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Unformatted text preview: Chapter 5 2-D Finite Element Method In the previous chapter, scalar volume integral equations have been solved by using the method of moments. In this chapter, we will focus on the finite element method for 2-D scalar problems arising in electromagnetics. 5.1 Weak Form of Helmholtz Equations for Anisotropic Media in 3D Here we will formulate the weak form of Helmholtz equations for anisotropic media. There are two main reasons for treating anisotropic media: (a) there are some natural materials whose permittivity and perme- ability are tensor, i.e., the materials behave differently in different direction; (b) to formulate the absorbing boundary condition to truncate the infinite physical domain into a finite computational domain, we may need to use the perfectly matched layer (PML), which is an anisotropic medium. For an anisotropic medium, Maxwell’s equations can be reformulated into vector Helmholtz equations-∇ × ( μ- 1 r ∇ × E ) + k 2 ² r E = jωμ J + ∇ × μ- 1 r M ≡ S e (5.1)-∇ × ( ²- 1 r ∇ × H ) + k 2 μ r H = jω² M- ∇ × ²- 1 r J ≡ S h (5.2) where ² r and μ r are relative permittivity and permeability tensors for the anisotropic medium, and the source terms are S e = jωμ J + ∇ × ( μ- 1 r M ) S h = jω² M- ∇ × ( ²- 1 r J ) We can now test equations (5.1) and (5.2) by a testing function w m ( r ). Note that w m ( r ) should have the same properties as the unknown fields; for example, if ˆ n × E = 0 on the outer boundary, we require ˆ n × w m ( r ) = 0 also on the boundary. By performing the dot-product of w m ( r ) with (5.1) and (5.2) and integrating over volume V in Figure 5.1, and using ∇ · ( A × B ) = B · ∇ × A- A · ∇ × B and identifying A = μ- 1 r ( ∇ × E ) and B = w m , we can derive the weak forms Z V [- ( ∇ × w m ) · μ- 1 r ( ∇ × E ) + k 2 w m · ² r E ] dv = Z S w m · [ˆ n × μ- 1 r ( ∇ × E )] ds + Z V w m · S e dv =- jωμ Z S w m · (ˆ n × H ) ds + Z V w m · S e dv (5.3) 89 CHAPTER 5. 2-D FINITE ELEMENT METHOD n S V Figure 5.1: A computational domain V with an outer surface in the finite-element method. Z V [- ( ∇ × w m ) · ²- 1 r ( ∇ × H ) + k 2 w m · μ r H ] dv = Z S w m · ˆ n × ²- 1 r ( ∇ × H ) ds + Z V w m · S h dv = jω² Z S w m · (ˆ n × E ) ds + Z V w m · S h dv (5.4) where S is the outer boundary of volume V . In the above, we have made use of Gauss’s theorem Z V ∇ · ( A × B ) dv = Z S ˆ n · ( A × B ) ds = Z S B · (ˆ n × A ) ds (5.5) The above weak-form equations can be solved together with the appropriate outer boundary conditions. Outer Boundary Conditions Depending on the applications, various boundary conditions can be applied to these equations. For examples, 1). Dirichlet BC for the electric field (Neumann BC for the magnetic field in isotropic media) ˆ n × E = f ( t ) , r ∈ S (5.6) In this case, ˆ n × E (tangential component of E ) is specified on S and thus does not require an unknown on the boundary. In the special case whereon the boundary....
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- Fall '09
- Electromagnet, ez, Boundary conditions, Dirichlet boundary condition, Neumann boundary condition, H. Liu, 2-d finite element