Solving Maxwell s Equations Using the Ultra Weak Variational Formulation

Transcription

1 Soving Maxwe s Equations Using the Utra Weak Variationa Formuation T. Huttunen, M. Mainen and P. Monk Department of Appied Physics, University of Kuopio, P.O.Box 1627, 7211 Kuopio, Finand Department of Mathematica Sciences, University of Deaware, Newark, Deaware, 19711, USA We investigate the utra weak variationa formuation for simuating time-harmonic Maxwe probems. This study has two main goas. First, we introduce a nove derivation of the UWVF method which shows that the UWVF is an unusua version of the standard upwind discontinuous Gaerkin (DG) method with a specia choice of basis functions. Second, we discuss the practica impementation of an eectromagnetic UWVF sover. In particuar, we propose a method to avoid the conditioning probems that are known to hamper the use of the UWVF for probems in genera geometries and inhomogeneous media. In addition, we show how to impement the PML in the UWVF to accuratey approximate physicay unbounded probems and discuss the paraeization of the UWVF. Three dimensiona numerica simuations are used to examine the feasibiity of the UWVF for simuating wave propagation in inhomogeneous media and scattering from compex structures. 1. INTRODUCTION We are concerned with deveoping a fexibe method for approximating the timeharmonic Maxwe system at resonance frequencies for compex geometries and media. Because of the need to aow a variabe refractive index, we decided to use a voume based method. In further narrowing down our requirements we think that a successfu numerica

2 2 HUTTUNEN, MALINEN, KAIPIO scheme for approximating the time-harmonic Maxwe system needs to address a number of sometimes conficting requirements: The scheme shoud be abe to hande compex geometry. Appications may incude scattering from aircraft, computation of antenna patterns or predicting the interaction of radiation with bioogica tissue (such as the human head). This requirement suggests the use of an unstructured grid. The agorithm shoud be abe to hande different materias (incuding sudden jumps in the eectromagnetic parameters at materia interfaces) incuding dieectrics and conductors. In addition it needs to be abe to hande surface coatings via the impedance or conducting boundary condition. Most scattering probems are posed on unbounded domains. Many probems require computing the eectromagnetic fied in domains (or for objects) that span mutipe waveengths. Thus good dispersion accuracy is desirabe. The focus of this paper is neither on very ow frequency probems in which the objects are a sma fraction of a waveength, or on very high frequency probems in which objects are many of waveengths ong. This is the so caed resonance region of scattering theory. The method shoud give a inear system that can be soved easiy, and the method shoud be easy to program! Of course it woud be easy to add to this ist (for exampe no mention is made of wires or adaptivity) but, for our project, the above summarizes the main design goas. There is no unique choice of method given the above constraints. However, taking into account these goas we decided to use a voume based finite eement ike procedure due to Cessenat and Després [5, 6] caed the Utra Weak Variationa Formuation (UWVF). This method can be impemented on a tetrahedra finite eement grid, so providing geometric fexibiity. It aso handes piecewise constant media and impedance boundary conditions and we have impemented a conducting boundary condition, athough we sha not discuss that here. The basic UWVF approximates unbounded domains via a ow order absorbing

3 SOLVING MAXWELL S EQUATIONS USING THE UWVF 3 boundary condition and this is a weakness. One goa of this paper is to show how to impement the perfecty matched ayer (PML) of Bérenger [2] in the UWVF to give a better absorbing boundary. The UWVF uses a pane wave basis on each eement and hence we hope has good dispersion behavior athough this has yet to be estabished rigorousy. Finay, due to the exceent appendices in Cessenat s thesis [5], the basic UWVF is reativey easy to program, and the inear system can be soved by a simpe iterative scheme (in our case BiCGStab) which makes paraeization reativey simpe. Of course there are many other more commony encountered competitors to the UWVF. Boundary Integra Equation (BIE) techniques hande the unbounded domain without troube, and can aso hande compex geometry (see for exampe [9]). Their main disadvantages are that penetrabe media need to be impemented either by systems of integra equations on each interface between piecewise constant regions or via voume integra equations (aternativey boundary eement and finite eement methods can be couped - see for exampe [14]). In addition, to sove the resuting inear system (which aso requires to evauate singuar integras in a carefu way) it is necessary to use a fast operator evauation strategy ike the Fast Mutipoe Method (FMM) [12]. On the basis of ease of impementation we eected not to pursue this direction athough such methods are justifiaby very popuar in computationa eectromagnetism. Esewhere, one of us (Monk) and E. Darrigrand have started to test couping boundary integra equations using the FMM and together with the UWVF to provide an aternative mesh truncation procedure [11]. Another competitor is the finite eement method (FEM). Typicay edge eements are used to discretize the eectromagnetic fied [19]. The FEM has the same advantages as the UWVF regarding geometric fexibiity and the handing of penetrabe scatterers (however, unike the UWVF, functionay graded materias where the refractive index varies continuousy can be handed by the FEM). It faces the same difficuties regarding modeing unbounded media. The theoretica anaysis of the convergence of the FEM is much better understood than for the UWVF and programming paradigms are we known. However, in order to obtain good dispersion behavior, it is necessary to use higher order eements which are

4 4 HUTTUNEN, MALINEN, KAIPIO quite compicated to program. In addition the soution of the resuting inear system is difficut (mutigrid is no onger an optima strategy due to imitations on the coarsest grid that can be used). This was the main reason for rejecting the FEM at the time this project was started. Recent advances in hierarchic edge basis functions [1] and non-overapping Schwarz soution methods (see for exampe [8]) makes the FEM approach increasingy usefu. Athough extremey compex, perhaps the utimate scheme of this type is an hp-finite eement method (see for exampe [13]). Finay we mention the popuar Finite Integration Method (FIT) [22, 23]. This is essentiay a voume based finite difference scheme so the representation of curved or compicated boundaries requires specia care. The impementation of impedance boundary conditions is then even more compex. Of course the simpe structured mesh of a finite difference method impies a more efficient code and potentiay specia soution methods when compared to simiar finite eement methods. Of course the UWVF is no panacea. First of a the theoretica convergence properties of the UWVF are not as we understood as for the the other methods mentioned above. Furthermore the method can suffer from bad conditioning probems if the basis or grid is not chosen carefuy. Finay the soution is not computed everywhere but ony on the faces of the tetrahedra mesh (termed the skeeton of the mesh) and requires a post-processing step to obtain the soution away from the skeeton. However the far fied pattern can easiy be cacuated directy from the UWVF soution [5]. A goa of this paper is to show practica ways to avoid or contro the above probems (we sha not address theoretica aspects here beyond presenting a nove and unifying derivation of the method), as we as showing how to hande unbounded media better than the basic UWVF. In particuar we sha present a nove derivation of the UWVF that shows it is a standard upwind Discontinuous Gaerkin (DG) method with a specia choice of degrees of freedom and approximating basis functions. After this derivation we sha address severa practica issues reated to using the UWVF as foows:

