if the local error distribution is uniform. The goal-oriented paradigm introduces new di±culties associated with the different possibilities for the spatial error representation and the signs of the local error contributions. A nodal error representation is introduced in order to improve the communication with the automatic mesh generation tool, precluding the transfer of information from elements to nodes. Numerical experiments demonstrate the ability of the introduced remeshing strategies to drive e±cient adaptive procedures and to control the error in quantities of interest. The results of the numerical tests fit the expected properties of the different remeshing strategies.
Abstract
A key ingredient in h-adaptivity pertains to the transformation of output data from a given error estimator into input data, usually in the form of an element-size distribution, [...]
- 0
- read
Subdomain-based flux-free a posteriori error estimators
N. Parés, P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (4-6), pp. 297-323

Abstract
A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks of standard residual type estimators, circumvents the need of flux-equilibration and results in a simple implementation that uses standard resources available in .nite element codes. This is specially interesting for 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. The bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. The presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest.
Abstract
A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks [...]
- 0
- read
Accurate upper and lower error bounds by solving flux-free local problems in “stars”
P. Diez, N. Parés, A. Huerta

European Journal of Finite Elements (2004). Vol. 13 (5-7), pp. 497-507

Abstract
Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced in [MAC 00, CAR 99, MOR 03, PRU 02]. The new approach introduced here is based on using the estimated error function rather than the estimated error norms. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.
Abstract
Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced [...]
- 0
- read
Convergence of finite elements enriched with meshless methods
S. Fernández-Méndez, P. Diez, A. Huerta

Numerische Mathematik (2003). Vol. 96 (1), pp. 43-59

Abstract
A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions based on a particle distribution. Convergence results are presented and proved.
Abstract
A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions [...]
- 0
- read
A note on the convergence of the secant method for simple and multiple roots
P. Diez

Applied Numerical Letters (2003). Vol. 16 (8), pp. 1211-1215

Abstract
The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is super linear: the rate of convergence is set by the gold number. Nevertheless, this property holds only for simple roots. If the multiplicity of the root is larger than one, the convergence of the secant method becomes linear. This communication includes a detailed analysis of the secant method when it is used to approximate multiple roots. Thus, a proof of the linear convergence is shown. Moreover, the values of the corresponding asymptotic convergence factors are determined and are found to be also related with the golden ratio.
Abstract
The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is super linear: the rate of [...]
- 0
- read
Goal-oriented adaptivity for shell structures: Error assessment and remeshig criteria
P. Diez, I. Morata, A. Huerta

European Journal of Finite Elements (2003). Vol. 12 (6), pp. 691-715

Abstract
The reliable computation of shell structures requires a tool to assess and control the quality of the finite element solution. For practical purposes, the quality of the numerical solution must be measured using a quantity of engineering interest rather than in the standard energy norm. However, the assessment of the error in an output of interest is based on a standard energy norm error estimator. The standard error estimator has to be applied to both the original problem (primal) and a dual problem related with the selected engineering quantity. In shells with assumed-strain models, the combination of primal and dual error estimation is performed differently than in the continuum mechanics case. Moreover, a part from the goal-oriented error estimator, the adaptive process requires a remeshing criterion. This work introduces an specific remeshing criterion for goal-oriented adaptivity and its particularization to the context of shell elements.
Abstract
The reliable computation of shell structures requires a tool to assess and control the quality of the finite element solution. For practical purposes, the quality of the numerical [...]
- 0
- read
Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates
P. Diez, N. Parés, A. Huerta

Int. J. Numer. Meth. Engng. (2003). Vol. 56 (10), pp. 1465-1488

Abstract
Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs. This work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates. This lower bound may also be used to assess the effectivity of the former estimate and to improve it.
Abstract
Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error [...]
- 0
- read
Enrichissement des interpolations d’éléments finis en utilisant des méthodes de particules
A. Huerta, S. Fernández-Méndez, P. Diez

Mathematical Modelling and Numerical Analysis (2002). Vol. 36 (6), pp. 1027-1042

Abstract
In the framework of meshless methods, the interpolation is based on a distribution of particles: it is not necessary to define connectivities. In these methods the interpolation can be easily enriched, increasing the number of particles (as in h-refinement of finite elements) or increasing the order of consistency (as in p-refinement of finite elements). However, comparing with finite elements, particle methods suffer from an increase in the computational cost, mainly due to the computation of the shape functions. In this paper, a mixed interpolation that combines finite elements and particles is presented. The objective is to take advantage of both methods. In order to define h- p refinement strategies an a priori error estimate is needed, and thus, some convergence results are presented and proved for this mixed interpolation
Abstract
In the framework of meshless methods, the interpolation is based on a distribution of particles: it is not necessary to define connectivities. In these methods the interpolation [...]
- 0
- read
Probabilistic analysis of an a posteriori error estimator for Finite Elements
P. Diez, J. Egozcue

Mathematical Models and Methods in Applied Sciences (2001). Vol. 11 (5), pp. 841-854

Abstract
A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive projections of the exact error on two finite-dimensional subspaces. The analysis introduced in this paper is based on a probabilistic approach, that is, the idea is to assess the average value of the effectivity index (the ratio estimated error over exact error) by assuming the randomness of the exact error. The average value characterizes the mean behavior of the estimator and it is found to be related with some geometric properties of the subspaces. These geometric properties are obtained from the standard matrices of the linear systems arising in the formulation of the finite element method.
Abstract
A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive [...]
- 0
- read
Adaptivity based on error estimation for viscoplastic softening materials
P. Diez, M. Arroyo, A. Huerta

Mechanics of Cohesive-Frictional Materials (2000). Vol. 5 (2), pp. 87-112

Abstract
This paper focuses on the numerical simulation of strain softening mechanical problems. Two problems arise: (1) the constitutive model has to be regular and (2) the numerical technique must be able to capture the two scales of the problem (the macroscopic geometrical representation and the microscopic behavior in the localization bands). The Perzyna viscoplastic model is used in order to obtain a regularized softening model allowing to simulate strain localization phenomena. This model is applied to quasistatic examples. The viscous regularization of quasistatic processes is also discussed: in quasistatics, the internal length associated with the obtained band width is no longer only a function of the material parameters but also depends on the boundary value problem (geometry and loads, specially loading velocity). An adaptive computation is applied to softening viscoplastic materials showing strain localization. As the key ingredient of the adaptive strategy, a residual type error estimator is generalized to deal with such highly nonlinear material model. In several numerical examples the adaptive process is able to detect complex collapse modes that are not captured by a first, even if fine, mesh. Consequently, adaptive strategies are found to be essential to detect the collapse mechanism and to assess the optimal location of the elements in the mesh.
Abstract
This paper focuses on the numerical simulation of strain softening mechanical problems. Two problems arise: (1) the constitutive model has to be regular and (2) the numerical [...]
- 0
- read
Error estimation including pollution assessment for nonlinear finite element analysis
A. Huerta, P. Diez

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 181 (1-3), pp. 21-41

Abstract
A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure. Pollution errors are taken into account by a feedback strategy, that is, an error estimate based on local computations is used as the input of the pollution analysis. This estimator is used in the frame of an adaptive procedure. Numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions. Moreover, one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms.
Abstract
A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps [...]
- 0
- read
Error estimation for adaptive computations on shell structures
P. Diez, A. Huerta

European Journal of Finite Elements (2000). Vol. 9 (1-3), pp. 49-66

Abstract
The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. The first is associated with the finite dimensional interpolation space and it is present in any finite element computation. The latter is associated with the piecewise polynomial approximation of a curved surface and is much more relevant in shell problems than in any other standard 2D or 3D computation. In the shells framework, formerly the quality control of the finite element solution has been carried out using flux projection a posteriori error estimators. This technique exhibits two main drawbacks: 1) the flux smoothing averages stress components over different elements that may have different physical meaning if the tangent planes are different and 2) the error estimation process uses only the approximate solution and hence, the discretized forces and the computational mesh: the data describing the real geometry and load is therefore not accounted for. In this work, a residual type error estimator introduced for standard 2D finite element analysis is generalized to shell problems. This allows to easily account for the real original geometry of the problem in the error estimation procedure and precludes the necessity of comparing generalized stress components between non coplanar elements. This estimator is based on approximating a reference error associated with a refined reference mesh. In order to build up the residual error equation the computed solution must be represented (projected) on the reference mesh. The use of thin shell finite elements requires a proper formulation in order to preclude shear locking. Following an idea of Donea and Lamain, the interpolation of the rotations is not unique and requires a particular technique to transfer the information from the computational mesh to the reference mesh. This technique is also developed in this work and may be used in any adaptive evolution problem where the solution must be transferred from one mesh to another.
Abstract
The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. [...]
- 0
- read
Adaptive finite element strategies based on error assessment
A. Huerta, A. Rodríguez-Ferran, P. Diez, J. Sarrate

Int. J. Numer. Meth. Engng. (1999). Vol. 46 (10), pp. 1803-1818

Abstract
Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.
Abstract
Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error [...]
- 0
- read
A unified approach to remeshing strategies for finite element h-adaptivity
P. Diez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (1999). Vol. 176 (1-4), pp. 215-229

Abstract
In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced different remeshing strategies yielding very different results. In this work these methods are included in a unified framework, emphasizing the role of the underlying hypotheses. Moreover, an objective tool to evaluate the accuracy of the resulting finite element solution is presented. Thus, a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions. The different remeshing strategies are compared with well-known numerical examples.
Abstract
In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced [...]
- 0
- read
A posteriori error estimation for standard finite element analysis
P. Díez, J. Egozcue, A. Huerta

Comput. Methods Appl. Mech. Engrg., (1998). Vol. 163 (1-4), pp. 141-157

Abstract
A new residual type estimator based on projections of the error on subspaces of locally-supported functions is presented. The estimator is defined by a standard element-by-element refinement. First, an approximation of the energy norm of the error is obtained solving local problems with homogeneous Dirichlet boundary conditions. A later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces. This estimate is a lower bound of the measure of the actual error. The estimator does not need to approximate local boundary conditions for the error equation. Therefore, computation of flux jumps is not necessary. Moreover, the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward. The presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error, as well as its usefulness in adaptive procedures.
Abstract
A new residual type estimator based on projections of the error on subspaces of locally-supported functions is presented. The estimator is defined by a standard element-by-element [...]
- 0
- read
Vulnerabilidad sísmica y toma de decisiones
J. Egozcue, G. Grande, P. Diez

Revista Internacional de Cálculo de Estructuras (1997). Vol. 2 (1), pp. 35-58

Abstract
Vulnerability of civil structures and systems under seismic actions have been extensively used and studied in the last decade. However, the goals of those studies are very diverse and it have caused some confusion on the meaning and definition of vulnerability. Naturally, different scopes of vulnerability lead to different methodologies in its study which may be incompatible. The present aim is to analyse the definition of seismic vulnerability in the framework of Bayesian decision making. Three different examples of decision making related to seismic vulnerability, which are qualitatively different, guide the discussion. They have an increasing complexity: decisions on normative seismic design; decisions on insurances against seismic risk and, finally, those decisions that should be taken to lead alarm, mitigation and rescue systems for seismic disasters. The three analised examples correspond to three different decision making schemes, respectively named: a priori, a posteriori and preposterior. In each case the definition of seismic vulnerability depends on the concept of considered utility (cost-benefit), on the involved random states and the scale in which the phenomena are to be considered. Most of the concepts associated with seismic vulnerability are reviewed through these three examples. A conclusion is that seismic vulnerability studies should consider the associated decision making problem. Simultaneously, the study of methodological and estimation aspects on vulnerability are remarked in order to improve both guidelines for seismic design and disaster mitigation.
Abstract
Vulnerability of civil structures and systems under seismic actions have been extensively used and studied in the last decade. However, the goals of those studies are very [...]
- 0
- read
La orientación profesional, la competencia y la Universidad del siglo XXI
B. Suarez

Revista de Obras Públicas (2007). Vol. 154 (3477), pp. 7-10

Abstract
The article refers to the role of the University, and particularly public universities, in the professional guidance of the public in the broadest sense of the term and, namely, its influence over higher education and the professional skills provided by the same. This is all considered within a setting of globalization and when taking into account the prevailing conditions defining the knowledge society and welfare.
Abstract
The article refers to the role of the University, and particularly public universities, in the professional guidance of the public in the broadest sense of the term and, namely, [...]
- 2
- read
Posibilidades de los ordenadores personales en el cálculo de estructuras por el método de los elementos finitos
E. Oñate, Á. Gaona, J. Oliver, B. Suarez

