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

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.