Numerical Methods : Scipedia

Some algorithms to correct a geometry in order to create a finite element mesh
R. Ribó, G. Bugeda, E. Oñate

Computers and Structures (2002). Vol. 80 (16-17), pp. 1399-1408

Abstract
One of the major difficulties in meshing 3D complex geometries is to deal with non proper geometrical definitions coming from CAD systems. Typically, CAD systems do not take care of the proper definition of the geometries for the analysis purposes. In addition, the use of standard CAD files (IGES, VDA,...) for the transfer of geometries between different systems introduce some additional difficulties. In this work, a collection of algorithms to repair and/or to improve the geometry definitions are provided. The aim of these algorithms is to make as easy as possible the generation of a mesh over complex geometries given some minimum requirements of quality and correctness. The geometrical model will be considered as composed of a set of NURBS lines and trimmed surfaces. Some examples of application of the algorithms and of the meshes generated from the corrected geometry are also presented in this work
Abstract
One of the major difficulties in meshing 3D complex geometries is to deal with non proper geometrical definitions coming from CAD systems. Typically, CAD systems do not take [...]
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Finite volumes and finite elements: Two ‘good friends’
E. Oñate, S. Idelsohn

Int. J. Numer. Meth. Engng. (1994). Vol. 37 (19), pp. 3323-3341

Abstract
In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. The paper shows that both procedures share a number of features, like mesh discretization and approximation. Moreover, it is shown that in many cases both techniques are completely equivalent.
Abstract
In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. [...]
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Partitioned Analysis of Coupled Systems
K. Park, C. Felippa

(1983). Computational Methods for Transient Analysis

Abstract
This paper reviews partitioned analysis procedures for the analysis of coupled-field dynamical problems. These problems involve two or more distinctive subsystems that are tightly coupled. Examples are provided by fluid-structure, soil-structure, thermal-structure and structure-magnetodynamic interactions. The computer analysis of such systems depends on the integration of analysis capabilities for the separate components. Partitioned analysis procedures provide an efficient and modular way of achieving that integration. These procedures advance the solution of the coupled problem in a staged fashion.
Abstract
This paper reviews partitioned analysis procedures for the analysis of coupled-field dynamical problems. These problems involve two or more distinctive subsystems that are [...]
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Neural networks for variational problems in engineering
R. Lopez, E. Balsa-Canto, E. Oñate

Int. J. Numer. Meth. Engng. (2008). Vol. 75 (11), pp. 1341-1360

Abstract
In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning problem for the multilayer perceptron lies in terms of finding a function, which is an extremal for some functional. Therefore, a variational formulation for NNs provides a direct method for the solution of variational problems. This proposed method is then applied to distinct types of engineering problems. In particular a shape design, an optimal control and an inverse problem are considered. The selected examples can be solved analytically, which enables a fair comparison with the NN results. Copyright © 2008 John Wiley & Sons, Ltd.
Abstract
In this work a conceptual theory of neural networks (NNs) from the perspective of functional analysis and variational calculus is presented. Within this formulation, the learning [...]
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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 [...]
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Polyhedrization of an arbitrary 3D point set
S. Idelsohn, N. Calvo, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 192, pp. 2649–2667

Abstract
Given a 3D point set, the problem of defining the volume associated, dividing it into a set of regions (elements) and defining a boundary surface is tackled. Several physical problems need to define volume domains, boundary surfaces and approximating functions from a given point distribution. This is for instance the case of particle methods, in which all the information is the particle positions and there are not boundary surfaces definition. Until recently, all the FEM mesh generators were limited to the generation of simple elements as tetrahedral or hexahedral elements (or triangular and quadrangular in 2D problems). The reason of this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements. Nowadays, there are several acceptable shape functions for a very large class of polyhedra. These new shape functions, together with a generalization of the Delaunay tessellation presented in this paper, gives an optimal marriage and a powerful tool to solve a large variety of physical problems by numerical methods. The domain partition into polyhedra presented here is not a standard mesh generation. The problem here is: for a given node distribution to find a suitable boundary surface and a suitable mesh to be used in the solution of a physical problem by a numerical method. To include new nodes or change their positions is not allowed.
Abstract
Given a 3D point set, the problem of defining the volume associated, dividing it into a set of regions (elements) and defining a boundary surface is tackled. Several [...]
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Multiscale computational analysis in mechanics using finite calculus. An introduction
E. Oñate

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 192 (28-30), pp. 3043-3059

Abstract
The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size. This introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters. The discrete nodal values obtained with the FIC formulation and the finite element method (FEM) can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched FEM. Typical multiscale problems in mechanics which can be solved with the FIC method include convection-diffusion-reaction problems with high localized gradients, incompressible problems in solid and fluid mechanics, localization problems such as prediction of shear bands in solids and shock waves in compressible fluids, turbulence, etc. The paper presents an introduction of the treatment of multiscale problems using the FIC approach in conjunction with the finite element method (FEM). Examples of application of the FIC/FEM formulation to the solution of simple multiscale convection-diffusion problems are given.
Abstract
The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC [...]
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A simple method for automatic update of finite element meshes
G. Chiandussi, G. Bugeda, E. Oñate

Commun. Numer. Meth. Engng (2000). Vol. 16 (1), pp. 1-19

Abstract
A simple method to automatically update the finite element mesh of the analysis domain is proposed. The method considers the mesh as a fictitious elastic body subjected to prescribed displacements at selected boundary points. The mechanical properties of each mesh element are appropriately selected in order to minimize the deformation and the distortion of the mesh elements. Different selection strategies have been used and compared in their application to simple examples. The method avoids the use of remeshing in the solution of shape optimization problems and reduces the number of remeshing steps in the solution of coupled fluid–structure interaction problems
Abstract
A simple method to automatically update the finite element mesh of the analysis domain is proposed. The method considers the mesh as a fictitious elastic body subjected to [...]
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Shape variable definition with C0 , C1 and C2 continuity functions
G. Chiandussi, G. Bugeda, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 188, pp. 727-742

Abstract
The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).
Abstract
The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of [...]
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An advancing front point generation technique
R. Löhner, E. Oñate

Commun. Numer. Meth. Engng (1998). Vol. 14 (12), pp. 1097-1108

Abstract
An algorithm to construct boundary‐conforming, isotropic clouds of points with variable density in space is described. The input required consists of a specified mean point distance and an initial triangulation of the surface. Borrowing a key concept from advancing front grid generators, one point at a time is removed and, if possible, surrounded by admissible new points. This operation is repeated until no active points are left. Timings show that the scheme is about an order of magnitude faster than volume grid generators based on the advancing front technique, making it possible to generate large (>106) yet optimal clouds of points in a matter of minutes on a workstation. Several examples are included that demonstrate the capabilities of the technique.
Abstract
An algorithm to construct boundary‐conforming, isotropic clouds of points with variable density in space is described. The input required consists of a specified mean point [...]
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To mesh or not to mesh. That is the question. . .
S. Idelsohn, E. Oñate

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195, pp. 4681–4696

Abstract
In the last decade a family of methods called meshless methods has been developed both for structural and fluid mechanics problems. After these ideas, a possible classification for numerical formulations may be to separate the methods that make use of a standard finite element mesh (such as those made of tetrahedra or hexahedra), from those that do not need a standard mesh, namely the meshless methods. For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. This paper discusses this issue to show that this choice is not, in the large majorities of the cases, the right question.
Abstract
In the last decade a family of methods called meshless methods has been developed both for structural and fluid mechanics problems. After these ideas, a possible classification [...]
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Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants
C. Felippa, E. Oñate, S. Idelsohn

Archives of Comp. Meths. Engng. (2018). Vol. 25 (4), pp. 919–963

Abstract
This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives of the classical fields of continuum mechanics integrated over control regions to construct stabilizing modification terms. Fields may include displacements, body forces, strains, stresses, pressure and volumetric strains. To support observer-invariant FIC formulations, we have catalogued field transformation equations as well as sets of linear and quadratic invariants of fields and of their derivatives up to appropriate order. Attention is focused on the two-dimensional case of a body in plane strain under the drilling-rotation transformation group. Results are presented for displacement and body-force derivatives of orders up to 4, and for stress, strain, pressure and volumetric strain derivatives of order up to 3. The material assembled here is self-contained because this catalog is believed to be useful beyond FIC applications; for example gradient-based, nonlocal constitutive models of multiscale mechanics and physics that involve finite characteristic dimensions analogous to FIC steplengths.
Abstract
This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives [...]
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A residual correction method based on finite calculus
E. Oñate, R. Taylor, O. Zienkiewicz, J. Rojek

Engineering Computations (2003). Vol. 20 (5-6), pp. 629-658

Abstract
In this paper, a residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar convection‐diffusion problem and the problem of elasticity at the incompressible or quasi‐incompressible limit. The formulation permits the use of equal interpolation for displacements and pressure on linear triangles and tetrahedra as well as any low order element type. To add additional stability in the solution, pressure gradient corrections are introduced as suggested from developments of sub‐scale methods. Numerical examples are included to demonstrate the performance of the method when applied to typical test problems.
Abstract
In this paper, a residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar [...]
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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 [...]
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The meshless finite element method
S. Idelsohn, E. Oñate, N. Calvo, F. Del Pin

Int. J. Numer. Meth. Engng. (2003). Vol. 58 (6), pp. 893-912

Abstract
A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C0 continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions.
Abstract
A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, [...]
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On the ‘most normal’ normal
R. Aubry, R. Lohner

Communications in numerical methods in engineering (2008). Vol. 24 (12), pp. 1641-1652

Abstract
Given a set of normals in ℛ3, two algorithms are presented to compute the ‘most normal’ normal. The ‘most normal’ normal is the normal that minimizes the maximal angle with the given set of normals. A direct application is provided supposing a surface triangulation is available. The set of normals may represent either the face normals of the faces surrounding a point or the point normals of the points surrounding a point. The first algorithm is iterative and straightforward, and is inspired by the one proposed by Pirzadeh (AIAA Paper 94‐0417 , 1994). The second gives more insight into the complete problem as it provides the unique solution explicitly. It would correspond to the general extension of the algorithm presented by Kallinderis (AIAA‐92‐2721 , 1992).
Abstract
Given a set of normals in ℛ3, two algorithms are presented to compute the ‘most normal’ normal. The ‘most normal’ normal is the normal that minimizes [...]
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The generalized finite point method
B. Boroomand, M. Najjar, E. Oñate

Computational Mechanics (2009). Vol. 44, pp. 173-190

Abstract
In this paper we propose a new mesh-less method based on a sub-domain collocation approach. By reducing the size of the sub-domains the method becomes similar to the well-known finite point method (FPM) and thus it can be regarded as the generalized form of finite point method (GFPM). However, unlike the FPM, the equilibrium equations are weakly satisfied on the sub-domains. It is shown that the accuracy of the results is dependent on the sizes of the sub-domains. To find an optimal size for a sub-domain we propose a patch test procedure in which a set of polynomials of higher order than those chosen for the approximations/interpolations are used as the exact solution and a suitable error norm is minimized through a size tuning procedure. In this paper we have employed the GFPM in elasto-static problems. We give the results of the size optimization in a series of tables for further use. Also the results of the integrations on a generic sub-domain are given as a series of library functions for those who want to use GFPM as a cheap and fast integral-based mesh-less method. The performance of GFPM has been demonstrated by solving several sample problems.
Abstract
In this paper we propose a new mesh-less method based on a sub-domain collocation approach. By reducing the size of the sub-domains the method becomes similar to the well-known [...]
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Advancing front techniques for filling space with arbitrary separated objects
R. Löhner, E. Oñate

Finite Elements in Analysis and Design (2010). Vol. 46 (1-2), pp. 140-151

Abstract
A review is given of advancing front techniques for filling space with arbitrary separated objects. Over the last decade, these techniques have reached a considerable degree of maturity and are being used to generate clouds of points for SPH and FPM simulations, as well as spheres, ellipsoids, objects defined by a collection of spheres or polyhedral objects for DEM simulations. Algorithmic as well as implementational aspects are discussed. Techniques to obtain maximum packing, such as closest object placement (during generation) and move/enlarge (after generation) are also considered. Several examples are included that demonstrate the capabilities developed.
Abstract
A review is given of advancing front techniques for filling space with arbitrary separated objects. Over the last decade, these techniques have reached a considerable degree [...]
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Traversing Critical Points with Penalty Springs
C. Felippa

(1987). Transient/Dynamic Analysis and Constitutive Laws for Engineering Materials

Abstract
We present aspects of computer implementation of finite element structural analysis capabilities into a series of concurrent machines. We approach the implementation issues from the viewpoint that the design of data structures for finite element computations will play a crucial role in the constructs of computational modules, data transfer mechanisms among processes, and, consequently, desirable non-numerical and numerical algorithms. We account for the essential architectural features of local memory machines, shared memory machines and connection machines that are pertinent to finite element computations. We outline a concurrent FEM system that we advocate, and report our activities on system design and algorithm development. We report also on our 'hands-on' experience in concurrent finite element computations using some of the existing multiprocessors such as Intel's hypercube iPSC, Encore's Multimax, Alliant/FX8, and the CRAY as a benchmark case.
Abstract
We present aspects of computer implementation of finite element structural analysis capabilities into a series of concurrent machines. We approach the implementation issues [...]
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Generation of three-dimensional unstructured grids by the advancing-front method
R. Lohner, P. Parikh

International Journal for Numerical Methods in Fluids (1988). Vol. 8 (10), pp. 1135-1149

Abstract
The generation of three-dimensional unstructured grids using the advancing-front technique is described. This technique has been shown to be effective for the generation of unstructured grids in two dimensions.1,2 However, its extension to three-dimensional regions required algorithms to define the surface and suitable data structures that avoid excessive CPU-time overheads for the search operations involved. After obtaining an initial triangulation of the surfaces, tetrahedra are generated by successively deleting faces from the generation front. Details of the grid generation algorithm are given, together with examples and timings.
Abstract
The generation of three-dimensional unstructured grids using the advancing-front technique is described. This technique has been shown to be effective for the generation [...]
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A study of mesh optimality criteria in adaptive finite element analysis
E. Oñate, G. Bugeda

Engineering Computations (1993). Vol. 10 (4), pp. 307-321

Abstract
The concepts of solution error and optimal mesh in adaptive finite element analysis are revisited. It is shown that the correct evaluation of the convergence rate of the error norms involved in the error measure and the optimal mesh criteria chosen are essential to avoid oscillations in the refinement process. Two mesh optimality criteria based on: (a) the equal distribution of global error, and (b) the specific error over the elements are studied and compared in detail through some examples of application.
Abstract
The concepts of solution error and optimal mesh in adaptive finite element analysis are revisited. It is shown that the correct evaluation of the convergence rate of the error [...]
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The extended Delaunay tessellation
N. Calvo, S. Idelsohn, E. Oñate

Engineering Computations (2003). Vol. 20 (5-6), pp. 583-600

Abstract
The extended Delaunay tessellation (EDT) is presented in this paper as the unique partition of a node set into polyhedral regions defined by nodes lying on the nearby Voronoï spheres. Until recently, all the FEM mesh generators were limited to the generation of tetrahedral or hexahedral elements (or triangular and quadrangular in 2D problems). The reason for this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements. Nowadays, there are several acceptable shape functions for a very large class of polyhedra. These new shape functions, together with the EDT, gives an optimal combination and a powerful tool to solve a large variety of physical problems by numerical methods. The domain partition into polyhedra presented here does not introduce any new node nor change any node position. This makes this process suitable for Lagrangian problems and meshless methods in which only the connectivity information is used and there is no need for any expensive smoothing process.
Abstract
The extended Delaunay tessellation (EDT) is presented in this paper as the unique partition of a node set into polyhedral regions defined by nodes lying on the nearby Voronoï [...]
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Some algorithms to correct a geometry in order to create a finite element mesh
R. Ribó, G. Bugeda, E. Oñate

Computers and Structures (2002). Vol. 80 (16-17), pp. 1399-1408

Abstract
One of the major difficulties in meshing 3D complex geometries is to deal with non-proper geometrical definitions coming from CAD systems. Typically, CAD systems do not take care of the proper definition of the geometries for the analysis purposes. In addition, the use of standard CAD files (IGES, VDA, …) for the transfer of geometries between different systems introduce some additional difficulties. In this work, a collection of algorithms to repair and/or to improve the geometry definitions are provided. The aim of these algorithms is to make as easy as possible the generation of a mesh over complex geometries given some minimum requirements of quality and correctness. The geometrical model will be considered as composed of a set of NURBS lines and trimmed surfaces. Some examples of application of the algorithms and of the meshes generated from the corrected geometry are also presented in this work.
Abstract
One of the major difficulties in meshing 3D complex geometries is to deal with non-proper geometrical definitions coming from CAD systems. Typically, [...]
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Possibilities of finite calculus in computational mechanics
E. Oñate

Int. J. Numer. Meth. Engng. (2004). Vol. 60 (1), pp. 255-281

Abstract
The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations.
Abstract
The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a [...]
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Reduced-order subscales for POD models
J. Baiges, R. Codina, S. Idelsohn

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 291; pp. 173-196

Abstract
In this work the Reduced-Order Subscales for Proper Orthogonal Decomposition models are presented. The basic idea consists in splitting the full-order solution into the part which can be captured by the reduced-order model and the part which cannot, the subscales, for which a model is required. The proposed model for the subscales is defined as a linear function of the solution of the reduced-order model. The coefficients of this linear function are obtained by comparing the solution of the full-order model with the solution of the reduced-order model for the same initial conditions, which, for convenience, are evaluated in the snapshots used to train the original reduced-order-model. The difference between both solutions are the subscales, for which a model can be built using a least-squares procedure. The subscales are then introduced as a correction in the reduced-order model, resulting in an important improvement in accuracy. The enhanced reduced-order model is tested in several numerical examples. These practical cases show that the use of the subscales leads to more accurate solutions, successfully corrects errors introduced by hyper-reduction, and allows to solve complex flow problems using a reduced number of degrees of freedom.
Abstract
In this work the Reduced-Order Subscales for Proper Orthogonal Decomposition models are presented. The basic idea consists in splitting the full-order solution into the part [...]
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A hierarchical finite element for composite laminated beams using a refined zigzag theory
L. Nallim, S. Oller, E. Oñate, F. Flores

Composite Structures (2017). Vol. 163, pp. 168-184

Abstract
In this work a kinematics for laminated beams enriched with a refined formulation ZigZag (RZT), originally presented by Tessler et al. in 2007, introduced in a hierarchical one dimensional type “p” finite element is presented. The finite element employs Lagrange polynomials for the approximation of the degrees of freedom of the ends (nodes) and orthogonal Gram-Schmidt polynomials to the internal degrees of freedoms. This finite element allows a very low discretization, is free of shear locking and behaves very well when the analysis of laminated composites with accurate determination of local stresses and strains at laminar level is necessary. This element has been validated in the analysis of laminated beams with various sequences of symmetric and asymmetric stacking, studying in each case its accuracy and stability.
Abstract
In this work a kinematics for laminated beams enriched with a refined formulation ZigZag (RZT), originally presented by Tessler et al. in 2007, introduced in a hierarchical [...]
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Parallel Delaunay triangulation for particle finite element methods
Y. Fragakis, E. Oñate