Revista de Obras Públicas (1983). Vol. 130 (3212), pp. 231-248
- 2
- read
El laboratorio de ensayo de estructuras en modelo reducido de la Escuela de Ingenieros de Caminos de Barcelona
J. Oliver, B. Suárez, E. Blanco, E. Oñate

Acero y Hormigón (1982). Vol. 145, pp. 65-78
- 1
- read
PFEM–DEM for particle-laden flows with free surface
A. Franci, I. de-Pouplana, G. Casas, M. Celigueta, J. González-Usúa, E. Oñate

Comp. Part. Mech (2019). pp. 1-20

Abstract
This work proposes a fully Lagrangian formulation for the numerical modeling of free-surface particle-laden flows. The fluid phase is solved using the particle finite element method (PFEM), while the solid particles embedded in the fluid are modeled with the discrete element method (DEM). The coupling between the implicit PFEM and the explicit DEM is performed through a sub-stepping staggered scheme. This work only considers suspended spherical particles that are assumed not to affect the fluid motion. Several tests are presented to validate the formulation. The PFEM–DEM results show very good agreement with analytical solutions, laboratory tests and numerical results from alternative numerical methods.
Abstract
This work proposes a fully Lagrangian formulation for the numerical modeling of free-surface particle-laden flows. The fluid phase is solved using the particle finite element [...]
- 2
- read
A finite element reduced order model based on adaptive mesh refinement and artificial neural networks
J. Baiges, R. Codina, I. Castañar, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2019). (prepint) pp. 1-14

Abstract
In this work a reduced order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN-ROM) is presented. The idea is to run a high-fidelity simulation by using an adaptively refined finite element mesh, and compare the results obtained with those of a coarse mesh finite element model. From this comparison, a correction forcing term can be computed for each training configuration. A model for the correction term is built by using an artificial neural network, and the final reduced order model is obtained by putting together the coarse mesh finite element model, plus the artificial neural network model for the correction forcing term. The methodology is applied to non-linear solid mechanics problems, transient quasi-incompressible flows, and a fluid-structure interaction problem. The results of the numerical examples show that the FAN-ROM is capable of improving the simulation results obtained in coarse finite element meshes at a reduced computational cost.
Abstract
In this work a reduced order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN-ROM) is presented. The [...]
- 4
- read
Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows
O. Guasch, R. Codina

Comput. Methods Appl. Mech. Engrg., (2013). Vol. 261-262, pp. 145-166

Abstract
Abstract Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus on the orthogonal subgrid scale (OSS) finite element method and make an analysis of the statistical behavior of its stabilization terms in the quasi static approximation. This is done by resorting to results from classical statistical fluid mechanics concerning two point velocity, pressure and combined correlation functions of various orders. Given a fine enough mesh with characteristic element size h in the inertial subrange of a turbulent flow, it is shown that the rate of transfer of subgrid kinetic energy provided by the OSS stabilization terms does not depend on h and that it equals the molecular physical dissipation rate (up to a dimensionless constant that only depends on the finite element shapes) for a proper redesign of the standard parameters of the formulation. This is a noteworthy fact taking into account that the subgrid stabilization terms do not arise from physical considerations, but from the mathematical necessity to allow equal interpolation for the pressure and velocity fields, as well as to control convection. Therefore, the obtained results contribute somehow to the line of reasoning supporting that pure numerical approaches (i.e., without introducing additional physical models) could probably suffice in the LES simulation of turbulent flows
Abstract
Abstract Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus [...]
- 2
- read
A general algorithm for compressible and incompressible flow—Part II. tests on the explicit form
O. Zienkiewicz, K. Morgan, B. Sai, R. Codina, M. Vasquez

Int. J. Numer. Meth. Fluids (1995). Vol. 20 (8-9), pp. 887-913

Abstract
The algorithm introduced in Part I of this paper is applied in its explicit form to a variety of problems in order to demonstrate its wide range of applicability and excellent performance. Examples range from nearly incompressible, viscous, flows through transonic applications to high speed flows with shocks. In most examples linear triangular elements are used in the finite element approximation, but some use of quadratic approximation, again in triangles, indicates satisfactory performance even in the case of severe shocks.
Abstract
The algorithm introduced in Part I of this paper is applied in its explicit form to a variety of problems in order to demonstrate its wide range of applicability and excellent [...]
- 5
- read
A bonded discrete element method for modeling ship–ice interactions in broken and unbroken sea ice fields
O. Jou Devesa, M. Celigueta, S. Latorre, F. Arrufat, E. Oñate

Computational Particle Mechanics (2019). Vol. 6 (4), pp 739–765

Abstract
This work investigates the failure patterns of ice cakes and floe-icewhen loaded by amoving and sloping structure (ice-breaking ships and cones). In the paper, we introduce the most frequently encountered ice infested scenarios, the main characteristics of ice-breaking ships and the predicted failure modes of floe-ice depending on the loading conditions, the structure type and the ice feature dimensions and thickness. For the simulations, a local bonded discrete element method (DEM) is used to model sea ice and its fractures. The packing of bonded spherical particles which reproduce the ice continuum can break due to ship–ice interactions and the failure modes are studied. A set of validation simulations are first carried out. A level ice sheet breaking against an installed ice-breaking cone with different slope angles is studied, and the results are compared with other DEM simulations. Then, a group of bonded DEM simulations are performed to predict the different failure modes produced when an ice-breaking ship bow contacts with ice cakes and floe-ice of different dimensions and thickness, typical in broken ice fields. Finally, the study of breaking a continuous level ice sheet is carried out by modeling with the bonded DEM an “infinite” large domain of sea ice and loaded by a single-degree-of-freedom model of an ice-breaking ship.
Abstract
This work investigates the failure patterns of ice cakes and floe-icewhen loaded by amoving and sloping structure (ice-breaking ships and cones). In the paper, we introduce [...]
- 4
- read
Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques
Ö. Türk, R. Codina

Applied Numerical Mathematics (2019). (preprint) Vol. 145, pp. 188-200

Abstract
Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method approach can successfully be adapted for the eigenproblem to rectify the associated problems such as the existence of zero diagonal entries in the resulting algebraic system. Two different schemes, namely, the standard penalisation with a small penalty parameter, and the iterative penalisation that enables relatively large parameters, are implemented. The employment of the latter leads to a so-called inhomogeneous generalised eigenvalue problem which requires a special attention. A feasible solution strategy is presented which is adapted from a procedure based on Newton's method proposed for the corresponding standard (inhomogeneous) eigenvalue problems. Concerning the spatial discretisation, among other possible options, the Chebyshev spectral collocation method based on expanding the unknown fields in tensor product of Chebyshev polynomials is employed. It is shown that the method constitutes a novel way of efficiently examining the approximate eigensolutions of the Stokes operator with the use of Chebyshev spectral collocation method directly, without a decoupling of velocity and pressure.
Abstract
Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method [...]
- 4
- read
Logarithmic conformation reformulation in viscoelastic flow problems approximated by a VMS-type stabilized finite element formulation
L. Moreno García, R. Codina, J. Baiges, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 354, pp. 706-731

Abstract
The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina (2014) for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids.
Abstract
The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers [...]
- 3
- read
Dynamic term-by-term stabilized finite element formulation using orthogonal subgrid-scales for the incompressible Navier–Stokes problem
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 349, pp. 701-721

Abstract
In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for the Navier–Stokes problem that can be viewed as a variational multiscale (VMS) method under some assumptions. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual-based formulations, the introduction of two velocity subscale components. They represent the components of the convective and the pressure gradient terms, respectively, of the momentum equation that cannot be captured by the finite element mesh. A key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field. The method ensures stability in anisotropic space–time discretizations, which is proved using numerical analysis for a linearized problem and demonstrated in classical numerical tests. The work includes a detailed description of the proposed formulation and several numerical examples that serve to justify our claims.
Abstract
In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for [...]
- 5
- read
Time-dependent semidiscrete analysis of the viscoelastic fluid flow problem using a variational multiscale stabilized formulation
G. Barrenechea, E. Castillo, R. Codina

Journal of Numerical Analysis (2019). Vol. 39 (2), pp. 792-819

Abstract
In this article we analyse a stabilized finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and robustness in the different benchmarks tested in the viscoelastic framework, and permitting the use of equal interpolation of the unknown fields. We first present results about a linearized subproblem, for which well-posedness and stability results can be proved. Then, the semidiscrete nonlinear time-dependent case is addressed using a fixed point theorem, which allows us to prove existence of a semidiscrete solution, along with error estimates.
Abstract
In this article we analyse a stabilized finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and [...]
- 2
- read
Solution of low Mach number aeroacoustic flows using a Variational Multi-Scale finite element formulation of the compressible Navier–Stokes equations written in primitive variables
C. Bayona, J. Baiges, R. Codina

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 344, pp. 1073-1103

Abstract
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic flows. We develop a Variational Multi-Scale formulation to stabilize the finite element discretization by including the orthogonal, dynamic and non-linear subscales, together with an implicit scheme for advancing in time. Three additional features define the proposed numerical scheme: the splitting of the pressure and temperature variables into a relative and a reference part, the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility, and the approximation of the dynamic stabilization matrix for the time dependent subscales. We also include a weak imposition of implicit non-reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes. The order of accuracy of the method is verified for two- and three-dimensional linear and quadratic elements using steady manufactured solutions. Several benchmark flow problems are studied, including transient examples and aeroacoustic applications.
Abstract
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic [...]
- 5
- read
Computational aeroacoustics to identify sound sources in the generation of sibilant /s/
A. Pont, O. Guasch, J. Baiges, R. Codina, A. Hirtum

Int. J. Numer. Meth. Biomed. Engng (2019). Vol. 35 (1), e3153

Abstract
A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between the upper and lower incisors and then impinges the space between the lower incisors and the lower lip. The flow eddies in that region become a source of direct aerodynamic sound, which is also diffracted by the speech articulators and radiated outwards. The numerical simulation of these phenomena is complex. The spectrum of an /s/ typically peaks between 4 and 10 kHz, which implies that very fine computational meshes are needed to capture the eddies producing such high frequencies. In this work, a large‐scale computation of the aeroacoustics of /s/ has been performed for a realistic vocal tract geometry, resorting to two different acoustic analogies. A stabilized finite element method that acts as a large eddy simulation model has been adopted to solve the flow dynamics. Also, a numerical strategy has been implemented that allows the determination, in a single computational run, of the separate contribution of the sound diffracted by the upper incisors from the overall radiated sound. Results are presented for points located close to the lip opening showing the relative influence of the upper teeth depending on frequency.
Abstract
A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between [...]
- 1
- read
Finite element modeling of nonlinear reaction-diffusion-advection systems of equations
S. Komala, R. Codina

Int. J. Numer. Meth. Heat and Fluid Flows (2018). Vol. 28 (11), pp. 2688-2715

Abstract
Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.
Abstract
Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear [...]
- 1
- read
Stationary and time-dependent numerical approximation of the lid-driven cavity problem for power-law fluid flows at high Reynolds numbers using a stabilized finite element formulation of the VMS type
A. Aguirre, E. Castillo, M. Cruchaga, R. Codina, J. Baiges

Journal of Non-Newtonian Fluid Mechanics (2018). Vol. 257, pp. 22-43

Abstract
In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian fluids, this benchmark case has been extensively studied by many authors for low and moderate Reynolds numbers (up to Re=10,000), giving place to steady flows, using stationary and time-dependent approaches. For more convective flows, the solution becomes unstable, describing an oscillatory behavior. The critical Reynolds number which gives place to this time-dependent fluid dynamics has been defined over a wide range 7, 300 ≲ Re ≲ 35, 000, using different numerical approaches. In the non-Newtonian case, the cavity problem has not been studied deeply for high Reynolds number (Re > 10, 000), specifically, in the oscillatory time-dependent case. A VMS formulation is presented to be validated using existing results, to determine flow conditions at which the instability appears, and lastly, to establish new benchmark solutions for high-Reynolds numbers fluid flows using the power-law model. Obtained results show a good agreement with those reported in the references, and new data related with the oscillatory behavior of the flow has been found for the non-Newtonian case. In this regard, time-dependent flows show dependence on both Reynolds number and power-law index, and the unsteady starting point has been determined for all studied cases. It is determined that the critical Reynolds number (Rec) that defines the first Hopf bifurcation for Newtonian fluid flow is ranged between 8,100 ≲ Rec ≲ 8, 250, whereas for power-law indexes n=0.5 and n=1.5, it is 7,100 ≲ Rec ≲ 7,200 and 18,250 ≲ Rec ≲ 18,500, respectively.
Abstract
In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian [...]
- 1
- read
On hp convergence of stabilized finite element methods for the convection–diffusion equation
R. Codina

