Author name used in this publication: S. L. HoAuthor name used in this publication: H. C. Wong2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Ho, SL

Hu, MQ

Wong, HCC

Yu, HT

IEEE Transactions on Magnetics

Author name used in this publication: S. L. HoAuthor name used in this publication: H. C. Wong2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 771Edge-Based FEM-BEM for Wide-BandElectromagnetic ComputationH. T. Yu1, S. L. Ho2, M. Q. Hu1, and H. C. Wong3Department of Electrical Engineering, Southeast University, Nanjing 210096, ChinaDepartment of Electrical Engineering, Hong Kong Polytechnic University, Kowloon, Hong KongIndustrial Centre, Hong Kong Polytechnic University, Kowloon, Hong KongEdge-based finite element method-boundary element method (FEM-BEM) for the computation of three-dimensional (3-D) electro-magnetic fields at high and low frequencies is presented. A novel H–A model that includes the differential and integral equations forwide-band electromagnetic problems is described. An edge element method is implemented to discretize the proposed formulation. Asurface edge element method (SEEM) for surface of arbitrary shape targets is derived according to the geometrical properties of thebasis functions. Unlike traditional electromagnetic formulations, the major coefficients of the proposed formulation in this model arefrequency independent and hence the model can be used to solve electromagnetic problems over a wide range of frequencies. The pro-posed method is implemented to compute: 1) the electromagnetic induction problems at low frequencies and 2) the scattering fields athigh frequency. Results from established methods are used to validate the proposed method.Index Terms—Edge-based FEM-BEM, electromagnetic induction, scattering field, surface edge element method.I. INTRODUCTIONTHE edge element method (EEM) [1] was proposed byNedelec in 1980. Only local elements are involved insuch formulations and the tangential components of the vectorsare continuous. Compared with the nodal-based finite elementmethod (NBFEM), EEM needs fewer unknowns and hence isintrinsically more accurate. With EEM, the field vectors areused directly as variables. In NBFEM, one needs to compute,firstly, the potential which is an intermediate variable. Bossavitet al. [2], [7] have also implemented EEM to discretize FEMformulations for solving eddy-current problems at low fre-quency and scattering field at high frequency.In addition to having the merits of edge elements, edge-basedfinite element method-boundary element method (FEM-BEM)can be used to solve electromagnetic (EM) problems in openareas and is more convenient to use when dealing with boundaryconditions than in edge-based FEM with nodal-based BEM.Wakao et al. [3] also solve eddy-current problems of rectan-gular targets using edge-based FEM-BEM using hexahedron asthe local element. Ahn solves high-frequency problems witha local tetrahedron element but the parameters in the expres-sion of the surface basis function are not presented [4]. More-over, the FEM-BEM formulation in [3] may not solve high-fre-quency EM problems because the effect of displacement currenthas been excluded. The coefficients in the electric field integralequation (EFIF) in [4] and (15) in [6] are frequency dependent.Such a coefficient in one field may become very dominant whenthe displacement current becomes negligible and the contribu-tion of the other EM fields in the same formula could then beovershadowed. Daveau et al. [6] also mention that such formu-lations are not applicable readily at low frequency.A novel edge-based FEM-BEM to study wide-band EM prob-lems is proposed. – formulation based on the differentialand integral equations is derived. A three-dimensional (3-D)Digital Object Identifier 10.1109/TMAG.2006.871457edge element method is used to discretize the differentiation for-mulation. SEEM is derived and implemented to discretize theintegral equation to facilitate the proposed FEM-BEM which isbeing used to solve arbitrarily shaped targets. The characteris-tics of the proposed formulation are that the coefficients of thediscretized formulation are insensitive to frequencies and there-fore it can be used to solve EM problems at high and low fre-quencies. Finally, the proposed FEM-BEM is used to solve EMinduction (EM-I) problems and to compute the scattering fields.The computational results are validated by either experimentsor other numerical methods.II. SURFACE EDGE ELEMENT METHODThe 3-D edge element method has been widely used in FEMcomputation and its basis function for any order is well known.Hence, the following discussion will focus on SEEM for tar-gets of arbitrate shapes. The surface basis functions of EEMfor the FEM-BEM computations in [4] have been presented butthe corresponding nodal basis function for surfaces of arbitraryshapes have not been presented. In this paper, the developmentof SEEM for arbitrary targets are discussed.Assume an arbitrary point in Fig. 1 is located in-side a triangular patch on the surfaces of a generalobject without any of its sides parallel to the , , or axis.The nodal basis function in this case is a linear function of co-ordinates and(1)where are coefficients related to the size and shapeof the triangle.From the property of nodal basis function, one obtains(2)0018-9464/$20.00 © 2006 IEEEAuthorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 07:48 from IEEE Xplore.  