5 SOLVING MAXWELL S EQUATIONS USING THE UWVF 5 We sha discuss the choice of the number of basis functions per eement to baance the competing needs of accuracy and conditioning. We sha show how to impement a PML in the UWVF and demonstrate that it provides enhanced performance compared to the basic ow order boundary condition in the standard UWVF. We sha show how to post-process the computationa resuts to approximate the soution away from the mesh skeeton. We sha show that the method can sove compex" scattering probems such as scattering in a ayered medium and scattering from a sphere where exact soutions are avaiabe. We sha aso show how to impement a point source in the method. As part of these investigations we sha aso exhibit the performance of the inear system sover (just the BiCGStab), and investigate the frequency dependence of the soution. We sha try to show that the UWVF is a viabe and usefu Maxwe sover. We finish this introduction by describing in detai the Maxwe system and boundary conditions approximated by the basic UWVF of Cessenat and Després [5, 6]. Suppose Ω is a bounded poyhedra domain in R 3. We want to approximate the eectric fied E and magnetic fied H that satisfy the foowing time-harmonic Maxwe system iωɛe H = in Ω, (1) iωµh + E = in Ω. (2) Here ω is the tempora frequency of the fied, and ɛ and µ are respectivey the permittivity (compex vaued in genera) and permeabiity (rea vaued) of the materia in Ω. In particuar, for ɛ, the rea part R(ɛ) is bounded and stricty positive. The imaginary part I(ɛ) is bounded and non-negative. Finay µ is rea, bounded and stricty positive. The fied is supposed to satisfy the foowing generaized impedance boundary condition E n + σ(h n) n = Q(E n + σ(h n) n) + g on Γ = Ω (3)

6 6 HUTTUNEN, MALINEN, KAIPIO where Q is a rea function of position on the boundary with Q 1, n is the unit outward norma on Γ and g is a tangentia vector fied giving the boundary condition. Finay we use a positive parameter σ defined on the boundary Γ (beware this is not the conductivity!). Often σ = µ / ɛ, but more generay we aow σ to be any bounded stricty positive rea function on Γ. Note that the boundary condition (3) is rather genera and for specia choices of Q can be used to impement severa standard boundary conditions. Choosing Q = 1 we get E n = 2g which is the standard perfect eectricay conducting boundary condition. If Q = we obtain the impedance boundary data. E n+σ(h n) n = g and with an appropriate choice of σ = µ / ɛ where ɛ and µ are the eectromagnetic parameters of free space (rea) we have the owest order absorbing boundary condition that can be used to truncate the computationa domain in a scattering cacuation. Finay Q = 1 gives a magnetic wa condition usefu for approximating surfaces with very high permeabiity (or to impement a symmetry boundary condition). 2. DERIVATION OF THE UWVF In this section we sha derive the UWVF for the basic Maxwe system (1) (3). Our derivation, which differs from that of Cessenat and Després [5, 6], highights the connection between the UWVF and the cassica fux spitting discontinuous Gaerkin method for symmetric hyperboic systems (see for exampe [18]). In fact we sha show that the UWVF can be viewed as a discontinuous Gaerkin method with a specia choice of test and tria space. Let τ h = {K} denote a reguar finite eement mesh of eements K of maximum diameter h covering Ω. In principe, this mesh can be quite genera aowing for mixing various eements (cubes, tetrahedra, prisms etc), but in our impementation we use a tetrahedra finite eement mesh since it is easy to generate and fits genera boundaries reasonaby we. Hence we sha assume that each eement K is a tetrahedron and hence has trianguar faces (so simpifying some integras that need to be performed during the cacuation).

7 SOLVING MAXWELL S EQUATIONS USING THE UWVF 7 We now proceed aong standard ines to derive a discontinuous Gaerkin method for the Maxwe system recaing first the integration by parts identity that for any a and b sufficienty smooth on K K a b dv = K a b dv + n K a b da K where the over-ine represents compex conjugate and n K is the unit outward norma to the boundary K of K. Now et ξ K and ψ K denote smooth test vector functions on an eement in the mesh. Mutipying (1) and (2) by the compex conjugate of ξ K and ψ K and integrating over K using the above integration by parts identity to move the cur off the tria functions we obtain K K ( iωɛe ξ K H ξ K) dv = ( iωµh ψ K + E ψ K) dv K = n K H ξ K da, Adding the two equations and reordering the eft hand side we obtain K K n K E ψ K da. ( ) E (iωɛξ K + ψ K ) + H (iωµψ K ξ K ) dv ( = n K H ξ K n K E ψ K) da K where we have used the fact that µ is assumed to be rea vaued. Usuay, in the derivation of the discontinuous Gaerkin method, we woud now specify how to compute the fuxes" or surface currents n K E and n K H from approximate discontinuous fieds, but in this case we now make an important assumption that is the essentia part of the UWVF. We assume that ξ K and ψ K satisfy the foowing adjoint Maxwe system on K: iωɛξ K + ψ K = in K, (4) iωµψ K ξ K = in K. (5) With this assumption the above identity for (E, H) on K reduces to K ( n K H ξ K n K E ψ K) da = (6)

8 8 HUTTUNEN, MALINEN, KAIPIO We now appy the usua discontinuous Gaerkin upwind spitting method to this identity. Let u K = E K H K and φ K = ξ K ψ K then (6) becomes K D K u K φ K da = (7) where the matrix D K is given by D K = (Z K ) T Z K and ZK = n K 3 n K 2 n K 3 n K 1 n K 2 n K 1. Note that Z K a = n K a for any vector a and Z K = (Z K ) T. Fux spitting now amounts to a suitabe factoring of D K into positive and negative semi-definite parts corresponding to eft and right going waves. To obtain the genera UWVF we use a sighty more genera factorization than usua. Let σ > be defined on the faces of the mesh (on the boundary faces it is the function σ appearing in (3), for other faces we sha discuss the choice of σ ater in this paper). To define the spitting of D K et L K,± = 1 2σ ( Z K, ±σ(z K ) 2) and define D K,± = ±(L K,± ) T (L K,± ). A simpe cacuation, using the fact that (Z K ) T (Z K ) 2 = (Z K ) T then shows that D K = D K,+ + D K, with D K,+ positive semidefinite and D K, negative semidefinite. An important property of the spitting that we sha use ater is that if eements K and K share a common face then on K K we have (using the fact that n K = n K ) L K, = 1 2σ ( Z K, σ(z K ) 2) = 1 2σ (Z K, σ(z K ) 2) = L K,+.