Sociedad Española de Matemática Aplicada (2018). Vol. 75 (4), pp. 591-606

Abstract
This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods discussed are classical-residual based stabilization techniques and also projection-based stabilization methods. The theoretical impossibility of obtaining an optimal convergence rate in terms of the polynomial order p for all possible Péclet numbers is explained. The key point turns out to be an inverse estimate that scales as p2. The use of this estimate is not needed in a particular case of (H1-)projection-based methods, and therefore the theoretical lack of convergence described does not exist in this case.
Abstract
This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods [...]
- 1
- read
Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations
C. Bayona, R. Codina, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2018). Vol. 337, pp. 501-526

Abstract
This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled L2-norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.
Abstract
This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. [...]
- 2
- read
Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows
A. Pont, R. Codina, J. Baiges, O. Guasch

Journal of Computational Physics (2018). Vol. 363, pp. 11-29

Abstract
The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. Moreover, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure time derivative. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries without using any tailored damping model will be presented.
Abstract
The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations [...]
- 1
- read
Stabilized mixed finite element method for the M1 radiation model
Q. Schmid, Y. Mesri, E. Hachem, R. Codina

Comput. Methods Appl. Mech. Engrg., (2018). Vol. 335, pp. 69-90

Abstract
In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial differential equations, the so-called M1 model. A Variational Multiscale method is developed for this system, and tested on some illustrative benchmarks. The question of dealing with heterogeneous physical properties is also considered, and treated by means of an immersed method. It combines a level-set approach for representation of interfaces, mixing laws to build effective physical properties and an anisotropic mesh adaptation process, all these ingredients leading to an accurate description of the interface.
Abstract
In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial [...]
- 1
- read
Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling
C. Bayona, J. Baiges, R. Codina

Int. J. Numer. Meth. Fluids (2018). Vol. 86 (5), pp. 313-328

Abstract
A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.
Abstract
A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. [...]
- 0
- read
Variational Multiscale error estimators for solid mechanics adaptive simulations: An Orthogonal Subgrid Scale approach
J. Baiges, R. Codina

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 325, pp. 37-55

Abstract
In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. The main idea is to consider a rich model for the subgrid scales as an error estimator. The subscales are considered to belong to a space orthogonal to the finite element space (Orthogonal Subgrid Scales) and we take into account their contribution both in the element interiors and on the element boundaries (Subscales on the Element Boundaries). A simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the Variational Multiscale Method. Numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non-linear cases.
Abstract
In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. [...]
- 3
- read
Interpolation with restrictions between finite element meshes for flow problems in an ALE setting
A. Pont, R. Codina, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 110 (13), pp. 1203-1226

Abstract
The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission of information between finite element meshes. Because the Lagrangian projection of results from one mesh to another is a dissipative method, a new conservative interpolation method has been developed. A series of constraints, such as the conservation of mass or energy, are applied to the interpolated arrays through Lagrange multipliers in an error minimization problem, so that the resulting array satisfies these physical properties while staying as close as possible to the original interpolated values in the L2 norm. Unlike other conservative interpolation methods that require a considerable effort in mesh generation and modification, the proposed formulation is mesh independent and is only based on the physical properties of the field being interpolated. Moreover, the performed corrections are neither coupled with the main calculation nor with the interpolation itself, for which reason the computational cost is very low.
Abstract
The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission [...]
- 1
- read
Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations
E. Castillo, R. Codina

Mathematical Models and Methods in Applied Sciences (2017). Vol. 51 (4), pp. 1407-1427

Abstract
In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.
Abstract
In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field [...]
- 2
- read
An adaptive Fixed-Mesh ALE method for free surface flows
J. Baiges, R. Codina, A. Pont, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 313, pp. 159-188

Abstract
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.
Abstract
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background [...]
- 1
- read
Finite element approximation of the viscoelastic flow problem: A non-residual based stabilized formulation
E. Castillo, R. Codina

Computers & Fluids (2017). Vol. 142, pp. 72-78

Abstract
In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns σ−u−p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.
Abstract
In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based [...]
- 3
- read
A pseudo-compressible variational multiscale solver for turbulent incompressible flows
L. Yang, S. Badia, R. Codina

Computational Mechanics (2016). Vol. 58 (6), pp. 1051-1069

Abstract
In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with 5123cells on a 12 cores 64 GB ram machine.
Abstract
In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. [...]
- 1
- read
Residual‐based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics
O. Guasch, P. Sánchez-Martín, A. Pont, J. Baiges, R. Codina

Int. J. Numer. Meth. Fluids (2016). Vol. 82, pp. 839-857

Abstract
The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf–sup compatibility condition can be built in the case of no mean flow, that is, for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual‐based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared‐solenoidal mean flow, and for the aeolian tone generated by flow past a two‐dimensional cylinder.
Abstract
The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics [...]
- 1
- read
A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
Ö. Türk, D. Boffi, R. Codina

Comput. Methods Appl. Mech. Engrg., (2016). Vol. 310, pp. 886-905

Abstract
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.
Abstract
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) [...]
- 1
- read
Variational multi-scale finite element approximation of the compressible Navier-Stokes equations
C. Bayona, J. Baiges, R. Codina

Int. J. Numer. Meth. Heat and Fluid Flows (2016). Vol. 26 (3-4), pp. 1240-1271

Abstract
Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value A complete investigation of the stabilized formulation of the compressible problem is addressed.
Abstract
Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written [...]
- 2
- read
Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics
O. Guasch, A. Pont, J. Baiges, R. Codina

Computers and Fluids (2016). Vol. 133, pp. 129-139

Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.
Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) [...]
- 1
- read
A stabilized finite element method for the mixed wave equation in an ALE framework with application to diphthong production
O. Guasch, M. Arnela, R. Codina, H. Espinoza

Acta Acustica united with Acustica (2016). (preprint) Vol. 102, pp. 94-106

Abstract
Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity field. Indeed, this becomes the natural option in many problems, such as those involving waves propagating in moving domains, because the equations can easily be set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Yet, when attempting a standard Galerkin finite element solution (FEM) for them, it turns out that an inf-sup compatibility constraint has to be satisfied, which prevents from using equal interpolations for the approximated acoustic pressure and velocity fields. In this work it is proposed to resort to a subgrid scale stabilization strategy to circumvent this condition and thus facilitate code implementation. As a possible application, we address the generation of diphthongs in voice production.
Abstract
Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity [...]
- 1
- read
Two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched shape functions
E. Castillo, J. Baiges, R. Codina

Journal of Non-Newtonian Fluid Mechanics (2015). Vol. 225, pp. 37-53

Abstract
The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we present a finite element method to solve the two immiscible fluid flow problem using the level set method. When the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The method is tested in two problems: the first example consists in a sloshing case that involves the interaction of a Giesekus and a Newtonian fluid. This example shows that the enriched pressure functions permit the exact resolution of the hydrostatic rest state. The second example is the classical jet buckling problem used to validate our method. To permit the use of equal interpolation between the variables, we use a variational multiscale formulation proposed recently by Castillo and Codina in Comput. Methods Appl. Mech. Engrg. 279 (2014) 579–605, that has shown very good stability properties, permitting also the resolution of the jet buckling flow problem in the the range of Weissenberg number 0 < We < 100, using the Oldroyd-B model without any sign of numerical instability. Additional ingredients of the work are the inclusion of a discontinuity capturing technique for the constitutive equation and some comparisons between a monolithic resolution and a fractional step approach to solve the viscoelastic fluid flow problem from the point of view of computational requirements
Abstract
The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we [...]
- 1
- read
On some time marching schemes for the stabilized finite element approximation of the mixed wave equation
H. Espinoza, R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 296, pp. 295-326

Abstract
In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved.
Abstract
In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, [...]
- 1
- read
A three‐field stabilized finite element method for fluid‐structure interaction: elastic solid and rigid body limit
E. Hachem, S. Feghali, T. Coupez, R. Codina

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 104 (7), pp. 566-584

Abstract
We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing an extra stress in the momentum equation. The obtained three‐field velocity, pressure and stress system is solved using a stabilized finite element method. The key feature of this unified formulation is the ability to describe different kind of interactions between the fluid and the structure, which can be either elastic or a perfect rigid body, without the need of treating this last case via penalization. The level‐set method combined with a dynamic anisotropic mesh adaptation is used to track the fluid‐solid interface.
Abstract
We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing [...]
- 1
- read
Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non‐matching meshes
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2015). Vol. 104 (7), pp. 624-654

Abstract
In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential.
Abstract
In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational [...]
- 1
- read
Numerical modeling of chlorine concentration in water storage tank
R. Codina, J. Principe, C. Muñoz, J. Baiges

Int. J. Numer. Meth. Fluids (2015). Vol. 79 (2), pp. 84-107

Abstract
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed.
Abstract
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution [...]
- 4
- read
Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics
S. Badia, R. Codina, R. Planas

Archives of Comp. Meths. Engng. (2015). Vol. 22 (4), pp. 621-636

Abstract
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.
Abstract
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]
- 1
- read
Finite element dynamical subgrid-scale model for low Mach number flows with radiative heat transfer
M. Ávila, R. Codina, R. Planas

Int. J. Numer. Meth. Heat and Fluid Flows (2015). Vol. 25, pp. 1361-1384

Abstract
Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields. Design/methodology/approach – The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations. Findings – As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show global conservation of thermal energy. Originality/value – The original contribution of the work is the proposal of keeping the VMS splitting into the nonlinear coupling between the low Mach number and the radiative transport equations, its numerical evaluation and the description of its properties.
Abstract
Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative [...]
- 3
- read
First, second and third order fractional step methods for the three-field viscoelastic flow problem
E. Castillo, R. Codina

Journal of Computational Physics (2015). Vol. 296, pp. 113-137

Abstract
In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier–Stokes problem. These methods can be understood as an inexact LU block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order fractional step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the fractional step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.
Abstract
In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and [...]
- 1
- read
Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows
O. Colomés, S. Badia, R. Codina, J. Principe

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 285, pp. 32-63

Abstract
In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. After a brief review of these models, we discuss some implementation aspects particularly relevant to the simulation of turbulent flows, namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used. We analyze the energy conservation (and numerical dissipation) of the alternative VMS formulations, which is numerically evaluated. In the numerical study, we have considered three well known problems: the decay of homogeneous isotropic turbulence, the Taylor–Green vortex problem and the turbulent flow in a channel. We compare the results obtained using different VMS models, paying special attention to the effect of using orthogonal subscale spaces. The VMS results are also compared against classical LES scheme based on filtering and the dynamic Smagorinsky closure. Altogether, our results show the tremendous potential of VMS for the numerical simulation of turbulence. Further, we study the sensitivity of VMS to the algorithmic constants and analyze the behavior in the small time step limit. We have also carried out a computational cost comparison of the different formulations. Out of these experiments, we can state that the numerical results obtained with the different VMS formulations (as far as they converge) are quite similar. However, some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different. The dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness.
Abstract
In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by [...]
- 1
- read
Variational MultiScale based dissipation models for the estimation of Atmospheric Seeing
J. Baiges, R. Codina

Computers and Fluids (2015). Vol. 107, pp. 141-154

Abstract
In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational Multiscale turbulence model, its main feature being that the numerical mechanisms which are used to deal with stability issues (convection and the inf–sup condition for incompressible flows) do also take care of the modeling of turbulence. Based on this turbulence model, we develop the expressions for the viscous and thermal dissipations, ∊∊num and χnum, which are later used for evaluating the constant of structure of the refraction index Cn2 following the classical model developed by Tatarski. Numerical examples show the behavior of the proposed numerical scheme when applied to turbulent flow practical cases, which include a convective boundary layer, the flow inside a transfer optics room, and a telescope enclosure.
Abstract
In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational [...]
- 1
- read
Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 579-605

