无网格方法在三维立方体中的应用

The applicable meshless method for three-dimensional applications.

近年来,在电磁散射等领域,有限元方法的使用越来越广泛,其优点在于准确性非常高,但在处理边界条件的时候比较复杂。而无网格方法中的MLPG方法,只需要局部的点的位置信息,在边界处理的时候,可以大大减少处理的难度,并且减少了网格的划分,提高了使用的效率。本文使用的MLPG方法,即无网格局部Petorv-Galerkin方法,在三维空间中的均匀介质立方体中,对亥姆霍兹方程进行变分,对本质边界条件采用罚函数法处理,再利用高斯点积分,组装刚度矩阵进行求解,并且通过赋予不同的相对介电常数来获得非均匀介质,以及通过程序实验进行验证,确保了有效性。

In recent years, the finite element method shows its outstanding performance in handling the electromagnetic scattering related problems. Though FEM provides great precision among estimation and approximation, the method can be complex on handling boundary conditions. Meshless method requires only small amount of information from few local points-Hence that, the complexity of handling boundaries can be simplified, the grid process can be subside, provides better efficiency in general. In this paper, we applied MLPG method in a medium cube of three-dimensional homogeneous. To such, applied 2 methods: 1. Variance the Helmholtz equation. 2. Apply penalty function to strengthen the essential boundary condition. Later, Gaussian quadrature method is applied to construct the stiffness matrix, which as our proposed solution. The in-homogeneous media is obtained from assigning different relative dielectric constants, which omits a series of complex operations such as meshing and boundaries defining. Finally, we programmed our proposed model and performed experiments to ensure effectiveness.