Second‐order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces

Convected particle domain interpolation (CPDI) is a recently developed extension of the material point method, in which the shape functions on the overlay grid are replaced with alternative shape functions, which (by coupling with the underlying particle topology) facilitate efficient and algorithmically straightforward evaluation of grid node integrals in the weak formulation of the governing equations. In the original CPDI algorithm, herein called CPDI1, particle domains are tracked as parallelograms in 2‐D (or parallelepipeds in 3‐D). In this paper, the CPDI method is enhanced to more accurately track particle domains as quadrilaterals in 2‐D (hexahedra in 3‐D). This enhancement will be referred to as CPDI2. Not only does this minor revision remove overlaps or gaps between particle domains, it also provides flexibility in choosing particle domain shape in the initial configuration and sets a convenient conceptual framework for enrichment of the fields to accurately solve weak discontinuities in the displacement field across a material interface that passes through the interior of a grid cell. The new CPDI2 method is demonstrated, with and without enrichment, using one‐dimensional and two‐dimensional examples. Copyright © 2013 John Wiley & Sons, Ltd.     

A convected particle least square interpolation material point method

Applying the convected particle domain interpolation (CPDI) to the material point method has many advantages over the original material point method, including significantly improved accuracy. However, in the large deformation regime, the CPDI still may not retain the expected convergence rate. The paper proposes an enhanced CPDI formulation based on least square reconstruction technique. The convected particle least square interpolation (CPLS) material point method assumes the velocity field inside the material point domain as nonconstant. This velocity field in the material point domain is mapped to the background grid nodes with a moving least squares reconstruction. In this paper, we apply the improved moving least squares method to avoid the instability of the conventional moving least squares method due to a singular matrix. The proposed algorithm can improve convergence rate, as illustrated by numerical examples using the method of manufactured solutions.     

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram‐shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4‐node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Enhancement of the material point method using B‐spline basis functions

The material point method (MPM) enhanced with B‐spline basis functions, referred to as B‐spline MPM (BSMPM), is developed and demonstrated using representative quasi‐static and dynamic example problems. Smooth B‐spline basis functions could significantly reduce the cell‐crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of large‐deformation problems for the improved accuracy and convergence, especially with the quadratic B‐splines. Moreover, the increase in the order of the B‐spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems.     

Implicit stress integration procedure for small and large strains of the Gurson material model

The Gurson material model has broad applications in fracture mechanics, large strain deformations and failure of metals. Void growth and void nucleation are included in the model considered in this paper. An implicit stress integration procedure with calculation of the consistent tangent moduli is developed for the Gurson model. The general 3D deformations and the plane stress conditions are considered. The procedure is robust, simple and computationally efficient, suitable for use within the finite element method (FEM). It represents an application of the governing parameter method (GPM) for stress integration in case of inelastic material deformation. A large strain formulation, based on the multiplicative decomposition of the deformation gradient for material with plastic change of volume and logarithmic strains, is used in the paper. The developed numerical procedure for stress integration is applicable to small and large strains conditions. Solved examples illustrate the main features of the developed numerical algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

Axisymmetric form of the generalized interpolation material point method

This paper reformulates the axisymmetric form of the material point method (MPM) using generalized interpolation material point (GIMP) methods. The reformulation led to a need for new shape functions and gradients specific for axisymmetry that were not available before. The new shape functions differ most from planar shape functions near the origin where r = 0. A second purpose for this paper was to evaluate the consequences of axisymmetry on a variety MPM extensions that have been developed since the original work on axisymmetric MPM. These extensions included convected particle domain integration (CPDI), traction boundary conditions, explicit cracks, multimaterial mode MPM for contact, thermal conduction, and solvent diffusion. Some examples show that the axisymmetric shape functions work well and are especially crucial near the origin. One real‐world example is given for modeling a cylinder‐penetration problem. Finally, a check list for software development describes all tasks needed to convert 2D planar or 3D codes to include an option for axisymmetric MPM. Copyright © 2014 John Wiley & Sons, Ltd.     

A 3D Lagrangian gradient smoothing method framework with an adaptable gradient smoothing domain-constructing algorithm for simulating large deformation free surface flows

