Abstract

One of the main drawbacks of all the time integration algorithms using an Eulerian formulations in Coupled Problems is the restricted time-step to be used to have acceptable results.

For the case of fluid-structure interactions (FSI) with or without free-surfaces or for the case of fluid with moving internal interfaces (multi-fluids), it is well known that in the explicit integrations, the time-step to be used in the solution is stable only for time-step smaller than two critical values: the Courant-Friedrichs-Lewy (CFL) number and the Fourier number. The first one is concerning with the convective terms and the second one with the diffusive ones. Both numbers must be less than one to have stable algorithms. For convection dominant problems the condition CFL<1 becomes crucial and limit the use of explicit methods or outdistance its to be efficient. On the other hand, implicit integrations using Eulerian formulations are restricted in the time-step size due to the lack of convergence of the non-linear terms. Both time integrations, explicit or implicit are, in most cases, limited to CFL no much larger than one.

In this lecture we will present a Particle Method to solve coupled problems like FSI or multi-fluid problems that use in all the domain (solid and fluid) a Lagrangian formulation with explicit or implicit time integration without the CFL<1 restriction. This allows large time-steps, independent of the spatial discretization, having equal or better precision that an Eulerian integration.

The proposal will be tested numerically for FSI and multi-fluid flows problems using the Particle Finite Element Method second generation (PFEM2). The results show than this Particle Method is largely more efficient compared as well in accuracy as in computing time with other more standard Eulerian formulations.

Abstract

The main idea of the Particle Finite Element Method in both versions: with moving mesh or with fixed mesh, are to have a set of particles that move in a Lagrangian frame convecting all the physical and mathematical variables (for instance, the density, the viscosity or the conductivity, but also the velocity, the pressure or/and the temperature). These physical and mathematical values are projected at the end of each time-step on a moving mesh or on a fixed mesh. The second possibility has been named PFEM-Second Generation or simply PFEM-2.

One of the main drawback of the time integrations using Eulerian formulations are the restricted time-step size that is necessary to use due to the lack of accuracy of the convective terms. Both time integrations, explicit or implicit are, in most cases, limited to small CFL numbers. The cases in which the problem to be solved include free-surfaces or moving internal interfaces, like multi-fluids of fluid-structure interactions this time-step limitation is even worse.

The objective of this presentation is to make an overview of recent examples solved using PFEM-2 and to demonstrate why this method based on particles that move in a Lagrangian frame projecting the results on a fixed mesh is faster than a classical Eulerian Finite Element Method. The authors claim that nowadays, the best way to improve the efficiency of the majority of the CFD problems is the use of a particle-based method like PFEM-2.

Abstract

Se describen los conceptos básicos del método de partículas y elementos finitos (PFEM) para análisis de flujos de fluidos en lámina libre y su interacción con objetos sólidos y estructuras flotantes o sumergidas. Se presentan diversas aplicaciones del PFEM en los ámbitos de la ingeniería de puertos e hidráulica y al estudio de la caída de una avalancha sobre un embalse.

Abstract

The paper presents an overview of the advances in recent years on the finite element method (FEM) and on particle-based methods for the simulation of industrial metal forming processes. More specifically, we present the evolution of the FEM in the field from the early plastic/viscoplastic flow approaches to the new stabilized FEM for analysis of multiphysics bulk forming processes. Also the paper describes the state of the art in the new rotation-free shell elements for simulation of sheet stamping processes. Finally, we present the so-called Particle Finite Element Method (PFEM), as a component of a family of new computational techniques integrating particle-based methods and mesh-based procedures. The PFEM is particularly suited for large deformation problems in solids and fluids involving nonlinear mechanical and geometrical effects, fluid-structure interactions and frictional contact situations. Applications of the FEM and the PFEM to several metal and material forming processes are presented.

Abstract

The aim of this work is to present a new procedure for modelling industrial processes
that involve granular material flows, using a numerical model based on the Particle
Finite Element Method (PFEM). The numerical results herein presented show
the potential of this methodology when applied to different branches of industry.
Due to the phenomenological richness exhibited by granular materials, the present
work will exclusively focus on the modelling of cohesionless dense granular flows.

The numerical model is based on a continuum approach in the framework of
large-deformation plasticity theory. For the constitutive model, the yield function is
defined in the stress space by a Drucker-Prager yield surface characterized by two
constitutive parameters, the cohesion and the internal friction coefficient, and
equipped with a non-associative deviatoric flow rule. This plastic flow condition is
considered nearly incompressible, so the proposal is integrated in a u- p mixed
formulation with a stabilization of the pressure term via the Polynomial Pressure
Projection (PPP). In order to characterize the non-linear dependency on the shear
rate when flowing a visco-plastic regularization is proposed.

The numerical integration is developed within the Impl-Ex technique, which increases
the robustness and reduces the iteration number, compared with a typical
implicit integration scheme. The spatial discretization is addressed within the
framework of the PFEM which allows treating the large deformations and motions
associated to granular flows with minimal distortion of the involved finite element
meshes. Since the Delaunay triangulation and the reconnection process minimize
such distortion but do not ensure its elimination, a dynamic particle discretization of
the domain is proposed, regularizing, in this manner, the smoothness and particle
density of the mesh. Likewise, it is proposed a method that ensures conservation of
material or Lagrangian surfaces by means of a boundary constraint, avoiding in this
way, the geometric definition of the boundary through the classic -shape method.

