HIGHER ORDER full-wave three-dimensional (3-D) large-domain techniques in

Transcription

1 FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 21, no. 2, August 2008, Comparison of Higher Order FEM and MoM/SIE Approahes in Analyses of Closed- and Open-Region Eletromagneti Problems Milan M. Ilić, Andjelija Ž. Ilić, and Branislav M. Notaroš Abstrat: We investigate the effiieny of the two most popular frequeny-domain approahes in omputational eletromagnetis (CEM); the finite element method (FEM) and the method of moments based on the surfae integral equation (MoM/SIE), both in the ontext of the higher order modeling. We ompare the performanes of the two approahes in two simple three-dimensional (3-D) problems with similar meshes, hosen as benhmark examples. The hosen examples demonstrate full-wave analysis of a wave-guiding struture (a losed-region problem) and a satterer in free spae (an open-region problem). Keywords: Computer-aided analysis, eletromagneti analysis, finite element method, higher order elements, inhomogeneous media, method of moments. 1 Introdution HIGHER ORDER full-wave three-dimensional (3-D) large-domain tehniques in omputational eletromagnetis (CEM) have proven themselves to be flexible and effiient in modeling of omplex strutures [1 6]. The redution in omputation osts is by up to one to two orders of magnitude when ompared to low order (small-domain) tehniques, for the same or better auray. Choie of a partiular numerial method suh as the finite element method (FEM) or the method of moments based on the surfae integral equation (MoM/SIE) is guided by the Manusript reeived on July 10, M. M. Ilić is with Shool of Eletrial Enigeering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade, Serbia ( milanili@etf.rs). A. Ž Ilić is with Laboratory of Physis 010, Vinča Institute of Nulear Sienes, Belgrade, Serbia ( iandja@vin.bg.a.yu). B. M. Notaroš is with Department of Eletrial and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA ( notaros@olostate.edu) 209

2 210 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: nature of the problem at hand. FEM, where system matries are banded and highly sparse, is often omputationally more effiient. Additionally, it is onsidered to be the method of hoie for modeling inhomogeneous media (possibly nonlinear and/or anisotropi) and geometrially versatile strutures. MoM/SIE, on the other hand, is onsidered to be superior in dealing with open-region problems. As an answer to dealing with the omplexities in modeling real world strutures, where various aspets mentioned above appear all at one, hybrid FEM-MoM approahes emerge; the fields within the hosen inhomogeneous or otherwise omplex domain are modeled by FEM and the domain itself is trunated by means of unknown surfae urrents (eletri and magneti) whih are evaluated by MoM [7, 8]. Our goal in this paper is to ompare the effiieny of higher order FEM and MoM/SIE tehniques in the analyses of strutures with inhomogeneous domains. Some pratial appliations where suh domains are involved inlude eletromagneti (EM) sattering from human tissues or radar absorbing oatings, as well as sattering and diffration from inhomogeneous dieletri lenses used in lens antennas. To the best of our knowledge, this kind of omparison has not been done before, probably beause pratially all of the ommerially available professional software pakages are based on the low order, small-domain approahes. A notable exeption is WIPL-D [9]. In setion 2 we briefly desribe the theoretial bakground of the higher order FEM and MoM software implementations. In setion 3 we analyze two simple problems involving inhomogeneous dieletri domains, as benhmark examples. For the omparison of approahes to be fair, we hoose the examples that have the same or similar meshes in both approahes. Additionally, our hoie of the two examples is suh that the first example is a losed-region problem, generally more onveniently analyzed by FEM, and the seond is an open-region problem, generally more onveniently analyzed by MoM. Hene, we hoose the first example to feature a disontinuity in retangular waveguide, whereas the seond example analyzes a satterer in free spae. 2 Theory and Implementation Our higher order FEM approah is based on the disretization of the url-url vetor wave equation µ 1 r E k 2 0ε r E = 0, (1) where ε r and µ r are omplex relative permittivity and permeability of the inhomogeneous (possibly lossy) medium, respetively, and k 0 = ω ε 0 µ 0 is the free-spae