Commun. Numer. Meth. Engng (2008). Vol. 24 (11), pp. 1009-1017

Abstract
Delaunay triangulation is a geometric problem that is relatively difficult to parallelize. Parallel algorithms are usually characterized by considerable interprocessor communication or important serialized parts. In this paper, we propose a method that achieves high speed‐ups, but needs information regarding locally maximum element circumspheres prior to the beginning of the algorithm. Such information is directly available in iterative methods, like the particle finite element methods. The developed parallel Delaunay triangulation method, has minimum communication requirements, is quite simple and achieves high parallel efficiency.
Abstract
Delaunay triangulation is a geometric problem that is relatively difficult to parallelize. Parallel algorithms are usually characterized by considerable interprocessor communication [...]
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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 [...]
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Solution of geometrically parametrised problems within a CAD environment via model order reduction
R. Sevilla, S. Zlotnik, A. Huerta

Computer methods in applied mechanics and engineering (2020). Vol. 358, pp. 112631:1-112631:33

Abstract
The main objective of this work is to describe a general and original approach for computing an off-line solution for a set of parameters describing the geometry of the domain. That is, a solution able to include information for different geometrical parameter values and also allowing to compute readily the sensitivities. Instead of problem dependent approaches, a general framework is presented for standard engineering environments where the geometry is defined by means of NURBS. The parameters controlling the geometry are now the control points characterising the NURBS curves or surfaces. The approach proposed here, valid for 2D and 3D scenarios, allows a seamless integration with CAD preprocessors. The proper generalised decomposition (PGD), which is applied here to compute explicit geometrically parametrised solutions, circumvents the curse of dimensionality. Moreover, optimal convergence rates are shown for PGD approximations of incompressible flows
Abstract
The main objective of this work is to describe a general and original approach for computing an off-line solution for a set of parameters describing the geometry of the domain. [...]
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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 [...]
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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 [...]
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Mesh projection between parametric surfaces
X. Roca, J. Sarrate, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 22 (6), pp. 591-603

Abstract
This paper presents a new algorithm to map a given mesh over a source surface onto a target surface. This projection is determined by means of a least-squares approximation of a transformation defined between the loops of boundary nodes of the cap surfaces in the parametric spaces. Once the new mesh is obtained on the parametric space of the target surface, it is mapped to the target surface according to its parameterization. Therefore, in contrast with the usual techniques, the developed algorithm does not require solving any root finding problem to ensure that the projected nodes are on the target surface. Finally, this projection algorithm is extended to three-dimensional cases and included in a sweep meshing tool in order to generate the inner layers of elements in the physical space.
Abstract
This paper presents a new algorithm to map a given mesh over a source surface onto a target surface. This projection is determined by means of a least-squares approximation [...]
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Inverse mass matrix via the method of localized lagrange multipliers
J. González, R. Kolman, S. Cho, C. Felippa, K. Park

International Journal for Numerical Methods in Engineering (2017). Vol. 113 (2), pp. 277-295

Abstract
An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces and constructs the free‐free inverse mass matrix using the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. In particular, the present paper constructs the kernel inverse matrix by using the standard FEM elemental mass matrices. It is shown that the accuracy of the present inverse mass matrix is almost identical to that of a conventional consistent mass matrix or a combination of lumped and consistent mass matrices. Numerical experiments with the proposed inverse mass matrix are conducted to validate its effectiveness when applied to vibration analysis of bars, beams, and plain stress problems.
Abstract
An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency [...]
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A general advancing front technique for filling space with arbitrary objects
R. Löhner, E. Oñate

Int. J. Numer. Meth. Engng. (2004). Vol. 61 (12), pp. 1977-1991

Abstract
An advancing front space‐filling technique for arbitrary objects has been developed. The input required consists of the specification of the desired mean point distance in space and an initial triangulation of the surface. One object at a time is removed from the active front, and, if possible, surrounded by admissible new objects. This operation is repeated until no active objects are left. Two techniques to obtain maximum packing are discussed: closest object placement (during generation) and move/enlarge (after generation). Different deposition or layering patterns can be achieved by selecting the order in which objects are eliminated from the active front. Timings show that for simple objects like spheres the scheme is considerably faster than volume mesh generators based on the advancing front technique, making it possible to generate large (> 106) yet optimal clouds of points in a matter of minutes on a PC. For more general objects, the performance may degrade depending on the complexity of the penetration checks. Several examples are included that demonstrate the capabilities of the technique.
Abstract
An advancing front space‐filling technique for arbitrary objects has been developed. The input required consists of the specification of the desired mean point distance [...]
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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 [...]
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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. [...]
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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 [...]
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NURBS-enhanced finite element method (NEFEM)
R. Sevilla, S. Fernández-Méndez, A. Huerta

Int. J. Numer. Meth. Engng. (2008). Vol. 76 (1), pp. 56-83

Abstract
An improvement to the classical finite element (FE) method is proposed. It is able to exactly represent the geometry by means of the usual CAD description of the boundary with non-uniform rational B-splines (NURBS). Here, the 2D case is presented. For elements not intersecting the boundary, a standard FE interpolation and numerical integration are used. But elements intersecting the NURBS boundary need a specifically designed piecewise polynomial interpolation and numerical integration. A priori error estimates are also presented. Finally, some examples demonstrate the applicability and benefits of the proposed methodology. NURBS-enhanced finite element method (NEFEM) is at least one order of magnitude more precise than the corresponding isoparametric FE in every numerical example shown. This is the case for both continuous and discontinuous Galerkin formulations. Moreover, for a desired precision, NEFEM is also more computationally efficient, as shown in the numerical examples. The use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details. The possibility of computing an accurate solution with coarse meshes and high-order interpolations makes NEFEM a more efficient strategy than classical isoparametric FE.
Abstract
An improvement to the classical finite element (FE) method is proposed. It is able to exactly represent the geometry by means of the usual CAD description of the boundary [...]
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FEM-FCT - Combining unstructured grids with high resolution
R. Lohner, K. Morgan, M. Vahdati, J. Boris, D. Book

Communications in numerical methods in engineering (1988). Vol. 4 (6), pp. 356-370

Abstract
We present the extension of flux‐corrected transport (FCT) schemes to unstructured grids. The spatial discretization is performed via finite elements. In particular, we have chosen triangular elements in two dimensions. The limiting procedure is based on Zalesak's extension to more than one dimension of the FCT schemes developed by Boris and Book. The resulting scheme, FEM‐FCT, is capable of resolving moving and stationary shocks within two elements, and several examples are given that demonstrate the accuracy attainable, even for complicated geometries.
Abstract
We present the extension of flux‐corrected transport (FCT) schemes to unstructured grids. The spatial discretization is performed via finite elements. In particular, [...]
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Solution of linear equations with skyline-stored symmetric matrix
C. Felippa

Computers and Structures (1975). Vol. 5 (1), pp. 13-29

Abstract
Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented. Such systems arise in a variety of applications, notably the numerical discretization of conservative physical systems by finite differences or finite element techniques. The routines can be used for processing constrained systems without need for prearranging equations. The application to ‘superelement’ condensation of large-scale systems is discussed.
Abstract
Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented. Such systems arise in [...]
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Numerically stable formulas for a particle-based explicit exponential integrator
P. Nadukandi, E. Oñate

Computational Mechanics (2015). Vol. 55 (5), pp. 903-920

Abstract
Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vector-valued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particle-based (Lagrangian-type) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the X-IVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using double-precision floating-point arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourth-order divided differences of the exponential function.
Abstract
Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential [...]
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An advancing front technique for filling space with arbitrary separated objects
R. Löhner, E. Oñate

Int. J. Numer. Meth. Engng. (2009). Vol. 78 (13), pp. 1618-1630

Abstract
An advancing front technique for filling space with arbitrary, separated objects has been developed. The input required consists of the specification of the desired object type, the mean object size, the distance between objects in space, as well as an initial triangulation of the surface. The objects are assumed to be described by a coarse mesh of tetrahedra. One face at a time is removed from the active front, and, if possible, surrounded by admissible new objects. This operation is repeated until no active faces are left. Two techniques to obtain maximum packing are discussed: closest object placement (during generation) and move/enlarge (after generation). Several examples are included that demonstrate the capabilities of the technique. Copyright © 2009 John Wiley & Sons, Ltd.
Abstract
An advancing front technique for filling space with arbitrary, separated objects has been developed. The input required consists of the specification of the desired object [...]
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General treatment of essential boundary conditions in reduced order models for non-linear problems
A. Cosimo, A. Cardona, S. Idelsohn

Adv. Model. and Simul. in Eng. Sci. (2016). Vol. 3, art 7

Abstract
Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.
Abstract
Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this [...]
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On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics
A. Limache, S. Idelsohn

Adv. Model. and Simul. in Eng. Sci. (2016). Vol. 3, art 13

Abstract
Discretization processes leading to numerical schemes sometimes produce undesirable effects. One potentially serious problem is that a discretization may produce the loss of validity of some of the physical principles or mathematical properties originally present in the continuous equation. Such loss may lead to uncertain results such as numerical instabilities or unexpected non-physical solutions. As a consequence, the compatibility of a discrete formulation with respect to intrinsic physical principles might be essential for the success of a numerical scheme. This paper addresses such type of issue. Its main objective is to demonstrate that standard Finite Element discretizations of the heat conduction equation violate Clausius’s postulate of the second law of thermodynamics, at nodal level. The problem occurs because non-physical, reversed nodal heat-fluxes arise in such discretizations. Conditions for compatibility of discrete nodal heat-fluxes with respect to Clausius’s postulate are derived here and named discrete thermodynamic compatibility conditions (DTCC). Simple numerical examples are presented to show the undesirable consequences of such failure. It must be pointed out that such DTCCs have previously appeared in the context of the study of the conditions that make discrete solutions to satisfy the discrete maximum principle (DMP). However, the present article does not put attention on such mathematical principle but on the satisfaction of a fundamental physical one: the second law of thermodynamics. Of course, from the presented point of view, it is clear that the violation of such fundamental law will cause, among different problems, the violation of the DMP.
Abstract
Discretization processes leading to numerical schemes sometimes produce undesirable effects. One potentially serious problem is that a discretization may produce the loss [...]
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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 [...]
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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 [...]
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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 [...]
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Numerical representation of the quality measures of triangles and triangular meshes
J. Sarrate, J. Palau, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 19 (7), pp. 551-561

Abstract
In this note a new procedure to represent the quality measure for triangles is proposed. The triangles are identified by their three angles and are represented in a bounded domain, called angle representation region, according to the area co-ordinates, which are common and well known by finite element users. The developed representation can also be used in order to visualize the characteristics of any quality measure. This new procedure is extended to graphically represent triangular meshes in the angle representation region.
Abstract
In this note a new procedure to represent the quality measure for triangles is proposed. The triangles are identified by their three angles and are represented in a bounded [...]
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Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods
M. Arroyo, M. Ortiz

Int. J. Numer. Meth. Engng. (2006). Vol. 65 (13), pp. 2167-2202

Abstract
We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum‐entropy (max‐ent) statistical inference. Local max‐ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max‐ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker‐delta property at the boundary. Local max‐ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max‐ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max‐ent approximation schemes is vastly superior to that of finite elements.
Abstract
We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important [...]
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Modelling unilateral frictionless contact using the null-space method and cubic B-Spline interpolation
J. Muñoz

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (9-12), pp. 979-993

Abstract
The analysis of unilateral sliding contact in elasticity is equivalent to a minimisation problem subjected to a set of inequality constraints. However, the presence of boundary discontinuities, such as those stemming from the spatial discretisation, appears as a major problem to determine the set of active constraints. This work introduces a smoothing technique of the master surface resorting to cubic B-Spline interpolation, which is C1 continuous in contact situations between elastic and rigid bodies, and G1 continuous in elastic–elastic contact problems. The technique is applied in conjunction with the null-space method, where the solution is searched in an unconstrained manifold. The resulting formulation eases the contact transition along the master surface, and recovers the quadratic convergence of the iterative Newton–Raphson process. The robustness of the method is demonstrated using 2D and 3D examples.
Abstract
The analysis of unilateral sliding contact in elasticity is equivalent to a minimisation problem subjected to a set of inequality constraints. However, the presence of boundary [...]
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Staggered transient analysis procedures for coupled mechanical systems: Formulation
C. Felippa, K. Park

Computer Methods in Applied Mechanics and Engineering (1980). Vol. 24 (1), pp. 61-111

Abstract
Coupled-field dynamic problems in mechanics have been traditionally solved by treating the entire system as one computational entity. More recently, increasing attention has been directed to an alternative approach; partition the governing equations into subsystems, which are treated by subsystem analyzers. The selection of the subsystems may be based on weak-coupling considerations, widely different time response characteristics, isolation of nonlinear effects, or more pragmatic reasons such as the availability of analyzer software. In a staggered solution procedure the solution state of the coupled system is advanced by sequentially executing the subsystem analyzers. Subsystem coupling terms are accounted for by temporal extrapolation techniques.This paper focuses on the formulation and computer implementation of staggered solution procedures for two-field problems governed by semidiscrete second-order coupled differential equations. Such equations find application in the modeling of structure-fluid, structure-soil and structure-structure interaction. Following an introductory description of candidate problems and general solution strategies, direct time integration methods are formulated and applied to the coupled system. Staggered solution procedures are constructed through two alternative approaches which are based upon partitioning at the difference and differential equation level, respectively. Characteristic equations that govern the stability of the resulting implementations are derived, and the selection of stable extrapolators discussed. Finally, possible extensions of staggered solution procedures to coupled-field static and eigenvalue problems are suggested.
Abstract
Coupled-field dynamic problems in mechanics have been traditionally solved by treating the entire system as one computational entity. More recently, increasing attention has [...]
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Supernatural QUAD4: A template formulation
C. Felippa

Computer Methods in Applied Mechanics and Engineering (2006). Vol. 195 (41-43), pp. 5316-5342

Abstract
A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated by J.H. Argyris since his Continua and Discontinua 1965 exposition, is further expanded. The qualifier ‘supernatural’ means that all governing equations: kinematic, constitutive and equilibrium, are expressed in both Cartesian and natural forms. These two sets are used to build different components of the template. With timely help from a computer algebra system, a template that include all possible quadrilateral elements that pass the Individual Element Test of Bergan and Hanssen emerges. It yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs. A striking example is the construction of a two-trapezoid macroelement that is bending exact about one direction, for any amount of distortion. This concludes a five decade search that begins with the formulation of the wing-cover rectangular panel in Argyris’ 1954 seminal serial on Energy Methods and Structural Analysis.
Abstract
A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated [...]
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Partitioned analysis of coupled mechanical systems
C. Felippa, K. Park, C. Farhat

Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (24-25), pp. 3247-3270

Abstract
This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.
Abstract
This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational [...]
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From the individual element test to finite element templates: Evolution of the patch test
C. Felippa, B. Haugen, C. Militello

International Journal for Numerical Methods in Engineering (1995). Vol. 38 (2), pp. 199-229

Abstract
This paper starts a sequence of three articles that follow an unconventional approach in finite element research. The ultimate objective is to construct high-performance elements and element-level error estimators for those elements. The approach takes off from our previous work in high-performance elements and culminates with the development of finite element templates. The present paper concentrates on the patch test and evolved versions of the test that have played a key role in this research. Following a brief review of the historical roots, we present the Individual Element Test (IET) of Bergan and Hanssen in an expanded context that encompasses several important classes of new elements. The relationship of the IET to the multielement forms A, B and C of the patch test and to the single-element test are investigated. An important consequence of the IET application is that the element stiffness equations decompose naturally into basic and higher-order parts. The application of this decomposition to the “sanitization” of the non-convergent BCIZ element is described and verified with numerical experiments. Two sequel papers in preparation are subtitled ‘the algebraic approach’ and ‘element-level error estimation’. These apply the fundamental decomposition to the derivation of templates for specific mechanical elements and to the construction of element-level error estimators, respectively.
Abstract
This paper starts a sequence of three articles that follow an unconventional approach in finite element research. The ultimate objective is to construct high-performance elements [...]
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Progress in grid generation via the advancing front technique
R. Lohner

Engineering with Computers (1996). Vol. 12, pp. 186-210

Abstract
We describe recent extensions and improvements to the advancing front grid generation technique. These improvements target a range of applicability, speed and user friendliness. The range of applicability is enlarged by the ability to produce volumetric grids around thin surfaces (such as shells, membranes, fabrics or surfaces with cusps), the generation of high aspect ratio grids for Navier-Stokes applications, the generation of higher order triangular and tetrahedral elements, and the generation of quadrilateral and hexahedral elements. Speed improvements are the result of reduced search overheads, as well as vectorization and parallelization. User friendliness is enhanced by the ability to grid directly discrete data and simpler ways of specifying the desired element size and shape in space. Numerous examples are included that demonstrate the versatility and maturity that advancing front grid generators have achieved.
Abstract
We describe recent extensions and improvements to the advancing front grid generation technique. These improvements target a range of applicability, speed and user friendliness. [...]
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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 [...]
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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 [...]
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PGD-based “Computational Vademecum” for efficient design, optimization and control
F. Chinesta, A. Leygue, F. Bordeu, E. Cueto, D. González, I. Ruiz, A. Ammar, A. Huerta

IACM expressions (2013). Vol. 33 (4), pp. 4-7

Abstract
D espite the impressive progresses attained by simulation capabilities and techniques, some challenging problems remain today intractable. These prob- lems, that are common to many branches of science and engineering, are of differ- ent nature. Among them, we can cite those related to high-dimensional models, on which mesh-based approaches fail due to the exponential increase of degrees of freedom. Other challenging scenarios concern problems requiring many direct solutions (optimization, inverse identifica- tion, uncertainty quantification ! ) or those needing very fast solutions (real time simulation, simulation based control ! ).
Abstract
D espite the impressive progresses attained by simulation capabilities and techniques, some challenging problems remain today intractable. These prob- lems, that are common [...]
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Mass Matrix Templates: General Description and 1D Examples
C. Felippa, Q. Guo, K. Park

Archives of Computational Methods in Engineering (2015). Vol. 22 (1), pp. 1-65

Abstract
This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.
Abstract
This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The [...]
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Partitioned analysis of coupled system
C. Felippa, K. Park, C. Farhat

Computer Methods in Applied Mechanics and Engineering (2001). Vol. 190 (s24-25), pp. 3247-3270

Abstract
This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.
Abstract
This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational [...]
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The solution of non‐linear hyperbolic equation systems by the finite element method
R. Lohner, K. Morgan, O. Zienkiewicz

International Journal for Numerical Methods in Fluids (1984). Vol. 4 (11), pp. 116-120

Abstract
The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co‐ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor‐Galerkin methods of Donea.17 Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.
Abstract
The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. [...]
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Partitioned analysis of coupled mechanical systems
C. Felippa, T. Geers

Engineering Computations (1988). Vol. 5 (2), pp. 123-133

Abstract
Partitioned analysis is a method by which sets of time-dependent ordinary differential equations for coupled systems may be numerically integrated in tandem, thereby avoiding brute-force simultaneous solution. The coupled systems addressed pertain to fluid—structure, fluid—soil, soil—structure, or even structure—structure interaction. The paper describes the partitioning process for certain discrete-element equations of motion, as well as the associated computer implementation. It then delineates the procedure for designing a partitioned analysis method in a given application. Finally, examples are presented to illustrate the concepts. It is seen that a key element in the implementation of partitioned analysis is the use of integrated, as opposed to monolithic software.
Abstract
Partitioned analysis is a method by which sets of time-dependent ordinary differential equations for coupled systems may be numerically integrated in tandem, thereby avoiding [...]
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On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled Problems
M. Cervera, R. Codina, M. Galindo

