Elementary Substructures
Almost any Finite Element book starts with the assembly of resistor-like finite
elements, without approximations (if you consider Ohm's law as being "exact").
Contained in [ NdV ] is a chapter about "Electrical Networks".
The matrix of an electrical resistor is derived directly, by applying the laws of Ohm
and Kirchhoff, giving:
$$
\left[ \begin{array}{cc} +1/R & -1/R \\
-1/R & +1/R \end{array} \right]
$$
where $R$ are the resistances. Defining admittances $A$ instead of resistances
$R$ turns out to be more convenient in this context, the relationship between
the two being simply $A = 1/R$. The finite element matrix of a resistor is then
given by:
$$
\left[ \begin{array}{cc} +A & -A \\
-A & +A \end{array} \right]
$$
Further define the connectivity (no coordinates!) of the resistor-network, and
apply two voltages. The standard FE assembly procedure can be carried out then
in a straightforward manner.
Confusion in 2-D
The purpose of this study is to re-establish some definite relationships
between Finite Difference and Finite Element Methods. As such, it may be
considered as a continuation of my 'Series on Unified Numerical
Approximations' (SUNA). A new result is the generalization of Patankar's
F.V. schemes for CONvection and difFUSION (also called confusion),
for meshes consisting of F.E. triangles.
- Triangle Algebra
- Conservation of Heat
- 2-D Resistor Model
- Voronoi Regions
Diffusion in 3-D
- Linear Tetrahedron
- FEM discretization
- Resistor model
Bibliography
- [NdV] Norrie D.H. and de Vries G.,
"An Introduction to Finite Element Analysis", Acad. Press 1978.
- [OC]
O.C. Zienkiewicz, "The Finite Element Method", 3th edition,
Mc.Graw-Hill U.K. 1977.
- [SV]
S.V. Patankar,
"Numerical Heat Transfer and Fluid Flow",
Hemisphere PublishingCompany U.S.A. 1980.