Restrictions apply.772 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006Fig. 1. Triangular patch.Fig. 2. Electromagnetic problem configuration.where are the three normalized components of anormal vector of the plane if , andif . The first three equations are obtained basedon the property of the nodal basis functions. The fourth rowof (2) is obtained from the geometrical property of nodal basisfunction, and it is equivalent to the normalized plane equationenclosing the corresponding edge on a triangular patch. How-ever, there are many such planes and hence the fourth row in(2) is imposed in order to select the plane, which is vertical tothe local triangle patch, as the basis plane for which (1) has aunique solution.Therefore, one of the edge based basis functions related toedge is obtained as follows:(3)As in volume edge elements, the following identity is alwaysheld:(4)which ensures that the tangential components of the vectors arealways continuous between two adjacent elements.The equivalent electrical and magnetic currents are related tothe corresponding tangential magnetic and electric fields on thesurface integration as follows:(5)(6)where is an edge variable corresponding to the magneticdensity, is the equivalent electric field. is a constantin a direction normal to the edge such that the equivalentcurrents are continuous in the normal direction of the adjacentelements to satisfy the surface current properties.III. FORMULATIONSFig. 2 shows the EM problem investigated in this paper. isa 3-D general target enclosed by the surface , in which the pa-rameters are: conductivity , permeability , permittivity .is the nonconducting region, where the parameters are: perme-ability , permittivity , and the current density in this regionis .A. Inside TargetsFEM is implemented inside the object and the magnetic fielddensity vector and magnetic potential vector are set asphysical variables. From Maxwell’s equations, the followingexpressions are derived:(7)(8)Equations (7) and (8) can be rewritten as(9)Using vector identities, one obtains(10)Setting(11)where is the conventional reduced magneticvector potential.By adding (11), (9) can be changed into(12)Integrating both sides of (12) gives the following governingequation:(13)where is the weighting function for the vector basis functionis the outward unit vector normal to the surface of thetarget.By implementing 3-D EEM, the vector is interpolatedwithin a tetrahedron as follows:(14)where the vector basis function for the volume edge elementmethod is expressed as(15)where is the 3-D nodal basis function.Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 07:48 from IEEE Xplore.  Restrictions apply.YU et al.: EDGE-BASED FEM-BEM FOR WIDE-BAND ELECTROMAGNETIC COMPUTATION 773As the third term in (13) exists only on the surface of targets,the surface basis function is used to interpolate(16)where is an unknown, stands for the surface basis func-tion as shown in (3).B. On the Surface of TargetsOn the surface of targets, edge-based BEM is incorporatedinto the integral equation. The integral formulation for generaltargets in this region is derived from the Green’s theorem(17)By setting and, together with (11), thegoverning function in this region can be rewritten as(18)where the constant is dependent on the position of the fieldpoint, and is the propagation constant in freespace. in (17) and (18) is the Green’s function.The first term on the right-hand side in (18) represents thedriving source excited by the sensor and the second term standsfor the contribution of displacement current, which can be ne-glected at low frequencies because is very small in this sit-uation. Another feature of (18) is that the coefficients of theother two terms are fairly constant and are practically frequencyindependent.Applying Galerkin method to (18) and setting the weightingfunction , one can obtain the linear equationsfor the surface of targets.The application of in (13) and (18) makes the values ofcoefficients corresponding to the terms in (13) and (18) com-parable to those of other terms, hence the characteristics of thediscretized equations are significantly improved.IV. EXAMPLESA. EMI Response From a CubeThe first example used to test the proposed FEM-BEMmethod is to simulate the electromagnetic response from aconducting target. EM induction (EM-I) is widely used forthe detection and discrimination of conducting and permeabletargets using specific sensors. In the context of subsurfacesensing, EM-I is used to simulate the signature of buried metalin land-mines and unexploded ordnance (UXO) [8]. As theconductivity of such targets is typically many orders of mag-nitude larger than that of the soil, the target can be analyzedwith the assumption that it is located, approximately, in freespace. The frequency-domain EM-I response of a conducting orferrous target is characterized by the lowest resonant frequencyat which the EM-I response is much stronger than that of othermodes.Fig. 3. Frequency feature of EMI response.Fig. 4. Electromagnetic Induction responses along the Y -direction.A numerical computation to investigate the EM-I responsefrom a simple model in the form of a cube, 0.07 m on each sidewith a conductivity of 1.3 10 S/m is reported below. Thedistance between the cube and the sensor center is 0.2 m. TheEM-I responses in the form of the terminal voltage at the end ofthe receiver over the frequency range