A novel smoothing particle hydrodynamics (SPH)-like Lagrangian meshfree method, named as Lagrangian gradient smoothing method (L-GSM), has been proposed to avoid the “tensile instability” issue in SPH simulation by replacing the SPH particle-summation gradient approximation technique with a local grid-based GSM gradient smoothing operator. The L-GSM model has been proven effective and efficient when applied to a wide range of large deformation problems for fluids and flowing solids in two-dimensional case. In this study, a three-dimensional (3D) L-GSM numerical framework is proposed for simulating large deformation problems with the existence of free surfaces through developing a widely adaptable 3D gradient smoothing domain (GSD) constructing algorithm. It includes three key novel ingredients: (i) the localized GSD based on an efficient distance-oriented particle-searching algorithm enabling both easy implementation and efficient computation; (ii) a novel algorithm for constructing 3D GSD to guarantee the effectiveness of the 3D GSM gradient operator adaptable to any extreme cases; (iii) a robust normalized 3D GSM gradient operator formulation that can restore the accuracy of gradient approximation even on boundary interface. The effectiveness of the proposed 3D GSD-constructing algorithm is first verified under various distribution conditions of particles. The accuracy of the proposed adaptable 3D GSM gradient algorithm is then examined through conducting a series of numerical experiments with different spacing ratios. Finally, the 3D L-GSM numerical framework is applied to solve a practical problem of free surface flows with large deformation: collapse of a soil column. The results reveal that the present adaptable 3D L-GSM numerical framework can effectively handle the large deformation problems, like flowing solids, with a constantly changing arbitrary free surface profile.     

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving dislocation and is an important extension (with advection term) of the celebrated Peierls‐Nabarro equation for a static dislocation. It belongs to the class of nonlinear reaction‐advection‐diffusion integro‐differential equations with Cauchy‐type kernel, thus involving an integration over an unbounded domain. In the Weertman problem, the unknowns are the shape of the core of the dislocation and the dislocation velocity. The proposed numerical method rests on a time‐dependent formulation that admits the Weertman equation as its long‐time limit. Key features are (1) time iterations are conducted by means of a new, robust, and inexpensive Preconditioned Collocation Scheme in the Fourier domain, which allows for explicit time evolution but amounts to implicit time integration, thus allowing for large time steps; (2) as the integration over the unbounded domain induces a solution with slowly decaying tails of important influence on the overall dislocation shape, the action of the operators at play is evaluated with exact asymptotic estimates of the tails, combined with discrete Fourier transform operations on a finite computational box of size L; (3) a specific device is developed to compute the moving solution in a comoving frame, to minimize the effects of the finite‐box approximation. Applications illustrate the efficiency of the approach for different types of nonlinearities, with systematic assessment of numerical errors. Converged numerical results are found insensitive to the time step, and scaling laws for the combined dependence of the numerical error with respect to L and to the spatial step size are obtained. The method proves fast and accurate and could be applied to a wide variety of equations with moving fronts as solutions; notably, Weertman‐type equations with the Cauchy‐type kernel replaced by a fractional Laplacian.     

Meshfree and finite element nodal integration methods

Nodal integration can be applied to the Galerkin weak form to yield a particle‐type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided. Copyright © 2007 John Wiley & Sons, Ltd.     

基于剪切非线性三维损伤本构模型的复合材料层合板失效强度预测

基于连续损伤力学,建立了同时考虑复合材料剪切非线性效应和损伤累积导致材料属性退化的三维损伤本构模型。模型能够区分纤维损伤、基体损伤和分层损伤不同的失效模式,并定义了相应损伤模式的损伤变量。复合材料层合板层内纤维初始损伤采用最大应力准则判定,基体初始损伤采用三维Puck准则中的基体失效准则判定,分层初始损伤采用三维Hou准则中的分层破坏准则判定,为了计算Puck失效理论中的基体失效断裂面角度,本文提出了分区抛物线法,通过Matlab软件编写计算程序并进行分析。结果表明,与Puck遍历法和分区黄金分割法对比,本文提出的分区抛物线法有效地降低了求解断裂面角度的计算次数,提高了计算效率和计算精度。推导了本构模型的应变驱动显式积分算法以更新应力和解答相关的状态变量,开发了包含数值积分算法的用户自定义子程序VUMAT,并嵌于有限元程序Abaqus v6.14中。通过对力学行为展现显著非线性效应的AS4碳纤维/3501-6环氧树脂复合材料层合板进行渐进失效分析,验证了本文提出的材料本构模型的有效性。结果显示,已提出的模型能够较准确地预测此类复合材料层合板的力学行为及其失效强度,为复合材料构件及其结构设计提供一种有效的分析方法。      

A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

This paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley & Sons, Ltd.     

Locally enhanced Voronoi cell finite element model (LE‐VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions

Ductile heterogeneous materials such as cast aluminum alloys undergo catastrophic failure that initiates with particle fragmentation, which evolves with void growth and coalescence in localized bands of intense plastic deformation and strain softening. The Voronoi cell finite element model (VCFEM), based on the assumed stress hybrid formulation, is unable to account for plastic strain‐induced softening. To overcome this shortcoming of material softening due to plastic strain localization, this study introduces a locally enhanced VCFEM (LE‐VCFEM) for modeling the very complex phenomenon of ductile failure in heterogeneous metals and alloys. In LE‐VCFEM, finite deformation displacement elements are adaptively added to regions of localization in the otherwise assumed stress‐based hybrid Voronoi cell finite element to locally enhance modeling capabilities for ductile fracture. Adaptive h‐refinement is used for the displacement elements to improve accuracy. Damage initiation by particle cracking is triggered by a Weibull model. The nonlocal Gurson–Tvergaard–Needleman model of porous plasticity is implemented in LE‐VCFEM to model matrix cracking. An iterative strain update algorithm is used for the displacement elements. The LE‐VCFEM code is validated by comparing with results of conventional FE codes and experiments with real materials. The effect of various microstructural morphological characteristics is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