3 Comparison of Higher Order FEM and MoM/SIE Approahes wave number. Galerkin-type weak form disretization of (1) yields µ r 1 ( e k ) ( E) dv k0 2 ε r e k E dv = V V jk 0 Z 0 e k n H ds, k = 1,...,N FEM, S where V is the volume of the FEM region, S is the boundary surfae of the region, n is the outward unit normal (ds = nds), Z 0 is the free spae impedane, and N FEM is the total number of the expansion funtions. Formal field-expansion used for the approximation of eletri-field in (2) is N FEM E = l=1 (2) γ l e l (3) where e l are the url-onforming hierarhial polynomial basis funtions [3,4], and γ l are unknown field-distribution oeffiients. The matrix form of (2) is [FEM]{γ} = jk 0 Z 0 e k (n H) ds. (4) Elements of the FEM matrix are the inner produts of the employed field-expansion basis funtions expliitly derived in [3] and [4]. The element type used for volume geometrial modeling in the FEM approah is the generalized urvilinear hexahedron, shown in Fig. 1. Lagrange interpolating polynomials [3] are used to define w S v u Fig. 1. Generalized urvilinear hexahedron, with loal u v w oordinates, used for volume geometri disretization in the FEM approah. the mapping from the ubial parent domain to the Lagrange-type urved parametri hexahedron. In our examples given in Setion 3, generalized hexahedral elements in FEM meshes redue to simple briks. The oupled eletri-field integral equation/magneti-field integral equation (EFIE/MFIE) system of equations to

4 212 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: be solved for eletri J S and magneti M S surfae urrents on eah of the MoM surfaes, generally residing in between the two homogeneous regions with parameters ε 1, µ 1 and ε 2, µ 2, in the higher order large-domain MoM/SIE approah is [E(J S,M S,ε 1, µ 1 )] tang +(E i ) tang = [E( J S, M S,ε 2, µ 2 )] tang, (5) [H(J S,M S,ε 1, µ 1 )] tang +(H i ) tang = [H( J S, M S,ε 2, µ 2 )] tang, (6) where sattered eletri field E is represented by E = E J (J S )+E M (M S ), E J (J S ) = jωa Φ, E M (M S ) = 1 ε F, (7) and sattered magneti field H is represented by H = H M (M S )+H J (J S ), H M (M S ) = jωf U, H J (J S ) = 1 µ A. (8) The system (5)-(6) is disretized by the substitution of the formal eletri and magneti urrent expansions N MoM J S = j=1 N MoM α j j s j, M S = j=1 β j j s j, (9) where j s j are the divergene-onforming hierarhial polynomial basis funtions [6], into the potential equations A = µ J S g ds, S F = ε M S g ds, S M S g ds, S Φ = j ωε U = j ωµ S S S J S g ds, S M S g ds, and solved for the values of unknown urrent distribution oeffiients α j and β j, j = 1,...,N MoM, where N MoM is the total number of expansion funtions. In the above potential equations, g is the Green s funtion to be evaluated separately for eah of the homogeneous regions. It is defined by (10) g = e γr 4πR, γ = jω εµ, (11) where ε and µ are permittivity and permeability of the medium, respetively, and ω is the angular frequeny of the implied time-harmoni variation.

5 Comparison of Higher Order FEM and MoM/SIE Approahes The element type used for surfae geometrial modeling in the MoM/SIE approah is the generalized urvilinear quadrilateral, shown in Fig. 2. Lagrange interpolating polynomials [6] are used to define mapping from the square parent domain to the Lagrange-type urved parametri quadrilateral. In our examples given in Setion 3, generalized quadrilateral elements in MoM meshes redue to simple retangles, whih also belong to the subset of bilinear quadrilaterals used for the higher order MoM/SIE modeling in WIPL-D [9]. v u Fig. 2. Generalized urvilinear quadrilateral, with loal u v oordinates, used for surfae geometri disretization in the MoM/SIE approah. 3 Results and Disussion Our extensive previous work [3,4,6], and [8], demonstrates exellent agreement of the simulation results with analytial results and/or measurements. This fat gave us onfidene to base the analysis in this paper only on numerial models. All numerial results reported here are obtained using an IBM Thinkpad T60p notebook omputer with Intel T7200 Core2 CPU running at 2.0 GHz under Mirosoft Windows XP operating system. As the first example, onsider a dieletrially loaded WR-15 retangular waveguide, shown in Fig. 3. The load is tightly fit inhomogeneous dieletri slab. The length of the slab and the lengths of the empty waveguide portions on both sides of the slab are the same. The slab is inhomogeneous and lossy. It onsists of three equally thik layers whose real and imaginary parts of the relative permittivity (ε r = ε r jε r ) approximate a linear hange from ε r = 2 to ε r = 4 and from ε r = 0.2 to ε r = 2, respetively, along the slab. The orresponding relative permittivities are depited in the inset of Fig. 3. The loss-tangent in this example is rather large and ranges from tan δ = 0.1 to tanδ = 0.5 along the slab. The slab is non-magneti. This example demonstrates analysis of a losed-region problem. For the higher order FEM approah [4], the omplete loaded waveguide struture