of 10 Hz to over 10 kHzand with different distances between the sensor and the targetare simulated by the proposed FEM-BEM method. Meanwhile,an experiment is carried out to validate the numerical results. Inthe experiment, the conducting cube is put on a plastic frame inorder to reduce the environmental effects. The sensor includesthe source and receiving coils. Due to the weakness of the re-sponse signal and as the source is very close to the receiver, oneneeds to detect the signal from the actual target and then the ex-periment is repeated again with the target removed in order toassess the environmental effects.The results shown in Fig. 3 are divided into real and imaginaryparts, and the imaginary part of the EM-I response is observedto reach its maximum when the frequency is close to the lowestresonant frequency. The real part of the EM-I response becomeslarger as frequency increases but its incremental change willAuthorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 07:48 from IEEE Xplore.  Restrictions apply.774 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006Fig. 5. Scattering field computed using two different methods.Fig. 6. Scattering field with different meshes.become increasingly small and it will eventually be close to zeroif the frequency is high enough. The frequency at the cross-point between the real and imaginary parts is taken as the lowestresonant frequency.Fig. 3 shows the comparison between the numerical result andthe experimental one. The maximum error between the compu-tational and experimental results is 5.02%.The results shown in Fig. 4 are obtained with the sensormoving along the axis from 0.5 to 0.5 m at 1 kHz and thepeak of the waveform is observed to appear above the target.From Figs. 3 and 4, one can see that the computational resultsmatch the experimental ones very well.B. Scattering Field at High FrequencyThe following example of FEM-BEM application is to com-pute the scattering field from a cylinder with its diameter andlength both equal to 0.254 m. Its relative permittivity isand the conductivity is 0.02 S/m.The incident electric field at 300 MHz is 1 V/m and it comesdirectly from the direction. The cylinder is subdivided into72 tetrahedrons. The number of unknowns is 197 with the edge-based FEM-BEM algorithm. The number of unknowns is 144with the method of moment (MoM). The scattering electric fieldshown in Fig. 5 is computed using the proposed FEM-BEMand MoM algorithms. In the case of homogenous targets, MoMrequires less computational time if both methods use the samealgorithm in solving linear equations. However, the coefficientmatrix of the discretized linear equation of MoM is full and thisoffsets the advantages of MoM.On the other hand, one can investigate the accuracy of the so-lution by computing the relative errors. Unlike [6], andinside the target cannot be calculated together, and therefore thequadratic relative errors [6] cannot be computed. However, wecan calculate the scattering electric field with different meshesas shown in Fig. 6. From this figure, one can see that the compu-tational results are closer when the meshes increase. The meanrelative error between 180 and 216 meshes is 2.63%.V. CONCLUSIONA generalized hybrid method has been proposed and imple-mented successfully for the 3-D EM computation of EM-I from10 Hz and the scattering fields at radio frequency. The –model overcomes the shortcomings of traditional mathematicalformulations for EM computations. SEEM for general targetshas also been derived and applied to interpolate the integralequation while the 3-D EEM is used in differential equations.Finally, the proposed method is applied in several case studiesand the errors of the numerical solutions have been investigated.Those results show that the proposed FEM-BEM is valid forwide-band frequency.ACKNOWLEDGMENTThis work was supported in part by NSFC.REFERENCES[1] J. C. Nedelec, “Mixed finite element in R ,” Numer. Math., vol. 35, pp.315–341, 1980.[2] A. Bossavit, “A rational for edge elements in 3-D field computations,”IEEE Trans. Magn., vol. 24, no. 1, pp. 74–79, Jan. 1988.[3] S. Wakao and T. Onuki, “Electromagnetic field computations by the hy-brid FE-BE method using edge elements,” IEEE Trans. Magn., vol. 28,no. 2, pp. 1487–1490, Mar. 1993.[4] C. H. Ahn, B. S. Jeong, and S. Y. Lee, “Efficient vectorial hybrid FE-BEmethod for electromagnetic scattering problem,” IEEE Trans. Magn.,vol. 30, no. 5, pp. 3136–3139, Sep. 1994.[5] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill,1941, pp. 250–254.[6] C. Daveau and F. Rioux-Damidau, “New formulation coupling a finiteelement method and a boundary integral method for the computation ofthe interaction of waves with a conducting domain,” IEEE Trans. Magn.,vol. 35, no. 2, pp. 1014–1018, Mar. 1999.[7] A. Bossavit and I. D. Mayergoyz, “Edge elements for scattering prob-lems,” IEEE Trans. Magn., vol. 25, no. 4, pp. 2816–2821, Jul. 1989.[8] N. Gang, C. B. Baum, and L. Carin, “On the low frequency natural re-sponse of conducting and permeable targets,” IEEE Trans. Geosci. Re-mote Sens., vol. 37, no. 1, pp. 347–359, Jan. 1999.Manuscript received June 20, 2005; revised November 20, 2005 (e-mail:eeslho@polyu.edu.hk).Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 07:48 from IEEE Xplore.  Restrictions apply.

Edge-based FEM-BEM for wide-band electromagnetic computation

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