Modellierungskonzept für die Versagensbewertung von Laserschweißverbindungen

Failure assessment of laser weldments based on numerical modelling Classical fracture mechanics based assessments are no more sufficient to provide realistic predictions of the deformation and failure behaviour of welded structures. This situation can be improved by numerical modelling based on damage mechanics. A new concept will be provided, which is based on a cohesive model for crack growth simulation. The determination of the relevant material parameters is also considered where testing is combined with numerical simulation. For a laser weld joint, the gradient of the material properties has to be properly characterized. With miniature sized specimens, the material properties can be discretized by homogeneous layers. A new method, based on the digital image technique, has been introduced to determine the stress‐strain curves also in the large strain region due to necking. Test results on small bend bars containing a thin laser weld and a crack like defect in the centre show different crack path developments resulting from a competitive fracture situation. Mainly shear fracture mode occurs, in some cases also a pure normal fracture mode or a combination of both were observed. The concept presented is able to consider the crack development, if all occuring fracture modes are included in the analysis. However, a complete simulation of an extensive crack extension through a heterogeneous structure has not yet been verified.     

Fragmentation algorithm for finite element failure simulation and analysis

An algorithm which combines the techniques of numerical analysis and numerical simulation in the study of transient dynamic structural response is proposed. This is achieved by incorporating the ability to create new surfaces and separate fragments according to a defined failure criterion into a finite element procedure which uses explicit time integration. Thus, the algorithm not only provides accurate prediction of the structural failure and fragmentation pattern, but also evaluates accurate stress, velocity, acceleration and displacement values of each fragmented component. To develop a viable fragmental algorithm, other than the expected modification in storage and node bookkeeping, we found it necessary to introduce an efficient algorithm to handle extremely large displacements resulting from the fracture. In that regard, an updated corotational approach is introduced.     

Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test

We present a geometrically non‐linear assumed strain method that allows for the presence of arbitrary, intra‐finite element discontinuities in the deformation map. Special attention is placed on the coarse‐mesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we first construct an enriched approximation for the deformation map with the non‐linear analogue of the extended finite element method (X‐FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise‐constant stress fields. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields. Importantly, the present formulation gives rise to a symmetric tangent stiffness matrix, even in localized elements. The present modification also allows for the satisfaction of a discontinuous patch test, wherein two different constant stress fields (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics. Copyright © 2003 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

3D NURBS‐enhanced finite element method (NEFEM)

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. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary element analysis in thermoelasticity with and without internal cells

In this paper, a set of internal stress integral equations is derived for solving thermoelastic problems. A jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations. The strongly singular domain integral is then regularized using a semi‐analytical technique. To avoid the requirement of discretizing the domain into internal cells, domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method (RIM). Two numerical examples for 2D and 3D, respectively, are presented to verify the derived formulations. Copyright © 2003 John Wiley & Sons, Ltd.     

A time‐integration method for the viscoelastic–viscoplastic analyses of polymers and finite element implementation

The present study introduces a time‐integration algorithm for solving a non‐linear viscoelastic–viscoplastic (VE–VP) constitutive equation of isotropic polymers. The material parameters in the constitutive models are stress dependent. The algorithm is derived based on an implicit time‐integration method (Computational Inelasticity. Springer: New York, 1998) within a general displacement‐based finite element (FE) analysis and suitable for small deformation gradient problems. Schapery's integral model is used for the VE responses, while the VP component follows the Perzyna model having an overstress function. A recursive‐iterative method (Int. J. Numer. Meth. Engng 2004; 59 :25–45) is employed and modified to solve the VE–VP constitutive equation. An iterative procedure with predictor–corrector steps is added to the recursive integration method. A residual vector is defined for the incremental total strain and the magnitude of the incremental VP strain. A consistent tangent stiffness matrix, as previously discussed in Ju (J. Eng. Mech. 1990; 116 :1764–1779) and Simo and Hughes (Computational Inelasticity. Springer: New York, 1998), is also formulated to improve convergence and avoid divergence. Available experimental data on time‐dependent and inelastic responses of high‐density polyethylene are used to verify the current numerical algorithm. The time‐integration scheme is examined in terms of its computational efficiency and accuracy. Numerical FE analyses of microstructural responses of polyethylene reinforced with elastic particle are also presented. Copyright © 2009 John Wiley & Sons, Ltd.