Complex variable element-free Galerkin method for viscoelasticity 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.

On the basis of the moving least-squares (MLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elastic large deformation problems, such as large displacement and large rotation, is presented in this paper. The advantage of the CVMLS approximation is that the trial function of a two-dimensional problem is formed with one-dimensional basis function. Then the CVMLS approximation is applied to solve two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the discretized system equation, the penalty method is used to apply the essential boundary conditions. And then the complex variable element-free Galerkin (CVEFG) method for two-dimensional large deformation problems based on the total lagrange formulation is formed, the corresponding formulae are obtained, and the Newton-Raphson iteration is used in the numerical implementation. Three numerical examples are given to show that this method in this paper is effective for elastic large deformation problems.

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

In this paper, the improved interpolating complex variable moving least squares (IICVMLS) method is applied, in which the complete basis function is introduced and combined with the singular weight function to achieve the orthometric basis function. Then, the interpolating shape function is achieved to construct the interpolating trial function. Incorporating the IICVMLS method and the Galerkin integral weak form, an improved interpolating complex variable element free Galerkin (IICVEFG) method is proposed to solve the 2D potential problem. Because the essential boundary conditions can be straightaway imposed in the above method, the expressions of final dispersed matrices are more concise in contrast to the non-interpolating complex variable meshless methods. Through analyzing four specific potential problems, the IICVEFG method is validated with greater computing precision and efficiency.

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

Stability of piezoelectric FGM rectangular plates subjected to non-uniformly distributed load, heat and voltage

This study aims to interpret the non-linear steady-state heat conduction for temperature-dependent thermal conductivity ($k$ ($T$)) using Element-Free Galerkin (EFG) method. In this present study, a one-dimensional heat conduction problem with uniform heat generation was explicated. Moving Least Squares (MLS) approximants were applied to estimate the unknown function of temperature $T$ ($x$) with $T^h$ ($x$) using linear basis and weight functions. The variational method has been used to develop discrete equations. Essential boundary conditions are enforced by using the Penalty method. The results have been obtained for the one-dimensional model using essential MATLAB codes. The results obtained by the EFG method are compared with the analytical and finite-element method results. The results are also studied by increasing the number of nodes to study the convergence which indicated that EFG has good convergence behavior. The results have also been obtained for different values of the scaling parameter ($α_s$) and any values of αs between 1.8 and 2.0 were found suitable for providing better results in the EFG method.

Shakedown theorems are exact theories of classical plasticity for the direct computation of the load-carrying capacity under varying loads. Based on the classical Melan’s theorem, a numerical solution procedure for determining the shakedown load of elasto-perfectly plastic structure is presented firstly making use of the element free Galerkin (EFG) method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced-basis technique is adopted here to solve the mathematical programming iteratively in a sequence of reduced self-equilibrium stress subspaces with very low dimensions. The self-equilibrium stress field is expressed by linear combination of several self-equilibrium stress basis vectors with parameters to be determined. These self-equilibrium stress basis vectors can be generated by performing equilibrium iteration procedure during elasto-plastic incremental analysis. The Complex method is used to solve nonlinear programming and determine the maximal load amplifier. The numerical results show that it is efficient and accurate to solve shakedown analysis problems by using the EFG method and nonlinear programming.

Meshfree analyses of cable-reinforced membrane structures by ALE–EFG method

A new formulation of the Element Free Galerkin (EFG) method is developed for the modelling of incompressible materials. Beginning with a mixed variational principle, a selective reduced integration procedure is developed by implementing nodal quadrature. Numerical examples are provided which compare the performance of the proposed technique to the standard EFG approximation. These studies illustrate the capability of the new formulation to eliminate volumetric locking. For the standard method, however, the degree of volumetric locking is shown to be a function of the local support sizes (domains of influence) of the EFG shape functions. Copyright © 1999 John Wiley & Sons, Ltd.

The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

In this paper, an improved interpolating moving least-square (IIMLS) method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker δ function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square (MLS) approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin (IIEFG) method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin (EFG) method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.

By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

The interpolating element-free Galerkin method for three-dimensional elastoplasticity problems

Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems