Mesh motion techniques for the ALE formulation in 3D large deformation problems

Efficient mesh motion techniques are a key issue to achieve satisfactory results in the arbitrary Lagrangian–Eulerian (ALE) finite element formulation when simulating large deformation problems such as metal‐forming. In the updated Lagrangian (UL) formulation, mesh and material movement are attached and an excessive mesh distortion usually appears. By uncoupling mesh movement from material movement, the ALE formulation can relocate the mesh to avoid distortion. To facilitate the calculation process, the ALE operator is split into two steps at each analysis time step: UL step (where deformation due to loading is calculated without convective terms) and Eulerian step (where mesh motion is applied). In this work, mesh motion is performed by new nodal relocation methods, developed for eight‐node hexahedral elements, which can move internal and boundary nodes, improving and concentrating the mesh in critical zones. After mesh motion, data is transferred from the UL mesh to the relocated mesh using an expansion of stresses in a Taylor's series. Two numerical applications are presented, comparing results of UL and ALE formulation with results found in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems

In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed‐mesh (FM)‐ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM‐ALE allows for the use of a single background mesh to solve both the fluid and the structure. Copyright © 2009 John Wiley & Sons, Ltd.     

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F ^e F ^p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A level set‐based topology optimization method targeting metallic waveguide design problems

In this paper, we propose a level set‐based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high‐frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary‐tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso‐contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso‐contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso‐contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz‐type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress integration and mesh refinement for large deformation in geomechanics

This paper first discusses alternative stress integration schemes in numerical solutions to large‐ deformation problems in hardening materials. Three common numerical methods, i.e. the total‐Lagrangian (TL), the updated‐Lagrangian (UL) and the arbitrary Lagrangian–Eulerian (ALE) methods, are discussed. The UL and the ALE methods are further complicated with three different stress integration schemes. The objectivity of these schemes is discussed. The ALE method presented in this paper is based on the operator‐split technique where the analysis is carried out in two steps; an UL step followed by an Eulerian step. This paper also introduces a new method for mesh refinement in the ALE method. Using the known displacements at domain boundaries and material interfaces as prescribed displacements, the problem is re‐analysed by assuming linear elasticity and the deformed mesh resulting from such an analysis is then used as the new mesh in the second step of the ALE method. It is shown that this repeated elastic analysis is actually more efficient than mesh generation and it can be used for general cases regardless of problem dimension and problem topology. The relative performance of the TL, UL and ALE methods is investigated through the analyses of some classic geotechnical problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A hierarchical sequential ALE poromechanics model for tire‐soil‐water interaction on fluid‐infiltrated roads

This paper introduces a hierarchical sequential arbitrary Lagrangian‐Eulerian (ALE) model for predicting the tire‐soil‐water interaction at finite deformations. Using the ALE framework, the interaction between a rolling pneumatic tire and the fluid‐infiltrated soil underneath will be captured numerically. The road is assumed to be a fully saturated two‐phase porous medium. The constitutive response of the tire and the solid skeleton of the porous medium is idealized as hyperelastic. Meanwhile, the interaction between tire, soil, and water will be simulated via a hierarchical operator‐split algorithm. A salient feature of the proposed framework is the steady state rolling framework. While the finite element mesh of the soil is fixed to a reference frame and moves with the tire, the solid and fluid constituents of the soil are flowing through the mesh in the ALE model according to the rolling speed of the tire. This treatment leads to an elegant and computationally efficient formulation to investigate the tire‐soil‐water interaction both close to the contact and in the far field. The presented ALE model for tire‐soil‐water interaction provides the essential basis for future applications, for example, to a path‐dependent frictional‐cohesive response of the consolidating soil and unsaturated soil, respectively. Copyright © 2017 John Wiley & Sons, Ltd.     