Engineering Computations (1996). Vol. 13 (6), pp. 4-30

Abstract
Outlines a general methodology for the solution of the system of algebraic equations arising from the discretization of the field equations governing coupled problems. Considers that this discrete problem is obtained from the finite element discretization in space and the finite difference discretization in time. Aims to preserve software modularity, to be able to use existing single field codes to solve more complex problems, and to exploit computer resources optimally, emulating parallel processing. To this end, deals with two well-known coupled problems of computational mechanics – the fluid-structure interaction problem and thermally-driven flows of incompressible fluids. Demonstrates the possibility of coupling the block-iterative loop with the nonlinearity of the problems through numerical experiments which suggest that even a mild nonlinearity drives the convergence rate of the complete iterative scheme, at least for the two problems considered here. Discusses the implementation of this alternative to the direct coupled solution, stating advantages and disadvantages. Explains also the need for online synchronized communication between the different codes used as is the description of the master code which will control the overall algorithm.
Abstract
Outlines a general methodology for the solution of the system of algebraic equations arising from the discretization of the field equations governing coupled problems. Considers [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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Discretization influence on regularization by two localization limiters
A. Huerta, G. Pijaudier

Journal of Engineering Mechanics (1994). Vol. 120 (6), pp. 1198-1218

Abstract
In materials with a strain‐softening characteristic behavior, classical continuum mechanics favors uncontrolled strain localization in numerical analyses. Several methods have been proposed to regularize the problem. Two such localization limiters developed to overcome spurious instabilities in computational failure analysis are examined and compared. A disturbance analysis, on both models, is performed to obtain the closed‐form solution of propagating wave velocities as well as the velocities at which the energy travels. It also shows that in spite of forcing the same stress‐strain response, the wave equation does not yield similar results. Both propagations of waves are dispersive, but the internal length of each model is different when equivalent behavior is desired. In fact, the previously suggested derivations of gradient models from nonlocal integral models were not completely rigorous. The perturbation analysis is pursued in the discrete space where computations are done, and the closed form solutions are also obtained. The finite‐element discretization introduces an added dispersion associated to the regularization technique. Therefore, the influence of the discretization on the localization limiters can be evaluated. The element size must be smaller than the internal length of the models in order to obtain sufficient accuracy.
Abstract
In materials with a strain‐softening characteristic behavior, classical continuum mechanics favors uncontrolled strain localization in numerical analyses. Several methods [...]
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NURBS-enhanced finite element method for Euler equations
R. Sevilla, S. Fernández-Méndez, A. Huerta

Int. J. Numer. Meth. Engng. (2008). Vol. 57 (9), pp. 1051-1069

Abstract
In this work, the NURBS-enhanced finite element method (NEFEM) is combined with a discontinuous Galerkin (DG) formulation for the numerical solution of Euler equations of gas dynamics. With NEFEM, numerical fluxes along curved boundaries are computed much more accurately due to the exact geometric representation of the computational domain. The proper implementation of the wall boundary condition and the exact geometry provide accurate results even with a linear approximation of the solution. A detailed comparison of NEFEM in front of isoparametric finite elements is presented, demonstrating the superiority of NEFEM approach for both linear and higher-order computations
Abstract
In this work, the NURBS-enhanced finite element method (NEFEM) is combined with a discontinuous Galerkin (DG) formulation for the numerical solution of Euler equations of [...]
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Comparison of high-order curved finite elements
R. Sevilla, S. Fernández-Méndez, A. Huerta

Int. J. Numer. Meth. Engng. (2011). Vol. 87 (8), pp. 719-734

Abstract
Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two-dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS-enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p-FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations.
Abstract
Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation [...]
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Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems
D. Canales, A. Leygue, F. Chinesta, D. González, E. Cueto, E. Feulvarch, J. Bergheau, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2016). Vol. 108 (9), pp. 971-989 (Preprint)

Abstract
This paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional finite element methods at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing proper generalized decompositions. Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics that depend on many parameters, such as welding. After introducing the vademecum-generalized finite element method formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes.
Abstract
This paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line [...]
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Are High-order and Hybridizable Discontinuous Galerkin methods competitive?
A. Huerta, A. Angeloski, X. Roca, J. Peraire

Oberwolfach Reports (2012). Vol. 9 (1), pp. 485-487

Abstract
The talk covered several issues motivated by a practical engineering wave propagation problem: real-time evaluation of wave agitation in harbors. The first part, presented the application of a reduced order model in the framework of a Helmholtz equation with non-constant coefficients in an unbounded domain. This problem requires large numbers of degrees of freedom (ndof) because relatively high frequencies with small (compared with the domain size) geometric features must be considered
Abstract
The talk covered several issues motivated by a practical engineering wave propagation problem: real-time evaluation of wave agitation in harbors. The first part, presented [...]
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Impact of short-sales in stock market efficiency
B. Llacay, G. Peffer

Algorithmic Finance (2019). Vol. 8 (1-2), pp. 5-26

Abstract
In the last two decades, the hedge fund sector has experienced a spectacular growth, up to the point that it is currently estimated to move more than 50% of the daily volume of stock markets. In contrast to other financial institutions, hedge funds are subject to less restrictive regulations which, in particular, allow them to sell short. As they exploit the asset mispricings, their action is thought to contribute to market efficiency. In this paper we aim at studying the impact that short sales have on the informational efficiency of a financial market. This can be used not only to assess the effect that hedge fund actions have in financial markets, but also the consequences of regulatory measures such as short-selling restrictions or bans. Building on an agent-based market, the simulation results indicate that short sales are beneficial to market efficiency, although the market does not become completely efficient even when all the population can sell short.
Abstract
In the last two decades, the hedge fund sector has experienced a spectacular growth, up to the point that it is currently estimated to move more than 50% of the daily volume [...]
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Maximum-principle preserving space–time isogeometric analysis
J. Bonilla, S. Badia

Computer Methods in Applied Mechanics and Engineering (2019). Vol. 354, pp. 422-440

Abstract
In this work we propose a nonlinear stabilization technique for convection–diffusion–reaction and pure transport problems discretized with space–time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space–time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space–time discretization can become highly costly. Therefore, we also propose a partitioned space–time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection–diffusion problems in one and two dimensions. -
Abstract
In this work we propose a nonlinear stabilization technique for convection–diffusion–reaction and pure transport problems discretized with space–time [...]
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Método de puntos finitos. Un análisis sobre el efecto de los parámetros que definen las nubes en aproximaciones de segundo y cuarto orden
E. Ortega

Mecánica Computacional (2004). Vol. 23 (30), pp. 3273-3291

Abstract
Se investiga en el presente trabajo la influencia de los distintos parámetros que definen la aproximación local en el Método de Puntos Finitos y su relación con la calidad de la misma. Este análisis se realiza sobre la solución de un problema de Poisson tridimensional el cual se emplea como caso testigo. Haciendo uso de este último se propone un método para ajustar los parámetros de la aproximación local mediante la definición individual de la función de ponderación en cada nube de puntos. Ello posibilita flexibilizar la definición de parámetros y lograr la mejor aproximación para el problema planteado. El análisis se extiende a funciones de aproximación de segundo y cuarto orden y distribuciones estructuradas y no estructuradas de puntos.
Abstract
Se investiga en el presente trabajo la influencia de los distintos parámetros que definen la aproximación local en el Método de Puntos Finitos y su relación [...]
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Some useful data structure for generation of unstructured grid
R. Lohner

Communications in numerical methods in engineering (1988). Vol. 4 (1), pp. 123-135

Abstract
Several data structures for the generation of unstructured grids are described. Their usefulness stems from the fact that they enable the necessary search operations to be performed in an optimal way. In particular, we describe heap lists, quad- and octrees, and linked lists. Combining these data structures, the important problem of interpolating information between unstructured grids is also solved.
Abstract
Several data structures for the generation of unstructured grids are described. Their usefulness stems from the fact that they enable the necessary search operations to be [...]
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Accelerated ‘relaxation’ or direct solution? Future prospects for fem
R. Lohner, O. Zienkiewicz

International Journal for Numerical Methods in Engineering (1985). Vol. 21 (1), pp. 1-11

Abstract
‘Dynamic’ or ‘viscous’ relaxation procedures have not gained much popularity in finite element analysis in which the direct (Gaussian elimination) solution dominates. Reasons for this are various—the most important being the rather slow convergence generally achieved for such procedures. However, it is possible to accelerate this quite dramatically and a method of doing so is shown in this paper. With the use of such acceleration and the inherent advantages of greatly reduced storage requirements and simplicity of programming, relaxation procedures promise an exciting possibility for the solution of large two‐ and three‐dimensional problems in both linear and nonlinear ranges.
Abstract
‘Dynamic’ or ‘viscous’ relaxation procedures have not gained much popularity in finite element analysis in which the direct (Gaussian elimination) [...]
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A compendium of FEM integration formulas for symbolic work
C. Felippa

Engineering Computations (2004). Vol. 21 (8), pp. 867-890

Abstract
This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating-point (numerical) form, or in exact symbolic form. The latter is useful for computer-algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non-product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM-useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non-product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low-level floating-point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before
Abstract
This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, [...]
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Penalty Spring Stabilization of Singular Jacobians
C. Felippa

Journal of Applied Mechanics (1987). Vol. 54 (3), pp. 728-729

Abstract
A limit-point treatment is presented which is a synthesis of two earlier techniques. To traverse limit points, Sharifi and Popov (1970) introduced fictitious stiffnesses in the form of rank-one updates but did not use auxiliary systems. Rheinboldt (1981) discussed the use of auxiliary systems in conjunction with the partition of the Jacobian matrix. The procedure presented in this paper avoids the need for partitioning and consequent special treatment of the elements in the ith row and column of K; only the diagonal entry is changed. It enjoys the added advantage that none of the auxiliary system right-hand sides requires access to elements of the K matrix.
Abstract
A limit-point treatment is presented which is a synthesis of two earlier techniques. To traverse limit points, Sharifi and Popov (1970) introduced fictitious stiffnesses in [...]
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The construction of free–free flexibility matrices for multilevel structural analysis
C. Felippa, K. Park

Computer Methods in Applied Mechanics and Engineering (2002). Vol. 191 (19-20), pp. 2139-2168

Abstract
This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.
Abstract
This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations [...]
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A simple algorithm for localized construction of non‐matching structural interfaces
K. Park, C. Felippa, G. Rebel

International Journal for Numerical Methods in Engineering (2002). Vol. 53 (9), pp. 2117-2142

Abstract
A simple and effective algorithm for the modular construction of non-matched interfaces is presented for the partitioned solution of large-scale structural problems. The formulation is based on a recently developed four-field variational principle, which introduces a connection frame between the interfaced partitions. A key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori. The method is demonstrated with several 2D and 3D example problems
Abstract
A simple and effective algorithm for the modular construction of non-matched interfaces is presented for the partitioned solution of large-scale structural problems. The formulation [...]
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An extended class of multilayer perceptron
R. López, E. Oñate

Neurocomputing (2008). Vol. 71 (13-15), pp. 2538-2543

Abstract
In this work an extended class of multilayer perceptron is presented. This includes independent parameters, boundary conditions and lower and upper bounds. In some cases, such extensions contain a priori information of the problem. On some other situations they are necessary in order to define a correct representation for the solution. The use of this augmented class of neural network is illustrated through a case study in the optimal control theory. The numerical results are compared against the analytical solution.
Abstract
In this work an extended class of multilayer perceptron is presented. This includes independent parameters, boundary conditions and lower and upper bounds. In some cases, [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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., [...]
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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 [...]
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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 [...]
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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. [...]
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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 [...]
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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 [...]
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On the solution of Lambert’s problem by regularization
D. Torre, R. Flores, E. Fantino, R. Flores

On the solution of Lambert’s problem by regularization (2018). Acta astronautica. 153, pp. 26 - 38

Abstract
Lambert’s problem is the two-point boundary-value problem resulting from a two-body orbital transfer between two position vectors in a given time. It lies at the very heart of several fundamental astrodynamics and space engineering problems and, as such, it has attracted the interest of scientists over centuries. In this work, we revisit the solution of Lambert’s problem based on Levi-Civita regularization developed by Carles Simó in 1973. We offer an exhaustive derivation of the theory, including proofs of all the results and the formulae employed, and we extend the algorithm to deal with multi-revolution transfers. Then, after investigating a range of initial guess search techniques and testing different numerical methods to approximate the solution, we propose a procedure in which the initial guess is assigned very efficiently by querying a pre-defined interpolating table. Then, in order to achieve both speed and robustness, we combine Newton-Raphson with safety checks to avoid out of boundary deviations to approximate the solution. We validate the method through several tests and applications, and we assess its convergence and performance. The algorithm presents no singularities, converges in all realistic scenarios and its computational cost is comparable with state of the art algorithms.
Abstract
Lambert’s problem is the two-point boundary-value problem resulting from a two-body orbital transfer between two position vectors in a given time. It lies at the very [...]
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Parametric solutions involving geometry: a step towards efficient shape optimization
A. Ammar, A. Huerta, F. Chinesta, E. Cueto, A. Leygue

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 268, pp. 178-193

Abstract
Optimization of manufacturing processes or structures involves the optimal choice of many parameters (process parameters, material parameters or geometrical parameters). Usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model. Then, an appropriate cost function is evaluated and its optimality checked. While the optimum is not reached, the process parameters should be updated by using an appropriate optimization procedure, and then the model must be solved again for the updated process parameters. Thus, a direct numerical solution is needed for each choice of the process parameters, with the subsequent impact on the computing time. In this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry. The main objective of this work is to describe an original approach for computing an off-line parametric solution. That is, a solution able to include information for different parameter values and also allowing to compute readily the sensitivities. The curse of dimensionality is circumvented by invoking the Proper Generalized Decomposition (PGD) introduced in former works, which is applied here to compute geometrically parametrized solutions
Abstract
Optimization of manufacturing processes or structures involves the optimal choice of many parameters (process parameters, material parameters or geometrical parameters). Usual [...]
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Efficient unstructured quadrilateral mesh generation
J. Sarrate, A. Huerta

Int. J. Numer. Meth. Engng. (2000). Vol. 49 (10), pp. 1327-1350

Abstract
This work is devoted to the description of an algorithm for automatic quadrilateral mesh generation. The technique is based on a recursive decomposition of the domain into quadrilateral elements. This automatically generates meshes composed entirely by quadrilaterals over complex geometries (there is no need for a previous step where triangles are generated). A background mesh with the desired element sizes allows to obtain the preferred sizes anywhere in the domain. The final mesh can be viewed as the optimal one given the objective function is defined. The recursive algorithm induces an efficient data structure which optimizes the computer cost. Several examples are presented to show the efficiency of this algorithm.
Abstract
This work is devoted to the description of an algorithm for automatic quadrilateral mesh generation. The technique is based on a recursive decomposition of the domain into [...]
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PGD-Based computational vademecum for efficient design, optimization and control
F. Chinesta, A. Leygue, F. Bordeu, J. Aguado, E. Cueto, D. González, I. Ruiz, A. Ammar, A. Huerta

Archives of Comp. Meths. Engng. (2013). Vol. 20 (1), pp. 31-59

Abstract
In this paper we are addressing a new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to high-dimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with high-dimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.
Abstract
In this paper we are addressing a new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies. Despite [...]
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A local multiple proper generalized decomposition based on the partition of unity
R. Ibañez, E. Abisset-Chavanne, F. Chinesta, A. Huerta, E. Cueto

Int. J. Numer. Meths. Engng. (2019). Vol 120 (2), pp. 139-152

Abstract
It is well known that model order reduction techniques that project the solution of the problem at hand onto alow-dimensional subspace present difficulties when this solution lies on a non-linear manifold. To overcomethese difficulties—notably, an undesirable augment in the number of requiredmodesin the solution—severalsolutions have been suggested. Among them we can cite the use of non-linear dimensionality reductiontechniques or, alternatively, the employ of local linear reduced order approaches. These last approachesusually present the difficulty of ensuring continuity between these local models. Here, a new method ispresented that ensures this continuity by resorting to the paradigm of the partition of unity, while employing Proper Generalized Decompositions at each local patch.
Abstract
It is well known that model order reduction techniques that project the solution of the problem at hand onto alow-dimensional subspace present difficulties when this solution [...]
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Real-time monitoring of thermal processes by reduced order modeling
J. Aguado, A. Huerta, F. Chinesta, E. Cueto

Int. J. Numer. Meth. Engng. (2015). Vol. 102 (5), pp. 991-1017 (Preprint)

Abstract
This work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity numerical simulations are costly and cannot be used, in general, for real-time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real-time the transient heat equation at the monitored point. In order to apply the reciprocity principle, harmonic analysis is used in the space-frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre-computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post-processing step. This last step can be performed online for real-time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems.
Abstract
This work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity [...]
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The aggregated unfitted finite element method for elliptic problems
S. Badia, F. Verdugo, A. Martín

Computer Methods in Applied Mechanics and Engineering (2018). Vol. 336, pp. 533-553

Abstract
Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.
Abstract
Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone [...]
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On the contact domain method: a comparison of penalty and Lagrange multiplier implementations
R. Weyler, J. Oliver, T. Sain, J. Cante

Computer Methods in Applied Mechanics and Engineering (2012). Vol. 205-208 (1), pp. 68-82

Abstract
This work focuses on the assessment of the relative performance of the so-called contact domain method, using either the Lagrange multiplier or the penalty strategies. The mathematical formulation of the contact domain method and the imposition of the contact constraints using a stabilized Lagrange multiplier method are taken from the seminal work (as cited later), whereas the penalty based implementation is firstly described here. Although both methods result into equivalent formulations, except for the difference in the constraint imposition strategy, in the Lagrange multiplier method the constraints are enforced using a stabilized formulation based on an interior penalty method, which results into a different estimation of the contact forces compared to the penalty method. Several numerical examples are solved to assess certain numerical intricacies of the two implementations. The results show that both methods perform similarly as one increases the value of the penalty parameter or decreases the value of the stabilization factor (in case of the Lagrange multiplier method). However there seems to exist a clear advantage in using the Lagrange multiplier based strategy in a few critical situations, where the penalty method fails to produce convincing results due to excessive penetration.
Abstract
This work focuses on the assessment of the relative performance of the so-called contact domain method, using either the Lagrange multiplier or the penalty strategies. [...]
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Error and work estimates for high‐order elements
R. Lohner

Int. J. Numer. Meth. Fluids (2013). Vol. 67 (12), pp. 2184-2188

Abstract
Error and work estimates for high‐order elements are derived. The comparison of error and work estimates shows that for relative accuracy in the 1% range, which is typical of engineering interest, it may prove very difficult to improve on linear elements. As expected, the estimates also show that the optimal order of element in terms of work and storage demands depends on the desired relative accuracy.
Abstract
Error and work estimates for high‐order elements are derived. The comparison of error and work estimates shows that for relative accuracy in the 1% range, which is typical [...]
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Linear Sources for Mesh Generation
R. Aubry, K. Karamete, E. Mestreau, S. Dey, R. Lohner