Abstract
In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to use equal interpolation for the problem unknowns σ–u–p (elastic deviatoric stress–velocity–pressure) and to stabilize dominant convective terms. Starting from residual-based stabilized formulations, the proposed method introduces a term-by-term stabilization which is shown to have a superior behavior when there are stress singularities. A general discontinuity-capturing technique for the elastic stress component is also proposed, which allows one to eliminate the local oscillations that can appear when the Weissenberg number We is high and the fluid flow finds an abrupt change in the geometry. The formulations are tested in the classical 4:1 planar contraction benchmark up to We=5 in the inertial case, with Reynolds number Re=1, and up to We=6.5 in the quasi non-inertial case, with Re=0.01. The standard Oldroyd-B constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation.
Abstract
In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to [...]
- 1
- read
Stabilized stress–velocity–pressure finite element formulations of the Navier–Stokes problem for fluids with non-linear viscosity
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 554-578

Abstract
The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability in the calculation of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf–sup conditions, whereas the dominant convection requires a stabilized formulation in any case. This paper proposes two stabilized schemes of Sub-Grid Scale (SGS) type, differing in the definition of the space of the sub-grid scales, and both allowing to use the same interpolation for the variables σ–u–p (deviatoric stress, velocity and pressure), even in problems where the convection component is dominant and the velocity–stress gradients are high. Another aspect considered in this work is the non-linearity of the viscosity, modeled with constitutive models of quasi-Newtonian type. This paper includes a description of the proposed methods, some of their implementation issues and a discussion about benefits and drawbacks of a three-field formulation. Several numerical examples serve to justify our claims.
Abstract
The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. [...]
- 2
- read
Stability, convergence and accuracy of stabilized finite element methods for the wave equation in mixed form
S. Badia, R. Codina, H. Espinoza

SIAM J. Numer. Anal. (2014). Vol. 52 (4), pp. 1729-1752

Abstract
In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns. The variational forms corresponding to different functional settings are treated in a unified manner through the introduction of length scales related to the unknowns. Stability and convergence analysis is performed together with numerical experiments. It is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods.
Abstract
In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element [...]
- 1
- read
Large eddy simulation of low Mach number flows using a dynamical and nonlinear finite element subgrid scale model
M. Ávila, R. Codina, J. Principe

Computers and Fluids (2014). Vol. 99, pp. 44-66

Abstract
Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is to analyze the behavior of our numerical method in the large eddy simulation (LES) of thermally coupled turbulent flows at low Mach number. Methods: Our numerical method is a stabilized finite element approximation based on the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account results in a stable formulation. The quality of the final approximation (accuracy, efficiency as LES model) depends on the particular subscale model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. Another important contribution of this work is the extension of the orthogonal subgrid scale method widely tested for incompressible flows to variable density flows, using a density-weighted L 2 product to define the orthogonality of the subscales and the finite element spaces. Results: Referring to numerical testing, we present numerical results for a laminar testcase validation that shows the dissipative behavior of the different stabilized methods. Then, we present results of the numerical simulation of two turbulent flow problems, the turbulent channel flow with large temperature differences in the wall normal direction at Res ¼ 180, and the turbulent thermally driven cavity with aspect ratio 4. The behavior of the method is evaluated by comparison against results available in the literature obtained using LES and direct numerical simulation (DNS). They are explained based on a careful analysis of the dissipative structure of the method, showing the physical interpretation of the subgrid scale method presented. Conclusion: The material presented here is a clear indication of the potential of the method to model all kinds of turbulent thermally coupled flows. The formulation is the same in laminar and turbulent regimes
Abstract
Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is [...]
- 4
- read
A Sommerfeld non-reflecting boundary condition for the wave equation in mixed form
H. Espinoza, R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 276, pp. 122-148

Abstract
In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D.
Abstract
In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial [...]
- 1
- read
Error analysis of discontinuous Galerkin methods for the Stokes problem under minimum regularity
S. Badia, R. Codina, T. Gudi, J. Guzmán

IMA Journal of Numerical Analysis (2014). Vol. 34 (2), pp. 800-819

Abstract
In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs to[H10(Ω)]d and the pressurep∈L20(Ω). First, we analyse standard DG methods assuming that the right-hand side f belongs to[H−1(Ω)∩L1(Ω)]d.A DG method that is well defined for f belonging to [H−1(Ω)]d is then investigated. The methods under study include stabilized DG methods using equal-order spaces and inf–sup stable ones where the pressure space is one polynomial degree less than the velocity space.
Abstract
In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs [...]
- 0
- read
On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations
R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2013). Vol. 263, pp. 158-168

Abstract
The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries. Rather than working with this three-field formulation, fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes, although other strategies could be devised. This procedure is applied to four elliptic problems, namely, the convection–diffusion equation (in the diffusion dominated regime), the Stokes problem, the Darcy problem and the Maxwell problem. We justify some well known dG methods with some modifications that in fact allow to improve the performance of the original methods, particularly when the physical properties are discontinuous.
Abstract
The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation [...]
- 1
- read
Immersed stress method for fluid structure interaction using anisotropic mesh adaption
E. Hachem, S. Feghali, R. Codina, T. Coupez

Int. J. Numer. Meth. Engng. (2013). Vol. 94 (9), pp. 805-825

Abstract
This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight.
Abstract
This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with [...]
- 1
- read
Anisotropic adaptive meshing and monolithic variational multiscale method for fluid-structure interaction
E. Hachem, S. Feghali, R. Codina, T. Coupez

Computers and Structures (2013). Vol. 122, pp. 88-100

Abstract
This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. The fluid–solid interfaces are captured using a level-set method. A new a posteriori error estimate, based on the length distribution tensor approach and the associated edge based error analysis, is then used to ensure an accurate capturing of the discontinuities at the fluid–solid interface. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The presence of the structure will be taken into account by means of an extra stress tensor in the Navier–Stokes equations. The system is solved using a stabilized three-field, stress, velocity and pressure finite element (FE) formulation. It consists in the decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales and also in the efficient enrichment of the extra constraint. We assess the accuracy of the proposed formulation by simulating 2D and 3D time-dependent numerical examples such as: falling disk in a channel, turbulent flows behind an airfoil profile and flow behind an immersed vehicle.
Abstract
This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. [...]
- 1
- read
A variational multiscale method with subscales on the element boundaries for the Helmholtz equation
J. Baiges, R. Codina

Int. J. Numer. Meth. Engng. (2013). Vol. 93 (6), pp. 664-684

Abstract
In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method.
Abstract
In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite [...]
- 1
- read
On unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics
S. Badia, R. Codina, R. Planas

Journal of Computational Physics (2013). Vol. 234, pp. 399-416

Abstract
In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation is the fact that it always converges to the physical solution, even for singular ones. A detailed set of numerical experiments have been performed in order to validate our approach.
Abstract
In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]
- 1
- read
A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes
J. Baiges, R. Codina, F. Henke, S. Shahmiri, W. Wall

Int. J. Numer. Meth. Engng. (2012). Vol. 90 (5), pp. 636-658

Abstract
In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element‐discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations.
Abstract
In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic [...]
- 1
- read
A third‐order velocity correction scheme obtained at the discrete level
H. Owen, R. Codina

Int. J. Numer. Meth. Fluids (2012). Vol. 69 (1), pp. 52-72

Abstract
In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed into a discrete pressure Poisson equation and a velocity extrapolation is used. In Badia et al. (IJNMF, 2008, p. 351), where the method was introduced, the discrete Laplacian that appears in the pressure Poisson equation is approximated by a continuous one using an extrapolation for the pressure. In this work we explore the possibility of actually solving the discrete Laplacian. This introduces significant differences because the pressure extrapolation is avoided and only a velocity extrapolation is needed. Our numerical results indicate that it is the second‐order pressure extrapolation which makes third‐order methods unstable. Instead, second‐order velocity extrapolations do not lead to instabilities. Avoiding the pressure extrapolation allows to obtain stable solutions in problems that become unstable when the Laplacian is approximated. A comparison with a pressure correction scheme is also presented to verify the well‐known fact that the use of a second order pressure extrapolation leads to instabilities. Therefore we conclude that it is the combination of a velocity correction scheme with a discrete Laplacian that allows to obtain a stable third‐order scheme by avoiding the pressure extrapolation.
Abstract
In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed [...]
- 1
- read
A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions
S. Badia, R. Codina

SIAM J. Numer. Anal. (2012). Vol. 50 (2), pp. 398-417

Abstract
A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl 0; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal-based interpolations are known to suffer from spurious convergence upon mesh refinement.
Abstract
A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous [...]
- 3
- read
Stokes, Maxwell and Darcy: A single finite element approximation for three model problems
S. Badia, R. Codina

Applied Numerical Mathematics (2012). Vol. 62 (4) pp. 246-263

Abstract
In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantages of our approach. In particular, for Darcyʼs problem we are able to design stabilized methods that yield optimal convergence both for the primal and the dual problems. In the case of Maxwellʼs problem, the formulation we propose allows one to use continuous finite element interpolations that converge optimally to the continuous solution even if it is non-smooth. Once the formulation is presented for the three model problems independently, we also show how it can be used for a problem that combines all the operators of the independent problems. Stability and convergence is achieved regardless of the fact that any of these operators dominates the others, a feature not possible for the methods of which we are aware.
Abstract
In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We [...]
- 1
- read
A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem
S. Badia, R. Codina

Applied Mathematics and Computation (2011). Vol. 218 (8), pp. 4276-4294

Abstract
Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG–dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functins continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.
Abstract
Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are [...]
- 1
- read
The Fixed‐Mesh ALE approach for the numerical simulation of floating solids
J. Baiges, R. Codina, H. Coppola-Owen

Int. J. Numer. Meth. Fluids (2011). Vol. 67 (8), pp. 1004-1023

Abstract
In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. We pay special attention to the tracking of the various interfaces and their intersections, and to the approximate imposition of coupling conditions between the solid and the fluid.
Abstract
In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature [...]
- 1
- read
Finite element dynamical subgrid-scale approximation of low Mach number flow equations
M. Ávila, J. Principe, R. Codina

Journal of Computational Physics (2011). Vol. 230, pp. 7988-8009

Abstract
In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number equations, which are obtained as a limit of the compressible Navier Stokes equations in the small Mach number. In contrast to the commonly used Boussinesq approximation, this model permits to take volumetric deformation into account. Although the former is more general than the later, both systems have similar mathematical structure and their numerical approximation can suffer the same type of instabilities. We propose a stabilized finite element approximation based on the the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account, results in a stable formulation. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. The first ingredient permits to obtain an improved time discretization scheme (higher accuracy, better stability, no restrictions on the time step size). The second ingredient permits to prove global conservation properties. It also allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view. Numerical tests show that nonlinear and dynamic subscales give more accurate solutions than classical stabilized methods.
Abstract
In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number [...]
- 0
- read
Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2011). Vol. 87 (1-5), pp. 386-411

Abstract
Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized finite element approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non‐conforming part, a hybrid formulation for the latter and a simple approximation for the unknowns involved in the hybrid problem.
Abstract
Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are [...]
- 1
- read
Thermal coupling of fluid and structural response of a tunnel induced by fire
B. Schrefler, R. Codina, F. Pesavento, J. Principe

Int. J. Numer. Meth. Engng. (2010). Vol. 87 (1-5), pp. 361-385

Abstract
In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling strategy is based on a Dirichlet/Neumann non‐overlapping domain decomposition of the problem, which is carried out by developing a master code that controls solvers dedicated to the fluid mechanics and to the solid mechanics simulation. The computational fluid dynamics formulation consists of a stabilized finite element approximation of the low Mach number equations based on the subgrid scale concept, that allows us to deal with convection‐dominated problems and to use equal order interpolation of velocity and pressure. The thermo‐structural model of the tunnel vault, that considers a multiphase porous material where pores are partly filled with liquid and partly by gas, is specially devised for the simulation of concrete at high temperatures and consists of balance equations for mass conservation of dry air, mass conservation of water species (both in the liquid and gaseous state), enthalpy conservation and linear momentum conservation taking phase changes into account. The developed algorithm is applied to the problem of the response of a tunnel to a fire. We consider the combustion process as a heat release which can vary usually from 1thinspaceMW for small car fires to 100 MW for catastrophic fires. The released heat is transferred to the concrete walls of the tunnel which could cause extensive and heavy damage of the structure.
Abstract
In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling [...]
- 1
- read
The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems
J. Baiges, R. Codina

Int. J. Numer. Meth. Engng. (2009). Vol. 81 (12), pp. 1529-1557

Abstract
In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed‐mesh (FM)‐ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM‐ALE allows for the use of a single background mesh to solve both the fluid and the structure.
Abstract
In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. [...]
- 0
- read
A free surface finite element model for low Froude number mould filling problems on fixed meshes
H. Coppola-Owen, R. Codina

