Proceedings - 49 - 12th International IGTE Symposium Perfectly Matched Layers for T, Φ Formulation Gernot Matzenauer, Oszkar´ B´ıro,´ Karl Hollaus, Kurt Preis and Werner Renhart Institute for Fundamentals and Theory in Electrical Engineering, Kopernikusgasse 24/3, A-8010 Graz, Austria E-mail: [email protected] Abstract: Perfectly Matched Layers (PMLs) are used for II. PROOF OF THE PERFECT MATCHING PROPERTY reflectionless truncation of the problem boundaries in FEM applications. The basic concept behind PMLs is to create an Based on [2] it is shown below that the PML interface artificial material with a complex and diagonal anisotropic also works with any lossless material in front of the PML. permittivity and permeability. For the A,V formulation Similarly to (2), the permittivity and permeability tensors PMLs are well known. In this paper the method of perfectly of the PML are assumed to be: matched layers is extended to truncate any lossless medium and the method is implemented for the T, Φ formulation. [] = 0r[Λ], Keywords: Perfectly Matched Layer, Finite Element Method, T, Φ formulation, Perfectly Matched Anisotropic [µ] = µ0µr[Λ]. (4) Absorber, Absorbing Boundary Condition The question is how to choose the parameters a, b, c, µr and r, so that no reflection occurs on an interface between I. INTRODUCTION ∗ the PML and a medium with permeability µr and permit- ∗ In the present paper, an artificial anisotropic lossy tivity r. The wave equations for the electric and the mag- material has been applied to a 3D edge finite-element T, Φ netic field in the PML are: formulation [1] to form perfectly matched layers. This −1 −1 2 material fully absorbs the electromagnetic field impinging [Λ] ∇ × [Λ] ∇ × E − ω µ0 µr0 rE = 0, on it without reflections on the PML-air interface [2]. −1 −1 2 [Λ] ∇ × [Λ] ∇ × H − ω µ0 µr0 rH = 0. The permittivity [] and the permeability [µ] in PMLs (5) are considered as complex diagonal tensors. The following General solutions for the electric and the magnetic field condition is required to match the intrinsic impedances of can be constructed from plane waves of the form vacuum and the PML[2]: j(ωt−kr) E = E0 e µ0 −1 [1] = [µ][] (1) j(ωt−kr) 0 H = H0 e (6) where [1] is the unit tensor. Thus, the permittivity [] and where k = kx ex + ky ey + kz ez , r = x ex + y ey + the permeability [µ] can be written as z ez and E0 and H0 are constant vectors. The dispersion relation of the PML leads to an equation of the form: a 0 0 k2 k2 k2 [] = 0 b 0 = [Λ], x y z 2 0 0 + + = k0. (7) 0 0 c bc ac ab where a 0 0 2 2 k0 = ω µ0 µr 0 r (8) [µ] = µ0 0 b 0 = µ0[Λ]. (2) 0 0 c is the propagation constant. The solution of the dispersion relation is: Here a, b and c are complex numbers describing the mate- √ rial properties of the anisotropic absorber. Consequently, kx = k0 bc sin θ cos φ Maxwell’s equations in the PML are √ ky = k0 ac sin θ sin φ ∇ × H = jω0[Λ]E √ kz = k0 ab cos θ (9) ∇ × E = −jωµ0[Λ]H To find out which component of the electric field belongs ∇ · [Λ]E = 0 to which component of the magnetic field, the solutions 0 for the electric and the magnetic field (6) are inserted into Maxwell’s equations. Using Faraday’s law yields ∇ · µ0[Λ]H = 0. (3) r ! 1 r c b The Maxwell equations are not modified with this ap- Hx = Ez sin θ sin φ − Ey cos θ , (10) proach and the implementation is straightforward in exist- Z0 a a ing solvers with finite element methods (FEM). It is suffi- 1 ra rc cient for the code to permit complex anisotropic materials. Hy = Ex cos θ − Ez sin θ cos φ , In this paper the method of perfectly matched layers is ex- Z0 b b tended to truncate any lossless medium and the method is r r ! implemented for the T, Φ formulation. The results are il- 1 b a Hz = Ey sin θ cos φ − Ex sin θ sin φ . lustrated by two numerical examples. Z0 c c 12th International IGTE Symposium - 50 - Proceedings Using Ampere’s law leads to r ! b r c E = Z H cos θ − H sin θ sin φ , (11) x 0 y a z a rc ra E = Z H sin θ cos φ − H cos θ , y 0 z b x b r ! ra b E = Z H sin θ sin φ − H sin θ cos φ . z 0 x c y c where r µ0 µr Z0 = (12) 0 r Figure 1. Plane wave in TE mode