SIAM J. on Scientific Computing (2013). Vol. 35 (2), pp. 886-907

Abstract
Sources offer a convenient way to prescribe a size distribution in space. For each newly created mesh point, the mesh generator queries the local size distribution, either to create a new point or element, depending on the underlying mesh generation method, to smooth the mesh, or to get a local relevant length scale. Sources may have different shapes such as points, edges, triangles, or boxes. They provide the size distribution given some user defined parameters and the distance of a point location to the source. Traditionally, the source strength is considered as constant. In this work, extensions to linear sources in space are proposed. It is shown that in the case of curvature refined mesh generation, substantial savings may occur due to the much better approximation of the curvature variation for a simple modification of traditional sources. Even though curvature refinement is the main application of this work, improvements through linear sources are relevant to other contexts such as user defined sources. The technique is very general as it deals with a fundamental aspect of mesh generation and can be easily incorporated into an existing mesh generator with traditional sources. Thorough details of source approximations and source filtering are provided. Relations with lower envelopes are highlighted. Practical examples illustrate the accuracy and efficiency of the method.
Abstract
Sources offer a convenient way to prescribe a size distribution in space. For each newly created mesh point, the mesh generator queries the local size distribution, either [...]
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An algebraically partitioned FETI method for parallel structural analysis: algorithm description
K. Park, M. Justino, C. Felippa

International Journal for Numerical Methods in Engineering (1997). Vol. 40 (15), pp. 2717-2737

Abstract
An algebraically partitioned FETI method for the solution of structural engineering problems on parallel computers is presented. The present algorithm consists of three attributes: an explicit generation of the orthogonal null‐space matrix associated with the interface nodal forces, the floating subdomain rigid‐body modes computed from the subdomain static equilibrium equation of the classical force method and the identification of redundant interface force constraint operator that emanates when the interface force computations are localized. Comparisons of the present method with the previously developed differentially partitioned FETI method are offered in terms of the saddle‐point formulations at the end of the paper. A companion paper reports implementation details and numerical performance of the proposed algorithm.
Abstract
An algebraically partitioned FETI method for the solution of structural engineering problems on parallel computers is presented. The present algorithm consists of three [...]
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Iterative solutions for improving penalty function solutions of algebraic systems
C. Felippa

International Journal for Numerical Methods in Engineering (1978). Vol. 12 (5), pp. 821-836

Abstract
The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large-scale finite element codes are offered.
Abstract
The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational [...]
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Hybrid-Game Strategies for multi-objective design optimization in engineering
D. Lee, L. González, J. Periaux, A. Srinivas, E. Oñate

Computers and Fluids (2011). Vol. 47 (1), pp. 189-204

Abstract
A number of Game Strategies (GS) have been developed in past decades. They have been used in the fields of economics, engineering, computer science and biology due to their efficiency in solving design optimization problems. In addition, research in multi-objective (MO) and multidisciplinary design optimization (MDO) has focused on developing robust and efficient optimization methods to produce a set of high quality solutions with low computational cost. In this paper, two optimization techniques are considered; the first optimization method uses multi-fidelity hierarchical Pareto optimality. The second optimization method uses the combination of two Game Strategies; Nash-equilibrium and Pareto optimality. The paper shows how Game Strategies can be hybridised and coupled to Multi-Objective Evolutionary Algorithms (MOEA) to accelerate convergence speed and to produce a set of high quality solutions. Numerical results obtained from both optimization methods are compared in terms of computational expense and model quality. The benefits of using Hybrid-Game Strategies are clearly demonstrated
Abstract
A number of Game Strategies (GS) have been developed in past decades. They have been used in the fields of economics, engineering, computer science and biology due to their [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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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 (Preprint)

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. [...]
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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 [...]
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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, [...]
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A highly scalable parallel implementation of balancing domain decomposition by constraints
S. Badia, A. Martín, J. Principe

SIAM J. Numer. Anal. (2014). (preprint) Vol. 36 (2), pp, C190-C218

Abstract
In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarse-grid duties in time. The global set of MPI tasks is split into those that have fine-grid duties and those that have coarse-grid duties, and the different computations and communications in the algorithm are then re-scheduled and mapped in such a way that the maximum degree of overlapping is achieved while preserving data dependencies among them. In many ranges of interest, the extra cost associated to the coarse-grid problem can be fully masked by fine-grid related computations (which are embarrassingly parallel). Apart from discussing code implementation details, the paper also presents a comprehensive set of numerical experiments, that includes weak scalability analyses, with structured and unstructured meshes, and exact and inexact solvers for the 3D Poisson and linear elasticity problems on a pair of state-of-the-art multicore-based distributed-memory machines. This experimental study reveals remarkable weak scalability in the solution of problems with thousands of millions of unknowns on several tens of thousands of computational cores.
Abstract
In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarse-grid [...]
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Smooth, second order, non-negative meshfree approximants selected by maximum entropy
C. Cyron, M. Arroyo, M. Ortiz

Int. J. Numer. Meth. Engng. (2009). Vol. 79 (13), pp. 1605-1632

Abstract
We present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent approximation schemes to second-order consistency. This method retains the fundamental properties of first-order max-ent schemes, namely the shape functions are smooth, non-negative, and satisfy a weak Kronecker-delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher-order scheme for function approximation from unstructured data in arbitrary dimensions with non-negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS-based meshfree methods and even B-Spline approximations, as shown through numerical experiments. When compared with usual MLS-based second-order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems.
Abstract
We present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent [...]
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Second order convex maximum entropy approximants with applications to high order PDE
A. Rosolen, D. Millán, M. Arroyo

Int. J. Numer. Meth. Engng. (2013). Vol. 94 (2), pp. 150-182 (Preprint)

Abstract
We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.
Abstract
We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for [...]
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Blending isogeometric analysis and local maximum entropy meshfree approximants
A. Rosolen, M. Arroyo

Comput. Methods Appl. Mech. Engrg., (2013). Vol. 264, pp. 95-107 (preprint)

Abstract
We present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME approximation, treats isogeometric and LME basis functions on an equal footing in the reproducibility constraints, but views the former as data in the constrained minimization. The resulting scheme exploits the best features and overcomes the main drawbacks of each of these approximants. Indeed, it preserves the high fidelity boundary representation (exact CAD geometry) of isogeometric analysis, out of reach for bare meshfree methods, and easily handles volume discretization and unstructured grids with possibly local refinement, while maintaining the smoothness and non-negativity of the basis functions. We implement the method with B-Splines in two dimensions, but the procedure carries over to higher spatial dimensions or to other non-negative approximants such as NURBS or subdivision schemes. The performance of the method is illustrated with the heat equation, and linear and nonlinear elasticity. The ability of the proposed method to impose directly essential boundary conditions in non-convex domains, and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples.
Abstract
We present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME [...]
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Cell-based maximum-entropy approximants
D. Millán, N. Sukumar, M. Arroyo

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 284, pp. 712-731 (preprint)

Abstract
In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R-2. More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighbor-ring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established.
Abstract
In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation [...]
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Topological obstructions in the way of data-driven collective variables
B. Hashemian, M. Arroyo

Journal of chemical physics (2015). Vol. 142 (4), pp. 044102-1/044102-6 (preprint)

Abstract
Nonlinear dimensionality reduction (NLDR) techniques are increasingly used to visualize molecular trajectories and to create data-driven collective variables for enhanced sampling simulations. The success of these methods relies on their ability to identify the essential degrees of freedom characterizing conformational changes. Here, we show that NLDR methods face serious obstacles when the underlying collective variables present periodicities, e.g., arising from proper dihedral angles. As a result, NLDR methods collapse very distant configurations, thus leading to misinterpretations and inefficiencies in enhanced sampling. Here, we identify this largely overlooked problem and discuss possible approaches to overcome it. We also characterize the geometry and topology of conformational changes of alanine dipeptide, a benchmark system for testing new methods to identify collective variables. Nonlinear dimensionality reduction (NLDR) techniques are increasingly used to visualize molecular trajectories and to create data-driven collective variables for enhanced sampling simulations. The success of these methods relies on their ability to identify the essential degrees of freedom characterizing conformational changes. Here, we show that NLDR methods face serious obstacles when the underlying collective variables present periodicities, e.g., arising from proper dihedral angles. As a result, NLDR methods collapse very distant configurations, thus leading to misinterpretations and inefficiencies in enhanced sampling. Here, we identify this largely overlooked problem and discuss possible approaches to overcome it. We also characterize the geometry and topology of conformational changes of alanine dipeptide, a benchmark system for testing new methods to identify collective variables.
Abstract
Nonlinear dimensionality reduction (NLDR) techniques are increasingly used to visualize molecular trajectories and to create data-driven collective variables for enhanced [...]
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Error estimation and adaptivity for nonlocal damage models
A. Huerta, A. Rodríguez-Ferran

Int. J. Solids and Structures (2000). Vol. 37 (48-50), pp. 7501-7528

Abstract
Nonlocal damage models are typically used to model failure of quasi-brittle materials. Due to brittleness, the choice of a particular model or set of parameters can have a crucial influence on the structural response. To assess this influence, it is essential to keep finite element discretization errors under control. If not, the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint. To ensure the quality of the FE solution, an adaptive strategy based on error estimation is proposed here. It is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. Another important consequence of brittleness is that it leads to structural responses of the snap-through or snap-back type. This requires the use of arc-length control, with a definition of the arc parameter that accounts for the localized nature of quasi-brittle failure. All these aspects are discussed for two particular nonlocal damage models (Mazars and modified von Mises) and for two tests: the Brazilian tensile splitting test and the single-edge notched beam test. For the latter test, the capability of the Mazars model to capture the curved crack pattern observed in experiments – a topic of debate in the literature – is confirmed.
Abstract
Nonlocal damage models are typically used to model failure of quasi-brittle materials. Due to brittleness, the choice of a particular model or set of parameters can have a [...]
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Stability of incompressible formulations enriched with X-FEM
G. Legrain, N. Moës, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (21-24), pp. 1835-1849

Abstract
The treatment of (near-)incompressibility is a major concern for applications involving rubber-like materials, or when important plastic ows occurs as in forming processes. The use of mixed nite element methods is known to prevent the locking of the nite element approximation in the incompressible limit. However, it also introduces a critical condition for the stability of the formulation, called the infsup or LBB condition. Recently, the nite element method has evolved with the introduction of the partition of unity. The eXtended Finite Element Method (XFEM) uses the partition of unity to remove the need to mesh physical surfaces or to remesh them as they evolve. The enrichment of the displacement eld makes it possible to treat surfaces of discontinuity inside nite elements. In this paper, some strategies are proposed for the enrichment of mixed nite element approximations in the incompressible setting. The case of holes, material interfaces and cracks are considered. Numerical examples show that for well chosen enrichment strategies, the nite element convergence rate is preserved and the inf-sup condition is passed.
Abstract
The treatment of (near-)incompressibility is a major concern for applications involving rubber-like materials, or when important plastic ows occurs as in forming processes. [...]
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Consistent tangent matrices for substepping schemes
A. Pérez-Foguet, A. Rodríguez-Ferran, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2001). Vol. 190 (35-36), pp. 4627-4647

Abstract
A very simple and general expression of the consistent tangent matrix for substepping time-integration schemes is presented. If needed, the derivatives required for the computation of the consistent tangent moduli can be obtained via numerical differentiation. These two strategies (substepping and numerical differentiation) lead to quadratic convergence in complex nonlinear inelasticity problems.
Abstract
A very simple and general expression of the consistent tangent matrix for substepping time-integration schemes is presented. If needed, the derivatives required for the computation [...]
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Imposing essential boundary conditions in mesh-free methods
S. Fernández-Méndez, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (12-14), pp. 1257-1275

Abstract
Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.
Abstract
Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition [...]
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Error estimation and adaptivity for nonlinear FE analysis
A. Huerta, A. Rodríguez-Ferran, P. Diez

Int. J. Applied Mathematics and Computer Science (2002). Vol. 12 (1), pp. 59-70

Abstract
An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type error estimator and an unstructured quadrilateral mesh generator. The error estimator is based on simple local computations over the elements and the so-called patches. In contrast to other residual estimators, no flux splitting is required. The adaptive strategy is illustrated by means of a complex nonlinear problem: the failure analysis of a single-edge notched beam. The quasi-brittle response of concrete is modelled by means of a nonlocal damage model.
Abstract
An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type [...]
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A second-order face-centred finite volume method for elliptic problems
L. Vieira, M. Giacomini, R. Sevilla, A. Huerta

Computer methods in applied mechanics and engineering (2020). Vol. 358, pp. 112655:1-112655:23

Abstract
A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed method defines the solution on the faces of the mesh (edges in two dimensions). The method is based on a mixed formulation and therefore considers the solution and its gradient as independent unknowns. They are computed solving an element-by-element problem after the solution at the faces is determined. The proposed approach avoids the need of reconstructing the solution gradient, as required by cell-centred and vertex-centred FV methods. This strategy leads to a method that is insensitive to mesh distortion and stretching. The current method is second-order and requires the solution of a global system of equations of identical size and identical number of non-zero elements when compared to the recently proposed first-order FCFV. The formulation is presented for Poisson and Stokes problems. Numerical examples are used to illustrate the approximation properties of the method as well as to demonstrate its potential in three dimensional problems with complex geometries. The integration of a mesh adaptive procedure in the FCFV solution algorithm is also presented
Abstract
A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed [...]
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Proper generalized decomposition solutions within a domain decomposition strategy
A. Huerta, E. Nadal Soriano, F. Chinesta

Int. J. Numer. Meth. Engng. (2019). Vol. 113 (13), pp. 1972-1994 (Preprint)

Abstract
Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.
Abstract
Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic [...]
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Point-to-point connectivity, an abstract concept or a key issue for risk assessment studies?
P. Trinchero, X. Sánchez-Vila, D. Fernàndez-García

Advances in Water Resources (2008). Vol. 31 (12), pp. 1742-1753

Abstract
Connectivity of high/low-permeability areas has been recognized to significantly impact groundwater flow and solute transport. The task of defining a rigorous quantitative measure of connectivity for continuous variables has failed so far, and thus there exist a suite of connectivity indicators which are dependent on the specific hydrodynamic processes and the interpretation method. Amongst the many existing indicators, we concentrate on those characterizing connectivity between the points involved in a hydraulic or tracer test. The flow connectivity indicator used here is based on the time elapsed for hydraulic response in a pumping test (e.g., the storage coefficient estimated by the Cooper–Jacob method, Sest). Regarding transport, we select the estimated porosity from the breakthrough curve (ϕest). According to Knudby and Carrera [Knudby C, Carrera J. On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 2005;28(4):405–21] these two indicators measure connectivity differently, and are poorly correlated. Here, we use perturbation theory to analytically investigate the intrinsic relationship between Sest and ϕest. We find that ϕest can be expressed as a weighted line integral along the particle trajectory involving two parameters: the transmissivity point values, T, and the estimated values of Sest along the particle path. The weighting function is linear with the distance from the pumping well, thus the influence of the weighting function is maximum at the injection area, whereas the hydraulic information close to the pumping well becomes redundant (null weight). The relative importance of these two factors is explored using numerical simulations in a given synthetic aquifer and tested against intermediate-scale laboratory tracer experiments. We conclude that the degree of connectivity between two points of an aquifer (point-to-point connectivity) is a key issue for risk assessment studies aimed at predicting the travel time of a potential contaminant.
Abstract
Connectivity of high/low-permeability areas has been recognized to significantly impact groundwater flow and solute transport. The task of defining a rigorous quantitative [...]
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On discrete maximum principles for discontinuous Galerkin methods
S. Badia, A. Hierro

Computer Methods in Applied Mechanics and Engineering (2015). Vol. 286, pp. 107-122 (Preprint)

Abstract
The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection–diffusion problems. To do so, a novel definition of discrete maximum principle (DMP) is proposed using the discrete variational setting of the problem, and we show that the fulfilment of this DMP implies that the minimum/maximum (depending on the sign of the forcing term) is on the boundary for multidimensional problems. Then, an artificial viscosity (AV) technique is designed for convection-dominant problems that satisfies the above mentioned DMP. The noncomplete stabilized interior penalty dG method is proved to fulfil the DMP property for the one-dimensional linear case when adding such AV with certain parameters. The benchmarks for the constant values to satisfy the DMP are calculated and tested in the numerical experiments section. Finally, the method is applied to different test problems in one and two dimensions to show its performance.
Abstract
The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection–diffusion problems. To do so, a novel [...]
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On a general implementation of h- and p-adaptive curl-conforming finite elements
M. Olm, S. Badia, A. Martín

Advances in Engineering Software (2019). Vol. 132, 74-91

Abstract
Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR . We cover into detail how to implement arbitrary order (i.e., p-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., h-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation.
Abstract
Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially [...]
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Impact of value-at-risk models on market stability
B. Llacay, G. Peffer

Journal of Economic Dynamics and Control (2017). Vol. 82, pp. 223-256

Abstract
Financial institutions around the world use value-at-risk (VaR) models to manage their market risk and calculate their capital requirements under Basel Accords. VaR models, as any other risk management system, are meant to keep financial institutions out of trouble by, among other things, guiding investment decisions within established risk limits so that the viability of a business is not put unduly at risk in a sharp market downturn. However, some researchers have warned that the widespread use of VaR models creates negative externalities in financial markets, as it can feed market instability and result in what has been called endogenous risk, that is, risk caused and amplified by the system itself, rather than being the result of an exogenous shock. This paper aims at analyzing the potential of VaR systems to amplify market disturbances with an agent-based model of fundamentalist and technical traders which manage their risk with a simple VaR model and must reduce their positions when the risk of their portfolio goes above a given threshold. We analyse the impact of the widespread use of VaR systems on different financial instability indicators and confirm that VaR models may induce a particular price dynamics that rises market volatility. These dynamics, which we have called `VaR cycles’, take place when a sufficient number of traders reach their VaR limit and are forced to simultaneously reduce their portfolio; the reductions cause a sudden price movement, raise volatility and force even more traders to liquidate part of their positions. The model shows that market is more prone to suffer VaR cycles when investors use a short-term horizon to calculate asset volatility or a not-too-extreme value for their risk threshold.
Abstract
Financial institutions around the world use value-at-risk (VaR) models to manage their market risk and calculate their capital requirements under Basel Accords. [...]
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A performance comparison of machine learning algorithms for arced labyrinth spillways
F. Salazar, B. Crookston

Water (2019). Vol. 11 (3), p. 544

Abstract
Labyrinth weirs provide an economic option for flow control structures in a variety of applications, including as spillways at dams. The cycles of labyrinth weirs are typically placed in a linear configuration. However, numerous projects place labyrinth cycles along an arc to take advantage of reservoir conditions and dam alignment, and to reduce construction costs such as narrowing the spillway chute. Practitioners must optimize more than 10 geometric variables when developing a head–discharge relationship. This is typically done using the following tools: empirical relationships, numerical modeling, and physical modeling. This study applied a new tool, machine learning, to the analysis of the geometrically complex arced labyrinth weirs. In this work, both neural networks (NN) and random forests (RF) were employed to estimate the discharge coefficient for this specific type of weir with the results of physical modeling experiments used for training. Machine learning results are critiqued in terms of accuracy, robustness, interpolation, applicability, and new insights into the hydraulic performance of arced labyrinth weirs. Results demonstrate that NN and RF algorithms can be used as a unique expression for curve fitting, although neural networks outperformed random forest when interpolating among the tested geometries.
Abstract
Labyrinth weirs provide an economic option for flow control structures in a variety of applications, including as spillways at dams. The cycles of labyrinth weirs are typically [...]
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Parameter-free, weak imposition of Dirichlet boundary conditions and coupling of trimmed and non-conforming patches
S. Kollmannsberger, A. Özcan, J. Baiges, M. Ruess, E. Rank, A. Reali

