HIGHER ORDER FINITE ELEMENT METHOD FOR INHOMOGENEOUS AXISYMMETRIC RESONATORS

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1 Pogess In Electomagnetics Reseach B, Vol. 21, , 2010 HIGHER ORDER FINITE ELEMENT METHOD FOR INHOMOGENEOUS AXISYMMETRIC RESONATORS X. Rui and J. Hu School of Electonic Engineeing Univesity of Electonic Science and Technology of China Chengdu, Sichuan , China Q. H. Liu Depatment of Electical and Compute Engineeing Duke Univesity Duham, NC 27708, USA Abstact To analyze esonances in an axisymmetic inhomogeneous cavity, a highe-ode finite element method (FEM) is developed. Mixed highe-ode node-based and edge-based elements ae applied to eigenvalue analysis fo the azimuthal component and meidian components of the field, espectively. Compaed with the lowe-ode FEM, the highe-ode FEM can impove accuacy with the same numbe of unknowns and can educe the CPU time and memoy equiement fo specified accuacy. Numeical esults ae given to demonstate the validity and efficiency of the poposed method. 1. INTRODUCTION Electomagnetic computation fo bodies of evolution (BOR) of abitay shape has been widely discussed fo many yeas. BOR objects of vaious types, including pefect electic conductos (PEC), homogeneous dielectic bodies, coated conducting bodies, combined dielectic and conducting bodies, multi-laye dielectic bodies and esonatos with axial symmety, have been studied [1 14]. Because of the axis-symmety of the geomety, only the 2-D coss section that foms the volume (i.e., the meidian plane) is needed fo solving the Coesponding autho: Q. H. Liu (qhliu@ee.duke.edu). Also with Depatment of Electical and Compute Engineeing, Duke Univesity, Duham, NC 27708, USA

2 190 Rui, Hu, and Liu scatteing, adiation and esonance poblems of the BOR by integal equations (IE) [1 5], finite element method (FEM) [12 14], hybid finite element method and bounday integation (FEM-BI) [15], hybid physical optics and method of moments (PO-MoM) [16], and the othe methods. Both the memoy equiement and CPU time ae educed compaed with the full thee-dimensional methods [17, 19]. In a BOR solution method, the fields and cuents ae decomposed into diffeent cylindical hamonic modes. These modes ae othogonal with each othe, so they can be solved sepaately. In a numeical method fo BORs, only the geneatix that foms the suface of the BOR is needed to solve the poblem. Fo the scatteing and adiation poblem fom a PEC o homogeneous BOR, it is convenient to use integal equations without the need to use any absobing bounday condition (ABC). The accuacy is bette than the FEM. Howeve, an suface integal equation solve cannot handle an inhomogeneous BOR poblem efficiently. The FEM, on the othe hand, can pefom bette fo a complex medium [21, 22] and is a good choice especially fo inhomogeneous axisymmetic cavity poblems with a PEC bounday. In this wok, we extend the highe-ode FEM [19, 23] to impove the accuacy fo the computation of inhomogeneous axisymmetic cavity poblems. Although the highe-ode FEM has been applied to 3-D poblems, the new contibution of this wok is that it develops and applies this method to BOR cavities. Both node- and edge-based elements ae used to discetize the azimuthal and meidian components of the field. Numeical esults ae given to demonstate the validity and efficiency of the highe-ode FEM. 2. FINITE ELEMENT METHOD FOR AXISYMMETRIC RESONATORS As shown in Figue 1, a BOR is in the cylindical coodinate system (ρ, φ, z), and the medium inside the body is inhomogeneous. Because of the axial symmety of the geomety, the field inside can be expessed in a Fouie seies [1] E(ρ, φ, z) = [E t,m (ρ, z) + ˆφE φ,m (ρ, z)]e jmφ (1) H(ρ, φ, z) = m= m= [H t,m (ρ, z) + ˆφH φ,m (ρ, z)]e jmφ (2) whee E t,m, E φ,m, H t,m and H φ,m ae the electic and magnetic fields in the meidian plane and the azimuthal component of the m-th Fouie mode, espectively. Only a 2-D mesh is needed fo analyzing the 3-D

