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Finite Elements in Electromagnetics 1. Introduction Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria

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Presentation on theme: "Finite Elements in Electromagnetics 1. Introduction Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria"— Presentation transcript:

1 Finite Elements in Electromagnetics 1. Introduction Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria

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

3 Maxwell‘s 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 

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

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

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) Characteristic function of a domain   Dirac function of a surface   Scalar product for ordinary functions:

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 Galerkin’s method: discrete 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) i = 1, 2,..., n n Shape functions:

22 Nodal finite elements (2) Shape functions Corner nodeMidside node

23 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

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 n n -1 (tree edges)

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

26 Edge finite elements (2) Basis functions Side edgeAcross edge

27 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

28 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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