6 214 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: is modeled by 5 trilinear hexahedra (briks) whih oinide with 5 waveguide setions shown in Fig. 3. Waveguide walls are onsidered to be lossless. The optimal polynomial orders of the field approximations, for whih the solution for the modal S-parameters aross the ports has onverged, in the hexahedral elements range from 2 to 4 in different elements in different diretions. This arrangement results in 231 FEM unknowns. The higher order FEM result will be ompared with the solution obtained by the higher order MoM/SIE approah. MoM/SIE model is not exited by single propagating modes, but by using 2-port feeding networks. Hene, the model requires appropriate feeds as the extensions on both waveguide ports. The analyzed waveguide struture with feeds is shown in Fig. 4. The feeds onsist of short-iruited waveguide setions with wire probes mounted some distane away from the short-iruited end in the middle of the waveguide. The wires have delta-funtion generators at the bottom ends onneted to the waveguide. These generators represent struture ports 1 and 2 for whih the S-parameters are firstly alulated and later de-embedded [9,10] to the planes of ports 1 and 2, respetively, for omparison with FEM results. Polynomial expansions for the surfae urrents in the MoM/SIE model are optimally hosen and they range from 1 to 6 on different surfaes in different diretions. The MoM/SIE analysis of the given example is arried out using WIPL D [9]. 0 b 3.67 j j j1.7 z r a /3 /3 /3 Port 2 b Port 1 a Fig. 3. Loaded WR-15 waveguide (a = 3.76 mm, b = 1.88 mm, and = 2.5 mm). The load is an inhomogeneous lossy dieletri slab modeled by three equally thik homogeneous layers optimally approximating linear hange of real and imaginary parts of ε r. This is also a omplete waveguide model for the FEM analysis. In Figs. 5(a) and 5(b), the results for modal S-parameters of the loaded waveguide obtained by the higher order FEM are ompared with those obtained by the higher order MoM/SIE approah. The results are alulated in 36 frequeny points in the band of interest. Only magnitudes of S 11 and S 21 parameters are shown, in Figs. 5(a) and 5(b), respetively, as it was onluded that the phase harateristis do not show signifiant differenes and thus do not arry relevant information

7 Comparison of Higher Order FEM and MoM/SIE Approahes Waveguide feed 2 b l a d Port 1' Port 1 Waveguide feed 1 Port 2 l Port 2' Fig. 4. Model of the loaded waveguide inluding short-iruited feeds with wire probes. This model is used in the MoM/SIE analysis ( d = /2 = 1.25 mm and l = 1 mm). for omparison. Corresponding number of unknowns and simulation times for the solutions are given in the legends. We observe an exellent agreement between the FEM solution and the MoM/SIE solution. However, a signifiantly large number of unknowns is employed in the MoM/SIE solution, 1088, ompared to only 231 unknowns in the FEM solution. This results in signifiantly longer simulation time required for the MoM/SIE smulation; 67 seonds ompared to only 1 seond required for the FEM simulation. Note that the presented MoM/SIE solution is optimal in terms of the polynomial expansion orders for urrents, it has onverged to the best possible solution for the given model, and it has proven to be invariant to hanges of the de-embedding parameters. Additionally, the presented number of unknowns and simulation time do not inlude the pre-simulations of the waveguide feeds, neessary for the de-embedding proess [10]. We reognize here that FEM inherently requires less omputational resoures than MoM/SIE for analysis of this kind of losed-region problems as it requires simpler model and yields modal S-parameters diretly. Additionally, FEM matrix is filled only one for all frequenies as medium parameters are frequeny independent. Nevertheless, we an onlude that the effiieny of the FEM approah is far more superior in this example; it requires almost 1/5 the number of unknowns and more than an order of magnitude less omputational time than the MoM/SIE approah. The seond example is an inhomogeneous ubial satterer in free spae. The ube is lossless. It onsists of three equally thik layers whose relative permittivities (ε r = ε r) approximate linear hange from ε r = 2 to ε r = 4 between two opposite faes. The ube is situated in vauum and it is illuminated by a uniform plane wave inident along the diretion of hange of ε r, as shown in Fig. 6. The ube is non-magneti. This example demonstrates analysis of an open-region problem. Analysis by FEM requires the domain of interest to be properly bounded and trunated from the rest of the (infinite) spae, whih we do by means of the hybrid

