Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions. Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results

Orthogonality constrained gradient reconstruction for superconvergent linear functionals

The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed over the years to improve the accuracy of post-processed results. Nonetheless such recovery methods usually deteriorate the convergence properties of linear functionals of the solution and, as a consequence, of the quantities of interest as well. The paper develops an enhanced gradient recovery scheme able to both preserve the good qualities of the recovered gradient and increase the accuracy and the convergence rates of linear functionals of the solution

Three-dimensional FE-EFGM adaptive coupling with application to nonlinear adaptive analysis

Three-dimensional problems with both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. Traditionally, these problems are simulated using an adaptive finite element method (FEM). However, the FEM faces many challenges in modeling these problems, such as mesh distortion and selection of a robust refinement algorithm. Adaptive meshless methods are a more recent technique for modeling these problems and can overcome the inherent mesh based drawbacks of the FEM but are computationally expensive. To take advantage of the good features of both methods, in the method proposed in this paper, initially the whole of the problem domain is modeled using the FEM. During an analysis those elements which violate a predefined error measure are automatically converted to a meshless zone. This zone can be further refined by adding nodes, overcoming computationally expensive FE remeshing. Therefore an appropriate coupling between the FE and the meshless zone is vital for the proposed formulation. One of the most widely used meshless methods, the element-free Galerkin method (EFGM), is used in this research. Maximum entropy shape functions are used instead of the conventional moving least squares based formulations'. These shape functions posses a weak Kronecker delta property at the boundaries of the problem domain, which allows the essential boundary conditions to be imposed directly and also helps to avoid the use of a transition region in the coupling between the FE and the EFG regions. Total Lagrangian formulation is preferred over the updated Lagrangian formulation for modeling finite deformation due to its computational efficiency. The well-established error estimation procedure of Zienkiewicz-Zhu is used in the FE region to determine the elements requiring conversion to the EFGM. The Chung and Belytschko error estimator is used in the EFG region for further adaptive refinement. Numerical examples are presented to demonstrate the performance of the current approach in thre

Moving least-squares in finite strain analysis with tetrahedra support

A finite strain finite element (FE)-based approach to element-free Galerkin (EFG) discretization is introduced, based on a number of simplifications and specialized techniques in the context of a Lagrangian kernel. In terms of discretization, a quadratic polynomial basis is used, support is determined from the number of pre-assigned nodes for each quadrature point and quadrature points coincide with the centroids of tetrahedra. Diffuse derivatives are adopted, which allow for the use of convenient non-differentiable weight functions which approximate the Dirac-Delta distribution. Due to the use of a Lagrangian kernel, recent finite strain elasto-plastic constitutive developments based on the Mandel stress are adopted in a direct form. These recent developments are especially convenient from the implementation perspective, as EFG formulations for finite strain plasticity have been limited by the previous requirement of updating the kernel. We also note that, although tetrahedra are only adopted for integration in the undeformed configuration, mesh deformation is of no consequence for the results. Four 3D benchmark tests are successfully solved

A cell-based smoothed finite element method for kinematic limit analysis

This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged

Deformation embedding for point-based elastoplastic simulation

pre-printWe present a straightforward, easy-to-implement, point-based approach for animating elastoplastic materials. The core idea of our approach is the introduction of embedded space-the least-squares best fit of the material's rest state into three dimensions. Nearest neighbor queries in the embedded space efficiently update particle neighborhoods to account for plastic flow. These queries are simpler and more efficient than remeshing strategies employed in mesh-based finite element methods.We also introduce a new estimate for the volume of a particle, allowing particle masses to vary spatially and temporally with fixed density. Our approach can handle simultaneous extreme elastic and plastic deformations. We demonstrate our approach on a variety of examples that exhibit a wide range of material behaviors

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells