Integral Equations And The Method Of Moments

Integral Equations and theMethod of Moments(Chapter 3)EC4630 Radar and Laser Cross SectionFall 2011Prof. D. Jennjenn@nps.milAY20111

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaE-field Integral Equation (EFIE)The EFIE is used derived from the radiation integrals and the boundary conditions.Consider a PEC with surface current J s (r ′) . The scattered field at an arbitrary observationpoint (not limited to the far field) is jEs (r ) jω A(r ) A(r )ωµε o oA(r ) µo J s (r ′)G (r , r ′)ds′(OBSERVATIONPOINTz)P ( x, y , z ) or P ( r,θ ,φ ) r R r r′Sexp( jkR ) G (r , r ′) 4π R R r r ′ ( x x′) xˆ ( y y′) yˆ ( z z′) zˆR RSxO r′ J s ( x ′, y ′, z ′)yDIFFERENTIALSURFACE CURRENTPATCHThe operator is taken with respect to the unprimed (observation)coordinates. Let P be on the surface, where the tangential component of the total E must be zero j Ei (r ) Es (r ) tan 0 Ei (r ) tan Es (r ) tan jω A(r ) ωµ ε A(r ) o o tan(AY2011)2

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaE-field Integral Equation (EFIE)Writing the equation in terms of the current gives the EFIE (called an integral equationbecause the unknown appears in the integrand) j ′′′′′′Ei (r )jJrGrrdsJrGrrdsωµ()(,)()(,) s o stanωεSS tano A form more suitable for numerical solution has the derivatives in terms of the primedcoordinates (i.e., ′ ) j ′′′′′′′′Ei (r )jωµJrGrrdsJrGrrds()(,)()(,) o sstan ωεSo S tanThe method of moments (MoM) is a technique used to solve for the current: 1. Expand J s (r ′) into a series with unknown expansion coefficients2. Perform a testing (or weighting) procedure to obtain a set of N linear equations (tosolve for N unknown coefficients)3. Solve the N equations using standard matrix methods4. Use the series expansion in the radiation integral to get the fields due to the currentAY20113

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaFourier Series Similarity to the MoMThe method of moments (MoM) is a general solution method that is widely used in all ofengineering. A Fourier series approximation to a periodic time function has a similarsolution process as the MoM solution for current. Let f (t ) be the time waveformao 2 f (t ) [an cos(ω n t ) bn sin (ω n t )]T T n 1For simplicity, assume that there is no DC component and that only cosines are necessaryto represent f (t ) (true if the waveform has the right symmetry characteristics)2 f (t ) an cos(ω n t )T n 1The constants are obtained by multiplying each side by the testing function cos(ω m t) andintegrating over a periodT /2 0, m n2 T /2 ()()() ()coscoscosfttdtattdtωωω n mn mT n1 a n , m n T / 2 T / 2This is analogous to MoM when f (t ) J s (r ′) , an I n , J n cos(ωnt ) , andWm cos(ωmt ) . (Since f (t ) is not in an integral equation, a second variable t ′ is notrequired.) The selection of the testing functions to be the complex conjugates of theexpansion functions is referred to as Galerkin’s method.AY20114

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaE-field Integral Equation (EFIE)N Current series expansion: J s (r ′) I n J n (r ′)n 1 J n basis functions (known, we get to select)I n complex expansion coefficientsTypes of basis functions: entire domainversus subdomain are illustrated belowfor a wire JJ1 2 J3.zSUBDOMAIN. JNLENTIRE DOMAIN J2PULSES (STAIRCASEAPPROXIMATION)TRIANGLES (LINEARAPPROXIMATION) J1L J3AY2011δ FUNCTIONS (POINTAPPROXIMATION)00Common subsectional basis functionsPIECEWISE SINUSOIDS(CURVEDAPPROXIMATION).z.z.z5