International Journal for Numerical Methods in Engineering (2015). Vol. 101 (9), pp. 670-699 (Preprint)

Abstract
We present a parameter-free domain sewing approach for low-order as well as high-order finite elements. Its final form contains only primal unknowns; that is, the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem-dependent parameters, which require an estimation. The presented approach is symmetry preserving; that is, the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization, and we demonstrate high-order accuracy for problems of non-matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces, which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p-version and B-spline version of the finite cell method, an embedded domain method of high order, and compared with more classical methods such as the penalty method or Nitsche's method.
Abstract
We present a parameter-free domain sewing approach for low-order as well as high-order finite elements. Its final form contains only primal unknowns; that is, the approach [...]
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Robust and scalable domain decomposition solvers for unfitted finite element methods
S. Badia, F. Verdugo

Journal of Computational and Applied Mathematics (2018). Vol. 344, pp. 740-759

Abstract
Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a customization of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments shows the good performance of the proposed preconditioners. -
Abstract
Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since [...]
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Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
S. Badia, J. Bonilla

Computer Methods in Applied Mechanics and Engineering (2017). Vol. 313, pp. 133-158 (Preprint)

Abstract
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).
Abstract
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, [...]
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A review of expansive phenomena in Wagenburg North Tunnel
I. Berdugo, E. Alonso, A. Gens, M. Albis

Revista de la academia Colombiana de Ciencias Exactas, Físicas y Naturales (2009). Vol. 31 (129), pp. 455-469

Abstract
The paper deals on tunnelling and swelling in anhydritic-gypsiferous claystones. The geology of the Gipskeuper in Baden-Württemberg (Germany) and the phenomenology of expansions in Wagenburg North tunnel are described. Consistent swelling triggering events, mechanisms and exhaustion causes were identified using a thermo-hydro-chemo-mechanical theoretical analysis. It is suggested that long-term swelling in tunnels excavated through anhydritic-gypsiferous claystones is a result of solvent-way gypsum crystal growth due to ventilation induced groundwater evaporation and rock drying; an opposite concept to the usual consequences of drying in argillaceous materials, which causes shrinkage strains.
Abstract
The paper deals on tunnelling and swelling in anhydritic-gypsiferous claystones. The geology of the Gipskeuper in Baden-Württemberg (Germany) and the phenomenology of [...]
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SIMP-ALL: a generalized SIMP method based on the topological derivative concept
A. Ferrer

International Journal for Numerical Methods in Engineering (2019). Vol. 120 (3), pp. 361-381 (Preprint)

Abstract
Topology optimization has emerged in the last years as a promising research fieldwith a wide range of applications. One of the most successful approaches, theSIMP method, is based on regularizing the problem and proposing a penaliza-tion interpolation function. In this work, we propose an alternative interpolationfunction, the SIMP-ALL method that is based on the topological derivative con-cept. First, we show the strong relation in plane linear elasticity between theHashin-Shtrikman (H-S) bounds and the topological derivative, providing anew interpretation of the last one. Then, we show that the SIMP-ALL interpo-lation remains always in between the H-S bounds regardless the materials tobe interpolated. This result allows us to interpret intermediate values as realmicrostructures. Finally, we verify numerically this result and we show the con-venience of the proposed SIMP-ALL interpolation for obtaining auto-penalizedoptimal design in a wider range of cases. A MATLAB code of the SIMP-ALLinterpolation function is also provided
Abstract
Topology optimization has emerged in the last years as a promising research fieldwith a wide range of applications. One of the most successful approaches, theSIMP method, [...]
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AAR-based decomposition method for lower-bound limit analysis
J. Muñoz, R. Syednima

Proceedings of the ICE-Engineering and Computational Mechanics (2015). Vol. 168 (4), pp. 169-177 (Preprint)

Abstract
Despite the recent progress in optimisation techniques, finite-element stability analysis of realistic three-dimensional problems is still hampered by the size of the resulting optimisation problem. Current solvers may take a prohibitive computational time, if they give a solution at all. The possible remedies to this are the design of adaptive de-remeshing techniques, decomposition of the system of equations or of the optimisation problem. This paper concentrates on the last approach, and presents an algorithm especially suited for limit analysis. Optimisation problems in limit analysis are in general convex but non-linear. This fact renders the design of decomposition techniques specially challenging. The efficiency of general approaches such as Benders or Dantzig–Wolfe is not always satisfactory, and strongly depends on the structure of the optimisation problem. This work presents a new method that is based on rewriting the feasibility region of the global optimisation problem as the intersection of two subsets. By resorting to the averaged alternating reflections (AAR) method in order to find the distance between the sets, the optimisation problem is successfully solved in a decomposed manner. Some representative examples illustrate the application of the method and its efficiency with respect to other well-known decomposition algorithms
Abstract
Despite the recent progress in optimisation techniques, finite-element stability analysis of realistic three-dimensional problems is still hampered by the size of the resulting [...]
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PedPIV: Pedestrian Velocity Extraction From Particle Image Velocimetry
M. Baqui, R. Lohner

IEEE Transactions on Intelligent Transportation Systems (2019). Vol. 21 (2), pp. 580-589

Abstract
The analysis of velocities from high-density pedestrian events may provide more information on pedestrian flow dynamics. A framework based on particle image velocimetry (PIV)-a technique commonly used in experimental fluid dynamics-has been developed to evaluate the pedestrian velocities from high-density pedestrian events. The framework takes a sequence of two or more images from a regular closed-circuit television camera and obtains the flow properties (speed, direction) of the pedestrians. A detailed analysis has been done in both the qualitative and quantitative aspects of adapting PIV to pedestrian flow. An investigation through the use of the fundamental diagram and real-time extraction of crowd velocities is also presented and discussed. The proposed PIV-based framework enables on-the-spot analysis of pedestrian flow via velocity extraction in a reliable, automated manner.
Abstract
The analysis of velocities from high-density pedestrian events may provide more information on pedestrian flow dynamics. A framework based on particle image velocimetry (PIV)-a [...]
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Adaptive Embedded/Immersed Unstructured Grid Techniques
R. Lohner, J. Cebral, F. Camelli, S. Appanaboyina, J. Baum, E. Mestreau, O. Soto

Computer Methods in Applied Mechanics and Engineering (2008). Vol. 197 (25-28), pp. 2173-2197

Abstract
Embedded mesh, immersed body or fictitious domain techniques have been used for many years as a way to discretize geometrically complex domains with structured grids. The use of such techniques within adaptive, unstructured grid solvers is relatively recent. The combination of body-fitted functionality for some portion of the domain, together with embedded mesh or immersed body functionality for another portion of the domain offers great advantages, which are increasingly being exploited. The present paper reviews the methodologies pursued so far, addresses implementational issues and shows the possibilities such techniques offer.
Abstract
Embedded mesh, immersed body or fictitious domain techniques have been used for many years as a way to discretize geometrically complex domains with structured grids. The [...]
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A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids
H. Luo, J. Baum, R. Lohner

Journal of Computational Physics (2007). Vol. 225 (1), pp. 686-713

Abstract
A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods.
Abstract
A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite [...]
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Surface triangulation over intersecting geometries
A. Shostko, R. Lohner, W. Sanderg

Int. J. Numer. Meth. Engng. (1999). Vol. 44 (9), pp. 1359-1376

Abstract
A method for the rapid construction of meshes over intersecting triangulated shapes is described. The method is based on an algorithm that automatically generates a surface mesh from intersecting triangulated surfaces by means of Boolean intersection/union operations. After the intersection of individual components is obtained, the exposed surface parts are extracted. The algorithm is intended for rapid interactive construction of non‐trivial surfaces in engineering design, manufacturing, visualization and molecular modelling applications. Techniques to make the method fast and general are described. The proposed algorithm is demonstrated on a number of examples, including intersections of multiple spheres, planes and general engineering shapes, as well as generation of surface and volume meshes around clusters of intersecting components followed by the computation of flow field parameters.
Abstract
A method for the rapid construction of meshes over intersecting triangulated shapes is described. The method is based on an algorithm that automatically generates a surface [...]
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Merging of intersecting triangulations for finite element modeling
J. Cebral, R. Lohner, P. Choyke, P. Yim

Journal of Biomechanics (2001). Vol. 34 (6), pp. 815-819

Abstract
Surface mesh generation over intersecting triangulations is a problem common to many branches of biomechanics. A new strategy for merging intersecting triangulations is described. The basis of the method is that object surfaces are represented as the zero-level iso-surface of the distance-to-surface function defined on a background grid. Thus, the triangulation of intersecting objects reduces to the extraction of an iso-surface from an unstructured grid. In a first step, a regular background mesh is constructed. For each point of the background grid, the closest distance to the surface of each object is computed. Background points are then classified as external or internal by checking the direction of the surface normal at the closest location and assigned a positive or negative distance, respectively. Finally, the zero-level iso-surface is constructed. This is the final triangulation of the intersecting objects. The overall accuracy is enhanced by adaptive refinement of the background grid elements. The resulting surface models are used as support surfaces to generate three-dimensional grids for finite element analysis. The algorithms are demonstrated by merging arterial branches independently reconstructed from contrast-enhanced magnetic resonance images and by adding extra features such as vascular stents. Although the methodology is presented in the context of finite element analysis of blood flow, the algorithms are general and can be applied in other areas as well.
Abstract
Surface mesh generation over intersecting triangulations is a problem common to many branches of biomechanics. A new strategy for merging intersecting triangulations is described. [...]
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Generalized Meshing Environment for Computational Mechanics
L. Spradley, R. Lohner, T. Chung

AIAA Journal (1998). Vol. 36 (9), pp. 1735-1737

Abstract
One of the most time-consuming tasks in performing any discrete analysis the construction of a suitable mesh to represent the computational domain. Mesh generation today remains one of the pacing items in computational mechanics and consumes too many work hours and too many computers resources. This critical need for more progress in three-dimensional resources. This critical need for more progress in three-dimensional mesh generation certainly requires efficient geometry definition systems and advances in meshing methods. However, another major factor toward this progress is efficient methodology and computer software for automation of the meshing process. It is precisely this aspect of meshing that’s is the focus of this Note.
Abstract
One of the most time-consuming tasks in performing any discrete analysis the construction of a suitable mesh to represent the computational domain. Mesh generation today remains [...]
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Parallel unstructured grid generation
R. Lohner, J. Camberos, M. Merriam

Computer Methods in Applied Mechanics and Engineering (1992). Vol. 95 (3), pp. 343-357

Abstract
A parallel unstructured grid generation algorithm is presented and implemented on the INTEL hypercube. Different processor hierarchies are discussed, and the appropriate hierarchies for mesh generation and mesh smoothing are selected. A domain-splitting algorithm for unstructured grids, which tries to minimize the surface-to-volume ratio of each subdomain, is described. This splitting algorithm is employed both for grid generation and grid smoothing. Results obtained on the INTEL hypercube demonstrate the effectiveness of the algorithms developed.
Abstract
A parallel unstructured grid generation algorithm is presented and implemented on the INTEL hypercube. Different processor hierarchies are discussed, and the appropriate hierarchies [...]
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Ray tracing with a space‐filling finite element mesh
J. Fave, R. Lohner

International Journal for Numerical Methods in Engineering (1994). Vol. 37 (20), pp. 3571-3580

Abstract
A new ray‐tracing technique is presented which does not work with ray–object intersections per se , but is based on the traversal of an unstructured tetrahedral mesh providing a convex enclosure of a scene of polyhedral objects. The tetrahedral mesh provides tight bounding and an adaptive subdivision of space. This non‐hierarchical data structure is traversed adaptively until one is led directly and unconditionally to the first object intersected. Rendering times are directly related to the average thickness of the enclosing mesh since all tetrahedra are traversed in constant time. Since the proposed algorithm operates directly with volume elements, it allows for volumetric rendering effects. Volume rendering or anisotropic media can be implemented without any further effort. This is an important advantage as compared to usual techniques, which only operate on surface data. Timings for several examples show that the use of this type of ray‐tracing technique, which is more suitable for general purpose visualization codes than traditional techniques, results in CPU times that are comparable with the best ray‐tracing techniques presently used. This is an unexpected and important result, as the vectorization and parallclization of the proposed technique are straightforward, in contrast with traditional ray‐tracing techniques
Abstract
A new ray‐tracing technique is presented which does not work with ray–object intersections per se , but is based on the traversal of an unstructured tetrahedral [...]
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Three‐dimensional parallel unstructured grid generation
A. Shostko, R. Lohner

International Journal for Numerical Methods in Engineering (1994). Vol. 38 (6), pp. 905-925

Abstract
An algorithm for the parallel generation of 3-D unstructured grids is presented. The technique is an extension of the algorithm presented in Reference 21 for the 2-D case. The method uses a background grid as the means to separate spatially different regions, enabling the concurrent, parallel generation of elements in different domains and interdomain regions. The parallel 3-D grid generator was implemented and tested on the INTEL hypercube and Touchstone Delta parallel computers. The results obtained demonstrate the effectiveness of the algorithm developed. The methodology is applicable to the parallel implementation of a wide range of problems that are, in principle, scalar by nature, and do not lend themselves to SIMD parallelization.
Abstract
An algorithm for the parallel generation of 3-D unstructured grids is presented. The technique is an extension of the algorithm presented in Reference 21 for the 2-D case. [...]
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An algebraically partitioned FETI method for parallel structural analysis: performance evaluation
M. Justino, K. Park, C. Felippa

International Journal for Numerical Methods in Engineering (1997). Vol. 40 (15), pp. 2739-2758

Abstract
This paper presents the algorithmic performance of an algebraically partitioned Finite Element Tearing and Interconnection (FETI) method presented in a companion paper. A simple structural assembly topology is employed to illustrate the implementation steps in a Matlab software environment. Numerical results indicate that the method is scalable, provided the iterative solution preconditioner employs the reduced interface Dirichlet preconditioner. A limited comparison of the present method with the differentially partitioned FETI method with corner modes is also offered. Based on this comparison and a reasonable extrapolation, we conclude the present algebraically partitioned FETI method possesses a similar iteration convergence property of the differentially partitioned FETI method with corner modes
Abstract
This paper presents the algorithmic performance of an algebraically partitioned Finite Element Tearing and Interconnection (FETI) method presented in a companion paper. A [...]
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Error analysis of penalty function techniques for constraint definition in linear algebraic systems
C. Felippa

International Journal for Numerical Methods in Engineering (1977). Vol. 11 (4), pp. 709-728

Abstract
The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.
Abstract
The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational [...]
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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 [...]
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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, [...]
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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 [...]
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A comparison between new adaptive remeshing strategies based on point wise stress error estimation and energy norm error estimation
G. Bugeda

Commun. Numer. Meth. Engng (2002). Vol. 18 (7), 469-482

Abstract
Traditionally, the most commonly used mesh adaptive strategies for linear elastic problems are based on the use of an energy norm for the measurement of the error, and a mesh refinement strategy based on the equal distribution of the error between all the elements. However, little attention has been paid to the study of alternative error norms and alternative refinement strategies. This paper studies the feasibility of using alternative mesh refinement strategies based on — the use of the classical error energy norm and an optimality criterion based on the equal distribution of the density of error, — the use of alternative error norms based on measurements of the point wise error contained in the main magnitudes that control the equilibrium problem and/or the material constitutive equations such as the stresses (e.g. the Von Mises stress). The feasibility of using all the described strategies is demonstrated through the solution of a benchmark example. This example is also used for comparison between the described refinement criteria.
Abstract
Traditionally, the most commonly used mesh adaptive strategies for linear elastic problems are based on the use of an energy norm for the measurement of the error, and a mesh [...]
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On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections
S. Badia, A. Martín, J. Principe

Parallel Computing (2015). (Preprint) Vol. 50, pp. 1-24

Abstract
In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.
Abstract
In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when [...]
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Computing the volume enclosed by a periodic surface and its variation to model a follower pressure
M. Rahimi, K. Zhang, M. Arroyo

Computational Mechanics (preprint) (2015). Vol. 55 (3), pp. 519-525

Abstract
In modeling and numerically implementing a follower pressure in a geometrically nonlinear setting, one needs to compute the volume enclosed by a surface and its variation. For closed surfaces, the volume can be expressed as a surface integral invoking the divergence theorem. For periodic systems, widely used in computational physics and materials science, the enclosed volume calculation and its variation is more delicate and has not been examined before. Here, we develop simple expressions involving integrals on the surface, on its boundary lines, and point contributions. We consider two specific situations, a periodic tubular surface and a doubly periodic surface enclosing a volume with a nearby planar substrate, which are useful to model systems such as pressurized carbon nanotubes, supported lipid bilayers or graphene. We provide a set of numerical examples, which show that the familiar surface integral term alone leads to an incorrect volume evaluation and spurious forces at the periodic boundaries.
Abstract
In modeling and numerically implementing a follower pressure in a geometrically nonlinear setting, one needs to compute the volume enclosed by a surface and its variation. [...]
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XLME interpolants, a seamless bridge between XFEM and enriched meshless methods
F. Amiri, C. Anitescu, M. Arroyo, S. Bordas, T. Rabezuk

Computational Mechanics (2014). Vol. 53 (1), pp. 45-57 (Preprint)

Abstract
In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.
Abstract
In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems [...]
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On the optimum support size in meshfree methods: a variational adaptivity approach with maximum-entropy approximants
D. Millán, A. Rosolen, M. Arroyo

Int. J. Numer. Meth. Engng. (2010). Vol. 82 (7), pp. 868-895

Abstract
We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions.
Abstract
We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming [...]
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Locking in the incompressible limit: pseudo-divergence-free element free Galerkin
Y. Vidal, P. Villon, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 19 (9), pp. 725-735

Abstract
Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the element free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the EFG method is proposed in order to alleviate volumetric locking.Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter), it is proved. Therefore, diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids corroborate this issue.
Abstract
Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This [...]
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Computing bounds for linear functionals of exact weak solutions to Poisson's equation
A. Sauer-Budge, J. Bonet, A. Huerta, J. Peraire

SIAM J. Numer. Anal. (2004). Vol. 42 (4), pp. 1610-1630

Abstract
We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact weak solution of the infinite-dimensional continuum problem with piecewise-polynomial forcing. The method results from exploiting the Lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem. Localization is achieved by Lagrangian relaxation and the bounds are computed by appeal to a local dual problem. The proposed method computes approximate Lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers. The computed bounds hold uniformly for any level of refinement, and in the asymptotic convergence regime of the finite element method, the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization. The elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators, as we demonstrate with a numerical example.
Abstract
We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact [...]
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An improved algorithm to smooth graded quadrilateral meshes preserving the prescribed element size
J. Sarrate, A. Huerta

Comput. Methods Appl. Mech. Engrg., (2001). Vol. 17 (2), pp. 89-99

Abstract
In the generation of quadrilateral unstructured meshes, special attention is focussed to the shape of the elements. This is because it is well known that the distortion of the elements and the accuracy of the analysis are closely related. However, in adaptive schemes it is also essential that the newly generated mesh meets the prescribed element sizes in order to obtain a solution with the desired precision. In 1982 Giuliani developed a robust rezoning algorithm based on geometrical criteria. It gives proven results in a smooth element size distribution, but elements do not verify the prescribed element size when sharp distributions appear. This paper presents a modification of the Giuliani method that generates non-distorted elements while preserving the element size. Similar to the original method, this modification can be extended to three-dimensional cases.
Abstract
In the generation of quadrilateral unstructured meshes, special attention is focussed to the shape of the elements. This is because it is well known that the distortion of [...]
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Locking in the incompressible limit: pseudo-divergence-free element free Galerkin
Y. Vidal, P. Villon, A. Huerta

