7.1 Where We Were, Where We’re Going

This very short chapter marks a break, ending the discussion of the method of moments and beginning discussion of the finite difference (FD) and finite element (FE) methods. This is a good place to compare and contrast these methods.

All three methods are approaches to solving for the electrostatic variables charge and voltage in a given geometric structure of electrodes and possibly dielectric interfaces. They are all based on developing a set of linear algebraic equations that approximate continuous variables with approximate locally defined variables.

The method of moments (MoM) takes as its solution domain all of (three-dimensional) space. Charge is constrained to exist on conducting electrodes. These electrodes can have thickness (i.e., they themselves can be three-dimensional bodies), but since the laws of electrostatics guarantee that all the charge will move to the surface of three-dimensional (3d) electrodes, it doesn’t matter whether if the electrodes have thickness or are simply arbitrarily shaped thin skins. This is very important mathematically, because even though we are dealing with 3d space, our variable — the charge — exists only on two-dimensional (2d) surfaces.

We divided the electrode surfaces into either square or triangular planar regions, or cells, each of which is assumed to have a uniform charge density. Since the cells each have a finite area (not necessarily all the same) and a linear equation will express the relationship between the (applied) voltage at each cell and the charge on all of the cells, the number of cells must be finite. This, in turn, implies that there must be a finite volume of conductors in our infinite space.

The set of equations is written by demanding that the known, applied, voltage on each cell is identically the voltage due to the charge on all of the cells. The equation set is solved for these charges; the voltage, and then the electric field, anywhere in space may be found by summing the contributions to this voltage (and field) of all the charges.

The capacitance between two electrodes falls out easily as simply the sum of the charge on an electrode divided by the voltage difference between two electrodes.

Mathematically, there are as many equations as there are cells. Since the voltage on any one cell depends on contributions from all of the cells, the coefficient matrix of the equation set is fully populated. This means that even though only 2d surfaces are being considered, the number of matrix terms grows very rapidly with the number of cells.

While writing an expression for each coefficient matrix term is very easy, accurate evaluation of these terms is not so easy. Except for the diagonal terms, numerical integration and/or approximation is necessary. A 1000-cell structure, for example, has approximately 1,000,000/2 coefficients to evaluate, 499,000 of them numerically. In the previous chapters empirical approximations were developed for these coefficients that, although not the most accurate approximations possible, allowed us to solve many problems and treat many situations in a reasonable and efficient manner.

The method of moments (MoM) is one case of a much more general and powerful approach called the boundary element method.¹ Further discussion of the boundary element method is outside the scope of this book but is certainly a worthwhile endeavor for anyone wishing to pursue these types of approximate solutions.²

The finite difference (FD) and finite element (FE) methods take a very different approach. With only a few special exceptions, only a finite region of space is considered. One-, two- and three-dimensional formulations are possible; one-dimensional (1d) formulations are sometimes useful for teaching purposes but for the most part not of any value for practical problems. A 2d structure is actually a cross section of a uniform infinitely long 3d structure, so in general we're always talking about 3d structures even when the third dimension isn't seen explicitly in the figures or the equations.

A finite region of (2d) space is enclosed by a boundary. At this boundary we specify boundary conditions that are either a voltage or the normal derivative of the voltage everywhere (on the boundary). There might also be electrode regions inside this boundary that have voltage boundary conditions. We then break up the region within the boundary into discrete regions (again, cells). In both the FD or FE methods we write a set of equations relating these cells to one another.

The FD method is derived from numerical approximations of Laplace's equation, and only the voltages at the corners of rectangular cells are defined. The FE method is derived from formal integral approximation techniques, and the voltage is defined throughout the cell. In both cases each cell communicates only with cells that share nodes. Also, in the FE method cell shape is arbitrary (although practical considerations will narrow our choices somewhat).

This latter statement has mathematical implications in that while we need an equation for each corner of a rectangular cell or vertex of a triangular cell, each of these points (nodes) is shared by only several cells. Also, and more important, this means that each cell “communicates” directly with only several of the many cells involved in a structure and hence the coefficient matrix that is developed is very sparse. We will utilize sparse matrix storage in our computer programs rather than carry around thousands of zeros (and limit our problem size and solution speed).

In the MoM method, calculating the coefficients was not trivial; however, in the FE and FD methods calculating the coefficients is easy — in the FD method, it is actually trivial.

The FD method, using rectangular cells, is very limited in the structural sophistication possible. However, the interconnection of the cells and generation of the coefficient equations is so simple that it pays to spend time with the FD method to see what useful problems can be solved and what electrostatics we can learn. The ties between the geometry and the resulting equations are very clear and facilitate understanding of the situation tremendously. The FE method allows for very generalized structure descriptions (which probably accounts for its huge popularity), but the tradeoff here is that there is a sophisticated “assembly” procedure necessary to describe (to the system of equations) the interconnection and relative locations of all the cells.

As is the case when dealing with the MoM, generation of the mesh for rectangular structures is easy but limited, while setting up the mesh for triangular problems is difficult but very powerful. For the same reasons, therefore, we will limit FE discussions to relatively simple structures (no Golden Gate Bridge frames) and leave the sophisticated mesh generation problem to specialized packages.

The principal purposes of introducing the FD method are to (1) demonstrate the basics of this technique and write useful programs without first engaging in the more involved mathematics of the FEM method and (2) introduce basic programs for generating the coefficient matrix, calculating stored energy, present techniques for calculating upper and lower bounds on the approximations, calculate properties of multielectrode and open-boundary systems, and so on. There is a tremendous amount of electrostatics to be shown here, and it is advantageous to show it using the simplest algorithms possible.

Because the FE method is so much more general than the FD method in its ability to handle complex problems, at the cost of computational complexity, two programming styles are adopted:

1. For the FD method, the structure is hardcoded into the programs themselves. Parameters and shapes must be changed by changing source code. This is very limiting in terms of general application of the technique, but keeps with the strategy of keeping the programming as simple as possible to demonstrate the FD technique and the electrostatics to be learned from the solution.
2. For the FE method, the programs are data-file-driven. Although mesh generation and zoning techniques were not the desired themes for this book, it was necessary to choose a software package to do the job and then to teach enough of its use so that examples could be generated and “getting the mesh data into the FEM program” could be demonstrated. Fortunately, a very capable free mesh generation package is available, as will be discussed in later chapters. Since all of the geometric description (including cell meshing) data reside outside the programs, these programs are very general. Within the limits of the FEM calculations being presented in any program, these programs can handle an infinite number of different structures.

As has been mentioned before, all of the programs have been written with the primary goal of showing how the analyses become algorithms, which then become (MATLAB) code. In most cases when it was necessary to choose between clarifying these correlations and writing the most compact and/or most efficient code possible, the former option was chosen.

References

1. 1. W.-T. Ang, A Beginner's Course in Boundary Element Methods, Universal Publishers, Boca Raton, FL, 2007.
2. 2. M. V. K. Chari and S. J. Salon, Numerical Methods in Electromagnetism, Academic Press, London, 2000.