incidence on a z-plane interface. is the wave impedance of the PML. Let us consider a plane wave incident on a PML as shown is the propagation vector and re and rr are the directions of in Fig. 1. The values of a, b, c, µr and r are to be deter- the exciting and the reflected wave. The transmitted wave mined so that independent of the incident angle αe no re- is given by: flection occurs on the interface. The reflection coefficient −j k·rt r and the transmission coefficient t are defined by: Ety = E0t e (16) r 1 b E −j k·rt r = 0r Htx = − E0t cos αt e Z0 a E0e r 1 b E0t −j k·rt t = (13) Htz = E0t sin αt e E0e Z0 c √ √ k · rt = k0(x bc sin αt + z ab cos αt) where E0e, E0r and E0t are the magnitudes of the excit- ing, the reflected and the transmitted electric field. The where r is the direction of the transmitted wave. Here Z∗ condition for no reflection is that the reflection coefficient t 0 is the wave impedance of the lossless linear space: vanishes r = 0 and the transmission coefficient is t = 1. The propagation vectors are restricted to the x − z-plane. s µ µ∗ For the TE mode, the excited and the reflected waves are Z∗ = 0 r . (17) 0 ∗ given by: 0 r Enforcing the tangential continuity of E on the interface at −j k·re Eey = E0e e (14) the position z = 0 yields the relation: √ 1 −jk∗ x sin α −jk∗ x sin α −jk x bc sin α −j k·re e 0 e + r e 0 r = t e 0 t . (18) Hex = − ∗ E0e cos αe e Z0 The phase matching condition of the tangential component 1 −j k·re of the electric field is given by: Hez = ∗ E0e sin αe e Z0 √ ∗ ∗ k0 sin αe = k0 sin αr = k0 bc sin αt. (19) k · re = k0(x sin αe + z cos αe) Provided −j k·rr Ery = E0r e √ bc = 1 (20) 1 −j k·rr Hrx = ∗ E0r cos αr e Z0 and 1 ∗ ∗ −j k·rr µr r = µr r, (21) Hrz = ∗ E0r sin αr e Z0 ∗ k · rr = k0 (x sin αr − z cos αr) αe = αr = αt (22) is obtained, i.e. no reflection occurs. Using (19), (18) be- where comes ∗2 2 ∗ ∗ k0 = ω µ0 µr 0 r (15) 1 + r = t. (23) Proceedings - 51 - 12th International IGTE Symposium Enforcing the tangential continuity of H and using (19) potential) realized by edge and nodal finite elements [1]. leads to Furthermore, the properties of the PML layer have been ∗ r extended to truncate any lossless linear material. Z0 b cos αe(1 − r) = t cos αt. (24) Z0 a The field quantities for the T, Φ formulation are derived The magnitude matching equations yield the reflection co- from the potentials as efficient H = T − gradΦ, ∗ q Z0 b cos αe − Z a cos αt J = curlT (32) r = 0 . (25) TE ∗ q Z0 b cos αe + cos αt Z0 a and satisfy the constitutive relations: B = [µ]H, D = []E and E = [ρ]J where [µ],[] and [ρ] are the tensors of For the TM mode the same procedure can be followed to permeability, permittivity and resistivity. The permeability determine the reflection coefficient: and conductivity may be written in complex form: ∗ q Z0 b Z a cos αt − cos αe [σ] = [σ ] + jω[], r = 0 (26) E TM ∗ q Z0 b cos αt + cos αe [σ ] Z0 a [µ] = M + [µ]. (33) jω Using the requirement (22) the reflection coefficient is in- dependent of the angle. By setting A magnetic conductivity tensor [σM ] is introduced to de- scribe the material properties of the PML. The complex a = b (27) material tensor for the electric resistivity can be obtained from and −1 ∗ [ρ] = ([σ]) . (34) Z0 = Z0 (28) the interface will be perfectly reflectionless for any fre- The tensors of permeability and resistivity have to be quency and angle of incidence. complex in the whole region to imply wave propagation with the T, Φ formulation. In summary, (20) and (27) lead to The governing differential equations for the T, Φ formu- 1 a = b = (29) lation are obtained from Faraday’s law and from the van- c ishing divergence of the magnetic induction: and the relations (21) and (28) yield ∇ × [ρ] ∇ × T + j ω [µ]T − j ω [µ]∇Φ = 0, ∗ r = r, ∇ · [µ]T − [µ]∇Φ = 0. (35) ∗ µr = µr. (30) The material properties of a perfectly matched layer can be Any arbitrarily polarized plane wave can be decomposed obtained from the complex PML tensor [Λ]: into a linear combination of TE and TM modes. So under the conditions (29) and (30) the PML will work with any e(1 − j) 0 0 polarization.