European Journal of Computational Mechanics (2002). Vol. 11 (7-8), pp. 869-892

Abstract
Locking in finite elements has been a major concern since its early developments and has been extensively studied. However, locking in mesh-free methods is still an open topic. Until now the remedies proposed in the literature are extensions of already developed methods for finite elements. Here a new approach is explored and an improved formulation that asymptotically suppresses volumetric locking for the EFG method is proposed. The diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids and fluids corroborate this issue.
Abstract
Locking in finite elements has been a major concern since its early developments and has been extensively studied. However, locking in mesh-free methods is still an open topic. [...]
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A new least-squares approximation of affine mappings for sweep algorithms
X. Roca, J. Sarrate, A. Huerta

Engineering with computers (2010). Vol. 26 (3), pp. 327-337

Abstract
This paper presents a new algorithm to generate hexahedral meshes in extrusion geometries. Several algorithms have been devised to generate hexahedral meshes by projecting the cap surfaces along a sweep path. In all of these algorithms the crucial step is the placement of the inner layer of nodes. That is, the projection of the source surface mesh along the sweep path. From the computational point of view, sweep methods based on a least-squares approximation of an affine mapping are the fastest alternative to compute these projections. Several functionals have been introduced to perform the least-squares approximation. However, for very simple and typical geometrical configurations they may generate low-quality projected meshes. For instance, they may induce skewness and flattening effects on the projected discretizations. In addition, for these configurations the minimization of these functionals may lead to a set of normal equations with singular system matrix. In this work we analyze previously defined functionals. Based on this analysis we propose a new functional and show that its minimization overcomes these drawbacks. Finally, we present several examples to assess the properties of the proposed functional.
Abstract
This paper presents a new algorithm to generate hexahedral meshes in extrusion geometries. Several algorithms have been devised to generate hexahedral meshes by projecting [...]
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Efficiency of high-order elements for continuous and discontinuous Galerkin methods
A. Huerta, A. Angeloski, X. Roca, J. Peraire

Int. J. Numer. Meth. Engng. (2013). Vol. 96 (6), pp. 529-560 (Preprint)

Abstract
To evaluate the computational performance of high-order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high-order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two-dimensional and three-dimensional domains for simplices and parallelotopes. Moreover, both element-wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high-order methods are more efficient than linear ones.
Abstract
To evaluate the computational performance of high-order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear [...]
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Étude de la stabilité d'une formulation incompressible traitée par X-FEM
G. Legrain, N. Moës, A. Huerta

European Journal of Computational Mechanics (2006). Vol. 15 (1, 2, 3), pp. 257-268

Abstract
The treatment of (near-)incompressibility is a major concern for the simulation of rubber-like parts, or forming processes. The use of mixed finite element methods is known to prevent the locking of the F.E. approximation in the incompressible limit. However, the stability of these formulations is conditionned by the fullfilment of the inf-sup condition. Recently, finite elements method has evolved with the introduction of the partition of unity. The X-FEM uses it to remove the need to mesh (and remesh) physical surfaces. In this paper, a strategy is proposed for the treatment of holes within X-FEM in the incompressible setting. Numerical examples show that F.E. convergence rate is preserved and that the inf-sup condition is passed.
Abstract
The treatment of (near-)incompressibility is a major concern for the simulation of rubber-like parts, or forming processes. The use of mixed finite element methods is known [...]
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Hybrid coupling of CG and HDG discretizations based on Nitsche’s method
A. Spina, M. Giacomini, A. Huerta

Computational mechanics (2020). Vol. 65 (2), pp. 311-330

Abstract
A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The con- tinuity of the solution is imposed in the CG problem via Nitsche’s method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann con- dition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors. This is a post-peer-review, pre-copyedit version of an article published in Computational mechanics.
Abstract
A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal [...]
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Tensor representation of non-linear models using cross approximations
J. Aguado, D. Borzacchiello, K. Kollepara, F. Chinesta, A. Huerta

Journal of scientific computing (2019). Vol. 81 (1), pp. 22-47

Abstract
Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed. This is a post-peer-review, pre-copyedit version of an article published in Journal of scientific computing
Abstract
Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution [...]
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A Bayesian approach to integrate temporal data into probabilistic risk analysis of monitored NAPL remediation
D. Fernàndez-García, D. Bolster, X. Sánchez-Vila, D. Tartakovsky

Advances in Water Resources (2012). Vol. 36, pp. 108-120

Abstract
Upon their release into the subsurface, non-aqueous phase liquids (NAPLs) dissolve slowly in groundwater and/or volatilize in the vadose zone threatening the environment and public health over extended periods of time. The failure of a treatment technology at any given site is often due to the unnoticed presence of dissolved NAPL trapped in low permeability areas and/or the remaining presence of substantial amounts of pure phase NAPL after remediation efforts. The design of remediation strategies and the determination of remediation endpoints are traditionally carried out without quantifying risks associated with the failure of such efforts. We conduct a probabilistic risk analysis (PRA) to estimate the likelihood of failure of an on-site NAPL treatment technology. The PRA integrates all aspects of the problem (causes, pathways, and receptors) without resorting to extensive modeling. It accounts for a combination of multiple mechanisms of failure of a monitoring system, such as bypassing, insufficient sampling frequency and malfunctioning of the observation wells. We use a Bayesian framework to update the likelihood of failure of the treatment technology with observed measurements of concentrations at nearby monitoring wells
Abstract
Upon their release into the subsurface, non-aqueous phase liquids (NAPLs) dissolve slowly in groundwater and/or volatilize in the vadose zone threatening the environment and [...]
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Impact of Basel III Countercyclical Measures on Financial Stability: An Agent-Based Model
B. Llacay, G. Peffer

Journal of Artificial Societies and Social Simulation (2019). Vol. 22 (1), pp. 1-35

Abstract
The financial system is inherently procyclical, as it amplifies the course of economic cycles, and precisely one of the factors that has been suggested to exacerbate this procyclicality is the Basel regulation on capital requirements. After the recent credit crisis, international regulators have turned their eyes to countercyclical regulation as a solution to avoid similar episodes in the future. Countercyclical regulation aims at preventing excessive risk taking during booms to reduce the impact of losses suffered during recessions, for example increasing the capital requirements during the good times to improve the resilience of financial institutions at the downturn. The Basel Committee has already moved forward towards the adoption of countercyclical measures on a global scale: the Basel III Accord, published in December 2010, revises considerably the capital requirement rules to reduce their procyclicality. These new countercyclical measures will not be completely implemented until 2019, so their impact cannot be evaluated yet, and it is a crucial question whether they will be effective in reducing procyclicality and the appearance of crisis episodes such as the one experienced in 2007-08. For this reason,we present in this article an agent-based model aimed at analysing the effect of two countercyclical mechanisms introduced in Basel III: the countercyclical buffer and the stressed VaR. In particular, we focus on the impact of these mechanisms on the procyclicality induced by market risk requirements and, more specifically, by value-at-risk models, as it is a issue of crucial importance that has received scant attention in the modeling literature. The simulation results suggest that the adoption of both of these countercyclical measures improves market stability and reduces the emergence of crisis episodes
Abstract
The financial system is inherently procyclical, as it amplifies the course of economic cycles, and precisely one of the factors that has been suggested to exacerbate this [...]
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Space-time balancing domain decomposition
S. Badia, M. Olm

SIAM journal on scientific computing (2017). Vol. 39 (2), pp. 194-213 (Preprint)

Abstract
In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the subassembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the subassembled to the original finite element space. With regard to the subassembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same total elapsed time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent wall clock time weak scalability is achieved for space-time parallel solvers on some thousands of cores No separate or additional fees are collected for access to or distribution of the work. -
Abstract
In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an [...]
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A robust algorithm for implicit description of immersed geometries within a background mesh
D. Baumgärtner, J. Wolf, R. Rossi, P. Dadvand, R. Wüchner

Advanced modeling and simularion in engineering sciences (2018). Vol. 5 (1), pp. 1-40

Abstract
The paper presents a robust algorithm, which allows to implicitly describe and track immersed geometries within a background mesh. The background mesh is assumed to be unstructured and discretized by tetrahedrons. The contained geometry is assumed to be given as triangulated surface. Within the background mesh, the immersed geometry is described implicitly using a discontinuous distance function based on a level-set approach. This distance function allows to consider both, “double-sided” geometries like membrane or shell structures, and “single-sided” objects for which an enclosed volume is univocally defined. For the second case, the discontinuous distance function is complemented by a continuous signed distance function, whereas ray casting is applied to identify the closed volume regions. Furthermore, adaptive mesh refinement is employed to provide the necessary resolution of the background mesh. The proposed algorithm can handle arbitrarily complicated geometries, possibly containing modeling errors (i.e., gaps, overlaps or a non-unique orientation of surface normals). Another important advantage of the algorithm is the embarrassingly parallel nature of its operations. This characteristic allows for a straightforward parallelization using MPI. All developments were implemented within the open source framework “KratosMultiphysics” and are available under the BSD license. The capabilities of the implementation are demonstrated with various application examples involving practice-oriented geometries. The results finally show, that the algorithm is able to describe most complicated geometries within a background mesh, whereas the approximation quality may be directly controlled by mesh refinement.
Abstract
The paper presents a robust algorithm, which allows to implicitly describe and track immersed geometries within a background mesh. The background mesh is assumed to be unstructured [...]
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On micro-to-macro connections in domain decomposition multiscale methods
O. Lloberas-Valls, D. Rixen, A. Simone, L. Sluys

Computer Methods in Applied Mechanics and Engineering (2012). Vol. 225-228, pp. 177-196

Abstract
Micro-to-macro connection techniques constitute a key ingredient in the formulation of multiscale strategies. In this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials. Particularly, these techniques are investigated in a domain decomposition strong coupling multiscale framework. The structural component under analysis is partitioned into a number of non-overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom-in technique. Mesh refinement is employed in the domains where crack coalescence and growth take place. The original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains. Standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links. The influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis
Abstract
Micro-to-macro connection techniques constitute a key ingredient in the formulation of multiscale strategies. In this contribution several connection methods are explored [...]
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Parabolic recovery of boundary gradients
R. Lohner, J. Cebral, S. Appanaboyina

Communications in numerical methods in engineering (2008). Vol. 24 (12), pp. 1611-1615

Abstract
A parabolic recovery procedure suited for shear stress and heat flux recovery on surfaces from linear element data is proposed. The information required consists of the usual unknowns at points, as well as gradients recovered at the points that are one layer away from the wall. The procedure has been in use for some time and has consistently delivered superior results as compared with the usual wall shear stress and heat flux obtained from linear finite element method shape functions.
Abstract
A parabolic recovery procedure suited for shear stress and heat flux recovery on surfaces from linear element data is proposed. The information required consists of the usual [...]
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On The Modeling Of Boundary Conditions For Embedded Schemes
E. Mestreau, J. Baum, R. Lohner, C. Charman

WIT Transtactions on The Built Environment (2003). Vol. 71 (12), pp. 229-239

Abstract
A simple embedded method for an unstructured grid is presented. An enhancement of the boundary treatments using ghost points is then described. Several examples are used to demonstrate the viability of the approach for inviscid flow simulations.
Abstract
A simple embedded method for an unstructured grid is presented. An enhancement of the boundary treatments using ghost points is then described. Several examples are used to [...]
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Generation of non‐isotropic unstructured grids via directional enrichment
R. Lohner, J. Cebral

Int. J. Numer. Meth. Engng. (2000). Vol. 49 (1-2), pp. 219-232

Abstract
A procedure for the generation of highly stretched grids suitable for Reynolds‐averaged Navier–Stokes (RANS) calculations is presented. In a first stage, an isotropic (Euler) mesh is generated. In a second stage, this grid is successively enriched with points in order to achieve highly stretched elements. The element reconnection is carried out using a constrained Delaunay approach. Points are introduced from the regions of lowest stretching towards the regions of highest stretching. The procedure has the advantages of not requiring any type of surface recovery, not requiring extra passes or work to mesh concave ridges/corners, and guarantees a final mesh, an essential requirement for industrial environments. Given that point placement and element quality are highly dependent for the Delaunay procedure, special procedures were developed in order to obtain optimal point placement.
Abstract
A procedure for the generation of highly stretched grids suitable for Reynolds‐averaged Navier–Stokes (RANS) calculations is presented. In a first stage, an isotropic [...]
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A parallel advancing front grid generation scheme
R. Lohner

Int. J. Numer. Meth. Engng. (2001). Vol. 51 (6), pp. 663-678

Abstract
A parallel advancing front scheme has been developed. The domain to be gridded is first subdivided spatially using a relatively coarse octree. Boxes are then identified and gridded in parallel. A scheme that resembles closely the advancing front technique on scalar machines is recovered by only considering the boxes of the active front that generate small elements. The procedure has been implemented on the SGI origin class of machines using the shared memory paradigm. Timings for a variety of cases show speedups similar to those obtained for flow codes. The procedure has been used to generate grids with tens of millions of elements.
Abstract
A parallel advancing front scheme has been developed. The domain to be gridded is first subdivided spatially using a relatively coarse octree. Boxes are then identified and [...]
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Regridding Surface Triangulations
R. Lohner

Journal of Computational Physics (1996). Vol. 126 (1), pp. 1-10

Abstract
An advancing front surface gridding technique that operates on discretely defined surfaces (i.e. triangulations) is presented. Different aspects that are required to make the procedure reliable for complex geometries are discussed. Notable among these are (a) the recovery of surface features and discrete surface patches from the discrete data, (b) filtering based on point and side-normals to remove undesirable data close to cusps and corners, (c) the proper choice of host faces for ridges, and (d) fast interpolation procedures suitable for complex geometries. Post-generation surface recovery or repositioning techniques are discussed. Several examples ranging from academic to industrial demonstrate the utility of the proposed procedure forab initiosurface meshing from discrete data, such as those encountered when the surface description is already given as discrete, the improvement of existing surface triangulations, as well as remeshing applications during runs exhibiting significant change of domain.
Abstract
An advancing front surface gridding technique that operates on discretely defined surfaces (i.e. triangulations) is presented. Different aspects [...]
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Extensions and improvements of the advancing front grid generation technique
R. Lohner

Comput. Methods Appl. Mech. Engrg., (1996). Vol. 5 (10), pp. 119-132

Abstract
We describe extensions and improvements to the advancing front grid generation technique that have proven useful over the years. The following areas are treated in detail: situations with thin or crossing surfaces, meshing of surfaces defined by triangulations, and ease of user input to define the desired element size in space. The first extension is important if one considers the generation of volumetric grids around shells, membranes, fabrics, or CAD-data that exhibit cusps. Traditional advancing front generators are likely to fail in these situations. We propose the introduction of a crossing environment variable attached to faces and points in order to filter out undesired or incorrect information during the grid generation process. The second extension is required for situations where the surfaces to be gridded are not defined analytically, but via a triangulation. Typical cases where such triangulations are used to define the domain are geophysical problems, climate modelling and medical problems. The third topic deals with the reduction of manual labour to specify element size in space. Sources, element size attached directly to CAD-data, and adaptive background grids are discussed. Adaptive background grids, in combination with surface deviation tolerances, are used to obtain surface triangulations that represent the geometry faithfully, and at the same time enable a smooth transition to volumetric meshes.
Abstract
We describe extensions and improvements to the advancing front grid generation technique that have proven useful over the years. The following areas are treated in detail: [...]
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A vectorized particle tracer for unstructured grids
R. Lohner, J. Ambrosiano

Journal of Computational Physics (1990). Vol. 91 (1), pp. 22-31

Abstract
A vectorized particle tracer for unstructured grids is described. The basic approach is to use elementary properties of the linear basis functions to search for particles on the grid using the element last occupied as an initial guess. To permit vectorization, a simple binary sort of the particles is performed every timestep such that all particles that have as yet not found their host element remain at the top of the list. In this way, vector-loops can be easily formed. Timings taken from a numerical example indicate that speed-ups of the order of 1:14 can be obtained on vector-machines when using this algorithm.
Abstract
A vectorized particle tracer for unstructured grids is described. The basic approach is to use elementary properties of the linear basis functions to search for particles [...]
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Unstructured multigrid methods for elliptic problems
R. Lohner, K. Morgan

International Journal for Numerical Methods in Engineering (1987). Vol. 24 (1), pp. 3323-3331

Abstract
A multigrid algorithm for implementation on unstructured meshes is proposed. The algorithm uses a sequence of unnested grids and requires the development of efficient inter-grid interpolation procedures. It is demonstrated how elliptic problems can be solved in this fashion by using Jacobi smoothers
Abstract
A multigrid algorithm for implementation on unstructured meshes is proposed. The algorithm uses a sequence of unnested grids and requires the development of efficient inter-grid [...]
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Comparison of strain recovery techniques for the mixed iterative method
J. Cervenka, S. Keating, C. Felippa

Computer Methods in Applied Mechanics and Engineering (1993). Vol. 9 (11), pp. 925-932

Abstract
Three strain recovery techniques are investigated in the framework of the mixed iterative method. The need for appropriate strain recovery for low-order elements is shown. The three techniques compared are: C-lumping, strain-smoothing and a new technique called C-splitting. It is shown that the use of C-lumping has negative effects on the convergence of the mixed iterative method. Both strain-smoothing and C-splitting deliver convergent results, but the new technique converges faster.
Abstract
Three strain recovery techniques are investigated in the framework of the mixed iterative method. The need for appropriate strain recovery for low-order elements is shown. [...]
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Customizing high performance elements by fourier methods
C. Felippa

(2001). Trends in computational structural mechanics.