3 Pogess In Electomagnetics Reseach B, Vol. 21, z X S ε(ρ,z) µ(ρ,z) φ ρ Y Figue 1. An abitay inhomogeneous axisymmetic cavity and the coodinate system. axisymmetic cavity. As all cylindical modes ae othogonal to each othe, they can be teated sepaately. On the axis (ρ = 0), the fields must etain the continuity fo any values of φ [13, 21, 22]. Thus, thee ae thee kinds of conditions fo diffeent cylindical modes: fo m = 0, fo m = ±1 and fo m > 1 E φ,0 = ( E) φ,0 = 0 (3) E ρ,0 = ( E) ρ,0 = 0 (4) E z,0 0, ( E) z,0 0 (5) E ρ,±1 = je φ,±1 (6) ( E) ρ,±1 = j( E) φ,±1 (7) E z,±1 = ( E) z,±1 = 0 (8) E φ,m = E ρ,m = E z,m = 0 (9) ( E) φ,m = ( E) ρ,m = ( E) z,m = 0 (10) Conside the vecto Helmholtz equation fo the electic field E ( µ 1 E ) + k0ɛ 2 E = jωµ 0 J + µ 1 M S e (11) whee µ and ɛ ae the elative complex pemeability and pemittivity, espectively. J and M ae the electic and magnetic cuent densities, espectively, and k 0 is the wave numbe in fee space. Fo a souce-fee

4 192 Rui, Hu, and Liu (S e = 0) cavity Ω with a pefect electic conducto oute bounday S, the weak fom of the vecto wave equation can be expessed as ( W l ) (µ 1 E)dΩ + k0w 2 l ɛ EdΩ = 0 (12) Ω whee W l is the testing function, and the oute bounday condition Ω ˆn E() = 0, S (13) has been utilized, whee S is the bounday of the cavity Ω, and ˆn is the unit outwad nomal vecto. In the geneal 3-D FEM, the basis functions fo electomagnetic fields ae the mixed-ode edge elements. Howeve, fo the BOR FEM, when pojected onto a meidian coss section, the azimuthal component E φ should use a nodal basis function, while the meidian components E t should use a mixed-ode edge element. Consideing the bounday conditions at ρ = 0 fo diffeent cylindical modes as shown in Equations (3) (10), the m-th cylindical mode can be expanded as N n E φ,m = e φ,i N i (14) i=1 N s N s N n j E t,m =δ m,0 e t,i N i +(1 δ m,0 ) e t,i ρn i (1 δ m,0 )ˆρ e φ,i m N i(15) i=1 i=1 whee N n is the numbe of nodes. N s is the numbe of segments (o edges), e φ,i. e t,i ae the unknown coefficients. N i and N i epesent the standad node-based and edge-based element basis functions, espectively. Note that because of the Konecke delta function, fo m = 0 the second and thid tems in E t,m become zeo. And the testing function is chosen as: ( Nn ) N s W l = N i ˆφ + N i e jlφ = W φ,l ˆφ + Wt,l (16) i=1 i=1 Afte substituting the basis and testing functions into Equation (12) and making use of the othogonality of cylindical modes, the wave equation can be ewitten as ρ 2π ( t W t, m ) ( t E t,m ) ds S µ ( ρ +2π t W φ, m + jm ρ W t, m + ˆρ ) ρ W φ, m S µ i=1

5 Pogess In Electomagnetics Reseach B, Vol. 21, ( t E φ,m + jm ρ E t,m + ˆρ ) ρ E φ,m ds = 2π k0ɛ 2 ρ (W t, m E t,m + W φ, m E φ,m ) ds (17) a system of equations [ K ee m Km ne S ] Km en Km nn [ ] E e E n = k0 2 [ M ee m Mm ne Mm en Mm nn ] [ ] E e E n (18) can be fomed, whee in the stiffness matix element Km αβ and mass matix element Mm αβ, α and β ae the choices fo the testing function and the basis function. m is the the mode index. e stands fo the edge unknowns. n stands fo node-based unknowns. k0 2 is the eigenvalue of the system, E e = (e t,1, e t,2,, e t,ns ) T and E n = (e φ,1, e φ,2,, e φ,nn ) T. Both the stiffness matix and mass matix ae spase. A seven-point numeical integation is used fo the impedance matix assembling. LAPACK outines ae used to compute eigenvalues and eigenvectos [18]. 3. HIGHER-ORDER FINITE ELEMENT METHOD In ode to impove the accuacy fom the conventional FEM with fist-ode nodal basis functions and fist-ode edge elements, a highe-ode FEM is applied to axisymmetic inhomogeneous dielectic esonatos. As shown in Figue 2(a), each tiangula element on the meidian plane has thee node-based and thee edge-based vecto basis functions fo the conventional lowe-ode basis functions. The thee (a) (b) (c) Figue 2. (a) Thee-noded tiangle fo fist-ode node-based and edge-based basis functions. (b) Six-noded tiangle fo second-ode node-based and edge-based basis functions. (c) Ten-noded tiangle fo thid-ode node-based and edge-based basis functions. 5 2