ALE stress update for transient and quasistatic processes

A key issue in Arbitrary Lagrangian–Eulerian (ALE) non-linear solid mechanics is the correct treatment of the convection terms in the constitutive equation. These convection terms, which reflect the relative motion between the finite element mesh and the material, are found for both transient and quasistatic ALE analyses. It is shown in this paper that the same explicit algorithms can be employed to handle the convection terms of the constitutive equation for both types of analyses. The most attractive consequence of this fact is that a quasistatic simulation can be upgraded from Updated Lagrangian (UL) to ALE without significant extra computational cost. These ideas are illustrated by means of two numerical examples. © 1998 John Wiley & Sons, Ltd.     

Automatic remeshing in 3‐D analysis of forming processes

Finite element modelling, employing updated Lagrangian techniques, is used extensively in the design and analysis of bulk forming processes. However, the full 3‐D capability has not seen widespread use in the automotive, aerospace, and, related industries due to, among other reasons, the need for remeshing, or, representation of the workpiece with a new finite element mesh as the analysis progresses. Automating the remeshing procedure of the deformed workpiece geometry would reduce the time required for a 3‐D analysis by several orders of magnitude. This paper discusses an algorithm for generating a new mesh to represent the deformed workpiece geometry during the analysis. The procedure is used to perform a 3‐D analysis of a valve forging problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Recent progress in ALEGRA development and application to ballistic impacts

ALEGRA is a multi-material, arbitrary-Lagrangian-Eulerian (ALE) code for solid dynamics being developed by the Computational Physics Research and Development Department at Sandia National Laboratories. It combines the features of modern Eulerian shock codes, such as CTH, with modern Lagrangian structural analysis codes. With the ALE algorithm, the mesh can be stationary (Eulerian) with the material flowing through the mesh, the mesh can move with the material (Lagrangian) so there is no flow between elements, or the mesh motion can be entirely independent of the material motion (Arbitrary). All three mesh types can coexist in the same problem, and any mesh may change its type during the calculation. In this paper we summarize several key capabilities that have recently been added to the code or are currently being implemented. As a demonstration of the capabilities of ALEGRA, we have applied it to the experimental data taken by Silsby.     

A three‐invariant hardening plasticity for numerical simulation of powder forming processes via the arbitrary Lagrangian–Eulerian FE model

In this paper, a three‐invariant cap plasticity model with an isotropic hardening rule is presented for numerical simulation of powder compaction processes. A general form is developed for single‐cap plasticity which can be compared with some common double‐surface plasticity models proposed for powders in literature. The constitutive elasto‐plastic matrix and its components are derived based on the definition of yield surface, hardening parameter and non‐linear elastic behaviour, as function of relative density of powder. Different aspects of the new single plasticity are illustrated by generating the classical plasticity models as special cases of the proposed model. The procedure for determination of powder parameters is described by fitting the model to reproduce data from triaxial compression and confining pressure experiments. The three‐invariant cap plasticity is performed within the framework of an arbitrary Lagrangian–Eulerian formulation, in order to predict the non‐uniform relative density distribution during large deformation of powder die pressing. In ALE formulation, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. This formulation introduces some convective terms in the finite element equations and consists of two phases. Each time step is analysed according to Lagrangian phase until required convergence is attained. Then, the Eulerian phase is applied to keep mesh configuration regular. Because of relative displacement between mesh and material, all dependent variables such as stress and strain are converted through the Eulerian phase. Finally, the numerical schemes are examined for efficiency and accuracy in the modelling of a rotational flanged component, an automotive component, a conical shaped‐charge liner and a connecting‐rod. Copyright © 2005 John Wiley & Sons, Ltd.     

Momentum advection on unstructured staggered quadrilateral meshes

The finite element method, as applied to problems in solid mechanics, typically uses a mesh with the velocities at the nodes and the remaining solution variables, including the density, located at the integration points. The arbitrary Lagrangian–Eulerian formulations used in solid mechanics are therefore faced with the challenge of transporting momentum, which is defined in terms of variables located at separate points in space, in a conservative manner. Two types of momentum transport methods have been developed over the years. The first constructs a dual mesh with the nodes as the integration points, a difficult task on an unstructured finite element mesh. The second uses the original mesh and constructs auxiliary variables for transport from which the final velocity may be recovered. An analysis demonstrates how the two methods are related. Simplified implementations of each type—dual mesh and element centered—are developed in detail and their performance is compared to verify the analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of artificial dissipation and limited flux schemes in Arbitrary Lagrangian–Eulerian finite element formulations

A comparison is made between Arbitrary Lagrangian–Eulerian (ALE) finite element formulations for simulation of forming processes based on an artificial dissipation scheme and a limited flux scheme. The first ALE algorithm is based on an averaging procedure used in post-processing of finite element calculations. The second ALE algorithm stems from a finite difference method for compressible fluid dynamics. Both approaches have complementary characteristics with respect to accuracy and implementation. © 1998 John Wiley & Sons, Ltd.     

