The improved element-free Galerkin method for three-dimensional wave equation

Abstract

The paper presents the improved element-free Galerkin (IEFG) method for three-dimensional wave propagation. The improved moving least-squares (IMLS) approximation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares (MLS) approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a discretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scaling parameter, number of nodes and the time step length are considered for the convergence study.

The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation

“…Therefore, for an arbitrary point in the domain, fewer nodes are required to cover that point by their influence domain. This leads to the improvement of the computational efficiency [35][36][37][38][39].…”

“…Besides, it is rather difficult to determine which algebraic equation system is ill-conditioned before the equation is solved, often leading to a poor or even an incorrect numerical solution. To overcome this drawback, based on the MLS approximation, many other methods are proposed to enhance the computational efficiency [36][37][38][39][40][41]. Among them, the improved complex variable moving least-squares (ICVMLS) approximation [40] is proposed to produce the trial function for two-dimensional problems in the form of one-dimensional basis function.…”

An improved moving least-squares Ritz method for two-dimensional elasticity problems

“…This drawback makes it difficult to obtain a correct numerical solution. To overcome this drawback, the IMLS approximation was proposed for construction of the shape functions [37][38][39][40][41][42].…”

Error estimates for the interpolating moving least-squares method in n -dimensional space

“…The EFG method has been the most important meshless method, and has been applied to solve many science and engineering problems [13,67]. Zhang et al presented the improved element-free Galerkin (IEFG) method for elasticity, fracture, elastodynamics, wave and transient heat conduction problems [74][75][76][77]. And Peng et al discussed the IEFG method for three-dimensional viscoelasticity problems [52].…”

“…Hence, this improved MLS approximation has great computing efficiency. Based on the improved MLS approximation, the improved EFG method [52,[73][74][75][76][77] and the BEFM [18,22,23,34,[44][45][46]51] are proposed.…”