6 194 Rui, Hu, and Liu nodal basis functions ae given by N e i = L e i, i = 1, 2, 3 (19) whee L e i denote the aea coodinates fo element e, and i is the index of the node. The thee edge-based basis functions can be expessed as N e ij = l e ij (L e i L e j L e j L e i ), i, j = 1, 2, 3, (i j) (20) whee lij e is the length of the edge fom node i to node j. And the vecto basis functions have the following popeties: N e ij = 0, N e ij = 2l e ij L e i L e j (21) The second-ode node-based and edge-based basis functions ae defined on a tiangula element with six nodes [19] in Catesian coodinates and ae adapted hee to the BOR poblem: thee of them ae cone nodes, and the est thee nodes ae mid-side nodes. As shown in Figue 2(b), each element has six nodal basis functions and eight edge-based basis functions. Thee of the nodal basis functions ae defined on the thee cone nodes, while the othe thee nodal basis functions ae defined on the thee mid-side nodes. Six of the vecto edge-based basis functions ae defined at the edges of the tiangula element, while the est two edge-based basis functions ae defined inside the tiangula element. The thid-ode node-based and edge-based basis functions ae defined on a tiangula element with ten nodes [20] in Catesian coodinates and ae adapted hee to the BOR poblem: As shown in Figue 2(c), each element has ten nodal basis functions and fifteen edgebased basis functions. Thee of the nodal basis functions ae defined on the thee cone nodes. Six nodal basis functions ae defined on the six side nodes, and the last one is defined inside the tiangula element. Nine of the vecto edge-based basis functions ae defined at the edges of the tiangula element, while the est six edge-based basis functions ae defined inside the tiangula element. 4. NUMERICAL RESULTS In this section, seveal numeical esults ae pesented to show the validity of the poposed method An Inhomogeneous BOR Cavity Lebaic and Kajfez [24] use the finite integation technique to analyze the cavity shown in Figue 3. The mesh of the poposed method contains 307 nodes and 555 tiangula elements. Table 1 lists the compaison of the esonant fequencies by measuement and two

7 Pogess In Electomagnetics Reseach B, Vol. 21, z PEC mm mm ε = 37.3 µ = mm 2.989mm ε = 3.78 µ = mm ε = 1 µ = 1 Figue 3. Kajfez [24] mm The stuctue of the cavity investigated by Lebaic and Table 1. Compaison of the esonant fequencies fo the cavity investigated by Lebaic and Kajfez [24], and the eo of diffeent method with measued data. Mode Measued Lebaic Relative This wok Relative [24] (GHz) [24] (GHz) Eo (%) (GHz) Eo (%) TE HEM HEM TM HEM TM HEM TE numeical methods. The fist subscipt epesents the azimuthal mode m, and the second subscipt epesents the index fo the esonance fequencies fo mode m. The elative eo is defined by eo = f 0 f Ref 0 2 f Ref 0 2 (22)

8 196 Rui, Hu, and Liu whee f Ref 0 is the measued fequency, and f 0 is the esonance fequency obtained by diffeent numeical methods Convegence of the Highe-ode BOR FEM To study the convegence of the highe-ode BOR FEM, a homogeneous PEC cavity filled with ai (ε = 1, µ = 1) is consideed. The cavity has a adius 1 m and a height 1 m. Thus, analytical esults can be obtained fo this homogeneous cavity to show the accuacy of the highe-ode BOR FEM. The eo in the numeical solution is defined in the L 2 -nom as k0 BOR k Ref eo = k Ref (23) 2 whee k Ref 0 is the efeence esult (hee obtained analytically), and k0 BOR is the esult by the BOR FEM implemented in this wok. Figues 4 5 show the elative eo of the eigenvalue fo diffeent modes (TM m11, m = 0, 2) vesus the sampling density (SD) in tems of the numbe of points pe wavelength (PPW) defined as SD = λ min (Nn + N s ) 2S (PPW) (24) Eo Axisymmetic FEM (m=0) 1st-ode 2nd-ode 3d-ode O(N 2 ) O(N 4 ) O(N 6 ) Eo Axisymmetic FEM (m=2) 1st-ode 2nd-ode 3d-ode O(N 2 ) O(N 4 ) O(N 6 ) SD (PPW) Figue 4. Relative eo of the eigenvalue ( ) of the axisymmetic cavity fo cylindical mode TM 011 vesus the SD SD (PPW) Figue 5. Relative eo of the eigenvalue ( ) of the axisymmetic cavity fo cylindical mode TM 211 vesus the SD.