Int. J. Numer. Meth. Fluids (2011). Vol. 66 (7), pp. 833-851

Abstract
The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position of the interface and the resulting flow field increases. The usual two‐phase flow model provides poor results for such flow. In order to overcome the difficulties, a free surface model that applies boundary conditions at the interface accurately is used. Moreover, the use of wall laws on curved boundaries also fails in the case of low Froude number flows. To solve this second problem, we combine wall laws with ‘do nothing’ boundary conditions. A special feature of our approach is that ‘do nothing’ boundary conditions are only applied in the normal direction. These two key ingredients together with the Level Set method allow us to simulate three‐dimensional mould filling problems borrowed directly from the foundry.
Abstract
The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position [...]
- 1
- read
Approximation of the inductionless MHD problem using a stabilized finite element method
R. Planas, S. Badia, R. Codina

Journal of Computational Physics (2011). Vol. 230 (8), pp. 2977-2996

Abstract
In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers.
Abstract
In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD [...]
- 2
- read
Approximation of the thermally coupled MHD problem using a stabilized finite element method
R. Codina, N. Hernández-Silva

Journal of Computational Physics (2011). Vol. 230 (4), pp. 1281-1303

Abstract
A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization terms, which serve several purposes. First, convective dominated flows in the Navier–Stokes and the heat equation can be dealt with. Second, there is no restriction in the choice of the interpolation spaces of all the variables and, finally, flows highly coupled with the magnetic field can be accounted for. Different aspects related to the design of the final fully discrete and linearized algorithm are also discussed.
Abstract
A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization [...]
- 1
- read
Spatial approximation of the radiation transport equation using a subgrid-scale finite element method
M. Ávila, R. Codina, J. Principe

Comput. Methods Appl. Mech. Engrg., (2011). Vol. 200 (5-8), pp. 425-438

Abstract
In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.
Abstract
In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition [...]
- 1
- read
Long-term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical sub-grid scale modeling
S. Badia, R. Codina, J. Gutiérrez-Santacreu

SIAM J. Numer. Anal. (2010). Vol. 48 (3), pp. 1013-1037

Abstract
Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L 2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation.
Abstract
Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure [...]
- 2
- read
Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling
R. Codina, J. Principe, M. Ávila

Int. J. Numer. Meth. Heat and Fluid Flow (2010). Vol. 20 (5), pp. 492-516

Abstract
The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. This strategy allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view and discuss important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation
Abstract
The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation [...]
- 2
- read
Stabilized continuous and discontinuous Galerkin techniques for Darcy flow
S. Badia, R. Codina

Comput. Methods Appl. Mech. Engrg., (2010). vol. 199 (25-28), pp. 1654-1667

Abstract
We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2-norm.
Abstract
We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. [...]
- 1
- read
On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes
J. Principe, R. Codina

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (21-22), pp. 1386-1402

Abstract
In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated. In order to incorporate the distortion of the mesh into this approximation, we transform the equation for the subgrid scale within each element to the shape-regular reference domain. The expression for the subgrid scale arises from an approximate Fourier analysis and the identification of the wave number direction where instabilities are most likely to occur. The final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation.
Abstract
In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting [...]
- 3
- read
The dissipative structure of variational multiscale methods for incompressible flows
J. Principe, R. Codina, F. Henke

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (13-16), pp. 791-801

Abstract
In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.
Abstract
In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. [...]
- 3
- read
A numerical strategy to compute optical parameters in turbulent flow. Application to telescopes
R. Codina, J. Baiges, D. Pérez-Sánchez, M. Collados

Computers and Fluids (2010). Vol. 39 (1), pp. 87-98

Abstract
We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier-Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.
Abstract
We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible [...]
- 3
- read
Approximate imposition of boundary conditions in immersed boundary methods
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2009). Vol. 80 (11), pp. 1379-1405

Abstract
We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition
Abstract
We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element [...]
- 2
- read
Computational aeroacoustics of viscous low speed flows using subgrid scale finite element methods
O. Guasch, R. Codina

Journal of Computational Acoustics (2009). Vol. 17 (3), pp. 309-330

Abstract
A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure is followed that makes use of Lighthill's acoustic analogy in the frequency domain. The procedure has been conceptually divided into three steps. In the first one, the incompressible Navier–Stokes equations are solved to obtain the flow velocity field. In the second step, Lighthill's acoustic source term is computed from this velocity field and then Fourier transformed to the frequency domain. Finally, the acoustic pressure field is obtained by solving the corresponding inhomogeneous Helmholtz equation. All equations in the formulation are solved using subgrid scale stabilized finite element methods. The main ideas of the subgrid scale numerical strategy are outlined and its benefits when compared to the Galerkin approach are described. As numerical examples, the aerodynamic noise generated by flow past a two-dimensional cylinder and by flow past two cylinders in parallel arrangement are addressed.
Abstract
A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure [...]
- 2
- read
Mathematical models for thermally coupled low speed flows
J. Principe, R. Codina

Adv. Theor. Appl. Mech. (2009). Vol. 2 (2), pp. 93-112

Abstract
In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat exchange (the isentropic regime) this limit is well understood and rigorous results are available. When heat exchange is considered different simplified models can be obtained, the most famous being the Boussinesq approximation. Here a unified formal justification of these models is presented, paying special attention to the relation between the low Mach number and the Boussinesq approximations. Precise conditions for their validity are given for classical problems in bounded domains.
Abstract
In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat [...]
- 2
- read
Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems
S. Badia, R. Codina

SIAM J. Numer. Anal. (2009). Vol. 47 (3), pp. 1971-2000

Abstract
In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior
Abstract
In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart [...]
- 2
- read
A numerical approximation of the thermal coupling of fluids and solids
J. Principe, R. Codina

Int. J. Numer. Meth. Fluids (2009). Vol. 59 (11), pp. 1181-1201

Abstract
In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context.
Abstract
In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including [...]
- 2
- read
On a multiscale approach to the transient Stokes problem: Dynamic subscales and anisotropic space–time discretization
S. Badia, R. Codina

Applied Mathematics and Computation (2009). Vol. 207 (2), pp. 415-433

Abstract
In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. We define an anisotropic time–space discretization as a family of time–space partitions that does not satisfy the condition h2⩽Cδt with C uniform with respect to h and δt. Standard residual-based stabilization techniques are motivated by a multiscale approach, approximating the effect of the subscales onto the large scales. One of the approximations is to consider the subscales quasi-static (neglecting their time derivative). It is well known that these techniques are unstable for anisotropic time–space discretizations. We show that the use of dynamic subscales (where the subscales time derivatives are not neglected) solves the problem, and prove optimal convergence and stability results that are valid for anisotropic time–space discretizations. Also the improvements related to the use of orthogonal subscales are addressed.
Abstract
In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. [...]
- 1
- read
The fixed-mesh ALE approach for the numerical approximation of flows in moving domains
R. Codina, G. Houzeaux, H. Coppola-Owen, J. Baiges

Journal of Computational Physics (2009). Vol. 228 (5), pp. 1591-1611

Abstract
In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation to be presented falls within the category of fixed-grid methods. Even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed-grid method, our approach can be applied together with any technique to impose approximately boundary conditions, although we also describe the one we actually favor. Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.
Abstract
In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation [...]
- 2
- read
Finite element approximation of the three field formulation of the Stokes problem using arbitrary interpolations
R. Codina

SIAM J. Numer. Anal. (2009). Vol. 47 (1), pp. 699-718

Abstract
The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis.
Abstract
The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of [...]
- 2
- read
Subscales on the element boundaries in the variational two-scale finite element method
R. Codina, J. Principe, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (5-8), pp. 838-852

Abstract
In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. In particular, we consider the scalar convection–diffusion–reaction equation, the Stokes problem and Darcy’s problem. For these problems the transmission conditions are the continuity of the unknown and its fluxes through element boundaries. The former is automatically achieved by introducing a single valued subscale on the boundaries (for the conforming approximations we consider), whereas the latter provides the effective condition for approximating these values. The final result is that the subscale on the interelement boundaries must be proportional to the jump of the flux of the finite element component and the average of the subscale calculated in the element interiors.
Abstract
In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. [...]
- 1
- read
A stabilized finite element approximation of low speed thermally coupled flows
J. Principe, R. Codina

Int. J. Num. Meths. for Heat and Fluid Flows. (2008). Vol. 18 (7-8), pp. 835-867

Abstract
Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed. Design/methodology/approach – The formulation, that allows us to consider convection dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity-pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed. Findings – A full Newtown linearization turns out to be very efficient for steady-state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity-pressure and temperature happens to be more appropriate for transient problems. Research limitations/implications – A fractional step scheme, splitting also momentum and continuity equations, could be further analysed. Practical implications – The results presented in the paper are useful to decide the solution strategy for a given problem. Originality/value – The numerical implementation of a stabilized finite element approximation of thermally coupled flows is described. The implementation algorithm is developed considering several possibilities for the solution of the discrete nonlinear problem.
Abstract
Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number [...]
- 1
- read
Algebraic pressure segregation methods for the incompressible Navier-Stokes equations
S. Badia, R. Codina

Archives of Comp. Meths. Engng. (2008). Vol. 15 (3), pp. 1-52

Abstract
This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled.
Abstract
This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block [...]
- 2
- read
Finite element approximation of the modified Boussinesq equations using a stabilized formulation
R. Codina, J. González-Ondina, G. Díaz-Hernández, J. Principe

Int. J. Numer. Meth. Fluids (2008). Vol. 57 (9), pp. 1305-1322

Abstract
In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system of equations, namely, the need to use stable velocity‐depth interpolations, which can be overcome by the use of a stabilization technique. The one described in this paper is based on the splitting of the unknowns into their finite element component and the remainder, which we call the subgrid scale. We also discuss the treatment of high‐order derivatives of the mathematical model and describe the time integration scheme
Abstract
In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system [...]
- 1
- read
Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscale
R. Codina

Applied Numerical Mathematics (2008). Vol. 58 (3), pp. 264-283

Abstract
In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf–sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. It is shown that this formulation is stable and optimally convergent for an adequate choice of the algorithmic parameters on which the method depends.
Abstract
In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility [...]
- 1
- read
Finite element approximation of the hyperbolic wave equation in mixed form
R. Codina

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (13-16), pp. 1305-1322

Abstract
The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector field to transform the problem into a first-order problem in space and time. We explain why the standard Galerkin method is inappropriate to solve this problem, and propose as alternative a stabilized finite element method that can be cast in the variational multiscale framework. The unknown is split into its finite element component and a remainder, referred to as subscale. As original features of our approach, we consider the possibility of letting the subscales to be time dependent and orthogonal to the finite element space. The formulation depends on algorithmic parameters whose expression is proposed from a heuristic Fourier analysis.
Abstract
The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector [...]
- 1
- read
On some fluid-structure iterative algorithms using pressure segregation methods. Applications to aeroelasticity
S. Badia, R. Codina

Int. J. Numer. Meth. Engng. (2007). Vol. 72 (1), pp. 46-71

Abstract
In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure segregation methods for the solution of the fluid problem and fixed point iterative algorithms for the fluid-structure coupling. These coupling algorithms are applied to the aeroelastic simulation of suspension bridges. We assess flexural and torsional frequencies for a given inflow velocity. Increasing this velocity we reach the value for which the flutter phenomenon appears.
Abstract
In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure [...]
- 1
- read
Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach
S. Badia, R. Codina

Int. J. Numer. Meth. Fluids (2008). Vol. 56 (4), pp. 351-382

Abstract
In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been replaced by a pressure Poisson equation obtained at the discrete level. In these methods it is the velocity instead of the pressure the extrapolated unknown. Moreover, predictor–corrector schemes are suggested, again motivated by the new monolithic system. Key implementation aspects are discussed, and a complete stability analysis is performed. We end with a set of numerical examples in order to compare these methods with classical pressure‐correction schemes.
Abstract
In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been [...]
- 1
- read
Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition
S. Badia, R. Codina

Numerische Mathematik (2007). Vol. 107 (4), pp. 533-557

Abstract
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.
Abstract
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using [...]
- 1
- read
Shockwaves in spillways with the particle finite element method
F. Salazar, J. San-Mauro, M. Celigueta, E. Oñate

Comp. Part. Mech. (2019).

Abstract
Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. These disturbances are propagated downstream and need to be considered in the design of the chute walls. In dam spillways, where flow rates are often high, this phenomenon can have significant implications for the cost and complexity of the solution. It has been traditionally analysed by means of reduced-scale experimental tests, as it has a clear three-dimensional character and therefore cannot be approached with two-dimensional numerical models. In this work, the ability of the particle finite element method (PFEM) to reproduce this phenomenon is analysed. PFEM has been successfully applied in previous works to problems involving high irregularities in free surface. First, simple test cases available in the technical bibliography were selected to be reproduced with PFEM. Subsequently, the method was applied in two spillways of real dams. The results show that PFEM is capable of capturing the shockwave fronts generated both in the contractions and in the expansions that occur behind the spillway piers. This suggests that the method may be useful as a complement to laboratory test campaigns for the design and hydraulic analysis of dam spillways with complex geometries.
Abstract
Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. [...]
- 1
- read
An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics
O. Guasch, R. Codina

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (45-48), pp. 4672-4689

Abstract
An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization.
Abstract
An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element [...]
- 2
- read
A finite element model for free surface flows on fixed meshes
H. Coppola-Owen, R. Codina

Int. J. Numer. Meth. Fluids (2007). Vol. 54 (10), pp. 1151-1171

Abstract
In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow.
Abstract
In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element [...]
- 2
- read
Dynamic subscales in the finite element approximation of thermally coupled incompressible flows
R. Codina, J. Principe

Int. J. Numer. Meth. Fluids (2007). Vol. 54 (6-8), pp. 707-730

Abstract
In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient. They are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. Apart from presenting the main properties of the formulation, we also discuss some computational aspects such as the linearization strategy or the way to integrate in time the equation for the subscales.
Abstract
In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to [...]
- 1
- read
Time dependent subscales in the stabilized finite element approximation of incompressible flow problems
R. Codina, J. Principe, O. Guasch, S. Badia

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (21-24), pp. 2413-2430

Abstract
In this paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point is that we explore the properties of the discrete formulation that results allowing the subgrid-scales to depend on time. This apparently “natural” idea avoids several inconsistencies of previous formulations and also opens the door to generalizations.
Abstract
In this paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point [...]
- 1
- read
A finite element method for the solution of rotary pumps
G. Houzeaux, R. Codina

Computers and Fluids (2007). Vol. 36 (4), pp. 667-679

Abstract
We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are rotating, the intersection between them changes in time. Therefore, the computational domain should be recomputed in some way at each time step. The strategy used here consists in dividing a cycle into a certain number of time steps and obtaining different computational meshes for each of these time steps. The coupling between two consecutive time steps is achieved by interpolating the flow unknowns in a proper way. This geometrical decomposition enables one to have a plain control over the mesh, particularly in the zones of interest, which are the gap between the gears and the casing, and the engagement and disengagement zones of the gears.
Abstract
We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are [...]
- 2
- read
Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ALE framework
S. Badia, R. Codina

SIAM J. Numer. Anal. (2006). Vol. 44 (5), pp. 2159-2197

Abstract
In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semidiscrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.
Abstract
In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) [...]
- 0
- read
Numerical approximation of the heat transfer between domains separated by thin walls
R. Codina, G. Houzeaux

Int. J. Numer. Meth. Fluids (2006). Vol. 52 (9), pp. 963-986

Abstract
In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin solid wall. First of all, we state the problem in the whole domain with discontinuous physical properties. As an alternative and under certain assumptions on the separating walls, a classical Robin boundary condition between the fluid domains is obtained, thus eliminating the solid wall, and according to which the heat flux is proportional to the temperature difference between the two subdomains. Apart from discussing the relation between both approaches, we consider their numerical approximation, considering different alternatives for the first case, that is, the case in which temperatures are also computed in the solid wall.
Abstract
In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin [...]
- 1
- read
Stabilized finite element approximation of the stationary magneto-hydrodynamics equations
R. Codina, N. Hernández-Silva

Computational Mechanics (2006). Vol. 38 (4-5), pp. 344-355

Abstract
In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational concept. The linearization that yields a well posed linear problem is first identified, and for this linear problem the stabilization method is designed. The key point is the correct behavior of the stabilization parameters on which the formulation depends. It is shown that their expression can be obtained only on the basis of having a correct error estimate. For the stabilization parameters chosen, a stability estimate is proved in detail, as well as the convergence of the numerical solution to the continuous one. The method is then extended to nonlinear problems and its performance checked through numerical experiments.
Abstract
In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational [...]
- 2
- read
Finite element modeling of the lost foam casting process tackling back‐pressure effects
G. Houzeaux, R. Codina

Int. J. Num. Meths. for Heat and Fluid Flow (2006). Vol. 16 (5), pp. 573-589

Abstract
Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – Back-pressure effects are due to the interactions of many physical processes. The strategy proposed herein tries to model all these processes within a simple formula. The main characteristic of the model consists of assuming that the back-pressure is a known function of the external parameters (coating, temperature, gravity, etc.) that affects directly the heat transfer coefficient from the metal to the foam. The general framework of the simulation is a finite element model based on an arbitrary Lagrangian Eulerian (ALE) approach and the use of level set function to capture the metal front advance. Findings – After experimental tunings, the model provides a way to include the back-pressure effects in a simple way. Research limitations/implications – The method is not completely predictive in the sense that a priori tuning is necessary to calibrate the model. Practical implications – Provides more realistic results than classical models. Originality/value – The paper proposes a theoretical framework of a finite element method for the simulation of LFC process. The method uses an ALE method on a fixed mesh and a level-set function to capture metal front advance. It proposes an original formula for the heat transfer coefficient that enables one to include back-pressure effects
Abstract
Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – [...]
- 1
- read
The Characteristic‐Based Split (CBS) scheme—a unified approach to fluid dynamics
P. Nithiarasu, R. Codina, O. Zienkiewicz

Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1514-1546

Abstract
This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the original characteristic schemes are discussed. The important features of the CBS scheme are brought out by studying several problems of compressible and incompressible flows. All special consideration necessary for solving these problems are thoroughly discussed. The CBS scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided. Several invicid and viscous flow examples are also provided to demonstrate the unified CBS approach. For sample two‐dimensional codes, input files and instructions, the readers are referred to ‘www.nithiarasu.co.uk’
Abstract
This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the [...]
- 2
- read
Numerical comparison of CBS and SGS as stabilization techniques for the incompressible Navier–Stokes equations
R. Codina, H. Coppola-Owen, P. Nithiarasu, C. Liu

Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1677-1689

Abstract
In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection‐dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The other two approaches are particular cases of the subgrid scale (SGS) method. The first, obtained after an algebraic approximation of the subgrid scales, is very similar to the popular Galerkin/least‐squares (GLS) method, whereas in the second, the subscales are assumed to be orthogonal to the finite element space. It is shown that all these formulations display similar stabilization mechanisms, provided the stabilization parameter of the SGS methods is identified with the time step of the CBS approach. This paper provides the numerical experiments for the comparison of formulations made by Codina and Zienkiewicz in a previous article.
Abstract
In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. The first is the characteristic‐based split (CBS). [...]
- 1
- read
On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems
R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (23-24), pp. 2900-2918

Abstract
In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities and the pressure. The first family of schemes consists of classical fractional step methods, of which we discuss several possibilities for the pressure extrapolation and the time integration of first and second order. The second family consists of schemes based on an explicit treatment of the pressure in the momentum equation followed by a Poisson equation for the pressure. It turns out that this “staggered” treatment of the velocity and the pressure is stable. Finally, we present predictor-corrector methods based on the above schemes that aim to converge to the solution of the monolithic time integration method. Apart from presenting these schemes and check its numerical performance, we also present a complete set of stability results for the fractional step methods that are independent of the space stability of the velocity-pressure interpolation, that is, of the classical inf-sup condition.
Abstract
In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities [...]
- 1
- read
Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions
A. Coppola‐Owen, R. Codina

Int. J. Numer. Meth. Fluids (2005). Vol. 49 (12), pp. 1287-1304

Abstract
In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. We propose a solution based on enriching the pressure shape functions on the elements cut by the interface. The enrichment is used to enable the pressure gradient to be discontinuous at the interface, thus improving the ability to simulate the behaviour of fluids with different density under a gravitational force. The additional shape function used is local to each element and the corresponding degree of freedom can therefore be condensed prior to assembly, making the implementation quite simple on any existing finite element code.
Abstract
In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. [...]
- 1
- read
Error estimates for an operator-splitting method for incompressible flows
J. Blasco, R. Codina

Applied Numerical Mathematics (2004). Vol. 51 (1), pp. 1-17

Abstract
In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain first order error estimates in the time step size, both for the intermediate and the end-of-step velocities of the method; we also give some error estimates for the pressure solution.
Abstract
In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity [...]
- 1
- read
A finite element model for the simulation of lost foam casting
G. Houzeaux, R. Codina

Int. J. Numer. Meth. Fluids (2004). Vol. 46 (2), pp. 203-226

Abstract
In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes in play during a casting. We also briefly comment on the possible physical and numerical models to envisage the numerical simulation. Next we present a model which aims at solving “part of” the complexities of the casting, together with a simple energy budget that enables to obtain an equation for the velocity of the metal front advance. Once the physical model is established we develop a finite element method to solve the governing equations. The numerical and physical methodologies are then validated through the solution of a two and a three-dimensional example. Finally, we briefly discuss some possible improvements of the numerical model in order to catch more physical phenomena.
Abstract
In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes [...]
- 1
- read
Approximation of the incompressible Navier–Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes
R. Codina, O. Soto

Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (15-16), pp. 1403-1419

Abstract
This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. The former consists of adding a least-square form of the component orthogonal to the finite element space of the convective and pressure gradient terms; this allows to deal with convection-dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Likewise, we put special emphasis on the use of anisotropic grids. In particular, we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples. The preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described.
Abstract
This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. [...]
- 5
- read
A stabilized finite element predictor–corrector scheme for the incompressible Navier–Stokes equations using a nodal‐based implementation
R. Codina, A. Folch

Int. J. Numer. Meth. Fluids (2004). Vol. 44 (5), pp. 483-503

Abstract
A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to segregate the pressure and a nodal based implementation is presented in this paper. The stabilization consists of adding a least‐squares form of the component orthogonal to the finite element space of the convective and pressure gradient terms, which allows to deal with convection‐dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Finally, the nodal‐based implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements and thus making the calculations much faster. Some issues related to this implementation for fractional step and our predictor–corrector scheme, which is the main contribution of this paper, are discussed
Abstract
A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to [...]
- 1
- read
A Dirichlet/Neumann domain decomposition method for incompressible turbulent flows on overlapping subdomains
G. Houzeaux, R. Codina

Computers and Fluids (2004). Vol. 33 (5-6), pp. 771-782

Abstract
When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility is the use of domain decomposition methods. The domain decomposition method we present in this work aims at coupling overlapping subdomains in relative motion using a Dirichlet/Neumann coupling. The method is applied to the solution of incompressible and turbulent flows.
Abstract
When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility [...]
- 2
- read
An iteration-by-subdomain overlapping Dirichlet/Robin domain decomposition method for advection–diffusion problems
G. Houzeaux, R. Codina

Journal of Computational and Applied Mathematics (2003). Vol. 158 (2), pp. 243-276

Abstract
We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning of the computational domain. We derive interface equations to study the convergence of the method and show its properties through four numerical examples. The mathematical framework is general and can be applied to derive overlapping versions of all the classical nonoverlapping methods.
Abstract
We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning [...]
- 1
- read
A Chimera method based on a Dirichlet/Neumann(Robin) coupling for the Navier–Stokes equations
G. Houzeaux, R. Codina

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 192 (31-32), pp. 3343-3377

Abstract
We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain method based on Dirichlet/Neumann(Robin) coupling. The DD method we propose is not only geometric but also algorithmic, for the solution on each subdomain is obtained on separate processes and the exchange of information between the subdomains is carried out by a master code. This strategy is very flexible as it requires almost no modification to the original numerical code. Therefore, only the master code has to be adapted to the numerical codes and the strategies used on each subdomain. As a basic flow solver, we a use stabilized finite element method.
Abstract
We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain [...]
- 1
- read
Stabilized finite element approximation of transient incompressible flows using orthogonal subscales
R. Codina

Comput. Methods Appl. Mech. Engrg., (2002). Vol. 191 (39-40), pp. 4295-4321

Abstract
In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.
Abstract
In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable [...]
- 1
- read
A numerical model to track two‐fluid interfaces based on a stabilized finite element method and the level set technique
R. Codina, O. Soto