9 SOLVING MAXWELL S EQUATIONS USING THE UWVF 9 Using the spitting of D K and the factorization of each term in the spitting we may rewrite (7) as K ( ) (L K,+ u K ) (L K,+ φ K ) (L K, u K ) (L K, φ K ) da =. (8) The discontinuous Gaerkin fux spitting approach is then to coupe the soution on adjacent eements using the second term in the above equation. Thus if K is an eement sharing a face with K we have (using the continuity properties of the soutions of Maxwe s equations across an interface in the absence of surface charges) L K, u K = L K,+ u K (9) on the common face. For faces on the boundary Γ we use the boundary condition (3) written in the convenient form L K, u K = QL K,+ u K ĝ on K Γ where ĝ = (1/ 2σ)g. Equation (8) then becomes (L K,+ u K ) (L K,+ φ K ) dv + (L K,+ u K ) (L K, φ K ) da K + K, K K=f K Γ=f f f (QL K,+ u K + ĝ) (L K, φ K ) da = (1) This is essentiay the UWVF of Cessenat and Després before discretization but to make the connection more obvious we define X K = L K,+ u K K, Y K = L K,+ φ K K and F K (Y K ) = L K, φ K K. Then (1) becomes the probem of finding X K on the faces K of each eement K such that X K Y K da X K F K (Y K ) da K K, K K=f QX K F K (Y K ) da = K Γ=f f K f ĝ F K (Y K ) da (11)

10 1 HUTTUNEN, MALINEN, KAIPIO for a appropriate Y K. This is the UWVF for Maxwe s equations before discretization. A natura question is what is the correct space for the tria and test functions. It turns out that if L 2 t ( K) denotes the space of square integrabe fieds on K that are tangent to K then we can seek X K L 2 t ( K) on each K such that (11) hods for a Y K L 2 t ( K) and a eements K. The ony difficuty is to see that in this case, given any Y K L 2 t ( K), we can find a soution φ K (in H(cur; K) 2 ) of (4) (5) that satisfies the generaized impedance boundary condition L K,+ φ K = Y K on K and furthermore that F K (Y K ) L 2 t ( K). This can be proved as usua by studying the impedance probem on a bounded domain (see [5, 19]). It can be shown [5] that (11) has exacty one soution for any ĝ L 2 t (Γ) and that if u K is computed eement-wise using the Maxwe system (1)-(2) together with the boundary condition L K,+ u K = X K, then the resuting piecewise defined function is exacty the soution u that satisfies (1) (3). We remark that the above derivation of the UWVF extends in a simpe way to the equations of easticity and to the Hemhotz equation written as a first order system (indeed to a genera cass of symmetric hyperboic equations). It remains ony to discretize the UWVF and here we foow exacty [5]. Hence we ony give enough detais to make this paper sef contained and refer the reader to the origina source for a detaied discussion of the discrete probem. The key to Cessenat and Després discrete UWVF is to choose the basis functions for X K and Y K in a way that aows the action of the operator F K to be computed easiy. Various choices are possibe, but for them a it is necessary to assume that ɛ and µ are piecewise constant on the mesh. We sha assume this from now on in the paper. Under this restriction we use pane wave soutions to approximate φ K. In practice, foowing [5], a suitabe famiy of pane waves are generated on K by choosing p K directions d K, d = 1, 1 p K (we use the optima spherica codes from the website [21]), and then defining a unit rea poarization vector ψ K, orthogona to d. From this we compute the

11 SOLVING MAXWELL S EQUATIONS USING THE UWVF 11 compex poarizations F K = ψ K, + iψ K, d K and G K = ψ K, iψ K, d K, 1 p K. It is then easy to verify that the functions (ξ F,K, ψ F,K ) given by ξ F,K = µ K F K exp(ikd K x) and ψ F,K = i ɛ K F K exp(ikd K x) where k = ω µ K ɛ K satisfy the adjoint Maxwe system (4)-(5) on K. Simiary, the pair (ξ G,K, ψ G,K ) given by ξ G,K = µ K G K exp(ikd K x) and ψ G,K = i ɛ K G K exp(ikd K x) are an independent set of soutions of the adjoint Maxwe system. These functions in turn generate pane waves φ F,K = ξ F,K ψ F,K and φ G,K = ξ G,K ψ G,K for 1 p K. The reason for this somewhat compex choice of pane wave is that ξ G,K ξ F,K = for any 1, p K which provides some extra sparseness in certain matrices to be defined shorty. We can now define the approximation to X K denoted X K h and given by X K h = p K =1 x K L K,+ φ F,K + x K p K +L K,+ φ G,K. To compute the unknown expansion coefficients {x K }pk =1 for each eement K in the mesh we foow the usua Gaerkin approach of substituting X K h in pace of X K in (11) and choosing the test functions Y K to be successivey the basis function Y K = L K,+ φ F,K, 1 p K and Y K = L K,+ φ G,K, 1 p K. We note that F K is easy to compute for these basis functions since F K (L K,+ φ F,K ) = L K, φ F,K and simiary for the G" basis functions. By enumerating the the tetrahedra we can form a vector of unknowns x of ength M = K τ h 2p K containing x K, 1 2p K in the

12 12 HUTTUNEN, MALINEN, KAIPIO same order as the eements are numbered. This vector satisfies that matrix equation (D C) x = f (12) where D is the bock diagona Hermitian M M matrix resuting from the first term on the eft hand side of (11), C is the sparse M M matrix resuting from the remaining two terms on the eft hand side and f is the oad vector given by the right hand side after choosing Y K to be each of the discrete basis functions. In order to compute these matrices, integras need to be performed over the faces of the tetrahedra in the mesh. For exampe, in order to compute the matrix D we must evauate integras of the form K (L K,+ φ F,K ) (L K,+ φ F,K ) da for 1, p K and simiar integras invoving the G" basis functions (as we as integras invoving both F" and G" basis functions for other terms in (11)). Using the definition of the operator L K,+ and the basis function φ F,K we see that L K,+ φ F,K = 1 2σ ( µz K F K ) + i ɛ K (Z K ) 2 F K exp(ikd K x) with a simiar expression for (L K,+ φ F,K ). On each panar face f of K the matrix Z K is constant, and we aso assume that σ is aso constant on f thus (L K,+ φ F,K ) (L K,+ φ F,K ) da = K 1 ( µz K F K + i ɛ 2σ K (Z K ) 2 F K f f K exp(ik(d K d K ) x) da f ) f ( µzk F K + i ɛ K (Z K ) 2 F K ) f The vector part of this expression is easy to compute and reativey cheap since the dot product must be done once per face. The main difficuty in computing this term is the compex exponentia integra. Fortunatey, Cessenat [5] shows that the integra over the face can be computed in cosed form as a difference of sinc functions. Athough some care needs to be taken to avoid canceation errors this integra is then easy to impement.

13 SOLVING MAXWELL S EQUATIONS USING THE UWVF 13 The remaining integras in (11) can be computed in the same way. The resut is that we can compute a bock diagona inner product matrix D corresponding to the first term on the eft hand side of (11). This matrix is Hermitian and positive definite (as ong as the wave directions are distinct), but may become severey i-conditioned if too many directions are used. We sha return to this point when we discuss the numerica impementation of the method in Section 3. We aso construct a matrix C corresponding to the remaining terms on the eft hand side of (11). This is a more genera matrix that coupes the expansion coefficients on a tetrahedron K to the expansion coefficients on tetrahedra K that share a face with K. The right hand side in the matrix equation (12) can aso be computed in the same way provided the source vector g invoves ony compex exponentias (such as is the case when soving scattering probems when a pane incident wave strikes a given scatterer and it is desired to compute the scattered fied). Otherwise quadrature is needed to compute the right hand side. This is rather expensive due to the osciatory nature of the basis functions and our current impementation does not aow genera boundary data for this reason. In other appications of the UWVF in two dimensions we have found that when the boundary of the domain is smooth, the accuracy of the UWVF can be improved by using curved edges to the eements adjacent to the boundary. This requires using quadrature to compute the integras on such curved edges. So far we have not impemented this scheme in 3D but a future improvement to our UWVF code woud be to improve the accuracy of the boundary representation in this way (at the cost of an increase in computer time to compute the matrices D and C). Currenty we use a refined grid near to curved boundaries and aow the eements to grow rapidy away from the boundary. 3. IMPLEMENTATION In the previous section we provided a nove derivation of the UWVF and summarized how, after choosing the number of pane wave directions on each eement, we can compute a matrix system (12) that must be soved in order to obtain the coefficients of the surface

