Introduction to Numerical Electrostatics Using MATLAB

16FEM in Three Dimensions

16.1 Creating Three-Dimensional Meshes

Our first task in developing three-dimensional (3d) FEM capability is to generate 3d meshes to describe our geometries. Fortunately, gmsh is well equipped to do this, with its two-dimensional (2d) meshing capabilities extending directly into three dimensions. As in the case of 2d meshes, there are many ways to generate a given geometry in three dimensions. In this chapter we will consider only one of these approaches. The gmsh documentation and many online sources should be consulted for a comprehensive discussion of this topic.

A problem that we will encounter using unstructured 3d meshes is that visualizing what we have done can be difficult; there is simply too much going on. In the examples in this chapter, many of the figures of meshed structures were generated using low-resolution meshing. This is necessary to make these figures at all useful. The .geo file accompanying these examples will show the resolution used to generate any results quoted from the FEM analysis. Also, the gmsh ability to rotate a figure in space was utilized in generating some of the figures. Since the rotation chosen for a figure was arbitrary (it looked good), details of the rotation are not given.

The gmsh script struct16_1.geo shows the construction of a cube:

// struct16_1.geo
// first experiments with 3D

lc = 1;

Point(1) = {-.5, -.5, -.5, lc};
Point(2) = {.5, -.5, -.5, lc};
Point(3) = {.5, .5, -.5, lc}; Point(4) = {-.5, .5, -.5, ...