Int. J. Numer. Meth. Fluids (2002). Vol. 40 (1-2), pp. 293-301

Abstract
The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. The stabilization technique employed allows us to use equal velocity–pressure interpolations and to deal with convection–dominated flows. The tracking of the fluid interface is based on the level set technique. A novel smoothing technique of this surface based on ideas of signal processing is presented.
Abstract
The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. [...]
- 1
- read
Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ale framework
S. Badia, R. Codina

Computing and Visualization in Science (2002). Vol. 4 (3), pp. 167-174

Abstract
In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semi-discrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.
Abstract
In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, [...]
- 2
- read
CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter
R. Codina, O. Zienkiewicz

Commun. Numer. Meth. Engng (2001). Vol 18 (2), pp. 99-112

Abstract
In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem.
Abstract
In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first [...]
- 1
- read
Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection
R. Codina, J. Blasco, G. Buscaglia, A. Huerta

Int. J. Numer. Meth. Fluids (2001). Vol. 37 (4), pp. 419-444

Abstract
We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the project of the pressure gradient and adding the difference between the pressure Laplacian and divergence of this new field to the incompressibility equations, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for stationary and the transient of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP.
Abstract
We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure [...]
- 1
- read
Numerical modeling of magma withdrawal during explosive caldera-forming eruptions
A. Folch, J. Martí i Castell, R. Codina

Journal of Geophysical Research (2001). Vol. 106 (B8), pp. 16 163-16 175

Abstract
We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such eruptions as a piston-like system in which the host rock is assumed to subside as a coherent rigid solid. Magma behaves as a Newtonian incompressible fluid below the exsolution level and as a compressible gas-liquid mixture above this level. We consider caldera-forming eruptions within the frame of fluid-structure interaction problems, in which the flow-governing equations are written using an arbitrary Lagrangian-Eulerian (ALE) formulation. We propose a numerical procedure to solve the ALE governing equations in the context of a finite element method. The numerical methodology is based on a staggered algorithm in which the flow and the structural equations are alternatively integrated in time by using separate solvers. The procedure also involves the use of the quasi-Laplacian method to compute the ALE mesh of the fluid and a new conservative remeshing strategy. Despite the fact that we focus the application of the procedure toward modeling caldera-forming eruptions, the numerical procedure is of general applicability. The numerical results have important geological implications in terms of magma chamber dynamics during explosive caldera-forming eruptions. Simulations predict a nearly constant velocity of caldera subsidence that strongly depends on magma viscosity. They also reproduce the characteristic eruption rates of the different phases of an explosive calderaforming eruption. Numerical results indicate that the formation of vortices beneath the ring fault, which may allow mingling and mixing of parcels of magma initially located at different depths in the chamber, is likely to occur for low-viscosity magmas. Numerical results confirm that exsolution of volatiles is an efficient mechanism to sustain explosive caldera-forming eruptions and to explain the formation of large volumes of ignimbrites.
Abstract
We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such [...]
- 2
- read
Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations
J. Blasco, R. Codina

Applied Numerical Mathematics (2001). Vol. 38 (4), pp. 475-497

Abstract
In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the time discretization we focus on a fully implicit, monolithic scheme. We provide some error estimates for the fully discrete solution which show that the velocity is first order accurate in the time step and attains optimal order accuracy in the mesh size for the given spatial interpolation, both in the spaces L2(Ω) and H10(Ω); the pressure solution is shown to be order 12 accurate in the time step and also optimal in the mesh size. These estimates are proved assuming only a weak compatibility condition on the approximating spaces of velocity and pressure, which is satisfied by equal order interpolations.
Abstract
In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the [...]
- 1
- read
Pressure stability in fractional step finite element methods for incompressible flows
R. Codina

Journal of Computational Physics (2001). Vol. 170 (1), pp. 112-140

Abstract
The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second order scheme in which part of the extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.
Abstract
The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection [...]
- 2
- read
Transmission conditions with constraints in finite element domain decomposition methods for flow problems
G. Houzeaux, R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol 17 (3), pp. 179-190

Abstract
This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD algorithm is based on the iterative update of the boundary conditions on the interfaces between the subregions, the so‐called transmission conditions. The transmission conditions involve the essential and natural boundary conditions of the weak form of the problem, and should ensure strong continuity of the velocity and weak continuity of the traction. As a first approach, the transmission conditions are interpolated using the classical Lagrange interpolation functions. Conservation problems might arise when two adjacent subdomains have a sensibly different mesh spacing. In order to conserve any desired quantity of interest, an interface constraining is introduced: continuity of the transmission conditions are constrained under a scalar conservation equation. An example of mass conservation illustrates the algorithm.
Abstract
This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD [...]
- 2
- read
A stabilized finite element method for generalized stationary incompressible flows
R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol. 190 (20-21), pp. 2681-2706

Abstract
In this paper we describe a finite element formulation for the numerical solution of the stationary Navier-Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection-diffusion-reaction equations and then we apply it to linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.
Abstract
In this paper we describe a finite element formulation for the numerical solution of the stationary Navier-Stokes equations including Coriolis forces and the permeability [...]
- 1
- read
Stabilization of incompressibility and convection through orthogonal sub-scale in finite element methods
R. Codina

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 190 (13-14), pp. 1579-1599

Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first of them is the stabilization thought the pressure gradient projection, which consists of adding a certain least-squares form of the difference between the pressure gradient and its L² projection onto the discrete velocity space in the variational equations of the problem. The second is a sub-grid scale method, whose stabilization effect is very similar to that of the Galerkin/least-squares method for the Stokes problem. It is shown here that the first method can also be recast in the framework of sub-grid scale method with a particular choice for the space of sub-scales. This leads to a new stabilization procedure, whose applicability to stabilize convection is also studied in this paper.
Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first [...]
- 1
- read
Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations
R. Codina, J. Blasco

Numerische Mathematik (2000). Vol. 87 (1), pp. 59-81

Abstract
The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the L2 norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this.
Abstract
The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. [...]
- 2
- read
Fourier analysis of an equal‐order incompressible flow solver stabilized by pressure gradient projection
G. Buscaglia, F. Basombrío, R. Codina

Int. J. Numer. Meth. Fluids (2000). Vol. 34 (1), pp. 65-92

Abstract
Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes equations, here denoted by pressure gradient projection (SPGP) method. The analysis is motivated by spurious waves that pollute the computed pressure in start‐up flow simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behaviour of stabilized methods with remarkable accuracy, even though the original Navier–Stokes setting must be significantly simplified to apply them. In the steady state case, good estimates for the stabilization parameters are obtained. In the transient case, spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though this implies additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons with Galerkin–least‐squares (GLS) method and Chorin's projection method are also addressed
Abstract
Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes [...]
- 1
- read
A nodal-based implementation of a stabilized Finite Element Method for Incompressible Flow Problems
R. Codina

Int. J. Numer. Meth. Fluids (2000). Vol. 33 (5), pp. 737-766

Abstract
The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are conducted to check its performance. This method is able to deal with all the instabilities that the standard Galerkin method presents, namely, the pressure instability, the instability arising in convection dominated situations and also the less popular instabilities found when the Navier-Stokes equations have a dominant Coriolis force, or there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal-based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over element and thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this given rise to questions regarding the consistency and the conservation properties of the final scheme that are addressed in this paper.
Abstract
The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are [...]
- 1
- read
On stabilized finite element methods for linear systems of convection-diffusion-reaction equations
R. Codina

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 188 (200), pp. 61-82

Abstract
A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a 1D model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner-Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.
Abstract
A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach [...]
- 1
- read
Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection
R. Codina, J. Blasco

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3-4), pp. 277-300

Abstract
In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the Laplacian of the pressure and the divergence of this new vector field. This leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures. In the case of the transient equations, we consider the possibility of treating the pressure gradient projection either implicitly or explicity. In the first case, the number of unknowns of the problem is substantially increased with respect to the standard Galerkin formulation. Nevertheless, iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns (velocity and pressure). When this vector field is treated explicitly, the increment of computational cost of the stabilized formulation with respect to the Galerkin method is very low. We provide a stability estimate for the case of the simple backward Euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection.
Abstract
In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown [...]
- 1
- read
Finite element simulation of the filling of thin moulds
R. Codina, O. Soto

Int. J. Numer. Meth. Engng. (1999). Vol. 46 (9), pp. 1559-1573

Abstract
In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, such as the Q2/P1 element (tri‐quadratic velocities and piecewise linear, discontinuous pressures) or stabilized finite element formulations. The tracking of the free surface is based on the Volume‐Of‐Fluid (VOF) method. The velocity profile is assumed to be parabolic in the direction normal to the mid‐plane, so that one element along the width of the mould is enough to reproduce this profile if this element is quadratic. The velocity is prescribed to zero on the upper and lower surfaces and the normal to the mid‐plane is also prescribed to zero. In the case of div‐stable elements, the pressure profile is prescribed to be constant along the width of the mould. This can be achieved by using as interpolation degrees of freedom the pressure values at the element centroid and its derivatives in the directions tangent and normal to the mid‐plane, and prescribing the latter to zero. No modifications are needed when stabilized formulations are employed. To advance in time the function used in the VOF technique, we use a constant velocity across the width of the mould, which is taken as the projection on the tangent plane of each element of the nodal velocities. This is needed in order to have mass conservation.
Abstract
In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, [...]
- 1
- read
The characteristic‐based‐split procedure: an efficient and accurate algorithm for fluid problems
O. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vázquez, P. Ortiz

Int. J. Numer. Meth. Fluids (1999). Vol. 31 (1), pp. 359-392

Abstract
In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self‐adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi‐implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic‐based‐split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised.
Abstract
In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them [...]
- 1
- read
Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows
R. Codina, O. Soto

Int. J. Numer. Meth. Fluids (1999). Vol. 30 (3), 309-334

Abstract
The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution.
Abstract
The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε [...]
- 2
- read
A fractional-step finite-element method for the Navier–Stokes equations applied to magma-chamber withdrawal
R. Folch, M. Vázquez, R. Codina, J. Martí i Castell

Computers and Geotechnics (1999). Vol. 25 (3), pp. 263-275

Abstract
We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows the use of equal interpolation spaces for the pressure and velocity fields and can be used to solve both compressible and incompressible flows. The standard Galerkin method is used to space-discretize the equations in which convective terms are dominant because the equations are reformulated in a characteristics co-moving frame providing thus the required artificial diffusion in a consistent way. The algorithm depends on four different parameters. Depending on their values, a fully implicit, a semi-implicit or a fully explicit solution can be obtained. The proposed algorithm may be useful in solving a wide spectrum of problems in geology, engineering and geosciences. Several flows frequently encountered in practical applications such as incompressible, slightly compressible or perfect gas are taken into account. As an example, we use it to model the withdrawal of magma from shallow chambers during explosive volcanic eruptions. Our model constitutes a first attempt to characterize the temporal evolution of the most revelant physical parameters during such a process.
Abstract
We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows [...]
- 1
- read
Numerical solution of the incompressible Navier–Stokes equations with Coriolis forces based on the discretization of the total time derivative.
R. Codina

Journal of Computational Physics (1999). Vol. 148 (2), pp. 467-496

Abstract
In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approximation is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the characteristic Galerkin technique for convection-dominated flows.
Abstract
In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on [...]
- 2
- read
Finite element solution of the Stokes problem with dominating Coriolis force
R. Codina, O. Soto

Comput. Methods Appl. Mech. Engrg., (1997). Vol. 142 (3-4), pp. 215-234

Abstract
The objective of this paper is to present a finite element formulation to solve the Stokes problem with Coriolis force. This force results in a skew-symmetric term in the weak formulation of the problem that deteriorates the stability of the standard Galerkin finite element method when the viscosity is small. We show that the stability is worsened due to the presence of the pressure gradient to enforce the incompressibility of the flow. The relevance of this effect depends on the relative importance of the viscous force and the Coriolis force, which is measured by the Ekman number. When it is small, oscillations occur using the Galerkin approach. To overcome them, we propose two different methods based on a consistent modification of the basic Galerkin formulation. Both methods eliminate the oscillations, keeping the accuracy of the formulation and enhancing its numerical stability.
Abstract
The objective of this paper is to present a finite element formulation to solve the Stokes problem with Coriolis force. This force results in a skew-symmetric term in the [...]
- 1
- read
Shock capturing viscosities for the general fluid mechanics algorithm
P. Nithiarasu, O. Zienkiewicz, B. Sai, K. Morgan, R. Codina, M. Vázquez