14 14 HUTTUNEN, MALINEN, KAIPIO unknown X K h. In this section we discuss severa practica choices needed to improve the basic UWVF Choice of the pane wave directions The choice of the number and directions of the pane waves on each eement has a critica infuence on the accuracy of the discrete UWVF. Choosing too many directions on a given eement can resut in a very poory conditioned matrix D (introduced in the previous section). Since our inversion scheme requires to compute D 1 this can cause the iterative method to fai to converge. Thus the choice of the number of directions on a given eement requires a baance between accuracy and conditioning. Cessenat and Després suggest the use of a fixed number of directions on a the eements in the mesh. In this case we reca the basic convergence resut due to Cessenat [5]. Theorem 3.1. Suppose ɛ = µ = 1 and that p directions are used on each eement in a quasi-uniform and reguar mesh. Then there is a set of directions {d n } p n=1, where p = (N + 1)(N + 3) such that if Q < 1 on Ω then X X h L 2 ( Ω) Ch N+1/2 as the mesh size h decreases, provided the soution is sufficienty smooth. This theorem tes us that, at east for smooth soutions to the Maxwe system, we can increase the order of convergence of the method by increasing the number of directions p per eement (simiar to an h-finite eement method where the degree of the poynomia basis governs the order of convergence for smooth soutions). In addition it was shown in [1] that for any fixed mesh, X X h V as p for any eectromagnetic fied in H(cur; Ω) (again assuming ɛ = µ = 1) provided the directions are chosen to be suitabe quadrature points on the unit sphere. These resuts suggest that choosing p arge may be advantageous for accuracy.

15 SOLVING MAXWELL S EQUATIONS USING THE UWVF 15 Using the UWVF for acoustic probems, we have found that a uniform choice of p across a eements may ead to poor accuracy on arge eements, or poor conditioning on sma eements [17]. In [17] we advocated the practica choice of setting a maximum aowabe condition number and choosing the number of directions p K for each eement K to be the argest number such that the bock of D associated with K has condition number at or beow the cutoff. Thus we emphasized the practica need for convergence of the iterative scheme at the expense of indirect contro over accuracy. This choice works we for a seria program, but when we come to impement a parae code we woud ike to predict approximatey the number of directions per eement quicky to hep oad baancing. This choice can ater be refined eement by eement after the parae job has been aocated to the processors as ong as the number of directions per eement does not change greaty. We have adopted a heuristic for choosing the number of directions per eement based on the anaysis of the error in using pane waves to approximate an independent pane wave not in the basis. This anaysis does not incude poarization effects at this stage and is therefore incompete, but it does suggest why pane waves are essentiay equivaent to Besse functions for buiding the basis. Another imitation of the anaysis is that it does not appy to evanescent waves. Suppose we wish to approximate a pane wave exp(ikx d), d = 1, using a sum of pane waves in the directions d 1,, d pk on an eement K. We choose the origin of the coordinate system to be at the center of the inscribed sphere (having radius ρ K ) and denote by h K the maximum distance of points on K from this inscribed center. In fact h K is roughy the radius of the eement if the eements are reguar. The assumption behind the anaysis we sha now give is that kρ K is arge so that the inscribed sphere is many waveengths across (and of course kh K is sti arger). For any point on the surface of the eement ρ K x h K Thus we can seek to approximate exp(ikd x) to error ɛ in the maximum norm for arge k x kρ K. Carayo and Coino [4] show that if L K kh K ( 2 3 ) 3/2 W 2/3 ( 2khK 3ɛ 2 ) (kh K ) 1/ terms vanishing in kh K. (13)

16 16 HUTTUNEN, MALINEN, KAIPIO where W is the Lambert W function defined on [1/e, ) by W (t) exp(w (t)) = t, then for arge kh K we can truncate the Jacobi-Anger expansion to error ɛ using (L K +1) 2 terms in the sum. Using the fact that L K is an increasing function of kh K and k x is aso arge on the surface of the eement we have exp(ikd x) 4π L K = m= i j (k x )Y m (ˆx)Y m(d) to error ɛ where ˆx = x/ x, j is the th spherica Besse function and Y m is the spherica harmonic of index and momentum m. The same hods for each of the directions d j and thus we can write exp(ikx d) 4π L K = m= p K n=1 c n exp(ikx d n ) i j (k x )Y m (ˆx) ( Y m (d) pk n=1 c n Y m (d n) to accuracy (1 + p K n=1 c n ) ɛ when k x is arge and where the c n, n = 1,, p K are suitabe expansion coefficients. The right hand side wi vanish if we can choose the coefficients c n such that Y m (d) = p K n=1 c n Y m (d n ) for L K and m. For a given choice of d and approximating directions this is a system of (L K + 1) 2 equations in p K unknowns (the c n s). We thus can choose p K = (L K + 1) 2 and then choose the directions {d n } pk n=1 so that they form a fundamenta system for the given set of spherica harmonics (this is aways possibe since the spherica harmonics are ineary independent) [24]. This guarantees the invertibiity of the matrix that woud arise if we wished to compute the coefficients c n. Hence the above equation is sovabe. Various tabes of fundamenta sets of directions for L K = 1,, 29 are given in [2] (optima with respect to different criteria). Note that the need for a fundamenta set of directions is aso suggested by the statement of the error estimate in Theorem 3.1 where ony certain choices of directions give a good error estimate. )

17 SOLVING MAXWELL S EQUATIONS USING THE UWVF 17 TABLE 1 Factor `1 + P p K n=1 c n appearing in the error anaysis as a function of L K computed from 1, randomy chosen d vectors on the unit sphere. Order L K Dimension p K Factor `1 + P p K n=1 c n There remains the possibiity that (1 + p K n=1 c n ) might grow very rapidy with p K and hence make the above estimates meaningess. We use the directions {d n } pk n=1 (from p K = 4 to p K = 121 or L K = 1,, 1) for the optima spherica codes from the website [21]. For these directions we have computed the expansion coefficients c n, n = 1,, p K for p K = (L K + 1) 2 and L K = 1, 1 tested at 1, randomy chosen points on the unit sphere. The resuts are shown in Tabe 1 and show that the factor generay does grow for the vaues of L K used here In fact the growth (ignoring L K = 4) is consistent with an exponentia growth (proportiona to exp(.35l K )). We have no expanation for the arge vaue for L K = 4. In concusion, if we compute L K via (13) and set p K = (L K + 1) 2 we can approximate the trace of the pane wave on the boundary of the eement K to accuracy approximatey ɛ provided the eement is non-degenerate and k x is arge on the faces of the eement (i.e. for exampe if the eement contains a sphere that many waveengths across) and the directions are chosen to be a fundamenta set. In practice we use estimate (13) to motivate a heuristic for cacuating L K even for eements that are sma compared to the waveength of the radiation, but this needs to be improved. Note that, even for arge eements, this procedure ony concerns the oca approximation of pane waves and does not guarantee good dispersion error or good approximation error for more genera fieds.

