Finite Elements in Electromagnetics 1. Introduction

1. Finite Elements in Electromagnetics1. 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 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 J+JD) on GH: H J+JD n

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

9. Nonhomogeneous Dirichlet boundary conditions

10. Scalar product for ordinary functions: Characteristic function of a domain W: QW Dirac function of a surface G: dG 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. 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) 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: n+1, n+2, ..., 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: n+1, n+2, ..., ne

28. Edge finite elements (4) Linear independence of edge basis functions i=1,2,...,nn-1. Taking the curl: i=1,2,...,nn-1. The number of linearly independent curls of the edge basis functions is ne-(nn-1) (co-tree edges)