Chapter 2. Input File Creation

In this chapter, we will study in detail the input file format and defining a physical problem to be solved by pdnmesh. I must apologise that I use a description and notation heavily biased to electrical engineering problems. But if you know what you are doing, you can adapt any problem to be solvable by pdnMesh.

2.1. Typical problems

It should be noted that (currently) only 2 Dimensional plane problems can be solved. Typical physical problems are:

The Poisson equation in rectangular coordinates can be written as:

Equation 2-1. Poisson Equation

x and y are the rectangular coordinates. phi is a scalar, generally called the potential. rho is also a scalar, dependent on x and y. epsilon is dependent on the medium properties and energy sources available. For instance, in Electrostatics, rho would actually mean the (negative) charge density and epsilon will be the permittivity of the medium. The first step should be to convert the PDE to be solved into above form.

The scalar, homogeneous Helmholtz equation, or Wave equation, can be written as:

Equation 2-2. Helmholtz Equation

This is an eigenvalue problem because phi appears in the derivative as well as in the stand alone term. We do not know the value of k in this equation as well. We can find possible values for k by solving an eigenvalue problem. The eigenvalues or k gives the cutoff wave number. The eigenvectors will give us the values for phi. The smallest k will give us the dominant mode.

Note that the Poisson's equation can be considered a special case of the Helmholtz equation. In the next sections, we will see how to prepare a problem suitable for input to pdnMesh.