1. Finite Elements in Electromagnetics 4. Wave problems
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24Graz, Austria

2. Overview Maxwell‘s equations Resonators Filters
Wave propagation in free space

3. Maxwell‘s equations
Time harmonic case

4. Resonators
GE:
GH:

5. Resonators, H-formulation
A problem without excitation:
Eigenvalue problem

6. Resonators, operator equation for H

7. Resonators, finite element Galerkin equations for H
Generalized algebraic eigenvalue problem

8. Filters
GH:
GE:
GH:

9. Filters, E-formulation
A problem with excitation:
Driven problem

10. Filters, operator equation for E

11. Filters, finite element Galerkin equations for E
conditioning of [A] strongly depends on frequency

12. Filters, A,V-formulation

13. Filters, operator equation for A,V

14. Filters, finite element Galerkin equations for A,V (1)

15. Filters, finite element Galerkin equations for A,V (2)
i = 1, 2, ..., n(n)

16. Filters, finite element Galerkin equations for A,V (3)
[A] is singular and its conditioning depends less on frequency
R. Dyczij-Edlinger and O. Biro, "A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements," IEEE Transactions on Microwave Theory and Techniques, vol. 44, pp , January 1996.

17. Wave propagation in free space
Finite element method needs closed domain
Modeling of infinite space necessary
Perfectly matched layers (PMLs)
PML

18. PMLs Nonphysical material properties z x y
No reflection on the interface between air and PML