9 Pogess In Electomagnetics Reseach B, Vol. 21, whee S is the aea of the meidian coss section, and λ min is the wavelength coesponding to the fist eigenvalue. Numeical esults show that the fist-ode FEM has second-ode accuacy. The secondode FEM has foth-ode accuacy, and the thid-ode FEM has sixth-ode accuacy fo all of the modes (m = 0, ±1, ±2,...). Figue 6 Eo Axisymmetic FEM (m=0) 1st-ode 2nd-ode 3d-ode O(N 1 ) O(N 2 ) O(N 3 ) Eo Axisymmetic FEM (m=0) 1st-ode 2nd-ode 3d-ode O(N 1 ) O(N 2 ) O(N 3 ) CPU time (s) Figue 6. Relative eo of the eigenvalue ( ) of the axisymmetic cavity fo cylindical mode TM 011 vesus the CPU time Memoy equiement (KB) Figue 7. Relative eo of the eigenvalue ( ) of the axisymmetic cavity fo cylindical mode TM 011 vesus the memoy equiement fo the impedance matix. z 1m PEC 1m 1m ε = 2 µ = 1 ε = 1 µ = 2 ε = 2 µ = 2 ε = 1 µ = 2 ε = 2 µ = 1 x 1m 1m Figue 8. The meidian coss section of the complex axisymmetic cavity. 1m

10 198 Rui, Hu, and Liu shows the elative eo of the eigenvalue fo the mode (TM 011 ) vesus the CPU time. Figue 7 shows the elative eo of the eigenvalue fo the mode (TM 011 ) vesus the memoy equiement fo assembling the impedance matix. It is obseved that with highe-ode FEM, both CPU time and memoy equiement ae significantly educed fom the low-ode FEM fo given accuacy. Table 2. The eigenvalue k 0 (m 1 ) of the complex inhomogeneous BOR esonato in Fig. 8 fo cylindical modes m = 0, 1, and 2. m = 0 TM m = 0 TE m = 1 m = Eo Axisymmetic FEM (m=0) 1st-ode 2nd-ode 3d-ode O(N 2 ) O(N 4 ) O(N 6 ) Eo Axisymmetic FEM (m=5) 1st-ode 2nd-ode 3d-ode O(N 2 ) O(N 4 ) O(N 6 ) SD (PPW) Figue 9. Relative eo of the eigenvalue ( ) of the complex axisymmetic cavity fo the cylindical mode m = 0 vesus the SD SD (PPW) Figue 10. Relative eo of the eigenvalue ( ) of the complex axisymmetic cavity fo the cylindical mode m = 5 vesus the SD.

11 Pogess In Electomagnetics Reseach B, Vol. 21, Table 3. The eigenvalue k 0 (m 1 ) of the complex inhomogeneous BOR esonato in Fig. 8 fo cylindical modes m = 3, 4, 5. m = 3 m = 4 m = A Complex Inhomogeneous Cavity As shown in Figue 8, a complex axisymmetic cavity filled with an inhomogeneous medium is simulated by the BOR FEM with diffeent odes. The cavity is filled with thee diffeent types of dielectic mateials as shown in the figue. The thid-ode FEM esult with a sampling density of 30 PPW (N n = 1147, N s = 2610) is set as the efeence esult, as listed in Tables 2 3. Figues 9 10 show the elative eo fo diffeent modes. The thid-ode FEM has the highest accuacy. The ode of accuacy fo the eigenvalues is 2, 4, and 6 fo the fist-, second-, and thid-ode BOR FEM. 5. CONCLUSION In this pape, a highe-ode FEM is applied to solve the eigenvalue poblem fo abitay inhomogeneous dielectic axisymmetic esonatos. The poposed method uses highe-ode node-based elements fo the azimuthal field component and highe-ode edge-based elements fo the meidian field components. It has been obseved that the highe-ode mixed basis functions can significantly impove the accuacy with the same numbe of unknowns when compaed with a lowe-ode FEM. Futhemoe, fo given accuacy, highe-ode FEM can significantly educe the memoy equiement and CPU time compaed with a lowe-ode FEM. ACKNOWLEDGMENT XR and JH thank the suppot by the National Science Foundation of China (No ), the Pogamme of Intoducing Talents of Discipline to Univesities unde Gant b07046, and the Joint Ph.D. Fellowship Pogam of the China Scholaship Council.

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