For modelling the interaction between the confinement boundaries and granular
material, it is advocated for a method, based on the Contact Domain Method (CDM)
that allows coupling of both domains in terms of an intermediate region connecting
the potential contact surfaces by a domain of the same dimension than the contacting
bodies. The constitutive model for the contact domain is posed similarly to that for
the granular material, defining a correct representation of the wall friction angle.
In order to validate the numerical model, a comparison between experimental
results of the spreading of a granular mass on a horizontal plane tests, and finite
element predictions, is carried out. These sets of examples allow us validating the
model according to the prediction of the different kinematics conditions of granular
materials while spreading – from a stagnant condition, while the material is at rest,
to a transition to a granular flow, and back to a deposit profile.

The potential of the numerical method for the solution and optimization of industrial
granular flows problems is achieved by focusing on two specific industrial
applications in mining industry and pellet manufacturing: the silo discharge and the
calculation of the power draw in tumbling mills. Both examples are representative
when dealing with granular flows due to the presence of variations on the granular
material mechanical response.

Abstract

The present work introduces a new application of the Particle Finite Element Method (PFEM) for the modeling of excavation problems. Lagrangian descriptions of motion of the continuum medium are the natural way of describing motion in solid mechanics. Particle finite element methods are based in these solid mechanics settings that can treat large material deformations and rapidly changing boundaries.

These capabilities are very suitable for modeling fluid motions and moving free surfaces. That is the reason that the most of the research and applications of PFEM can be found in the context of computational fluid dynamics (CFD) instead of solid mechanics.

The satisfying results in modeling fluids have been the motivation to use this method in dynamic solid mechanics.
The simulation of an excavation process is a non-linear dynamic problem. It contains geometrical, material and contact non-linearities. Modeling the contact process can be classiffied as the main difficulty. The simulation face the problems of: the detection of a changing geometry, the detection of contact between several solid domains, the estimation of correct interacting forces, the computation of the wear related to this contact forces and the removing of the material that has been excavated from the model.

The PFEM have its fundamentals in the classical non linear finite element analysis. The formulation provides a foundation in the updated lagrangian formulation for solids.

The dynamic problem is integrated using an implicit scheme. Remeshing strategies are employed in order to identify the boundary surfaces. A remeshing of the domain is introduced for the detection of rapidly changing boundaries. The Delaunay tessellation and the alpha-shape concept together, are used as a methodology to de_ne the boundary from a cloud of points. At the same time an interface mesh is created for contact recognition. A particular constitutive contact law has been developed for capturing contact normal and frictional forces using this mesh. By means of an Archard-type law, the excavation and damage caused in the ground is quantified. The erosion and wear parameters of the grounds under study determine the excavability. The behavior of the geomaterials is described using a Damage Model which model fracture in the continuum. The update and storage of the variables in particles have especial schemes for its correct treatment. The method and the meshing process is adapted for the simulation of an excavation problem.
PFEM is presented as a very suitable tool for the treatment of excavation problem.
The method gives solution for the analysis of all processes that derive from it. The method has a high versatility and a reasonable computational cost. The obtained results are really promising.

Abstract

We present a general Lagrangian formulation for treating elastic solids and quasi/fully incompressible fluids in a unified form. The formulation allows to treat solid and fluid subdomains in a unified manner in fluid-structure interaction (FSI) situations. In our work the FSI problem is solved via the Particle Finite Element Method (PFEM). The PFEM is an effective technique for modeling complex interactions between floating and submerged bodies and free surface flows, accounting for splashing of waves, large motions of the bodies and frictional contact conditions. Applications of the unified Lagrangian formulation to a number of FSI problems are given.

Abstract

We present some advances in the formulation of the Particle Finite Element Method (PFEM) for solving complex fluid-structure interaction problems with free surface waves. In particular, we present extensions of the PFEM for the analysis of the interaction between a collection of bodies in water allowing for frictional contact conditions at the fluid-solid and solid-solid interfaces via mesh generation. An algorithm to treat bed erosion in free surface flows is also presented. Examples of application of the PFEM to solve a number of fluid-multibody interaction problems involving splashing of waves, large motions of floating and submerged bodies and bed erosion situations are given.

Abstract

The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown.

Abstract

We present some developments in the particle finite element method (PFEM) for analysis of complex coupled problems in mechanics involving fluid–soil–structure interaction (FSSI). The PFEM uses an updated Lagrangian description to model the motion of nodes (particles) in both the fluid and the solid domains (the later including soil/rock and structures). A mesh connects the particles (nodes) defining the discretized domain where the governing equations for each of the constituent materials are solved as in the standard FEM. The stabilization for dealing with an incompressibility continuum is introduced via the finite calculus method. An incremental iterative scheme for the solution of the non linear transient coupled FSSI problem is described. The procedure to model frictional contact conditions and material erosion at fluid–solid and solid–solid interfaces is described. We present several examples of application of the PFEM to solve FSSI problems such as the motion of rocks by water streams, the erosion of a river bed adjacent to a bridge foundation, the stability of breakwaters and constructions sea waves and the study of landslides.