Abstract
This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling problems. The example focuses on the improvement of the mass and geometric stiffness matrices of plane beams. Similar methods can be used for Kirchhoff plate bending elements, but this material is omitted because of space constraints. The process interweaves classical tools: Fourier analysis and orthogonal polynomials, with newer ones: finite element templates and computer algebra systems. Templates are parametrized algebraic forms that uniquely characterize an element population by a "genetic signature" defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design custom elements. For the beam example, Legendre polynomials are used to construct templates for the material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and stiffness that deliver matrices with desirable attributes for specific target situations. Three objectives are noted: high accuracy for vibration and buckling analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. This paper examines in some detail only the first objective
Abstract
This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling [...]
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A survey of parametrized variational principles and applications to computational mechanics
C. Felippa

Computer Methods in Applied Mechanics and Engineering (1994). Vol. 113 (1-2), pp. 109-139

Abstract
This survey paper describes recent developments in the area of parametrized variational principles (PVPs) and selected applications to finite-element computational mechanics. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations. The theory of single-field PVPs, based on gauge functions (also known as null Lagrangians) is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multifield PVPs are more interesting from theoretical and practical standpoints. Following a tutorial introduction, the paper describes the recent construction of multifield PVPs in several areas of elasticity and electromagnetics. It then discusses three applications to finite-element computational mechanics: the derivation of high-performance finite elements, the development of element-level error indicators, and the construction of finite element templates. The paper concludes with an overview of open research areas.
Abstract
This survey paper describes recent developments in the area of parametrized variational principles (PVPs) and selected applications to finite-element computational mechanics. [...]
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A localized version of the method of Lagrange multipliers and its applications
K. Park, C. Felippa, U. Gumaste

Computational Mechanics (2000). Vol. 24 (6), pp. 476-490

Abstract
This paper describes a novel version of the method of Lagrange multipliers for an improved modeling of multi-point constraints that emanate from contact-impact problems, partitioned structural analysis using parallel computers, and structural inverse problems. It is shown that the classical method of Lagrange multipliers can lead to a non-unique set of constraint conditions for the modeling of interfaces involving more than two or multi-point substructural interface nodes. The proposed version of the method of Lagrange multipliers leads not only to unique construction of constraints but also encounters no singularity in modeling an arbitrary number of multi-point constraints. An important utilization of the present method is in the regularized modeling of interfaces whose rigidities are radically different from one to another. The present approach is demonstrated via several examples for its simplicity in modeling constraints, ease of implementation and computational advantages
Abstract
This paper describes a novel version of the method of Lagrange multipliers for an improved modeling of multi-point constraints that emanate from contact-impact problems, partitioned [...]
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Construction of Customized Mass-Stiffness Pairs Using Templates
C. Felippa

Journal of Aerospace Engineering (2006). Vol. 19 (4), pp. 241-258

Abstract
This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition focuses on adjusting the mass matrix while a separately provided stiffness matrix is kept fixed. Two well known kinetic-energy discretization methods described in finite-element method (FEM) textbooks since the mid-1960s lead to diagonally lumped and consistent mass matrices, respectively. These two models are sufficient to cover many engineering applications. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. This approach is discussed in this paper using one-dimensional structural elements as examples. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance, and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Thus those standard models are not excluded. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes.
Abstract
This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition [...]
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Partitioned Transient Analysis Procedures for Coupled-Field Problems: Accuracy Analysis
K. Park, C. Felippa

Journal of Applied Mechanics (1980). Vol. 47 (4), pp. 919-926

Abstract
Partitioned solution procedures for direct time integration of second-order coupled-field systems are studied from the standpoint of accuracy. These procedures are derived by three formulation steps: implicit integration of coupled governing equations, partitioning of resulting algebraic systems and extrapolation on the right-hand partition. It is shown that the combined effect of partition, extrapolation, and computational paths governs the choice of stable extrapolators and preservation of rigid-body motions. Stable extrapolators for various computational paths are derived and implementation-extrapolator combinations which preserve constant-velocity and constant-acceleration rigid-body motions are identified. A spectral analysis shows that the primary error source introduced by a stable partition is frequency distortion. Finally, as a guide to practical applications, the advantages and shortcomings of five specific partitions are discussed.
Abstract
Partitioned solution procedures for direct time integration of second-order coupled-field systems are studied from the standpoint of accuracy. These procedures are derived [...]
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Trends in computational engineering
C. Felippa

Advances in Engineering Software (1981). Vol. 3 (2), pp. 50-54

Abstract
This article presents the author's views on computational engineering. This is an emerging branch of the applied sciences shaped by the interaction of engineering, mathematics, and computer sciences, and paced by present advances in software technology and computer hardware. Impact of these advances on various aspects of engineering software are discussed, and predictions as to future effects are ventured. An annotated multidisciplinary reference list is provided for curious readers.
Abstract
This article presents the author's views on computational engineering. This is an emerging branch of the applied sciences shaped by the interaction of engineering, mathematics, [...]
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Programming the isoparametric six-node triangle
C. Felippa

Engineering Computations (1990). Vol. 7 (2), pp. 173-177

Abstract
The formulation and programming of the six‐node, curved isoparametric triangle in plane stress is presented. This is intended to fill an apparent vacuum in the open finite element literature
Abstract
The formulation and programming of the six‐node, curved isoparametric triangle in plane stress is presented. This is intended to fill an apparent vacuum in the open finite [...]
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Penalty‐function iterative procedures for mixed finite element formulations
C. Felippa

International Journal for Numerical Methods in Engineering (1986). Vol. 22 (1), pp. 267-279

Abstract
Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. This paper looks at the problem from the standpoint of allowing fairly arbitrary approximations to be made on both the stiffness and compliance matrices used in solving for the corrections. The resulting iterative processes usually diverge unless stabilized with Courant penalty terms. An iterative procedure previously constructed for equality-constrained displacement models is recast to fit the mixed finite element formulation in which displacements play the role of Lagrange multipliers. The penalty function iteration is shown to reduce to an ordinary staggered stress-displacement iteration if the weight is set to zero. Convergence conditions for these procedures are stated and the potentially troublesome effect of prestress modes noted.
Abstract
Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. [...]
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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 [...]
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A new adaptive remeshing scheme based on the sensitivity analysis of the SPR point wise error estimation
G. Bugeda

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (4-6), pp. 462-478

Abstract
This paper presents a formulation for the obtainment of the sensitivity analysis of a point wise error estimator with respect to the nodal coordinates using the adjoint state method. The proposed point wise error estimator is based on the SPR method. The numerical accuracy of the presented sensitivity analysis has been tested by perturbing a mesh. The capability of the presented sensitivity analysis for the detection of pollution error has also been tested. A new adaptive remeshing strategy based on the sensitivity analysis of the point wise error estimation has been developed and tested. This strategy produces very cheap meshes for the accurate evaluation of stresses at specific points.
Abstract
This paper presents a formulation for the obtainment of the sensitivity analysis of a point wise error estimator with respect to the nodal coordinates using the adjoint state [...]
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Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants
C. Peco, D. Millán, A. Rosolen, M. Arroyo

Computers and Structures (2015). Vol. 150, pp. 52-62 (Preprint)

Abstract
In Galerkin meshfree methods, because of a denser and unstructured connectivity, the creation and assembly of sparse matrices is expensive. Additionally, the cost of computing basis functions can be significant in problems requiring repetitive evaluations. We show that it is possible to overcome these two bottlenecks resorting to simple and effective algorithms. First, we create and fill the matrix by coarse-graining the connectivity between quadrature points and nodes. Second, we store only partial information about the basis functions, striking a balance between storage and computation. We show the performance of these strategies in relevant problems.
Abstract
In Galerkin meshfree methods, because of a denser and unstructured connectivity, the creation and assembly of sparse matrices is expensive. Additionally, the cost of computing [...]
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A comparison of two formulations to blend finite elements and mesh-free methods
A. Huerta, S. Fernández-Méndez, W. Liu

Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (12-14), pp. 1105-1117

Abstract
Mesh-free methods have since their early developments been blended to the finite element formulation in order to benefit from the advantages of both numerical techniques. In this paper, two recently proposed formulations to couple mesh-free and finite element methods are discussed and compared.
Abstract
Mesh-free methods have since their early developments been blended to the finite element formulation in order to benefit from the advantages of both numerical techniques. [...]
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Benchmarks for the validation of a non local model
L. Jason, S. Ghavamian, G. Pijaudier, A. Huerta

Revue française de génie civil (2004). Vol. 8 (2-3), pp. 303-328

Abstract
The aim of this contribution is to present a series of organised benchmarks that helps at validating the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of structural elements. It is applied to an isotropic damage model used for concrete, coupled with a non local gradient formulation to avoid a spurious description of strain localisation. After the elementary (uniaxial monotonic, cyclic or triaxial loading) and structural (three point bending tests) simulations, an industrial application is presented in the form of a representative structural volume of a containment building for French nuclear power plants.
Abstract
The aim of this contribution is to present a series of organised benchmarks that helps at validating the robustness of the FE implementation of a constitutive relation, its [...]
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Numerical differentiation for non-trivial consistent tangent matrices: an application to the MRS-lade model
A. Pérez-Foguet, A. Rodríguez-Ferran, A. Huerta

Int. J. Numer. Meth. Engng. (2000). Vol. 48 (2), pp. 159-184

Abstract
In a companion paper Pérez-Foguet, A., Rodríguez-Ferran, A. and Huerta, A. Numerical differentiation for local and global tangent operators in computational plasticity. Computer Methods in Applied Mechanics and Engineering, 2000, in press, the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS-Lade model. This plastic model has a cone-cap yield surface and exhibits strong coupling between the flow vector and the hardening moduli. Because of this, differentiating these quantities with respect to stresses and internal variables - the crucial step in obtaining consistent tangent matrices - is rather involved. Numerical differentiation is used here to approximate these derivatives. The approximated derivatives are then used to (1) compute consistent tangent matrices (global problem) and (2) integrate the constitutive equation at each Gauss point (local problem) with the Newton-Raphson method. The choice of the stepsize (i.e. the perturbation in the approximation schemes), based on the concept of relative stepsize, poses no difficulties. In contrast to previous approaches for the MRS-Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational efficiency (CPU time) and robustness of the proposed approach is illustrated by means of several numerical examples, where the major relevant topics are discussed in detail.
Abstract
In a companion paper Pérez-Foguet, A., Rodríguez-Ferran, A. and Huerta, A. Numerical differentiation for local and global tangent operators in computational [...]
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Locking in the incompressible limit for the element-free Galerkin method
A. Huerta, S. Fernández-Méndez

Int. J. Numer. Meth. Engng. (2001). Vol. 51 (11), pp. 1361-1383

Abstract
Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis developed here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non-physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element-free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example.
Abstract
Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis [...]
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NURBS-Enhanced Finite Element Method (NEFEM)
R. Sevilla, S. Fernández-Méndez, A. Huerta

Archives of Comp. Meths. Engng. (2011). Vol. 18 (4), pp. 441-484

Abstract
The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.
Abstract
The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain [...]
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A note on upper bound formulations in limit analysis
J. Muñoz, A. Huerta, J. Bonet, J. Peraire

Int. J. Numer. Meth. Engng. (2012). Vol. 91 (8), pp. 896-908

Abstract
In this paper we study some recent formulations for the computation of upper bounds in limit analysis. We show that a previous formulation presented by the authors does not guarantee the strictness of the upper bound, nor does it provide a velocity field that satisfies the normality rule everywhere. We show that these deficiencies are related to the quadrature employed for the evaluation of the dissipation power. We derive a formulation that furnishes a strict upper bound of the load factor, which in fact coincides with a formulation reported in the literature. From the analysis of these formulations we propose a post-process which consists in computing exactly the dissipation power for the optimum upper bound velocity field. This post-process may further reduce the strict upper bound of the load factor in particular situations. Finally, we also determine the quadratures that must be used in the elemental and edge gap contributions so that they are always positive and their addition equals the global bound gap.
Abstract
In this paper we study some recent formulations for the computation of upper bounds in limit analysis. We show that a previous formulation presented by the authors does not [...]
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3D NURBS-enhanced finite element method (NEFEM)
R. Sevilla, S. Fernández-Méndez, A. Huerta

Int. J. Numer. Meth. Engng. (2011). Vol. 88 (12) pp. 109-125

Abstract
This paper presents the extension of the recently proposed NURBS-enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non-uniform rational B-splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features.
Abstract
This paper presents the extension of the recently proposed NURBS-enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the [...]
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Goal oriented error estimation for the element free Galerkin method
Y. Vidal, A. Huerta

Lecture Notes in Computational Sciences and Engineering (2007). Vol. 57, pp. 265-282

Abstract
A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute 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.
Abstract
A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with [...]
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A multidimensional data-driven sparse identification technique: the sparse proper generalized decomposition
R. Ibañez, E. Abisset-Chavanne, A. Ammar, D. González, E. Cueto, A. Huerta, J. Duval, F. Chinesta

Complexity (2018). Vol. 2018, pp. 1-11

Abstract
Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.
Abstract
Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount [...]
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A face-centred finite volume method for second-order elliptic problems
R. Sevilla, M. Giacomini, A. Huerta

Int. J. Numer. Meth. Engng. (2018). Vol. 115 (8), pp. 986-1014 (preprint)

Abstract
This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in 2D) to construct locally-conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin (HDG) formulation with constant degree of approximation, thus inheriting the convergence properties of the classical HDG. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element-by-element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived and numerical evidence of optimal convergence in 2D and 3D is provided. Numerical examples are presented to illustrate the accuracy, efficiency and robustness of the proposed methodology. The results show that, contrary to other FV methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy and robustness using simplicial elements, facilitating its application to problems involving complex geometries in 3D.
Abstract
This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume [...]
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Particle density estimation with grid-projected and boundary-corrected adaptive kernels
G. Sole-Mari, D. Bolster, D. Fernàndez-García, X. Sánchez-Vila

Advances in Water Resources (2019). Vol. 131, pp. 103382:1-103382:13

Abstract
The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the improved nature of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the “true” particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a “hybrid” approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE.
Abstract
The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) [...]
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Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes
S. Badia, J. Bonilla, A. Hierro

Computer Methods in Applied Mechanics and Engineering (2017). Vol. 320, pp. 582-605 (Preprint)

Abstract
This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.
Abstract
This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport [...]
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Higher-order simulations: Strategic investment under model-induced price patterns
G. Peffer, B. Llacay

Journal of Artificial Societies and Social Simulation (2007). Vol. 10 (2), p. 6

Abstract
The trading and investment decision processes in financial markets become ever more dependent on the use of valuation and risk models. In the case of risk management for instance, modelling practice has become quite homogeneous and the question arises as to the effect this has on the price formation process. Furthermore, sophisticated investors who have private information about the use and characteristics of these models might be able to make superior gains in such an environment. The aim of this article is to test this hypothesis in a stylised market, where a strategic investor trades on information about the valuation and risk management models used by other market participants. Simulation results show that under certain market conditions, such a'higher-order'strategy generates higher profits than standard fundamental and momentum strategies that do not draw on information about model use.
Abstract
The trading and investment decision processes in financial markets become ever more dependent on the use of valuation and risk models. In the case of risk management for instance, [...]
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Physics-based balancing domain decomposition by constraints for multi-material problems
S. Badia, A. Martín, H. Nguyen

Journal of Scientific Computing (2019). Vol. 79 (2), pp. 718-747

Abstract
In this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced subdomain partition, and based on an aggregation of elements according to their physical coefficients, we end up with a finer physics-based (PB) subdomain partition. Next, we define corners, edges, and faces for this PB partition, and select some of them to enforce subdomain continuity (primal faces/edges/corners). When the physical coefficient in each PB subdomain is constant and the set of selected primal faces/edges/corners satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient across subdomains. An extensive set of numerical experiments for 2D and 3D for the Poisson and linear elasticity problems is provided to support our findings. In particular, we show robustness and weak scalability of the new preconditioner variant up to 8232 cores when applied to 3D multi-material problems with the contrast of the physical coefficient up to 108 and more than half a billion degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced inter-level overlapped implementation of the preconditioner that deals very efficiently with the coarse problem computation. The proposed preconditioner is compared against a state-of-the-art implementation of an adaptive BDDC method in PETSc for thermal and mechanical multi-material problems.
Abstract
In this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced [...]
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Hidden Markov Models for indirect classification of occupant behaviour
J. Liisberg, J. Moller, H. Bloem, J. Cipriano, G. Mor Martinez, H. Madsen

Sustainable Cities and Society (2016). Vol. 27, pp. 83-98

Abstract
Even for similar residential buildings, a huge variability in the energy consumption can be observed. This variability is mainly due to the different behaviours of the occupants and this impacts the thermal (temperature setting, window opening, etc.) as well as the electrical (appliances, TV, computer, etc.) consumption. It is very seldom to find direct observations of occupant presence and behaviour in residential buildings. However, given the increasing use of smart metering, the opportunity and potential for indirect observation and classification of occupants’ behaviour is possible. This paper focuses on the use of Hidden Markov Models (HMMs) to create methods for indirect observations and characterisation of occupant behaviour. By applying homogeneous HMMs on the electricity consumption of fourteen apartments, three states describing the data were found suitable. The most likely sequence of states was determined (global decoding). From reconstruction of the states, dependencies like ambient air temperature were investigated. Combined with an occupant survey, this was used to classify/interpret the states as (1) absent or asleep, (2) home, medium consumption and (3) home, high consumption. From the global decoding, the average probability profiles with respect to time of day were investigated, and four distinct patterns of occupant behaviour were observed. Based on the initial results of the homogeneous HMMs and with the observed dependencies, time dependent HMMs (inhomogeneous HMMs) were developed, which improved forecasting. For both the homogeneous and inhomogeneous HMMs, indications of common parameters were observed, which suggests further development of the HMMs as population models.
Abstract
Even for similar residential buildings, a huge variability in the energy consumption can be observed. This variability is mainly due to the different behaviours of the occupants [...]
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Dimensional hyper-reduction of nonlinear finite element models via empirical cubature
J. Hernández, M. Caicedo, A. Ferrer

Computer Methods in Applied Mechanics and Engineering (2017). Vol. 313, pp. 687-722 (Preprint)

Abstract
We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy
Abstract
We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), [...]
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A consistent relaxation of optimal design problems for coupling shape and topological derivatives
S. Amstutz, C. Dapogny, A. Ferrer

Numerische mathematik (2018). Vol. 140 (1), pp. 35-94

Abstract
In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiments
Abstract
In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization [...]
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On the numerical resolution of the discontinuous material bifurcation problem
J. Oliver, A. Huespe, J. Cante, G. Díaz

International Journal for Numerical Methods in Engineering (2010). Vol. 83, pp. 768-804

Abstract
The work focuses on the numerical resolution of the discontinuous material bifurcation problem as a relevant ingredient in computational material failure mechanics. The problem consists of finding the conditions for the strain localization onset in terms of the so-called bifurcation time, localization directions and localization modes. A numerical algorithm, based on the iterative resolution of a coupled eigenvalue problem in terms of the localization tensor, is proposed for such purpose. The algorithm is shown to be always convergent to the exact solution for the symmetric case (major and minor symmetries of the tangent constitutive operator). In the unsymmetric case (only minor symmetries), the solution is no longer exact, although it is shown that using the symmetric part of the localization tensor in the proposed algorithms provides enough accurate solutions for most of cases. Numerical examples illustrate the benefits of the proposed methodology in terms of accuracy and savings in the computational cost associated with the problem.
Abstract
The work focuses on the numerical resolution of the discontinuous material bifurcation problem as a relevant ingredient in computational material failure mechanics. The problem [...]
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Post processing-based interpolation schemes for nested Cartesian finite difference grids of different size
A. Figueroa, R. Lohner

Int. J. Numer. Meth. Fluids (2018). Vol. 89 (6), pp. 196-215

Abstract
Nested Cartesian grid systems by design require interpolation of solution fields from coarser to finer grid systems. While several choices are available, preserving accuracy, stability, and efficiency at the same time require careful design of the interpolation schemes. Given this context, a series of interpolation algorithms based on postprocessing halo information on nested Cartesian finite difference grids of different size were developed and tested. The results obtained indicate that most of these do not yield the expected improvement, and some even tend to make the solver unstable. However, some third‐ and fourth‐order interpolation functions do yield considerable accuracy improvement, are stable, and are worth implementing.
Abstract
Nested Cartesian grid systems by design require interpolation of solution fields from coarser to finer grid systems. While several choices are available, preserving accuracy, [...]
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Generating seamless surfaces for transport and dispersion modeling in GIS
F. Camelli, J. Lien, D. Shen, D. Wong, M. Rice, R. Lohner, C. Yang

