Introduction to Numerical Electrostatics Using MATLAB

15Some FEM Topics

15.1 Symmetries

One of the principal reasons for the popularity of the FEM is the ease with which boundary conditions are handled. All of the FEM examples presented thus far have utilized boundary conditions formed by setting boundary nodes to an assigned voltage. Since we have dealt exclusively with triangular shapes with simple linear shape functions, this means that the lines connecting these nodes (an edge of the triangle) are set to a linear interpolation between these two node voltages.

A second type of boundary condition occurs when the node voltages are not assigned but the electric flux along line edges is assigned. The electric flux at a region boundary is defined as the integral of the normal component of the electric field lines crossing that boundary (in this case the triangle’s edge).

There are elegant derivations of how to translate these boundary conditions into FEM equations.¹ These derivations, in turn, require some background in variational calculus and vector calculus. We will consider only a subset of the general problem, so a simpler explanation of how to set up our equations will suffice.

A symmetry boundary, or a magnetic wall, is a boundary which has zero electric flux across it. That is, the electric field at the boundary is parallel to the boundary wall (and the normal field across the boundary wall is zero).

Figure 15.1 shows an example of a symmetric structure. It may also be regarded as a subset of a larger symmetric structure.