Finite Elements in Electromagnetics 1. Introduction Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria
Overview Maxwell‘s equations Boundary value problems for potentials Nodal finite elements Edge finite elements
Maxwell‘s equations
Potentials Continuous functions Satisfy second order differential equations Neumann and Dirichlet boundary conditions E.g. magnetic vector and electric scalar potential (A,V formulation):
Differential equations in a closed domain
Dirichlet boundary conditions Prescription of tangential E (and normal B) on E : n is the outer unit normal at the boundary n E B
Neumann boundary conditions Prescription of tangential H (and normal J+J D ) on H : n H J+JDJ+JD
General boundary value problem Differential equation: Boundary conditions: Dirichlet BC Neumann BC
Nonhomogeneous Dirichlet boundary conditions
Formulation as an operator equation (1) Characteristic function of a domain Dirac function of a surface Scalar product for ordinary functions:
Formulation as an operator equation (2) Define the operators A, B and C as (with the definition set Equivalent operator equation:
Formulation as an operator equation (3) Properties of the operators: Symmetry: Positive property:
Operators of the A,V formulation (1)
A,V formulation: symmetry of A
A,V formulation: positive property of A
A,V formulation: symmetry of B and C
Weak form of the operator equation
Galerkin’s method: discrete counterpart of the weak form Set of ordinary differential equations
Galerkin equations [A] is a symmetric positive matrix [B] and [C] are symmetric matrices
Finite element discretization
Nodal finite elements (1) i = 1, 2,..., n n Shape functions:
Nodal finite elements (2) Shape functions Corner nodeMidside node
Nodal finite elements (3) Basis functions for scalar quantities (e.g. V): Shape functions Number of nodes: n n, number of nodes on D : n Dn nodes on D : n+1, n+2,..., n n
Nodal finite elements (4) Linear independence of nodal shape functions Taking the gradient: The number of linearly independent gradients of the shape functions is n n -1 (tree edges)
Edge finite elements (1) Edge basis functions: i = 1, 2,..., n e
Edge finite elements (2) Basis functions Side edgeAcross edge
Edge finite elements (3) Basis functions for vector intensities (e.g. A): Edge basis functions Number of edges: n e, number of edges on D : n De edges on D : n+1, n+2,..., n e
Edge finite elements (4) Linear independence of edge basis functions Taking the curl: The number of linearly independent curls of the edge basis functions is n e -(n n -1) (co-tree edges) i=1,2,...,n n -1.