Jiun-Shyan Chen

A stabilized conforming nodal integration for Galerkin mesh-free methods

Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples. Copyright

Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures

Abstract Large deformation analysis of non-linear elastic and inelastic structures based on Reproducing Kernel Particle Methods (RKPM) is presented. The method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis. The current formulation considers hyperelastic and elasto-plastic materials since they represent path-independent and path-dependent material behaviors, respectively. In this paper, a material kernel function and an RKPM material shape function are introduced for large deformation analysis. The support of the RKPM material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation. The essential boundary conditions are introduced by the use of a transformation method. The transformation matrix is formed only once at the initial stage if the RKPM material shape functions are employed. The appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions. In transient problems with an explicit time integration method, the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles. Several hyperelasticity and elasto-plasticity problems are studied to demonstrate the effectiveness of the method. The numerical results indicated that RKPM handles large material distortion more effectively than finite elements due to its smoother shape functions and, consequently, provides a higher solution accuracy under large deformation. Unlike the conventional finite element approach, the nodal spacing irregularity in RKPM does not lead to irregular mesh shape that significantly deteriorates solution accuracy. No volumetric locking is observed when applying non-linear RKPM to nearly incompressible hyperelasticity and perfect plasticity problems. Further, model adaptivity in RKPM can be accomplished simply by adding more points in the highly deformed areas without remeshing.

Overview and applications of the reproducing Kernel Particle methods

SummaryMultiple-scale Kernel Particle methods are proposed as an alternative and/or enhancement to commonly used numerical methods such as finite element methods. The elimination of a mesh, combined with the properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and high modal density. The Reproducing Kernel Particle Method (RKPM) utilizes the fundamental notions of the convolution theorem, multiresolution analysis and window functions. The construction of a correction function to scaling functions, wavelets and Smooth Particle Hydrodynamics (SPH) is proposed. Completeness conditions, reproducing conditions and interpolant estimates are also derived. The current application areas of RKPM include structural acoustics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity. The effectiveness of RKPM is extended through a new particle integration method. The Kronecker delta properties of finite element shape functions are incorporated into RKPM to develop a Cm kernel particle finite element method. Multiresolution and hp-like adaptivity are illustrated via examples.

New boundary condition treatments in meshfree computation of contact problems

Due to the loss of Kronecker delta properties in the meshfree shape functions, the imposition of essential boundary conditions consumes significant CPU time in meshfree computation. In this work, two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems. The mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub-degrees of freedom associated with the boundary group. The boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values, and consequently kinematic constraints can be imposed directly. The effectiveness of the proposed methods is demonstrated in several numerical examples.

Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua

Abstract The fundamental arbitrary Lagrangian-Eulerian (ALE) mechanics and its finite element formulation are given. The tangential stiffness matrix, which is shown to be composed of the linarized material response matrix, the geometrical stiffness matrix, and the ALE transport matrix are derived from a consistent linearization procedure. Various numerical methods for the ALE finite element equations are then presented, and several examples are analyzed to examine some features of the method.

An improved reproducing kernel particle method for nearly incompressible finite elasticity

The previously developed reproducing kernel particle method (RKPM) employs a high-order quadrature rule for desired domain integration accuracy. This leads to an over-constrained condition in the limit of incompressibility, and volumetric locking and pressure oscillation were encountered. The employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking. However, this approach consumes high CPU and it cannot effectively resolve pressure oscillation difficulty. In this paper, a pressure projection method is introduced by locally projecting the pressure onto a lower-order space to reduce the number of independent discrete constraint equations. This approach relieves the over-constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in RKPM. The method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable. To further reduce the computational cost, a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced. The resulting stiffness matrix and force vector of RKPM are obtained explicitly without numerical integration.

Analysis of metal forming process based on meshless method

Abstract Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion. Since 1993, meshless methods have been considerably developed for structural applications. The main feature of these methods is that the domain of the problem is represented by a set of nodes, and a finite element mesh is unnecessary. This new generation of computational methods reduces time-consuming model generation and refinement effort, and it provides a higher rate of convergence than that of the conventional finite element methods. A meshless method based on the reproducing kernel particle method (RKPM) is applied to metal forming analysis. The displacement shape functions are developed from a reproducing kernel approximation that satisfies consistency conditions. The use of smooth shape functions with large support size are particularly effective in dealing with large material distortion in metal forming analysis. In this work, a collocation formulation is used in the boundary integral of the contact constraint equations formulated by a penalty method. Metal forming examples, such as ring compression test and upsetting, are analyzed to demonstrate the performance of the method.

Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched reproducing Kernel particle method

A formulation is proposed for homogenization of magnetostrictive particle-filled elastomers (MPFE) based on an interface-enriched reproducing kernel particle method. A variational equation for obtaining the local fluctuating deformation of MPFE is introduced. The magnetostrictive effect in the metal inclusion is modeled as an eigen-deformation. An interface-enriched reproducing kernel approximation with embedded derivative discontinuities on the material interface is presented. This approach does not require additional degrees of freedom in the approximation of displacement field for the interface conditions compared to the conventional reproducing kernel approximation. Microscopic solution and homogenized constitutive behavior of uniaxial tension and simple shear deformation of MPFE are presented.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory

A least-squares-based pressure projection method is proposed for the nonlinear analysis of nearly incompressible hyperelastic materials. The strain energy density function is separated into distortional and dilatational parts by the use of Penns invariants such that the hydrostatic pressure is solely determined from the dilatational strain energy density. The hydrostatic pressure and hydrostatic pressure increment calculated from displacements are projected onto appropriate pressure fields through the least-squares method. The method is applicable to lower and higher order elements and the projection procedures can be implemented into the displacement based nonlinear finite element program. By the use of certain pressure interpolation functions and reduced integration rules in the pressure projection equations, this method can be degenerated to a nonlinear version of the selective reduced integration method.

The sandia fracture challenge: Blind round robin predictions of ductile tearing

Existing and emerging methods in computational mechanics are rarely validated against problems with an unknown outcome. For this reason, Sandia National Laboratories, in partnership with US National Science Foundation and Naval Surface Warfare Center Carderock Division, launched a computational challenge in mid-summer, 2012. Researchers and engineers were invited to predict crack initiation and propagation in a simple but novel geometry fabricated from a common off-the-shelf commercial engineering alloy. The goal of this international Sandia Fracture Challenge was to benchmark the capabilities for the prediction of deformation and damage evolution associated with ductile tearing in structural metals, including physics models, computational methods, and numerical implementations currently available in the computational fracture community. Thirteen teams participated, reporting blind predictions for the outcome of the Challenge. The simulations and experiments were performed independently and kept confidential. The methods for fracture prediction taken by the thirteen teams ranged from very simple engineering calculations to complicated multiscale simulations. The wide variation in modeling results showed a striking lack of consistency across research groups in addressing problems of ductile fracture. While some methods were more successful than others, it is clear that the problem of ductile fracture prediction continues to be challenging. Specific areas of deficiency have been identified through this effort. Also, the effort has underscored the need for additional blind prediction-based assessments.