18 18 HUTTUNEN, MALINEN, KAIPIO Note that if kh K is arge, due to the sower than ogarithmic growth of W, we have that L K Ckh K + 1/2 for any fixed C > 1 and for kh K arge enough. This avoids the (simpe) task of computing W. Thus, in practice, we aow the user to choose coefficients A j, j =, 1, 2, then compute p K = A 2 (kh K ) 2 + A 1 kh K + A and set p K to be the smaest integer greater than ( L K + 1) 2. This gives the number of directions on K and the actua directions are drawn from the tabe of spherica codes as mentioned above. No restriction is paced on the choice of p K other than the practica restriction that 3 p K 13 so adjacent eements may have widey different number of basis eements. We sha present more detais of this approach and some numerica tests in Section Adding a PML ayer The first order absorbing boundary condition obtained by setting Q = and σ = µ / ɛ in (3) requires the absorbing boundary to be far from the scatterer to obtain reasonabe accuracy. It is thus desirabe to be abe to use more efficient mesh termination methods. In [16] we showed how to impement the Perfecty Matched Layer (PML) in the UWVF for acoustics. A simiar approach can be taken for the Maxwe system and we now outine that approach here. The PML is appied to the Maxwe system in free space where ɛ = ɛ and µ = µ. For simpicity we sha assume that the PML is appied when x i = x,i > so the standard Maxwe system governs the fied in the box x i x,i, i = 1, 2, 3 which contains the scatterer. The PML wi occupy the region outside the box and within the box x,i L i < x i < x,i +L i, i = 1, 2, 3. Thus the PML has thickness L i in the direction x i aong the ith coordinate axis. Our experience with the acoustic UWVF is that, unike standard finite eement methods, the UWVF works we with a constant absorption in the ayer. As we sha see this aows the anaytic cacuation of certain integras in the theory, and the constant PML does not cause unacceptabe refections at the

19 SOLVING MAXWELL S EQUATIONS USING THE UWVF 19 PML boundary. A key assumption is that the panes x i = ±x,i, i = 1, 2, 3 at the interface between the PML and the ordinary Maxwe region of the computationa domain are the union of faces of eements (i.e. the panes coincide with boundaries between eements - any eement is either entirey in the PML or entirey in the Maxwe part of the computationa domain). In order to define the PML we use a compex stretching of the spatia variabes [7, 19] so that we define x + iσ x i x,i for x i > x,i x i = k L i (14) x i for x i < x,i for i = 1, 2, 3 where k = ω ɛ µ, L i is the previousy defined thickness of the absorbing ayer in the ith direction and σ i, > is the constant PML absorption parameter in the ith direction. Repacing x i, formay, by x i in (1)-(2) defines the non-physica eectromagnetic fied denoted Ẽ and H which satisfy the Maxwe system with respect to the tide" variabes: iωɛ Ẽ H =, iωµ H + E =, in the PML where denotes the cur in tide variabes. We now use the definition of the tide" variabes to change variabes back to rea coordinates x i, i = 1, 2, 3. Define d i = 1 + iσ i, /(kl i ) and et the matrices A and B be given by 1/(d 2 d 3 ) d 1 A = 1/(d 1 d 3 ) and B = d 2 1/(d 1 d 2 ) d 3 then after the change of variabes the above equations become iωɛ Ẽ A B H =, iωµ H + A BE =.

20 2 HUTTUNEN, MALINEN, KAIPIO Defining the computed fieds in the PML denoted, in an abuse of notation, by E = BẼ and H = B H we obtain the foowing system for the non-physica eectromagnetic fied in the PML: iωɛ A 1 B 1 E H =, (15) iωµ A 1 B 1 H + E =. (16) Thus defining the non-physica anisotropic eectromagnetic parameters in the PML by ɛ B = ɛ A 1 B 1 and µ B = µ A 1 B 1 we see that in the PML the fieds satisfy the Maxwe system (1)-(2) with ɛ and µ repaced by ɛ B and µ B respectivey (the subscript B refers to J.P. Bérenger who first proposed the PML in 1996 [2]). Note that now µ B is compex vaued, and ɛ B and µ B are symmetric but not Hermitian matrices. Thus, within the PML region, the derivation of the UWVF in Section 2 appies. The ony change is to aow µ to be compex so that the adjoint probem becomes iωɛ B ξ K ψ K =, (17) iωµ B ψ K + ξ K =, (18) in each eement K in the PML. The same boundary condition can be used on the outer surface of the PML, and since the matrix B is continuous within the PML, the same matching condition and fux computation can be used across inter-eement boundaries. At the boundary between the PML and Maxwe (vacuum) regions the same matching of fieds between adjacent eements can aso be performed. This is because the boundaries between the PML and Maxwe regions are coordinate panes that are the union of faces in the mesh. For exampe suppose we have one tetrahedron K in the vacuum region, and another K in the PML meeting at a common face on the surface x 1 = x 1,. Across the interface (i.e. on f) the change of variabes approach impies that E K n K = ẼK nk. But d 2 = d 3 = 1 in K and ony d 1 1. Since ony tangentia components of the fied are continuous across f we have aso E K n K = (BẼK ) nk = E K n K. Simiary H n K is aso continuous across f. Thus the fux matching equation (9) hods between K

21 SOLVING MAXWELL S EQUATIONS USING THE UWVF 21 and K since both sides ony invove tangentia components of the reevant fieds. We can concude that the UWVF equation (11) hods throughout the Maxwe region and the PML regions of the computationa domain, provided the modified adjoint equations (17)-(18) are used in cacuating F K for eements in the PML. The PML can be discretized, because, by construction, pane wave soutions of the adjoint system (17)-(18) can be derived from standard pane wave soutions in the tide" coordinates via the change of variabes (14) Point sources In Section 4 we sha investigate fieds originating from an eectric dipoe source at the point x Ω. The dipoe point source can be defined as the soution of the Maxwe system iωɛe H = j in Ω, (19) iωµh + E = in Ω., (2) where j = Iaδ x and where I and a, a = 1 denote the ampitude and poarization of the dipoe. In addition, δ x denotes the Dirac deta function. Foowing the procedure of Section 2, it is easy to show that the right hand side of equation (19) provides a term to the right hand side of the UWVF equation (11). For the point source in the eement x K, the additiona term is simpy as 2 j E K = 2Ia E(x ). (21) K 3.4. Reconstruction within eements Since the UWVF method provides an approximation for the function X K (which is a function of E and H) on each eement face K, but not a direct soution for the eectric fied E and the magnetic fied H, a post-processing step is needed to resove E and H within eements. The approximation X K h for the function X K is constructed by using pane