GeoInformatica (2012). Vol. 16, pp. 307-327

Abstract
A standard use of triangulation in GIS is to model terrain surface using TIN. In many simulation models of physical phenomena, triangulation is often used to depict the entire spatial domain, which may include buildings, landmarks and other surface objects in addition to the terrain surface. Creating a seamless surface of complex building structures together with the terrain is challenging and existing approaches are laborious, time-consuming and error-prone. We propose an efficient and robust procedure using computational geometry techniques to derive triangulated building surfaces from 2D polygon data with a height attribute. We also propose a new method to merge the resultant building surfaces with the triangulated terrain surface to produce a seamless surface for the entire study area. Using Oklahoma City data, we demonstrate the proposed method. The resultant surface is used as the input data for a simulated transport and dispersion event in Oklahoma City. The proposed method can produce the seamless surface data to be used for various types of physical models in a fraction of the time required by previous methods.
Abstract
A standard use of triangulation in GIS is to model terrain surface using TIN. In many simulation models of physical phenomena, triangulation is often used to depict the entire [...]
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Recent Advances in Parallel Advancing Front Grid Generation
R. Lohner

Archives of Comp. Meths. Engng. (2014). Vol. 21 (2), pp. 127-140

Abstract
The quest for scalable, parallel advancing front grid generation techniques now spans more than two decades. A recent innovation has been the use of a so-called domain-defining grid, which has led to a dramatic increase in robustness and speed. The domain-defining grid (DDG) has the same fine surface triangulation as the final mesh desired, but a much coarser interior mesh. The DDG renders the domain to be gridded uniquely defined and allows for a well balanced work distribution among the processors during all stages of grid generation and improvement. In this way, most of the shortcomings of previous techniques are overcome. Timings show that the approach is scalable and able to produce large grids of high quality in a modest amount of clocktime. These recent advances in parallel grid generation have enabled a completely scalable simulation pipeline (grid generation, solvers, post-processing), opening the way for truly large-scale computations using unstructured, body-fitted grids.
Abstract
The quest for scalable, parallel advancing front grid generation techniques now spans more than two decades. A recent innovation has been the use of a so-called domain-defining [...]
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Selective edge removal for unstructured grids with Cartesian cores
R. Lohner, H. Luo, J. Baum

Journal of Computational Physics (2005). Vol. 26 (1), pp. 208-226

Abstract
Several rules for redistributing geometric edge-coefficient obtained for grids of linear elements derived from the subdivision of rectangles, cubes or prisms are presented. By redistributing the geometric edge-coefficient, no work is carried out on approximately half of all the edges of such grids. The redistribution rule for triangles obtained from rectangles is generalized to arbitrary situations in 3-D, and implemented in a typical 3-D edge-based flow solver. The results indicate that without degradation of accuracy, CPU requirements can be cut considerably for typical large-scale grids. This allows a seamless integration of unstructured grids near boundaries with efficient Cartesian grids in the core regions of the domain.
Abstract
Several rules for redistributing geometric edge-coefficient obtained for grids of linear elements derived from the subdivision of rectangles, cubes or prisms are presented. [...]
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Adaptive methodology for meshless finite point method
F. Perazzo, R. Lohner, L. Perez-Pozo

Advances in Engineering Software (2008). Vol. 39 (3), pp. 156-166

Abstract
In this work, a posteriori error estimator and an adaptive refinement process for the meshless finite point method (FPM), which is based on point collocation, are presented. The error indicator is formulated by the least-squares functional evaluation, used in the shape function development. New degrees of freedom or additional points can be incorporated without difficulty, in zones where the error estimator presents a high value, by means of h–p refinement processes. The validity of the proposed error estimator can be demonstrated by developments of numerical problems in mechanics of solids, using an adaptive refinement process of the solution.
Abstract
In this work, a posteriori error estimator and an adaptive refinement process for the meshless finite point method (FPM), which is based on point collocation, are presented. [...]
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Adaptive Embedded Unstructured Grid Methods
R. Lohner, J. Baum, E. Mestreau, D. Sharov, C. Charman, D. Pelessone

International Journal for Numerical Methods in Engineering (2004). Vol. 60 (3), pp. 641-660

Abstract
A simple embedded domain method for node-based unstructured grid solvers is presented. The key modification of the original, edge-based solver is to remove all geometry-parameters (essentially the normals) belonging to edges cut by embedded surface faces. Several techniques to improve the treatment of boundary points close to the immersed surfaces are explored. Alternatively, higher-order boundary conditions are achieved by duplicating crossed edges and their endpoints. Adaptive mesh refinement based on proximity to or the curvature of the embedded CSD surfaces is used to enhance the accuracy of the solution. User-defined or automatic deactivation for the regions inside immersed solid bodies is employed to avoid unnecessary work. Several examples are included that show the viability of this approach for inviscid and viscous, compressible and incompressible, steady and unsteady flows, as well as coupled fluid–structure problems.
Abstract
A simple embedded domain method for node-based unstructured grid solvers is presented. The key modification of the original, edge-based solver is to remove all geometry-parameters [...]
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Automatic unstructured grid generators
R. Lohner

Finite Elements in Analysis and Design (1997). Vol. 25 (1-2), pp. 111-134

Abstract
A review of automatic unstructured grid generators is given. These types of grids have found widespread use in computational fluid dynamics, computational structural dynamics, computational electro-magnetics and computational thermodynamics. The following topics are treated: the methods most commonly used, the specification of desired element size/shape and surface definition/meshing. Finally, the use of automatic grid generators as an enabling technology for moving body simulations and adaptive remeshing techniques is discussed.
Abstract
A review of automatic unstructured grid generators is given. These types of grids have found widespread use in computational fluid dynamics, computational structural dynamics, [...]
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A feature‐preserving volumetric technique to merge surface triangulations
J. Cebral, F. Camelli, R. Lohner

Int. J. Numer. Meth. Engng. (2002). Vol. 55 (2), pp. 177-190

Abstract
Several extensions and improvements to surface merging procedures based on the extraction of iso-surfaces from a distance map defined on an adaptive background grid are presented. The main objective is to extend the application of these algorithms to surfaces with sharp edges and corners. In order to deal with objects of different length scales, the initial background grids are created using a Delaunay triangulation method and local voxelizations. A point enrichment technique that introduces points into the background grid along detected surface features such as ridges is used to ensure that these features are preserved in the final merged surface. The surface merging methodology is extended to include other Boolean operations between surface triangulations. The iso-surface extraction algorithms are modified to obtain the correct iso-surface for multi-component objects. The procedures are demonstrated with various examples, ranging from simple geometrical entities to complex engineering applications. The present algorithms allow realistic modelling of a large number of complex engineering geometries using overlapping components defined discretely, i.e. via surface triangulations. This capability is very useful for grid generation starting from data originated in measurements or images.
Abstract
Several extensions and improvements to surface merging procedures based on the extraction of iso-surfaces from a distance map defined on an adaptive background grid are presented. [...]
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Minimization of indirect addressing for edge‐based field solvers
R. Lohner, M. Galle

Commun. Numer. Meth. Engng (2002). Vol. 18 (5), pp. 335-343

Abstract
A point renumbering scheme that halves the number of indirect addressing operations required for edge‐based field solvers has been developed. At the same time, an alternative (and entirely serendipitous) assembly of right‐hand side vectors was found to enhance performance considerably. The new loop structure is especially attractive for vector‐parallel machines with direct memory access. Timings on the CRAY‐SV1 and NEC‐SX5 show that for Laplacian loops, which appear in many CFD and CEM codes, CPU can be reduced considerably. The new renumbering scheme does not require any new data structures or arrays, allowing for progressive porting of loops in large‐scale production codes.
Abstract
A point renumbering scheme that halves the number of indirect addressing operations required for edge‐based field solvers has been developed. At the same time, an alternative [...]
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Three-dimensional space-marching algorithm on unstructured grids
M. McGrory, R. Lohner, R. Walters

AIAA Journal (1991). Vol. 29 (1), pp. 1844-1849

Abstract
A three-dimensional space-marching algorithm using an unstructured discretization is proposed. The method utilizes a two-dimensional unstructured grid generator to construct grids in a crossflow plane while maintaining structure in the marching direction. The spatial discretization is obtained by applying a characteristic-based, upwind, finite volume scheme for the solution of the Euler equations. Solutions are presented for several different geometries comparing the results with existing numerical techniques and experiment.
Abstract
A three-dimensional space-marching algorithm using an unstructured discretization is proposed. The method utilizes a two-dimensional unstructured grid generator to construct [...]
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Some useful renumbering strategies for unstructured grids
R. Lohner

International Journal for Numerical Methods in Engineering (1993). Vol. 36 (19), pp. 3259-3270

Abstract
Several renumbering strategies for unstructured grids are discussed. They lead to a minimization of eache-misses and an optimal grouping of elements for different computer platforms, from superscalar workstations to multiprocessor register-to-register vector machines. Timings for a typical computational fluid dynamics (CFD) code that employs these renumbering strategies indicate that CPU requirements may be halved by applying them. The renumbering strategies discussed are all of linear time complexity, making them ideally suited for applications requiring frequent mesh changes. Furthermore, these renumbering strategies are not only valid for element-based codes but carry over to edge-based or face-based field solvers
Abstract
Several renumbering strategies for unstructured grids are discussed. They lead to a minimization of eache-misses and an optimal grouping of elements for different computer [...]
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Edges, stars, superedges and chains
R. Lohner

Computer Methods in Applied Mechanics and Engineering (1994). Vol. 111 (3-4), pp. 255-263

Abstract
Several possibilities for the reduction of indirect addressing operations within edge-based solvers are discussed. It is shown that indirect addressing operations may be reduced by a factor in excess of 1 : 4 if edges are grouped and linked together. Tests of benchmark loops on two common engineering computer platforms reveal that CPU saving factors ranging from 1.3 to 3.5 may be realized when switching from edge-based to superedge or chain-based loops.
Abstract
Several possibilities for the reduction of indirect addressing operations within edge-based solvers are discussed. It is shown that indirect addressing operations may be reduced [...]
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The construction of free–free flexibility matrices as generalized stiffness inverses
C. Felippa, K. Park, M. Justino

Computers and Structures (1998). Vol. 68 (4), pp. 411-418

Abstract
We present generalizations of the classical structural flexibility matrix. Direct or indirect computation of flexibilities as ‘influence coefficients’ has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions. Here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This entity is called a free–free flexibility matrix. It preserves exactly the rigid body modes. The definition is element independent. It only involves access to the stiffness generated by a standard finite element program as well as a separate geometric construction of the rigid body modes. With this information, the computation of the free–free flexibility can be done by solving linear equations and does not require the solution of an eigenvalue problem or performing a singular value decomposition. Flexibility expressions for symmetric and unsymmetric free–free stiffnesses are studied. For the unsymmetric case two flexibilities, one preserving the Penrose conditions and the other the spectral properties, are examined. The two versions coalesce for symmetric matrices
Abstract
We present generalizations of the classical structural flexibility matrix. Direct or indirect computation of flexibilities as ‘influence coefficients’ has [...]
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Variational formulation of high-performance finite elements: Parametrized variational principles
C. Felippa, C. Militello

Computers and Structures (1990). Vol. 36 (1), pp. 1-11

Abstract
High-performance (HP) elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This paper is part of a series on the variational basis of HP elements, with emphasis on those constructed with the free formulation (FF) and assumed natural strain (ANS) methods. The present paper studies parametrized variational principles that provide a foundation for the FF and ANS methods, as well as for a combination of both methods.
Abstract
High-performance (HP) elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This paper is part of a series on [...]
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Symmetrization of coupled eigenproblems by eigenvector augmentation
C. Felippa

Communications in Applied Numerical Methods (1988). Vol. 4 (4), pp. 561-563

Abstract
The problem of elastoacoustic vibrations of a fluid enclosed in a flexible container leads to a generalized unsymmetric eigensystem of special form. This system can be symmetrized in eight different ways by eigenvector augmentation followed by static condensation.
Abstract
The problem of elastoacoustic vibrations of a fluid enclosed in a flexible container leads to a generalized unsymmetric eigensystem of special form. This system can be symmetrized [...]
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Recent developments in parametrized variational principles for mechanics
C. Felippa

Computational Mechanics (1996). Vol. 18 (3), pp. 159-174

Abstract
This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions. The theory of single-field PVPs, based on gauge functions, is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multified PVPs are more interesting from both theoretical and practical standpoints. The two-dimensional Poisson equation is used to present, in a tutorial fashion, the formulation of parametrized mixed functionals. This treatment is then extended to internal interfaces, which are useful in treatment of discontinuities, subdomain linkage and construction of parametrized hybrid functionals. This is followed by a similar but more compact treatment of three-dimensional classical elasticity, and a parametrization of nonlinear hyperelasticity.
Abstract
This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected [...]
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An alphanumeric finite element mesh plotter
C. Felippa

International Journal for Numerical Methods in Engineering (1972). Vol. 5 (2), pp. 217-236

Abstract
A FORTRAN subroutine name MESHPL is described. This routine generates an alphanumeric image of an arbitrary two-dimensional finite element mesh on the line printer. This plotting technique can be used for fast on-line display of nodal points, elements, boundary conditions and loads during the input data preprocessing stage of a finite element analysis.
Abstract
A FORTRAN subroutine name MESHPL is described. This routine generates an alphanumeric image of an arbitrary two-dimensional finite element mesh on the line printer. This plotting [...]
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A contour-line printplotter for finite element work
C. Felippa

Advances in Engineering Software (1979). Vol. 1 (2), pp. 51-60

Abstract
A Fortran subroutine for producing countour line printplots of functions defined at the nodes of an arbitrary two-dimensional finite element mesh is presented. This display technique is useful for quick inspection of analysis results and as guide for downstream use of more sophisticated graphics equipment.
Abstract
A Fortran subroutine for producing countour line printplots of functions defined at the nodes of an arbitrary two-dimensional finite element mesh is presented. This display [...]
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Numerical experiments in finite element grid optimization by direct energy search
C. Felippa

Applied Mathematical Modelling (1977). Vol. 1 (5), pp. 239-244

Abstract
A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total potential energy of the discrete model with respect to the node locations. An implementation relying upon non-linear programming techniques is found to be numerically reliable and to lead to improved grids in accord with engineering intuition. This rigorous approach is hampered, however, by the excessive computational effort required by the energy minimization process. A combination of related techniques is therefore proposed to make dynamic node distribution a useful tool within the framework of large-scale finite element analysis. The combined strategy involves use of substructuring methods, application of a local energy-balancing optimality criterion for fast node distribution, and automatic refinement of previously-improved coarse grids.
Abstract
A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total [...]
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Optimization of finite element grids by direct energy search
C. Felippa

Applied Mathematical Modelling (1976). Vol. 1 (2), pp. 93-96

Abstract
One of the possible approaches to the reduction of computational costs in finite element analysis is the selection of ‘optimal grids’, which produce the ‘best’ answers, in the sense of minimizing a discretization error measure, for a fixed level of computational effort. The grid optimization problem is studied in the case of grids of similar topology having a fixed number of degrees of freedom per node. A general formulation based on weighted-residual error measures is specialized to field problems associated with a positive-definite energy functional, the minimization of which, with respect to variable node locations, is adopted as a grid optimality criterion. The problem is then embedded in the framework of the general nonlinear programming problem, and desirable computational features of candidate search algorithms are described.
Abstract
One of the possible approaches to the reduction of computational costs in finite element analysis is the selection of ‘optimal grids’, which produce the [...]
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Interfacing Finite Element and Boundary Element Discretizations
C. Felippa

Applied Mathematical and Mechanics (1981). Vol. 5 (5), pp. 383-386

Abstract
A review is provided of various methods for coupling a mechanical finite element model to an infinite external acoustic domain. The associated physical problem involves a three-dimensional structure, submerged in an acoustic fluid, and a pressure shock impinging on the structure. Of primary concern is the structural response and, in particular, the structure's survivability. The governing matrix equation of motion for the dynamic response of a discrete structure is considered along with the fluid equations of the interaction equations. Three approaches for solving the obtained coupled finite element/boundary element system are reported, giving attention to the 'field elimination' method, the procedure of simultaneous integration, and a partitioned integration process.
Abstract
A review is provided of various methods for coupling a mechanical finite element model to an infinite external acoustic domain. The associated physical problem involves a [...]
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Software for testing stability polynomials
C. Felippa

Applied Mathematical Modelling (1982). Vol. 6 (3), pp. 209-215

Abstract
Fortran 77 routines for testing root-locus stability conditions on real polynomials with the Routh and Schur-Cohn algorithms are presented. The use of the routines is illustrated by two case studies.
Abstract
Fortran 77 routines for testing root-locus stability conditions on real polynomials with the Routh and Schur-Cohn algorithms are presented. The use of the routines is illustrated [...]
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Architecture of a distributed analysis network for computational mechanics
C. Felippa

Computers and Structures (1981). Vol. 13 (1), pp. 405-413

Abstract
An integrated software system called NICE (Network of Interactive Computational Elements) is presently under development at the Applied Mechanics Laboratory. The overall purpose of this development is to further the formulation, implementation and application of advanced computational methods in fluid and solid mechanics. From a software engineering standpoint, system design stresses functional modularity, decentralization, adaptability to interactive computing environments, and ability to accommodate research, development, analysis and testing activities. A clear separation is established between architectural elements such as analysis control and data management, and result-productive computational components called processors. This paper gives a functional overview of the NICE architecture, for which the author is responsible. Control functions that shape the user's perception of the system are emphasized.
Abstract
An integrated software system called NICE (Network of Interactive Computational Elements) is presently under development at the Applied Mechanics Laboratory. The overall purpose [...]
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Multiscale proper generalized decomposition based on the partition of unity
R. Ibañez, A. Ammar, E. Cueto, J. Duval, F. Chinesta

Int. J. Numer. Meths. Engng. (2019). Vol. 120 (6), pp. 727-747

Abstract
Solutions of partial differential equations could exhibit a multiscale behavior. Standard discretization techniques are constraints to mesh up to the finest scale to predict accurately the response of the system. The proposed methodology is based on the standard proper generalized decomposition rationale; thus, the PDE is transformed into a nonlinear system that iterates between microscale and macroscale states, where the time coordinate could be viewed as a 2D time, representing the microtime and macrotime scales. The macroscale effects are taken into account because of an FEM-based macrodiscretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the domain. The proposed methodology can be seen as an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors.
Abstract
Solutions of partial differential equations could exhibit a multiscale behavior. Standard discretization techniques are constraints to mesh up to the finest scale to predict [...]
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Enrichment and coupling of the finite element and meshless methods
A. Huerta, S. Fernández-Méndez

Int. J. Numer. Meth. Engng. (2000). Vol. 48 (11), pp. 1615-1636

Abstract
A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown.
Abstract
A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements [...]
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Adapting Broyden method to handle linear constraints imposed via Lagrange multipliers
A. Rodríguez-Ferran, A. Huerta

Int. J. Numer. Meth. Engng. (1999). Vol. 46 (12), pp. 2011-2026

Abstract
Various non-linear equation solvers are adapted to handle linear constraints via the Lagrange-multiplier technique. This adaptation process turns out to be quite straightforward for Newton-Raphson methods and rank-two Quasi-Newton methods (BFGS and DFP), but rather more involved for Broyden method. In fact, two Broyden methods can be obtained: the standard one and a modified one, better adapted to the Lagrange-multiplier environment. Some numerical examples are used to assess the relative performance of the various adapted solvers. These tests illustrate the superiority of the modified Broyden method over the standard one.
Abstract
Various non-linear equation solvers are adapted to handle linear constraints via the Lagrange-multiplier technique. This adaptation process turns out to be quite straightforward [...]
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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 [...]
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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 [...]
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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 [...]
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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 [...]
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