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaTesting Procedure Define weighting (testing) functions Wm (r ) We get to choose, but generally select the complex conjugates of the expansion * functions (Galerkin’s method): Wm (r ) J m (r ) The testing functions are defined at P (i.e., we are testing the field at observationpoints on the surface) To test we multiply the EFIE by each testing function and integrate over the surface(inner product) Wm (r ) Ei (r ) tan dsSm for m 1, 2, , N N j ′ J n (r ′) ′G (r , r ′) ds′ ds Wm (r ) I n jωµo J n (r ′)G (r , r ′) ωεSmSn o tan n 1 Physically, this is equivalent to measuring the effect of current J n (r ′) at r ′ at the location of the test function Wm (r ) at rAY20116

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaTesting ProcedureNow take the summation outside: Vm Wm (r ) Ei (r ) tan ds Sm j ′ J n (r ′) ′G (r , r ′) ds′ ds I n Wm (r ) jωµo J n (r ′)G (r , r ′) ωε on 1 Sm tan Sn N Z mnor,NVm I n Z mn for m 1, 2, , Nn 1The N equations can be put in matrix form: V Z IV N by 1 excitation vector (elements with units of volts)Z N by N impedance matrix (elements with units of ohms) 1 I Z V N by 1 current vector (elements with units of amps)AY20117

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaMoM Matrix and VectorsImpedance matrix elements: jZ mn ds ds′ jωµoWm (r ) J n (r ′) ′ J n ( r ′) Wm ( r ′) G ( r , r ′)ωε oSm Sn Excitation vector elements: θˆ Vm , Ei θ Eiθ (TM)Vm Wm (r ) Ei (r ) tan ds φSm Vm , Ei φˆEiφ (TE)The measurement vector gives the field at an observation point due to each basis function.These elements are obtained by using each basis function in the radiation integral (see thebook for details): θˆ , for TM pθ or φ , pˆ Rn J n (r ′) pˆ E p ( r ′) ds′ where p Sn φˆ , for TE jkηo jkr N θEθ e Rn I n4π rn 1 jkηo jkr N θEφ e Rn I n4π rn 1AY20118

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaBasis Functions for SurfacesThe two-dimensional extension of the step is a pedestal. The two-dimensional extensionof the triangle is the rooftop. A minimum of two orthogonal components are required torepresent an arbitrary current vector.PedestalsRooftopzzRECTANGULARPATCH xx yJ mnn xJ mn y yRECTANGULARPATCHym yJ mn xxy xJ mnmn Rectangular shaped subdomains are notsuited to surfaces with curved edges The discretized shape does not representthe true edge contour -- computed edgescattered fields are not accurate Trianglular subdomains are more accurate.AY20119

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaRWG Basis FunctionsThe RWG (Rao, Wilton, Glisson) triangularsubdomains are the most common.The surface is meshed into triangularsubdomains.FREEVERTEXFREEVERTEXGLOBAL ORIGIN, OAY2011A “rooftop” basis function is associated witheach interior edge: Ln ρ n ,inrT n 2 An J s (r ) Ln ρ n ,inrTn 2 A nThe current at a point in a subdomain is thevector sum of the current crossing the threeedges, weighted by the current coefficients.10

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaSurface MeshingCAD (computer aided design) models can beused as the basis for surface meshing modelsMany models are available cheaply fromweb sites (e.g., www.3dcadbrowser.com)Common formats:1. International Graphics ExchangeStandard (*.iges or *.igs)2. 3D design studio (*.3ds)3. Stereolithography (*.stl)4. Autocad (*.dwg)5. Solidworks, Catia (*.step, *.stp)6. Rhino (*.3dm)CAD software packages have their owntranslators that are not always compatible.Further (substantial) modification is usuallyrequired to make model electrically realistic.AY201111

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaCurrent State of the ArtState of the art for computer processing for rigorous integral equation solvers (December2009)Induced current on a A380 aircraft in dBµA/m A380 aircraft1.2 GHz32 million unknowns960 GB memory11.5 hoursSee Taboada, et al in IEEE Antennas &Propagation Magazine, V. 51, No. 6,December 2009AY201112

