Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

An improved complex variable element-free Galerkin method for two-dimensional elasticity problems

In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.

A new complex variable element-free Galerkin method for two-dimensional potential problems

In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square （ICVMLS） approximation, a new complex variable meshless method （CVMM） for two-dimensional （2D） transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

Complex variable element-free Galerkin method for viscoelasticity problems

Based on the complex variable moving least-square （CVMLS） approximation, the complex variable element-free Galerkin （CVEFG） method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin （EFG） method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

New complex variable meshless method for advection-diffusion problems

In this paper,an improved complex variable meshless method（ICVMM） for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square（ICVMLS） approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for twopoint boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency.

A complex variable meshless method for fracture problems

Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-conditioned equations, and has greater precision and computational efficiency. Using the analytical solution near the tip of a crack, the trial functions in the complex variable moving least-square approxi- mation are extended, and the corresponding approximation function is obtained. And from the minimum potential energy principle, a complex variable meshless method for fracture problems is presented, and the formulae of the complex variable meshless method are obtained. The complex variable meshless method in this paper has greater precision and computational efficiency than the conventional meshless method. Some examples are given.

移动最小二乘法研究进展与述评

为使移动最小二乘法能更好地应用到无网格方法中,详细阐述移动最小二乘逼近法、移动最小二乘插值法、MUKHERJEE改进的移动最小二乘法以及程玉民等提出的改进的移动最小二乘法和复变量移动最小二乘法等的研究进展,述评各种移动最小二乘法的优缺点,并概述各种移动最小二乘法形成的无网格方法的研究进展.

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method(CVBEFM)for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers.To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative,a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures.To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables,a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure.The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers,even for extremely large wavenumbers such as k=10000.

黏弹性问题的改进的复变量无单元Galerkin方法

基于改进的复变量移动最小二乘法,提出了二维黏弹性问题的改进的复变量无单元Galerkin方法.采用改进的复变量移动最小二乘法建立形函数,根据Galerkin积分弱形式建立求解方程,并用罚函数法施加本质边界条件,推导了二维黏弹性问题的改进的复变量无单元Galerkin方法的计算公式.最后,通过实际算例,将计算结果与复变量无单元Galerkin方法及有限元法的结果进行了对比,说明了本文方法具有更高的计算精度和计算效率.

正交各向异性稳态热传导问题的ICVEFG方法

论文将改进的复变量无单元Galerkin方法(Improved Complex Variable Element-free Galerkin method,ICVEFG)应用于求解正交各向异性介质中的稳态热传导问题,提出了正交各向异性稳态热传导问题的ICVEFG方法.采用罚函数法引入本质边界条件,推导了正交各向异性介质中的稳态热传导问题的Galerkin积分弱形式.采用改进的复变量移动最小二乘近似(Improved Complex Variable Moving least-squares approximation,ICVMLS)建立二维温度场问题的逼近函数,推导了相应的计算公式.编制了计算程序,对三个正交各向异性介质中的热传导问题进行了分析,说明了论文方法的有效性.

Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法

基于改进的复变量移动最小二乘法,建立了Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法.相对于移动最小二乘法,改进的复变量移动最小二乘法采用一维基函数建立二维问题的逼近函数,提高了形函数计算效率.由改进的复变量移动最小二乘法建立Kirchhoff板的挠度逼近函数,根据Kirchhoff板弯曲问题的Galerkin弱形式建立离散方程,并应用罚函数法施加本质边界条件,推导了Kirchhoff板弯曲问题的改进的复变量无单元Galerkin法的公式.通过对4个典型算例进行计算和分析,说明了本文建立的Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法的有效性,并通过分析数值解的精度对本文方法中如何选取合适的基函数、权函数、影响域比例参数、节点分布和罚因子进行了讨论.数值算例说明了本文方法具有较好的收敛性和较高的计算精度.