22 22 HUTTUNEN, MALINEN, KAIPIO waves φ F,K and φ G,K which in turn are soutions of the adjoint Maxwe system (4) and (5). It is cear that in the absence of absorption, the permittivity ɛ and the wave number k are rea vaued and the adjoint system (4) and (5) is the same as the physica Maxwe equations (1) and (2). Consequenty, for rea vaued ɛ, the pane wave basis functions φ F,K and φ G,K are soutions of the oca Maxwe equations in the corresponding eement K. Therefore, it is easy to observe that for any eement K in a non-absorbing medium, the approximation u K h for uk is u K h = p K =1 x K φ F,K + x K M k +φ G,K. (22) Resoving the fied u K for eements in an absorbing medium (i.e. ɛ is compex vaued) or within the PML requires a different approach. The method used here is anaogous with the UWVF post-processing technique introduced for the Hemhotz probem in [16]. Namey, we want to approximate the fieds E and H in a pane wave basis which is a soution of the actua Maxwe system (1) and (2) (or (15) and (16) in the PML), rather than using the adjoint pane wave basis of the discrete UWVF. Therefore, we define a new set of pane waves basis functions as where the pairs (ˆξ F,K, ˆφ F,K = ˆψ F,K ˆξ F,K ˆψ F,K ) and (ˆξ G,K, and ˆφG,K = system (1) and (2) ((15) and (16) in the PML) so that ˆξ G,K ˆψ G,K G,K ˆψ ) are soutions of the physica Hemhotz ˆξ F,K = µ K F K exp(iˆkd K x), ˆψF,K = i ɛ K F K exp(iˆkd K x, ) ˆξ G,K = µ K G K exp(iˆkd K x), ˆψG,K = i ɛ K G K exp(iˆkd K x), where ˆk = ω µ K ɛ K. For the eements in the PML, the permittivity ɛ K and permeabiity µ K are taken as the modified parameters ɛ K = ɛ B and µ K = µ B which aso eads to a matrix vaued wave number ˆk.

23 SOLVING MAXWELL S EQUATIONS USING THE UWVF 23 The next step is to compute new coefficients y K, 1 2p K for this new basis corresponding to the coefficients x K of the discrete UWVF probem. In particuar, in each eement in an absorbing medium (a physica interpretation of the PML is an anisotropic absorbing medium) we want to approximate the soution u K as u K h = p K y K =1 F,K ˆφ + ym K G,K k + ˆφ. (23) In the context of the UWVF, the coefficients y K for the eement K are naturay obtained as a soution of the equation y K = ˆD 1 K D K x K, (24) where the vectors y K and x K contain the coefficients y K and x K, 1 2p K for the eement K. The 2p K 2p K matrices ˆD k and D k are assembed as the diagona bocks of D in the discrete UWVF equation (12). However, due to the two types of pane wave bases invoved in the post-processing, the matrix ˆD K is computed using integras of the form K (L K,+ ˆφF,K ) (L F,K K,+ ˆφ ) da, i.e. using the physica (non-adjoint) pane waves ony. Simiar integras are needed for the basis functions invoving G basis functions as we as both F and G functions. The integras for the matrix D K incude both adjoint and non-adjoint basis functions being of the form K (L K,+ φ F,K ) (L F,K K,+ ˆφ ) da. In essence, the post-processing step for a non-absorbing medium is trivia since the soution for E and H can be extended directy within eements using the same pane basis functions and coefficients needed to approximate the UWVF function X, see Eq. (22). For eements in an absorbing medium or in the PML, the extension of the soution in the

24 24 HUTTUNEN, MALINEN, KAIPIO eements can be computed simiary (see Eq. (23)) but by defining a new non-adjoint pane wave basis and by resoving coefficients for the new basis using a reativey simpe UWVF-type fitting (24) Iterative soution and paraeization In his thesis Cessenat [5] suggests to sove the UWVF equation (12) by writing it as (I D 1 C) x = D 1 f (25) and appying a damped Richardson scheme. Note that D 1 is easy to compute since D is bock diagona, so the action of D 1 C on any vector can be computed at the expense, essentiay, of mutipying by C. We have found that the stabiized bi-conjugate Gradient scheme (BiCGStab) is faster, and we use that method in a the exampes presented here. The UWVF has been paraeized using the same technique used to paraeize the acoustic UWVF in [15]. A domain decomposition strategy is used. The mesh is decomposed into coections of eements using METIS. Because the number of basis functions per eement differs widey, the predicted number of basis functions per eement is used to weight the METIS graph nodes to improve oad baancing. Note that eements are ony connected through faces which simpifies the connectivity graph and decreases the number of eements in one sub-domain that are connected to another compared to a FEM soution. Once the mesh is partitioned, each partition is sent to a processor (using MPI) and the processor performs the conditioning check and adjusts the number of unknowns per eement (as described in Section 3.1). The oca matrix D is aso computed. This requires no communication. Then the matrix C is computed requiring communication to determine the directions on eements neighboring each partition (through faces). Finay f is computed ocay. Then the bi-conjugate gradient scheme is paraeized in the usua way using parae matrix mutipy. 4. NUMERICAL RESULTS

25 SOLVING MAXWELL S EQUATIONS USING THE UWVF 25 To investigate the UWVF method for simuating actua probems invoving the timeharmonic Maxwe probems, we study the method for three different mode cases for which the exact soution is known. In the first case we approximate the fied emitted by an eectric dipoe in free-space. Since the domain of the probem is physicay unbounded, an absorbing boundary condition (ABC) is need on the exterior boundary of the computationa domain. We compare two ABCs of which the first is obtained by choosing Q = in (3). This condition is referred to as ABC in the foowing sections. The second method to truncate the domain is the perfecty matched ayer (PML) outined in Section 3.2. Second, we approximate the fied emitted by the dipoe in an inhomogeneous medium. In particuar, when the dipoe is ocated over a ayered materia for which an exact soution is avaiabe via the Sommerfed integra [19]. The third mode probem is the scattering of a pane wave from a perfecty conducting sphere. In this case, the principa interest is in the computation of the far-fied pattern and in the efficiency of the paraeized UWVF code. However, prior to proceeding to specific mode cases we sha outine the method for seecting a stabe basis for the discrete UWVF. Finay we provide some soutions using the NASA Amond and compare the resuts to resuts in the iterature The choice of basis Since it is known from the previous UWVF studies that the method can suffer from instabiity if the pane wave basis is not carefuy chosen [17], we begin this study by examining a method for seecting a basis on each eement (i.e. a possiby different number of directions on each eement) which eads to stabe soution of the UWVF probem. The stabiity of the probem is in particuar importance since we use the Bi-CGstab iteration for soving the UWVF matrix system. As was noted in 2D UWVF simuations of the Hemhotz equation in [17], if the number of basis directions is too arge, the matrix (I D 1 C) may become i-conditioned. It was aso observed that by controing the condition number of matrix bocks D K it is