8 216 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: S Method Unknowns Time[s] FEM MOM f [GHz] (a) S Method Unknowns Time[s] FEM MOM f [GHz] (b) Fig. 5. S-parameters of the loaded WR-15 waveguide shown in Fig. 3; omparison of FEM (blak urve) and MoM/SIE (red irles) results. Magnitudes of (a) S 11 parameter and (b) S 21 parameter. higher order FEM-MoM; the fields within the ube are modeled by FEM and the domain is trunated at the ube faes by means of unknown surfae urrents whih are evaluated by MoM [7,8]. The three layers of the ube are modeled by the three trilinear hexahedra in the FEM domain, visible in Fig. 6. In MoM domain, they are modeled by 14 bilinear quadrilaterals oiniding with the outer faes of the layers. The polynomial orders of the field approximations in FEM domain are 5 and

9 Comparison of Higher Order FEM and MoM/SIE Approahes along the longer and shorter edges of the layers, respetively. The orresponding polynomial orders of the urrent expansion in MoM domain are 4 and 3 along the longer and shorter edges for all quadrilaterals (outer faes of the layers). The higher order hybrid FEM-MoM results will be ompared with the solution obtained by the higher order MoM/SIE approah [6]. The MoM/SIE model onsists of 16 bilinear quadrilaterals; the same 14 bilinear quadrilaterals omprising the outer boundary surfae from Fig. 6 and additional two quadrilaterals (squares) positioned between the adjaent layers. Polynomial expansions for the surfae urrents in these models are optimally hosen and they range from 1 to 6 on different surfaes in different diretions r a n in H in a/3 a/3 a/3 a E in Fig. 6. Inhomogeneous lossless ubial satterer in free spae. The ube onsists of three equally thik homogeneous layers optimally approximating linear hange of ε r. The figure desribes models for both FEM-MoM and MOM/SIE analyses. Fig. 7 shows the normalized radar ross setion (RCS) of the ube against the normalized length of the ube side 4a/λ, λ being the wavelength in the dieletri where ε r = 4. It an be observed that the solution obtained by means of the higher order MoM/SIE is pratially idential to the higher order FEM-MoM solution. The numbers of unknowns used in eah of the analyses, shown in the figure legend, are as follows: hybrid FEM-MoM analysis employed 924 unknowns in the FEM region and 704 unknowns in the MoM region, MoM/SIE analysis employed 776 unknowns. Thus, in this example, the FEM-MoM solution employs approximately twie as many unknowns as the MoM/SIE solution (1628 ombined FEM- MoM unknowns in the hybrid solution are needed ompared to 776 unknowns in the MoM/SIE solution). The simulation times are also in favor of the MoM/SIE approah in this ase. The MoM/SIE simulation lasted 85 s. The FEM-MoM simulation required 2.4 times more time, namely 209 s. This is primarily due to high FEM polynomial orders used for modeling of the fields in the ube interior and due to the omplex nature of the hybrid FEM-MoM approah whih simultaneously yields high-auray solution of the fields in the FEM domain. Nevertheless, we an onlude that the MoM/SIE approah is indeed superior to the FEM approah

10 218 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: in this example; it requires one half the number of unknowns and it is more than twie as fast as the FEM approah Method Unknowns Time [s] FEM-MOM MOM σ/λ a/λ Fig. 7. Normalized RCS of the layered ubial satterer shown in Fig. 6 against the normalized length of the ube side; omparison of hybrid FEM-MoM (blak urve) and MoM/SIE (red irles) results. 4 Conlusions We have investigated the effiieny of the higher order FEM and MoM/SIE modeling approahes in CEM by omparing their performanes in two simple 3-D benhmark problems with similar meshes. The hosen examples demonstrate full-wave analysis of a wave-guiding struture (a losed-region problem) and a satterer in free spae (an open-region problem). Suh a omparison has not been found in the open literature, probably beause all of the ommerially available FEM and MoM software pakages (apart from WIPL-D) are essentially low order tehniques. We have evaluated the effiieny of the presented modeling approahes in two simple examples of losed-region and open-region EM problems involving inhomogeneous (and possibly lossy) dieletri bodies. The results obtained by the higher order large-domain FEM (hybridized with MoM for analysis of the openregion problem) have been ompared with numerial results obtained by the higher order MoM/SIE approah. The examples have demonstrated that effetive higher order FEM hexahedral meshes and MoM quadrilateral meshes, onstruted from a very small number of large finite elements, ould be applied even in the presene of the inhomogeneities. This has always been one of the strongest points

11 Comparison of Higher Order FEM and MoM/SIE Approahes of the higher order modeling paradigm. It has been observed that, for the same auray, the presented FEM modeling approah required almost 1/5 the number of unknowns and more than an order of magnitude less omputational time than MoM/SIE in the losed-region (loaded waveguide) example. In the open-region (satterer in free spae) example, it has been observed that the number of unknowns used by the presented hybrid FEM-MoM modeling approah was twie as high as that of the MoM/SIE and its omputational time was 2.4 times longer than that of the MoM/SIE. Note, however, that signifiantly longer omputational time of the hybrid FEM-MoM simulation in this example an partially be ontributed to the omplex nature of the hybrid method (whih simultaneously yields high auray solution of the fields inside the FEM domain), but primarily to high FEM polynomial orders used for modeling of the fields in the satterer interior. Based on the presented analysis, we an onlude that higher order FEM approah represents a method of hoie in modeling of losed-region problems when effetive largedomain volumetri meshing is available. At the same time, higher order MoM/SIE approah is the method of hoie in modeling of open-region problems. Our future work will inlude a omparison of the two higher order approahes in analysis of partially losed and partially opened EM strutures, suh as avity baked antennas and similar strutures with radiating apertures, where the potential benefits of the hybrid method ould be more pronouned. Aknowledgments This work was supported by the Serbian Ministry of Siene and Tehnologial Development under grant ET Referenes [1] R. D. Graglia, D. R. Wilton, and A. F. Peterson, Higher order interpolatory vetor bases for omputational eletromagnetis, IEEE Transations on Antennas and Propagation, vol. 45, no. 3, pp , Mar [2] J. M. Jin, J. Liu, Z. Lou, and C. S. T. Liang, A fully high-order finite-element simulation of sattering by deep avities, IEEE Transations on Antennas and Propagation, vol. 51, no. 9, pp , Sept [3] M. M. Ilić and B. M. Notaroš, Higher order hierarhial urved hexahedral vetor finite elements for eletromagneti modeling, IEEE Transations on Mirowave Theory and Tehniques, vol. 51, no. 3, pp , Mar [4] M. M. Ilić, A. Z. Ilić, and B. M. Notaroš, Higher order large domain FEM modeling of 3-D multiport waveguide strutures with arbitrary disontinuities, IEEE Transations on Mirowave Theory and Tehniques, vol. 52, no. 6, pp , June 2004.

12 220 M. M. Ilić, A. Ž Ilić, and B. M. Notaroš: [5] B. M. Kolundžija, Eletromagneti modeling of omposite metalli and dieletri strutures, IEEE Transations on Mirowave Theory and Tehniques, vol. 47, no. 3, pp , July [6] M. Ðor dević and B. M. Notaroš, Double higher order method of moments for surfae integral equation modeling of metalli and dieletri antennas and satterers, IEEE Transations on Antennas and Propagation, vol. 52, no. 8, pp , Aug [7] X. Yuan, Three-dimensional eletromagneti sattering from inhomogeneous objets by the hybrid moment and finite element method, IEEE Transations on Mirowave Theory and Tehniques, vol. 38, no. 8, pp , Aug [8] B. M. Notaroš, M. Ðor dević, and M. M. Ilić, Hybrid higher order tehniques for CEM analysis and design, in North Amerian Radio Siene Meeting - URSI 2007 digest, Ottawa, Canada, [9] B. M. Kolundžija, J. S. Ognjanović, and T. K. Sarkar, WIPL-D: Eletromagneti Modeling of Composite Metalli and Dieletri Strutures, Software and User s Manual. Boston: Arteh House, [10] B. Kolundžija, B. Janić, and M. Rakić, Novel tehnique for deembeding S- parameters in eletromagneti modeling of arbitrary iruits, in Proeedings of IEEE AP-S International Conferene, vol.3, Monterey, California, USA, 2004, pp