Int. J. Numer. Meth. Fluids (1998). Vol. 28 (9), pp. 1325-1353

Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous and inviscid compressible flow problems. Results show that the methods based on the second gradient of pressure give better performance in all situations. For instance, the method constructed from the nodal pressure values and consistent and lumped mass matrices is an excellent choice for inviscid problems. The method based on L2 projection is better than any other method in viscous flow computations. The residual based anisotropic method gives excellent performance in the supersonic range and gives better results in the hypersonic regime if a small amount of residual smoothing is used
Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous [...]
- 2
- read
A fractional-step method for the incompressible navier-stokes equations related to a predictor-multicorrector algorithm
J. Blasco, R. Codina, A. Huerta

Int. J. Numer. Meth. Fluids (1998). Vol. 28 (10), pp. 1391-1419

Abstract
An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied in this paper. The method, which is first order accurate in the time-step, is shown to convergence to an exact solution of the equations. By adequately splitting the viscous term, it allows to enforce full Dirichlet boundary conditions on the velocity in all substeps of the scheme, while needing no boundary condition at all for the pressure. It is also shown to be related to an iterative predictor-multicorrector algorithm for evolution equations, when this is applied to the incompressible Naiver-Stokes system. A new derivation of the algorithm in a general setting is provided. Two different finite element interpolations are considered for the implementation of the algorithm; numerical results obtained with them for standard benchmark cases are presented.
Abstract
An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied [...]
- 2
- read
A numerical model for temporal variations during explosive central vent eruptions
R. Folch, J. Marti, R. Codina, M. Vázquez

Journal of Geophysical Research: Solid Earth (1998). Vol. 103, 20883-20899

Abstract
An axisymmetrical numerical model has been developed in order to find the temporal evolution of pressure, the position of the exsolution level, the velocity field, the eruption rate, and the amount of erupted material of a shallow, volatile‐rich, felsic magma chamber during a Plinian central vent eruption. The overpressure necessary to trigger the eruption is assumed to result from crystallization‐driven volatile oversaturation. We solve the resulting set of equations using a finite element method. The results obtained show that the pressure at the conduit entrance decreases exponentially as the eruption proceeds. This produces a shifting of the exsolution level, so that deeper parts of the chamber become progressively volatile oversaturated during the eruption. We assess the influence of chamber geometry and the physical properties of the magma on the computed parameters using several numerical examples. The results are also compared with those predicted by previous models from the literature and are found to be in good agreement with documented eruptions. The model constitutes a first attempt to numerically model the dynamics and the temporal evolution of the most relevant physical parameters during withdrawal from a closed magma chamber.
Abstract
An axisymmetrical numerical model has been developed in order to find the temporal evolution of pressure, the position of the exsolution level, the velocity field, the eruption [...]
- 2
- read
Comparison of some finite element methods for solving the diffusion-convection-reaction equation
R. Codina

Comput. Methods Appl. Mech. Engrg., (1998). Vol. 156, (1-4), pp. 185-210

Abstract
In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard in an effort to emphasize the similarities and differences between them. In particular, it is shown that the classical SUPG method is very similar to an explicit version of the Characteristic-Galerkin method, whereas the Taylor-Galerkin method has a stabilization effect similar to a sub-grid scale model, which is un turn related to the introduction of bubbles functions.
Abstract
In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard [...]
- 3
- read
A general algorithm for compressible and incompressible flows. Part III: The semi‐implicit form
R. Codina, M. Vázquez, O. Zienkiewicz

Int. J. Numer. Meth. Fluids (1998). Vol. 27 (1-4), 13-32

Abstract
In this paper we consider some particular aspects related to the semi‐implicit version of a fractional step finite element method for compressible flows that we have developed recently. The first is the imposition of boundary conditions. We show that no boundary conditions at all need to be imposed in the first step where an intermediate momentum is computed. This allows us to impose the real boundary conditions for the pressure, a point that turns out to be very important for compressible flows. The main difficulty of the semi‐implicit form of the scheme arises in the solution of the continuity equation, since it involves both the density and the pressure. These two variables can be related through the equation of state, which in turn introduces the temperature as a variable in many cases. We discuss here the choice of variables (pressure or density) and some strategies to solve the continuity equation. The final point that we study is the behaviour of the scheme in the incompressible limit. It is shown that the method has an inherent pressure dissipation that allows us to reach this limit without having to satisfy the classical compatibility conditions for the interpolation of the velocity and the pressure
Abstract
In this paper we consider some particular aspects related to the semi‐implicit version of a fractional step finite element method for compressible flows that we have developed [...]
- 1
- read
A finite element formulation for Stokes proble allowing equal velocity-pressure interpolation
R. Codina, J. Blasco

Comput. Methods Appl. Mech. Engrg., (1997). Vol. 143 (3–4), pp. 373-391

Abstract
In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of this method relies on the analysis of a class of fractional-step methods for the Navier-Stokes equations for which it is known that equal interpolations yield good numerical results. The reason for this turns out to be the difference between two discrete Laplacian operators computed in a different manner. The formulation of the Stokes problem considered here aims to reproduce this effect. From the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity-pressure interpolations satisfying of the standard formulation. In particular, this condition is fulfilled by the most common equal order interpolations.
Abstract
In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of [...]
- 1
- read
Split, characteristic based semi‐implicit algorithm for laminar/turbulent incompressible flows
O. Zienkiewicz, B. Sai, K. Morgan, R. Codina

Int. J. Numer. Meth. Fluids (1996). Vol. 23, 787-809

Abstract
In an earlier paper, Zienkiewicz and Codina (Int. j. numer. methods fluids, 20, 869–885 (1995)) presented a general algorithm for the solution of both compressible and incompressible Navier–Stokes equations. The algorithm, based on operator splitting, permits arbitrary interpolation functions to be used while avoiding the Babŭska–Brezzi restriction. In addition, its characteristic based approach introduces a form of rational dissipation. Zienkiewicz et al. (Int. j. numer. methods fluids, 20, 887–913 (1995)) presented the application of this algorithm in its fully explicit form to various inviscid compressible flow problems. They also presented two incompressible flow problems solved by the fully explicit form, employing a pseudo compressibility. The present work deals with the application of the above algorithm it its semi‐implicit form to some incompressible flow benchmark problems. Further, it extends the methodology to turbulent flows by employing both one, and two equation turbulence models. A comparison of results with earlier investigations is presented. Other issues addressed in this study include the effect of additional diffusion terms present in the scheme for both laminar and turbulent flow problems and some practical difficulties associated with local time stepping.
Abstract
In an earlier paper, Zienkiewicz and Codina (Int. j. numer. methods fluids, 20, 869–885 (1995)) presented a general algorithm for the solution of both compressible [...]
- 1
- read
A general algorithm for compressible and incompressible flow—Part I. the split, characteristic‐based scheme
O. Zienkiewicz, R. Codina

Int. J. Numer. Meth. Fluids (1995). Vol. 20, 869-885

Abstract
The paper outlines the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier‐Stokes or Euler equations. Full incompressibility can be dealt with if the algorithm is used in its semi‐explicit form and its structure permits arbitrary interpolation functions to be used avoiding the Babuška‐Brezzi restriction. In a fully explicit version it introduces a rational form of balancing dissipation avoiding the use of arbitrary parameters and forms for this.
Abstract
The paper outlines the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier‐Stokes or Euler equations. Full [...]
- 2
- read
An investigation on thermal performance of wollastonite and bentonite reinforced intumescent fire-retardant coating for steel structures
F. Ahmad, S. Ullah, N. Merican, E. Oñate, A. Al-Sehemi, G. Yeoh

Construction and Building Material (2019). Vol. 228, 116734

Abstract
This research emphases on the synthesis of new formulations of intumescent coating with improved thermal performance for steel structures. The coating formulations were based on the expandable graphite reinforced with wollastonite and bentonite. Ten samples of five formulations were synthesized by varying grinding time duration between 1 and 2 min. To analyse the substrate temperature of coated steel, fire test was performed according to ASTM-E119. The char morphology was observed by Field Emission Scanning Electron (FESEM) and Transmission Electron Microscopy (TEM). FTIR and X-ray Diffraction (XRD) test is conducted to analyse the composition of the residual char. The residual char mass was perceived by Thermogravimetric analysis (TGA) of the coating. X-ray Photo Electron Spectroscopy (XPS) was utilized for binding energy and elemental composition of the char. One-hour fire protection test showed 166 _C, the lowest substrate temperature of IFRC5-2 and 40.46% residual mass was obtained by TGA analysis. XRD analysis showed that residual char has aluminum borate and borophosphate and confirmed by functional group analysis using and FTIR. FESEM and TEM illustrated that char relates to hexagonal particles of wollastonite. XPS analysis of IFRC5-2 showed the carbon and oxygen contents were 41.40% and 51.20%. Pyrolysis-Gas Chromatography–Mass Spectrometry (Pyrolysis GC–MS) results showed IFRC-5 produced less concentration of the gaseous products compared to IFRC-C. The formulations developed by grinding solid ingredients for 2 min showed improved thermal performance compared with the formulation produced by grinding solid ingredients for 1 min. Longer grinding time and higher amount of filler improved the thermal properties of the intumescent coating.
Abstract
This research emphases on the synthesis of new formulations of intumescent coating with improved thermal performance for steel structures. The coating formulations were based [...]
- 3
- read
A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation
R. Codina

Comput. Methods Appl. Mech. Engrg., (1993). Vol. 110 (3-4), pp. 325-342

Abstract
To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of a nonlinear crosswind dissipation is proposed. It is shown that the method is less overdiffusive than other discontinuity-capturing techniques and has better numerical behavior. The design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases.
Abstract
To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of [...]
- 1
- read
Stability analysis of the forward Euler scheme for the convection‐diffusion equation using the SUPG formulation in space
R. Codina

Int. J. Numer. Meth. Engng. (1993). Vol. 36, 1445-1464

Abstract
The stability and accuracy of the forward Euler scheme for the semidiscrete problem arising from the space discretization of the convection‐diffusion equation using the SUPG formulation are analysed. Both linear and quadratic finite elements are considered. The stability limits are derived for the one‐dimensional problem and a heuristic criterion is proposed for the multidimensional equation. Some numerical experiments are conducted in order to assess the performance of different finite elements
Abstract
The stability and accuracy of the forward Euler scheme for the semidiscrete problem arising from the space discretization of the convection‐diffusion equation using the [...]
- 1
- read
An iterative penalty method for the finite element solution of the stationary Navier-Stokes equations
R. Codina

Comput. Methods Appl. Mech. Engrg., (1993). Vol. 110 (3-4), pp. 237-262

Abstract
The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, the usual condition on the viscosity and the data that ensures uniqueness is assumed. The method is based on the iterative imposition of the incompressibility condition via penalization. Theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations. Two particular iterative schemes are analysed, namely those based on the Picard and Newton-Raphson algorithms.
Abstract
The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, [...]
- 1
- read
Numerical simulation and visualization of material flow in friction stir welding via particle tracing
N. Dialami, M. Chiumenti, M. Cervera, C. Agelet de Saracibar, J. Ponthot, P. Bussetta

(2014). S. Idelsohn (Ed.), Numerical Simulations of Coupled Problems in Engineering, Part III, Chapter 7 Computational Methods, Vol. 33, Springer, pp. 157-169

Abstract
This work deals with the numerical simulation and material flow visualization of Friction Stir Welding (FSW) processes. The 4-th order Runge-Kutta (RK4) integration method is used for the computation of particle trajectories. The particle tracing method is used to study the effect of input process parameters and pin shapes on the weld quality. The results show that the proposed method is suitable for the optimization of the FSW process.
Abstract
This work deals with the numerical simulation and material flow visualization of Friction Stir Welding (FSW) processes. The 4-th order Runge-Kutta (RK4) integration method [...]
- 4
- read
Challenges to be tackled in the computational modeling and numerical simulation of FSW processes
C. Agelet de Saracibar

Metals (2019). Vol. 9 (5), pp. 573

Abstract
The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes is an extremely challenging task due to the highly nonlinear and coupled nature of the physical problem and the complex computational issues that need to be properly tackled in the numerical model.
Abstract
The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes is an extremely challenging task due to the highly nonlinear and coupled nature [...]
- 3
- read
Conserving algorithms for frictionless and full stick friction contact dynamic problems using the direct elimination method
C. Agelet de Saracibar, D. Di Capua

Int. J. Numer. Meth. Engng. (2017). (preprint) Vol. 113 (6), pp. 910-937

Abstract
In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.
Abstract
In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless [...]
- 5
- read