26 26 HUTTUNEN, MALINEN, KAIPIO possibe to have contro over the conditioning of the overa UWVF matrix system. More precisey, a arge toerance is set and the computation of the matrix bocks D K is begun by using a reativey sma number of basis functions per eement. After the assemby of D K, its condition number is computed. If the condition number is beow the predetermined toerance, the oca basis dimension (number of directions) is increased and the matrix D K is recomputed. This procedure is repeated unti the argest number of basis functions giving a condition number beow the toerance found. Our experience is that this approach ensures that the iterative scheme for soving (25) converges provided the toerance is not chosen too arge. Accuracy can be improved by choosing the toerance arger within the overa constraint of requiring the iterative method converge. Whie in 2D it is possibe to make a reativey poor initia guess for the basis dimension, due to the wider range of possibe basis dimension in 3D, a better approach for 3D probems is needed. The anaysis of Section 3.1 showed that when kh K is arge, the approximation error for p K pane waves is reativey a simpe function of k K h K. On the other hand, numerica experiments for the 3D Hemhotz probem in [15] show that by constraining the condition number of D K eads to amost inear reation of scaed wave number k K h K and the basis dimension p K. Despite the fact that the reationship between the conditioning and the error is not yet propery understood, we focus on the controing the conditioning since i-conditioning eads to divergent Bi-CG iterations. Let k re denote the rea part of the wave number k. In Fig. 1, the basis dimension p K is potted as a function of k re h av K when the maximum condition number of the matrix bocks D K is imited by the toerances 1 5, 1 7 and 1 9. The eement size parameter h av K defined as a mean distance of the eement vertices from its centroid by is h av K = x K CM x K j, j=1 where x K CM is the position of the centroid of the tetrahedron K and xk j, j = 1,...4 are the coordinates of the vertices.

27 SOLVING MAXWELL S EQUATIONS USING THE UWVF max(cond(d K )) 1e5 max(cond(d K )) 1e7 max(cond(d K )) 1e9 8 p K FIG av k K h K The number of basis functions p K as a function of k K h av K when the basis dimension is chosen by constraining the maximum condition number of D k. As in the Hemhotz case [15], we see an amost inear reationship between the basis dimension and the eement size scaed wave number k re h av K. Motivated by this observation and our accuracy considerations in Section 3.1, we suppose that the basis dimension can be approximated by using a quadratic poynomia of the form p K = round(a 2 (k re h av K ) 2 + A 1 k re h av K + A ). (26) The coefficient A 2, A 1 and A computed using a east-squares fit to data of Fig. 1 are isted in Tabe 2. TABLE 2 Parameters for the basis poynomias Max(Cond(D K )) A 2 A 1 A 1e e e We note that since the estimate (26) uses the rea part of the wave number ony, the absorption is not taken into account. However, numerica simuations show that the absorption has a strong effect on the condition number of D K. Since we want to investigate the performance of the PML (which generates an absorption), the estimate (26) for the

28 28 HUTTUNEN, MALINEN, KAIPIO eements in the PML is miseading. Consequenty, the basis estimate (26) is used mainy for choosing the initia basis for the conditioning based seection. In particuar, to give a fair comparison between the PML and ABC, the same strategy for choosing the basis must be used. We sha show, however, in Section 4.6 that in the absence of absorption, the estimates of (26) are directy appicabe Eectric dipoe in free-space The first mode probem we investigate is to compute the fied due to an eectric dipoe which we get by choosing equation (21) as the right hand side of the UWVF. In the free-space, the exact soution of the probems is where E ex = iωiφ(x, x )a φ(x, x ) = exp(iω ɛ x x ). 4π x x I iωɛ x( x φ a), (27) The geometry and the mesh used in the free-space dipoe simuations are shown in Fig. 2. To avoid possibe spurious accuracy due to symmetry, the point source is ocated at the point (.2,.2,.2) of the cube centered at the origin. The cube is surrounded by.1 thick PML. Due to the presence of the singuarity at the ocation of the point source, the mesh is refined near the point (.2,.2,.2). In a simuations for this probem, the basis is chosen by imiting the condition number of the matrix bocks D K. The initia guess for the basis is made using the poynomias of Fig. 1. During the assemby of matrix D, the argest number of pane waves which give the condition number of D K beow the predetermined imit is chosen. The Bi-CGstab is terminated when the reative residua is beow 1 5. Fig. 3 shows that a three condition number imits used in Fig. 1 ead to convergent Bi- CGstab iteration when the anguar frequency is ω = 1π and the ABC is used. Subsequent simuations show that the condition numbers beow 1 5 give a sufficienty arge basis dimension for accuracy in the frequency range used in this study (to be quantified shorty).

29 SOLVING MAXWELL S EQUATIONS USING THE UWVF 29 FIG. 2. Left: The domain encosing the point source at (.2,.2,.2) (mark by a sma sphere). The actua region of interest is the cube surrounded by a.1 thick PML. Right: The mesh for the probem consisting of 2262 eements and 443 vertices. The mesh is refined near the ocation of the point source so that eements size increases with distance from the source r as 5r 2. Reative residua max(cond(d K )) 1e5 max(cond(d K )) 1e7 max(cond(d K )) 1e Iteration number FIG. 3. The convergence of the Bi-CGstab for different condition numbers at ω = 1π. Therefore, it is used in the rest of the simuations. Hence, the number of basis functions in each eement are approximatey the same as shown in the owest graph of Fig. 1, depending on the oca wave number and size of the eement. We want to note, however that the actua number of basis functions can have sma variation between eements, despite the same oca wave number k K and eement size parameter h av K, since the conditioning is aso affected by the shape of the eements. And as noted earier, the absorption in the PML eements reduces the number of basis functions as compared to the estimates of Fig. 1 which are computed for a non-absorbing medium.

30 3 HUTTUNEN, MALINEN, KAIPIO The effect of the PML One objective of this study is to investigate the performance of the PML in the UWVF as a method for eiminating spurious refections arising from the truncation of wave probems on unbounded domains. Fig. 4 shows the soution for the free-space dipoe at the anguar frequency ω = 3π. rea( E 3 ), Exact rea( E 3 ), UWVF ABC rea( E 3 ), UWVF PML z z z y y y 15 E, Exact E, UWVF ABC E, UWVF PML z 1 z 1 z y y y E E ex E E ex z.6 z.6 FIG y y UWVF approximation for the free-space dipoe at ω = 3π. The exact soution is computed using equation (27). The UWVF-ABC corresponds to the absorbing boundary condition (3) with Q = and g =. The approximation with the PML is for the decay parameter σ =. The top row shows the rea parts of the z-component of the eectric fied. The midde row shows the fu ampitude of the eectric fied E. The bottom row presents the distribution of the error in the UWVF-ABC and UWVF-PML soutions. The errors for the ABC and PML approximations are 6.69% and 2.63%, respectivey. The effect of the PML decay parameter σ on the accuracy of the UWVF approximation is investigated in Fig. 5. Since the boundary condition on the exterior boundary of the PML is equation (3) with Q = and g =, the case σ = corresponds to the ow order

