Advantages and Disadvantages of Various Numerical Methods:
For computing the electric fields, various methods have been used, viz. Finite Difference Method, Finite Element Method, Charge Simulation Method and Boundary Element Method. Each of these methods has its own advantages for solving a particular problem.
With the FDM, the numerical evaluation of the difference equation is simple but time consuming. For treating a given field problem, it is necessary to subdivide the finite plane of the field problem into a predominantly regular net of polygons which is supplemented with irregular elements at the boundaries. However, in this method, all difference equations are approximation to the field equation by neglecting the higher order terms. Thus, the resulting error can be large.
On the other hand, Finite Element Method is a very general method and has been used for solving a variety of problems. Any non-linearity/iehomogeneity can be modelled and the solution will be available on the entire surface of the domain. Material interface conditions are automatically satisfied. However, it needs a powerful graphic user interface for processing.
Open geometry does not pose any problem with the Charge Simulation Method since the surface of the conductor is the only one that is discretized. In addition, as the solution satisfies the Laplace’s/Poisson’s equation, it will be very smooth, and always gives a small but dense matrix and therefore can be easily handled using personal computers. However, due to the application of superposition principle, non-linearities and non-homogeneity cannot be modelled using this method.
On the other hand a unique feature of Boundary Element Method is that the electric fields are proportional to the charge densities on an enclosed electrode which is simulated by real charges. This direct field derivation is based on a well known Gauss’s area integral. Although BEM is sufficiently developed for use in two-dimensional axi-symmetric problems, some difficulties still exist. They are, the programming complexity and the need for large amount of computational time to execute an improper integral.
Of the above methods, the choice of a particular method depends on the specific problem on hand. In general, the construction of Finite Element model requires considerable effort. since the entire field region should be meshed. while the Charge Simulation and Boundary Element Methods require only the outer surface of the electrode and the outer layer of the dielectric to be meshed. In practice, an important difference between the various numerical methods is that the Finite Element Method can be used only with fields which are bounded while the Charge Simulation method and the Boundary Element Methods can also deal with unbounded fields.
A comparison of the accuracies of using the various computational methods shows a good agreement between the results of BEM and FEM, for two-dimensional problems a discrepancy is about 1% while in the 3-D case it is about 2%. On the other hand, descrepancies between BEM and FEM are 2% in the case of 2D calculations and 3% in the case of 3D problems.
For FEM applications, there are a few commercially available software packages like ANSYS (Ansoft Corporation Inc.) and NISA (Engineering Mechanics Research Centre). However, the electric field computations based on other methods like FDM, CSM and BEM generally require programmes to be developed individually by the user.