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Comput. Methods Appl. Mech. Engrg. 280 (2014) 84–116 www.elsevier.com/locate/cma

A four-point integration scheme with quadratic exactness for three-dimensional element-free Galerkin method based on variationally consistent formulation

Qinglin Duan∗, Xin Gao, Bingbing Wang, Xikui Li, Hongwu Zhang

The State key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China

Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China

Received 3 February 2014; received in revised form 13 June 2014; accepted 16 July 2014

Available online 23 July 2014

Abstract

The formulation of three-dimensional element-free Galerkin (EFG) method based on the Hu–Washizu three-field variational principle is described. The orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions. This leads to a variationally consistent formulation which has a similar form as the formulation of standard Galerkin weak form. Based on this formulation, an integration scheme which employs only four cubature points in each background tetrahedral element (cell) is rationally developed for three-dimensional EFG with quadratic approximation. The consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved. Extension of the proposed method to small strain elastoplasticity is also presented. The proposed method can exactly pass quadratic patch test, that is, quadratic exactness is achieved, and thus it is named as quadratically consistent 4-point (QC4) integration method. In contrast, EFG with standard tetrahedral cubature and the existing linearly consistent 1-point (LC1) integration fail to exactly pass quadratic patch test. Numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy, convergence, efficiency and stability. The capability of the proposed QC4 scheme in solving elastoplastic problems is also demonstrated by numerical examples. c⃝ 2014 Elsevier B.V. All rights reserved.

Keywords: Meshfree/Meshless; EFG; Three-dimensional; Variational principle; Integration 1. Introduction

Meshfree methods which emerge and develop in the past twenty years possess several appealing advantages against the traditional finite element method (FEM) which dominates engineering analysis for a long time. For instance, they are easier to implement h-adaptivity since only nodes (not elements) are needed to be added or removed. This merit is due to the fact that meshfree approximation is constructed entirely in terms of a set of scattered nodes and no element ∗ Corresponding author at: Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China. Tel.: +86 411 84705515.

E-mail addresses: qinglinduan@gmail.com, qinglinduan@dlut.edu.cn (Q. Duan). http://dx.doi.org/10.1016/j.cma.2014.07.015 0045-7825/ c⃝ 2014 Elsevier B.V. All rights reserved.

Q. Duan et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 84–116 85 is needed. This is also the major difference between meshfree methods and FEM. More importantly, the meshfree approximation function is much smoother than that of FEM, that is, the element-based Lagrangian interpolation which only has C0 continuity. This is an important reason why meshfree methods usually show higher convergence rates than FEM. In addition, it is much easier to construct high order approximation in meshfree methods than in

FEM since the mesh of high order elements is not needed. Furthermore, it is very convenient to enrich meshfree approximations with closed-form solutions such as the near-tip crack fields. Due to these reasons, intensive interest has developed in this field and many meshfree methods have been invented, such as the reproducing kernel particle method (RKPM) proposed by Liu et al. [1], the h-p Cloud by Duarte and Oden [2] and the point interpolation method (PIM) by Liu et al. [3]. Among them, the element-free Galerkin (EFG) method invented by Belytschko et al. [4] is one of the most successful meshfree methods and is popular in various practical applications such as plate analysis [5], dynamic fracture [6] and large-scale simulations [7].

However, meshfree methods also show some disadvantages. Taking EFG for example, the moving least-square (MLS) approximation employed in EFG does not possess the so called Kronecker delta property and this complicates the enforcement of essential boundary conditions. Several methods such as the penalty method [8], Nitsche’s method [9] and the coupling methods [9,10] are developed to address this issue. More severe drawback of meshfree methods is their low efficiency and this is still an open issue. Since most meshfree approximants are rational functions with overlapping supports, a large number of quadrature points are required in the domain integration of the Galerkin weak form. For example, in standard EFG with quadratic MLS approximation [11], 16 quadrature points have to be used in each background triangle cell to result stable solutions, whereas in FEM only 3 evaluation points per element can make the 6-node triangle element stable for elastostatic problems.

Many efforts have been devoted to reducing the number of sampling points and, as a consequence, improving the computational efficiency of meshfree Galerkin methods. Beissel and Belytschko [12] initiated the study of nodal integration which integrates the Galerkin weak form only at the approximation nodes. They found that direct nodal integration is unstable and, therefore, a least-square stabilization term with an artificial parameter is introduced into the weak form to remove the spurious oscillations. However, the selection of the stabilization parameter depends on numerical experiments and improper parameters may cause oscillated or over-diffusive numerical results. Nagashima [13] stabilized the nodal integration in EFG by using a Taylor’s expansion for the stiffness matrix. In such a way, the stabilization terms are introduced in a rational manner without using artificial parameters. However, Duan and Belytschko [14] reported that it has poor stabilization effect for EFG with quadratic approximations. Rabczuk et al. [15] added additional quadrature points between approximation nodes to improve the stability of nodal integration. This method originates from the concept of stress points proposed by Dyka et al. [16] to remove the tensile instabilities in the method of smoothed particle hydrodynamics (SPH). However, Fries and Belytschko [17] found that such stress-point integration still shows mild oscillation and its convergence is not satisfactory. Duan and Belytschko [14] further stabilized stress-point integration by introducing additional terms to the weak form, similar to the least-square stabilization given in [12]. Stability is improved, but proper choice of the stabilization parameters is still required.