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaMoM Issues The integral equations and the method of moments are “rigorous:”o Includes all scattering mechanisms and can be applied at any frequencyo The current series converges to the true value as the number of basis functions isincreased Subdomain basis functions are the most flexible and robust Triangle functions are the most common (good tradeoff between accuracy andcomplexity); the Rao-Wilton-Glisson (RWG) functions are the standard To obtain a converged result the subdomains are limited to approximately 0.1λo This implies large matrices for electrically large targetso The large impedance matrices tend to become ill-conditioned (large spread inelement order magnitudes; small values on the diagonal) Numerical integrations must be performed; the integrations must be converged The surface impedance approximation can be applied to obtain a modified EFIE The EFIE can be extended to volume currents and magnetic currents The EFIE suffers from “resonances” for closed bodies at low frequencies The magnetic field integral equation (MFIE) is the dual to the EFIE it can be used intandem with the EFIE for a more robust solution Other integral equations can be derived based on equivalence principles and inductiontheoremsAY201113

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaExample: Scattering From a Thin WireExample 3.1: TM scattering from a thin wire of length L using pulse basis functions. Athin wire satisfies the condition radius, a λ and a L . There is only an axialcomponent of current and the problem reduces to one dimension: 1, zn / 2 z′ zn / 2I ( z′) zˆ N′′′J s ( z ) zˆ n (z ) I n pn ( z ), where p 2π a 2π a n 1 0, else2n ( N 1) N / L is the segment length and zn is the center of segment n .2 xEikˆi2aθz L / 2Since ′ zˆ d / dz′ the EFIE reduces to L/2 2 d 2 e jk z z′ˆ iz zˆ zEdz′ I ( z′) k 2 ωε o L / 2dz 4π z z′ jL/2G ( z , z ′)AY201114

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaExample: Scattering From a Thin WireUse identities to convert the unprimed derivatives to primed onesL/2Eiz jωµo I ( z′)G ( z , z′)dz′ L / 2The derivative of the pulse I ( z′) G ( z , z′) ′ dz′ωε o L / 2 z′ z jL/2zn zn pn′ ( z′) δ z′ ( zn / 2 ) δ z′ ( zn / 2 ) δ n δ n Pulse basis functionDerivativepn′ ( z )pn ( z )11zn / 2zzn / 2zn / 2zzn / 2 1AY201115

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaExample: Scattering From a Thin WireAfter the testing procedure, the impedance elements are found to be zmzn jk z z′jηo zmzn ()() jk z z′ee 2 dz dz′ δ n δ n δ m δ m Z mn jkηo dz dz′4π z z′ k z z 4π z z′ zmzn mnTo avoid a singularity when m n the testing is performed on the surface jk z z′e4π z z′ e4π jk( z z′ )2 a 2( z z′ )2 a 2aRz′zzThis method of handling the singularity is not very robust (see book for discussion).The impedance integrals can be evaluated numerically. To reduce computation time oneof the double integrals in each term can be approximated by using the simple “rectangularrule”zn f ( z′)dz′ f ( zn )zn which leads to Equations (3.56), (3.57) and (3.58) in the book (see the next page).AY201116

Naval Postgraduate SchoolDepartment of Electrical & Computer EngineeringMonterey, CaliforniaExample: Scattering From a Thin WireFinal result:Z mn zmzn jkRmjηo 2e jkR1 e jkR2 e jkR3 e jkηo dz dz′ 2 444πRπRπR 4π Rmk123 zmzn where Rm( zm z′) 2 a 2 , R1 R3( zm zn ) 2 a 2 . The TM excitation elements are( zm zn ) 2 a 2 , R2 2π zm() 2π zn()( zm zn ) 2 a 2 , and pm ( z ) jkzm cosθjkz cosθˆVmθ zadφdz θkθsinsinccos θˆe e 2 2π a 0 zmThe receive elements for TM polarization are the same as the excitation elements pn ( z ) jkzn cosθjkz cosθˆˆθφθRnθ ezaddzsinsinc k cosθ e 2 2π a 0 zn AY201117