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.     

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.     

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.     

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.     

Arbitrary Lagrangian–Eulerian finite element analysis of strain localization in transient problems

Non-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian–Eulerian (ALE) formulation, well known in hydrodynamics and fluid–structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.     

A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

飞机风挡夹层结构耐撞性数值分析

用LS-DYNA3D软件,建立了由钢化玻璃和PU、PVB塑料薄膜组成的风挡夹层结构鸟撞数值模型。鸟体采用ALE格式和与应变率相关的随动硬化材料模型,夹层结构用Lagrange格式和双线性材料模型,对11种风挡夹层结构进行了数值计算,分析了撞击过程的损伤和应力,讨论了夹层结构耐撞性的评估方法,为风挡夹层结构设计提供理论依据。     

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.     

Advection approaches for single- and multi-material arbitrary Lagrangian–Eulerian finite element procedures

The essential feature in arbitrary Lagrangian–Eulerian (ALE) based finite element approaches is the additional requirement to consider flow effects of the materials and the solution variables through the computational domain. These flow effects are commonly known as advective effects. The present paper examines different advection strategies for the application of the ALE finite element method in a finite deformation solid mechanics framework, where especially micromechanical problems are referred to. The global solution algorithm is based on the well-known fractional step method that provides an operator splitting approach for the solution of the coupled ALE equations. Distinguishing the so-called single-material and the multi-material ALE method, different advection schemes based on volume- and material-associated advection procedures are required. For the latter case, the material interfaces are not resolved explicitly by the finite element mesh. Instead a volume-of-fluid interface tracking approach in terms of the volume fractions of the different material phases is applied.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization

This work provides a robust variational‐based numerical implementation of a phase field model of ductile fracture in elastic–plastic solids undergoing large strains. This covers a computationally efficient micromorphic regularization of the coupled gradient plasticity‐damage formulation. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture‐specific computational structures such as finite element design of crack discontinuities or intricate crack‐tracking algorithms. The proposed gradient‐extended plasticity‐damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post‐critical ranges. The proposed setting is rooted in a canonical variational principle. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient‐extended internal variables that enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient‐extended dissipative solids with generalized internal variables that are passive in nature. It is specified for a conceptual model of von Mises‐type elasto‐plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate‐type minimization principle, which fully determines the coupled multi‐field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. An important aspect of this work is the regularization towards a micromorphic gradient plasticity‐damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms. This enhances the robustness of the finite element implementation, in particular, on the side of gradient plasticity. The performance of the formulation is demonstrated by means of some representative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient solution method for finite element ring‐rolling simulation

Finite element ring‐rolling simulation gives rise to poor conditioned non‐linear equations that require repeated solution. The associated computational costs are extreme making analysis impracticable in industry. This paper is concerned with a solution strategy that addresses this problem and involves the combined use of an arbitrary Lagrangian–Eulerian (ALE) formulation and a successive preconditioned conjugate gradient method (SPCGM). This approach, coupled to a finite element flow formulation, is shown to offer considerable computational savings. Through the combined use of the ALE flow formulation and the SPCGM the stability and condition of the non‐linear systems is enhanced. This purely iterative approach takes advantage of the slowly evolving velocity field and the self‐preconditioning offered by the SPCGM. The performance of the solver is compared against well‐known alternatives for varying problem sizes. The approach is shown to be generic but in particular makes ring‐rolling simulation a more practicable proposition. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixed finite element method for saturated poroelastoplastic media at large strains

A mixed finite element for hydro‐dynamic analysis in saturated porous media in the frame of the Biot theory is proposed. Displacements, effective stresses, strains for the solid phase and pressure, pressure gradients, and Darcy velocities for the fluid phase are interpolated as independent variables. The weak form of the governing equations of coupled hydro‐dynamic problems in saturated porous media within the element are given on the basis of the Hu–Washizu three‐field variational principle. In light of the stabilized one point quadrature super‐convergent element developed in solid continuum, the interpolation approximation modes for the primary unknowns and their spatial derivatives of the solid and the fluid phases within the element are assumed independently. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of pressure‐dependent non‐associated plasticity. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elastoplastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is used. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization due to strain softening in poroelastoplastic media subjected to dynamic loading at large strain. Copyright © 2003 John Wiley & Sons, Ltd.