Mesh adaptive finite element formulation for moisture transfer in materials with a critical moisture content

Wetting or drying of most open porous building materials is characterized by a sharp moving waterfront. Due to the high moisture gradients at the waterfront, an accurate finite element simulation requires a very fine mesh. To reduce computational costs a mesh adaptive method based on the Arbitrary Lagrangian Eulerian (ALE) technique is proposed. To continuously relocate the nodes on the computational domain a remesh‐indicator is equally distributed. In problems of water imbibition or drying of open porous building materials specific attention has been paid to the zone of critical moisture content. To this extent the traditional jump‐based indicator, quantifying the jump of a selected state variable, has been modified into an area‐based remesh‐indicator. An error analysis of an academic example shows that the area‐based indicator is superior to the jump‐based one. To illustrate the capabilities of the remeshing method based on an area‐based indicator, one and two‐dimensional examples of water imbibition of ceramic brick and drying of cellulose fibre cement are included. Copyright © 1999 John Wiley & Sons, Ltd.     

Arbitrary Lagrangian–Eulerian method in plasticity of pressure-sensitive material: application to powder forming processes

In this paper, an application of Arbitrary Lagrangian–Eulerian (ALE) method is presented in plasticity behavior of pressure-sensitive material, with special reference to large deformation analysis of powder compaction process. In ALE technique, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. The convective term is used to reflect the relative motion between the mesh and the material. Each time-step is divided into the Lagrangian phase and Eulerian phase. The convection term is neglected in the material phase, which is identical to a time-step in a standard Lagrangian analysis. The stresses and plastic internal variables are converted to account the relative mesh-material motion in the convection phase. The ALE formulation is then performed within the framework of a three-invariant cap plasticity model in order to predict the non-uniform density distribution during the large deformation of powder die pressing. The plasticity model is based on a hardening rule with the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. Finally, the numerical examples are performed to illustrate the applicability of the computational algorithm in modeling of powder forming process and the results are compared with those obtained from Lagrangian simulation in order to demonstrate the accuracy of proposed model.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

A particle‐in‐cell solution to the silo discharging problem

The problem of flow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elastic–plastic solid with the Drucker–Prager yield condition and the non‐associative flow rule. The phenomenon of friction between the stored material and the silo walls is taken into account—the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle‐in‐cell method—a variant of the finite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian–Eulerian formulation of the finite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time‐integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with flat bottoms and conical hoppers. Some results are compared with experimental ones. Copyright © 1999 John Wiley & Sons, Ltd.     

The ALE‐method with triangular elements: direct convection of integration point values

The arbitrary Lagrangian–Eulerian (ALE) finite element method is applied to the simulation of forming processes where material is highly deformed. Here, the split formulation is used: a Lagrangian step is done with an implicit finite element formulation, followed by an explicit (purely convective) Eulerian step. The purpose of this study is to investigate the Eulerian step for quadratic triangular elements. To solve the convection equation for integration point values, a new method inspired by Van Leer is constructed. The new method is based on direct convection of integration point values without intervention of nodal point values. The Molenkamp test and a so‐called block test were executed to check the performance and stability of the convection scheme. From these tests it is concluded that the new convection scheme shows accurate results. The scheme is extended to an ALE‐algorithm. An extrusion process was simulated to test the applicability of the scheme to engineering problems. It is concluded that direct convection of integration point values with the presented algorithm leads to accurate results and that it can be applied to ALE‐simulations. Copyright © 2000 John Wiley & Sons, Ltd.