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Unformatted text preview: Chapter 8 Vector Finite Element Method Previously, 1-D and 2-D scalar finite-element method has been discussed for electromagnetic fields where there is only one non-zero field component in the governing equations. For example, in the 1-D case, the EM fields are TEM, so the only non-zero components are ( E y , H z ) or the uncoupled counterpart ( E z , H y ), assuming that the medium is only a function of x . Similarly, in the 2-D TM z case, E = ˆ zE z , thus the scalar FEM can be used for the electric field; in the 2-D TE z case, H = ˆ zH z , thus the scalar FEM can be for the magnetic field. In this chapter, we will discuss the vector FEM for vectorial electromagnetic fields. Examples for such fields include electric field and magnetic field in the general 3-D case, magnetic field H = ˆ xH x + ˆ yH y in the 2-D TM z case, and electric field E = ˆ xE x + ˆ yE y in the 2-D TE z case. In such cases, the fields have more than one non-zero components, thus the scalar basis functions are not appropriate because the basis functions used in scalar basis functions do not satisfy some basic properties of the vectorial fields, as discussed in more detail in the first section. For these vectorial fields, a new set of basis functions, namely the curl-conforming basis functions, will be used to represent E and H since they satisfy the boundary condition that the tangential components are continuous between adjacent elements. Such basis functions also facilitate the outer boundary conditions on the tangential components of the fields. 8.1 Weak-Form Vector Helmholtz Equations The general 3-D weak-form equations have been given in Chapter 5. Here we first summarize these equations for the 3-D case, and then specialize them to 2-D TM and TE cases. 8.1.1 3-D Weak-Form Equations For a 3-D anisotropic medium, Maxwell’s equations are-∇ × ( μ- 1 r ∇ × E ) + k 2 ² r E = S e (8.1)-∇ × ( ²- 1 r ∇ × H ) + k 2 μ r H = S h (8.2) where the source terms are S e = jωμ J + ∇ × ( μ- 1 r M ) S h = jω² M- ∇ × ( ²- 1 r J ) 175 CHAPTER 8. VECTOR FINITE ELEMENT METHOD We have derived the weak form of Maxwell’s equations for anisotropic media as 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 (8.3) 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 V w m · (ˆ n × E ) ds + Z V w m · S h dv (8.4) where w m is the testing function. These equations must be solved in conjuction with the outer boundary conditions. Note that for both PEC and PMC outer boundary conditions (including PML backed by outer PEC or PMC boundary conditions), the surface integral terms in the above two equations are equal to zero. For the special case of the PML for an isotropic medium, we have ² r = ² r Λ , μ r = μ r Λ , Λ = Diag( e...
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- Spring '09
- Electromagnet, Covariance and contravariance of vectors, Wm, Vector field, Dirichlet boundary condition, basis functions, H. Liu