31 SOLVING MAXWELL S EQUATIONS USING THE UWVF 31 absorbing boundary condition referred to as ABC in this study. Simuation are computed for three different anguar frequencies ω = 1π, 2π and 3π. The corresponding waveengths are λ =.2,.1 and.667, so the thickness of the PML in terms of waveengths is.5λ, λ and 3/2λ. The same figure aso shows the number of Bi-CGstab iterations needed to reach the reative residua beow 1 5. Reative error ( % ) ω = 1π ω = 2π ω = 3π Number of iterations FIG σ σ Left: Error as a function of the PML decay parameter σ at the point (-.2,-.2,-.2). Right: The number of BiCGstab iterations as a function of σ in the same probem. These resuts suggest that the UWVF approximation using the PML becomes unstabe at ow frequency causing an increased number of iterations. On the other hand, there is a window of vaues for σ which improve the accuracy of the UWVF-PML in comparison to the UWVF-ABC. When the decay parameter is too arge, the error increases. This is caused by the reduced number of basis functions in the PML eements resuting from the condition number based criterion for choosing the basis. More precisey, we imit the maximum condition number of bocks D K beow 1 5. At arge σ this criterion is met or even exceeded when the number of pane waves in the PML eements is ony three. The instabiity of the UWVF-PML approximation at ow frequencies is seen more ceary in Fig. 6 in which the error and the number of degrees of freedom (DOF) are potted a function of the frequency. The axis on the top of the error pot shows the ratio of the maximum eement size h max and the waveength λ. The PML eads to poorer accuracy than the ABC at the owest frequency ω = 5π after which the performance of the PML improves. Resuts suggest that the use of the PML at higher frequencies has two advantages, First, it reduces the error. Second, due to the smaer number of basis functions needed for

32 32 HUTTUNEN, MALINEN, KAIPIO h max /λ Reative error ( % ) ABC PML, σ = 2. DOF x FIG. 6. 1pi 2pi 3pi 4pi ω.5 1pi 2pi 3pi 4pi ω Left: Error as a function of the anguar frequency ω at the point (-.2,-.2,-.2). The top axis of the figure shows the number of waveengths per argest eement edge ength in the non-pml region h =.17. Right: The number of degrees of freedom (DOF) as the function of ω. the eements in the PML, it aso reduces the size of the probem (of course, the PML woud aso aow us to reduce the size of the computationa domain which woud further reduce the size of the probem) Fied near the singuarity Since the fied of the dipoe has a singuarity at its origin, it is important to investigate the error of the UWVF approximations as a function of the distance from the singuarity. Fig. 7 presents the error for approximating the soution using the UWVF-ABC and the UWVF-PML aong the diagona of the cubic computationa domain. The error is potted on the ine from the point (-.5,-.5,-.5) to (.5,.5,.5). As is to be expected, the error peaks strongy at ocation of the dipoe where the true soution is unbounded. Whie the soution with the ABC has wavy spurious refections, the error for the PML is smoother within the computationa domain. However, the PML error aso increases rapidy near the corner of the actua computationa domain. This suggests that the PML sti induces weak refections back to the computationa domain Layered media The UWVF method can easiy be used for probems in an inhomogeneous medium. Fig. 8 shows the UWVF approximations for a dipoe source above a ayered medium. The

33 SOLVING MAXWELL S EQUATIONS USING THE UWVF 33 5 ω = 1π 5 ω = 2π Reative error (%) Reative error (%) x.5.5 x 5 ω = 3π Reative error (%) FIG x The error on the diagona of the computationa domain (aong the ine from (-.5,-.5,-.5) to (.5,.5,.5)). The dashed ine is the error for the ABC and soid ine shows the error when the PML is used. The PML simuations are computed using the decay parameter σ = 2.. The peak in the error is at the ocation of the dipoe at (.2,.2,.2). upper domain z > has n = 1 and in the region z < n = 2. Resuts are shown for ω = 3π. As in the case of the homogeneous medium, the use of the PML reduces spurious refections from the exterior boundary. This can be shown by comparing the resuts for the ABC and the PML with a anaytica soution of the probem which is outined in [19]. In Fig. 9 we pot the fied E aong the ine z =.2 in the x = pane. The wavy spurious refection of the ABC are amost extinguished when the PML is used. Consequenty, the error is reduced from from 3.4% to.8% Scattering from a sphere The third mode probem we study is the scattering of a pane wave from a perfecty conducting sphere with radius R =.5. The actua region of interest is a cube with

34 34 HUTTUNEN, MALINEN, KAIPIO rea( E 3 ), UWVF ABC E, UWVF ABC z y z y 1 5 rea( E 3 ), UWVF PML E, UWVF PML z y z y FIG. 8. The UWVF-approximations for a dipoe in a ayered medium when ω = 3π. The PML improves the soution Exact UWVF ABC UWVF PML E y FIG. 9. The fied E aong the ine z =.2 in the x = pane. The PML reduces the error from 3.4% to.8%. The exact soution for the probem is formuated in [19]. side ength.55. This domain is surrounded by a.2 thick PML. The mesh used in a simuations of this section is shown in Fig. 1. It has maximum ength of an eement edge h max =.196. To ensure an accurate geometric representation of the surface of the sphere, the mesh is refined near the scatterer. We compute soution at ω = 3π which gives λ/h max =.34, i.e. approximatey three waveengths per eement.

35 SOLVING MAXWELL S EQUATIONS USING THE UWVF 35 FIG. 1. The mesh used in the scattering from the sphere simuations. The mesh consists of vertices and tetrahedra. To better approximate the sphere, the mesh is reativey fine on near the surface of the scatterer. The probem can be decomposed into the scattered part E sc and the incident part E in. The incident fied is a y-direction poarized pane wave propagating in the direction of positive x-axis. The probem is formuated for the scattered fied ony. In the UWVF, we set Q = 1 and g = (1/2)n E in (see Eq. (3)) on the surface of the sphere. On the exterior boundary we have Q =, σ = µ / ɛ and g = which corresponds to the ow-order ABC. We aso compute the eectric far-fied pattern E defined in spherica coordinates (r, θ, φ) as µɛ r E(r, θ, φ) E (θ, φ) eiω, r when r. The computation of the far-fied from the UWVF approximation is presented in [5]. In a far-fied simuations of this study, the integras for the resoving the far-fied are computed over the exterior boundary of the computationa domain. Fig. 11 shows the near-fied UWVF-PML approximationswith σ = 2. for the scattered fied E sc. The tota fied E sc + E in is shown on the right. The eectric far-fied pattern for the same probem is potted in Fig. 12. The figure shows UWVF approximations using the ABC and the PML which both compare we with the anaytica Mie series soution (the PML soution is amost indistinguishabe from the exact soution).

36 36 HUTTUNEN, MALINEN, KAIPIO E sc, UWVF E, UWVF z.8.6 z x x FIG. 11. The UWVF-PML approximation for the scattering probem at ω = 3π. The ampitude of the scattered fied is shown on the eft. On the right is the ampitude of the tota fied (giving an idea of the radius of the sphere in waveengths). Far fied E (db) Mie UWVF ABC UWVF PML Ange (degrees) FIG. 12. The far-fied pattern E for the scattered fied from the sphere at ω = 3π. The UWVF approximations are in good agreement with the Mie series soution. The error in the far-fied and the number of degrees of freedom (DOF) as a function of the anguar frequency ω is shown in Fig. 13. The axis on the top of the error pot shows the ratio of the maximum eement size h max and the waveength λ NASA amond The ast mode probem is the scattering of a pane from the NASA amond for which experimenta data is pubished in [25]. The perfecty conducting metaic amond-shaped scatterer is 25.2 cm ong. The coordinate system is chosen so that the ongest dimension of the amond is in the x-direction and the smaest in the z-direction. The computationa domain is cm 3 cube. The object is iuminated by pane waves which propagate in (x, y)-pane and are verticay poarized (i.e. the VV-poarization in [25]). Hence, the