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Finite Elements in Electromagnetics 1. Introduction Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria email: biro@igte.tu-graz.ac.at

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Overview Maxwell‘s equations Boundary value problems for potentials Nodal finite elements Edge finite elements

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Maxwell‘s equations

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Potentials Continuous functions Satisfy second order differential equations Neumann and Dirichlet boundary conditions E.g. magnetic vector and electric scalar potential (A,V formulation):

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Differential equations in a closed domain

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Dirichlet boundary conditions Prescription of tangential E (and normal B) on E : n is the outer unit normal at the boundary n E B

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Neumann boundary conditions Prescription of tangential H (and normal J+J D ) on H : n H J+JDJ+JD

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General boundary value problem Differential equation: Boundary conditions: Dirichlet BC Neumann BC

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Nonhomogeneous Dirichlet boundary conditions

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Formulation as an operator equation (1) Characteristic function of a domain Dirac function of a surface Scalar product for ordinary functions:

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Formulation as an operator equation (2) Define the operators A, B and C as (with the definition set Equivalent operator equation:

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Formulation as an operator equation (3) Properties of the operators: Symmetry: Positive property:

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Operators of the A,V formulation (1)

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A,V formulation: symmetry of A

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A,V formulation: positive property of A

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A,V formulation: symmetry of B and C

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Weak form of the operator equation

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Galerkin’s method: discrete counterpart of the weak form Set of ordinary differential equations

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Galerkin equations [A] is a symmetric positive matrix [B] and [C] are symmetric matrices

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Finite element discretization

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Nodal finite elements (1) i = 1, 2,..., n n Shape functions:

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Nodal finite elements (2) Shape functions Corner nodeMidside node

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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

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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)

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Edge finite elements (1) Edge basis functions: i = 1, 2,..., n e

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Edge finite elements (2) Basis functions Side edgeAcross edge

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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

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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.

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