The FEM approach to the 2D Poisson equation in 'meshes' optimized with the Metropolis algorithm

TL;DR

A 2D mesh generation routine optimized with the Metropolis algorithm that can solve the 2D Poisson problem and serve as standard discrete patterns for the Finite Element Method (FEM).

Abstract

The presented article contains a 2D mesh generation routine optimized with the Metropolis algorithm. The procedure enables to produce meshes with a prescribed size h of elements. These finite element meshes can serve as standard discrete patterns for the Finite Element Method (FEM). Appropriate meshes together with the FEM approach constitute an effective tool to deal with differential problems. Thus, having them both one can solve the 2D Poisson problem. It can be done for different domains being either of a regular (circle, square) or of a non--regular type. The proposed routine is even capable to deal with non--convex shapes.

Figures (12)

• Figure 1: Figure presents the domain $\Omega$ and its boundary $\Gamma$. The whole domain $\Omega$ could be divided into subdomains $\Omega^i$ with corresponding line segments $\Gamma^i$ being part of the boundary. The idea of division into subdomains (elements) constitutes the main concept of the finite element method.
• Figure 2: Figure presents the domain $\widetilde{\Omega}$ and its boundary $\widetilde{\Gamma}$ after projection to the polygonal domain. It has eight boundary nodes and one central node. Comparing both the initial $\Omega$ and the polygonal $\widetilde{\Omega}$ domain one can notice that such a simple projection gives rather rough correspondence between them a), however, in some cases it could be a sufficient one i. e. when an integrated function changes very slowly in some $\delta$-thick neighbourhood of the boundary $\Gamma$ b).
• Figure 3: Figure shows four domains $\Omega$ having different shapes. In brackets, finally established set of parameters is written: $N_{p}$ -- number of mesh points, $N_{divisions}$ -- number of divisions (according to Sec. 3.2), $\overline{S}_{N}$ -- a normalized average element area are presented; a) regular polygon -- square (258, 8, 1.002); b) regular polygon with 16 nodes (376, 6, 1.026) which approximates circular shape well; c) non-regular, convex figure (315, 8, 1.01); d) non-regular, semi-convex figure (247, 6, 1.071); and two non--regular, non--convex figures e) (245, 7, 0.993) and f) (164, 6, 1.0003) both with weight = [0.25, 0.75].
• Figure 4: Figure presents a division process of non-regular and circular domains together with their boundaries. Pictures a) and c) show meshes with new nodes. Some of them are of the illegal type (defined in Sec. 3.2). These nodes constitute starting points for next complementary division that transforms such not well--defined elements into the correct ones, see pictures b) and d).
• Figure 5: Figure presents the square domain divided into a set of new elements $\widetilde{\Omega}^i$ with corresponding set of line segments $\widetilde{\Gamma}^i$ being its boundary. A way of finding new nodes constitutes the main point of the mesh generation process (see Sec. 3.2) while a selection of nodes is perform according to the algorithm from Sec. 3.3 a) nodes a,b,c,d have been classified as boundary nodes whereas b) nodes e,f,g have been determined as internal nodes.