Finite Elements in Electromagnetics 1' Introduction

1
Finite Elements in Electromagnetics1.
Introduction

• Oszkár Bíró
• IGTE, TU Graz
• Kopernikusgasse 24, Graz, Austria
• email biro_at_igte.tu-graz.ac.at

2
Overview

• Maxwells equations
• Boundary value problems for potentials
• Nodal finite elements
• Edge finite elements

3
Maxwells equations
4
Potentials

• Continuous functions
• Satisfy second order differential equations
• Neumann and Dirichlet boundary conditions

E.g. magnetic vector and electric scalar
potential (A,V formulation)
5
Differential equations
in a closed domain W
6
Dirichlet boundary conditions
Prescription of tangential E (and normal B) on
GE
E
B
n
n is the outer unit normal at the boundary
7
Neumann boundary conditions
Prescription of tangential H (and normal JJD)
on GH
H
JJD
n
8
General boundary value problem
Differential equation
Boundary conditions
Dirichlet BC
Neumann BC
9
Nonhomogeneous Dirichlet boundary conditions
10
Formulation as an operator equation (1)
11
Formulation as an operator equation (2)
Define the operators A, B and C as
(with the definition set
Equivalent operator equation
12
Formulation as an operator equation (3)
Properties of the operators Symmetry
Positive property
13
Operators of the A,V formulation (1)
14
A,V formulation symmetry of A
15
A,V formulation positive property of A
16
A,V formulation symmetry of B and C
17
Weak form of the operator equation
18
Galerkins methoddiscrete counterpart of the
weak form
Set of ordinary differential equations
19
Galerkin equations
A is a symmetric positive matrix
B and C are symmetric matrices
20
Finite element discretization
21
Nodal finite elements (1)
Shape functions
i 1, 2, ..., nn
22
Nodal finite elements (2)
Shape functions
Corner node
Midside node
23
Nodal finite elements (3)
Basis functions for scalar quantities (e.g. V)
Shape functions
Number of nodes nn, number of nodes on GD nDn
nodes on GD n1, n2, ..., nn
24
Nodal finite elements (4)
Linear independence of nodal shape functions
Taking the gradient
The number of linearly independent gradients of
the shape functions is nn-1 (tree edges)
25
Edge finite elements (1)
Edge basis functions
i 1, 2, ..., ne
26
Edge finite elements (2)
Basis functions
Side edge
Across edge
27
Edge finite elements (3)
Basis functions for vector intensities (e.g. A)
Edge basis functions
Number of edges ne, number of edges on GD nDe
edges on GD n1, n2, ..., ne
28
Edge finite elements (4)
Linear independence of edge basis functions
i1,2,...,nn-1.
Taking the curl
i1,2,...,nn-1.
The number of linearly independent curls of the
edge basis functions is ne-(nn-1) (co-tree edges)