Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 67

literature, these solutions are referred to as fundamental solutions , Green's functions or
Kernels . Obviously, these solutions can only be found for linear material behaviour and
for a homogeneous domain.
The fundamental solutions have to satisfy three conditions:

x Constitutive law
x Equilibrium or conservation of energy
x Compatibility or continuity
The last condition will be automatically satisfied for solutions which are continuous
in the domain. In the following, we will first derive the governing differential equations
and then present fundamental solutions for potential problems (heat flow and seepage)
and for elasticity problems in two and three dimensions.

4.2. STEADY STATE POTENTIAL PROBLEMS

Heat conduction in solids and flow in porous media (seepage) are diffusion problems
and can be treated concurrently, because they are governed by the same differential
equation (Laplace).
Steady state heat flux or fluid flow q per unit area is related to temperature or
potential u by

q D u

(4.1)

where the negative sign is due to the fact that the flow is always from higher to lower
temperature/potential. The flow vector is defined as:

- q ½

x

® ¾

q q (4.2)

y

¯ ¿

q

z

The conductivity/pemeabilty tensor D is given by

ª k k k º

xx xy xz

« »

D k k k

(4.3)

yx yy yz

« »

k k k

¬ ¼

zx zy zz

where k xx etc, are conductivities measured in [W/°K-m] in the case of thermal problems
and permeabilities measured in [m/sec], in the case of seepage problems. The
coefficients in D represent flow values for unit values of temperature gradient or
potential gradient. It can be shown that D has to be symmetric and positive definite.