Analyzing three-dimensional potential problems with the improved element-free Galerkin method

Abstract

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.

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].…”

The dimension splitting reproducing kernel particle method for three‐dimensional potential problems

“…This means that the number of planes in the direction x 1 should be large. In this numerical example, we choose the number of planes to be 30,35,40,45, and the corresponding results are shown in Figure 14. From the results, we can see that the results converge as the number of planes increase, and the DSRKPM has obvious advantages of computational efficiency.…”

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

“…A convergence study of the proposed method is carried out by analyzing the final function values under different discretization schemes and different scaling factors for the nodes of the studied field, and the time step length [25,26,[31][32][33][34][35].…”

“…Moreover, it is difficult to obtain the correct numerical solution. Using the weighted orthogonal basis functions, we present the IMLS approximation [22][23][24][25][26].…”