24  Introduction

In the previous chapters we have studied solutions to the Poisson and Laplace equations for simple geometries with homogeneous density distributions.

The following chapters introduce techniques which will enable us to calculate potentials and their gradients in a subdomain \(\Omega\) that does not contain the sources.

However, we incorporate the effect of the sources by enforcing boundary conditions along the boundaries \(\partial\Omega\) of the subdomain.

The ultimate goal is to find a solution of the Poisson (or similar) PDE.

The Poisson equation

\[ \Delta U(\vb r) = f(\vb r) \]

with \(\vb r \in \Omega \subset \mathbb R^3\) has no unique solution, if no further conditions on \(U(\vb r)\) are enforced.

Example

The function \[ \Phi(x,y,z) = e^{\alpha x + \beta y} \sin(\sqrt{\alpha^2 + \beta^2}z) \]

is a solution of the PDE \(\Delta \Phi = 0\) for arbitrary choices of \(\alpha\) and \(\beta\).

Hence, the solution is non-unique.

However, the solution space can be restricted by proper choices of \(\alpha\) and \(\beta\).

Check using SymPy:

Show the code 
import sympy as sp
from IPython.display import display, Math
alpha, beta, x, y, z = sp.symbols('alpha beta x y z') 
Phi = sp.exp(alpha * x + beta * y) * sp.sin(sp.sqrt(alpha**2 + beta**2) * z)
display(Math('\\Phi = ' + sp.latex(Phi)))
Phi

\(\displaystyle \Phi = e^{\alpha x + \beta y} \sin{\left(z \sqrt{\alpha^{2} + \beta^{2}} \right)}\)

\(\displaystyle e^{\alpha x + \beta y} \sin{\left(z \sqrt{\alpha^{2} + \beta^{2}} \right)}\)

Show the code 
LPhi = sp.diff(Phi, x, 2) + sp.diff(Phi, y, 2) + sp.diff(Phi, z, 2)
display(Math('\\Delta\\Phi = ' + sp.latex(LPhi.simplify())))

\(\displaystyle \Delta\Phi = 0\)

24.1 Classification of Boundary Value Problems (BVP)

In potential theory we will study the following boundary value problems:

  • Dirichlet problem
  • Neumann problem
  • Robin or Mixed Dirichlet-Neumann problem
Typical problems
  • Point source within a conducting half-space
  • Point source embedded in one of two adjacent conducting half-spaces
  • Point source in the top layer of a 2-layer case
  • Potential of a conductive semi-sphere embedded in a homogeneous half-space
  • Horizontal cylinder in a homogeneous electrical field
  • DC resistivity sounding over a half-space consisting of \(N\) horizontal layers

For the solution of these problems, we will use the following tools:

  • Green’s function
  • Method of images
  • Finite Element Method

24.2 Notation

Domain: \(\Omega\)

Boundary: \(\partial\Omega\)

We distinguish between the open domain \(\Omega\) and its boundary \(\partial\Omega\).

The closed domain \(\overline\Omega\) is the union of the interior and the boundary.