and stable computations the change of discretization length l *

Transcription

1 Computation of scattering from N spheres using mutipoe reexpansion Nai A. Gumerov a) and Ramani Duraiswami b) Perceptua Interfaces and Reaity Laboratory, Institute for Advanced Computer Studies, University of Maryand, Coege Park, Maryand Received 3 Apri 2002; revised 30 August 2002; accepted 6 September 2002 A computationa technique for the soution of probems of wave scattering from mutipe spheres is deveoped. This technique, based on the T-matrix method, uses the theory for the transation and reexpansion of mutipoe soutions of the Hemhotz equation for fast and exact recursive computation of the matrix eements. The spheres can have prescribed radii, impedances, and ocations. Resuts are vaidated by comparison with boundary eement cacuations, and by convergence anayses. The method is much faster than numerica methods based on discretization of space, or of the sphere surfaces. An even faster method is presented for the case when the spheres are aigned coaxiay Acoustica Society of America. DOI: / PACS numbers: Fn LLT I. INTRODUCTION a Eectronic mai: gumerov@umiacs.umd.edu; URL: b Eectronic mai: ramani@umiacs.umd.edu; URL: Numerous practica probems of acoustic and eectromagnetic wave propagation require computation of the fied scattered by mutipe objects. Exampes incude the scattering of acoustic waves by objects e.g., the scattering of sound by humans and the environment, ight scattering by couds and the environment, eectromagnetic waves in composite materias and the human body, pressure waves in disperse systems aerosos, emusions, bubby iquids, etc. Our interest is in modeing the cues that arise due to scattering of sound and ight, and to use this information in simuating audio and video reaity Duda and Martens, 1998; Duraiswami et a., In many cases the scatterers are spheres, or can be modeed as such. Such modeing is convenient for parametrization of arge probems, since each sphere can be characterized by a few quantities such as the coordinates of its center, its radius, and its impedance. This impedance wi in genera be a compex quantity, and characterizes the absorbing/ refecting properties of the body/surface. For exampe, we are exporing the head and the torso Gumerov et a., 2002; Agazi et a., In fuid mechanica probems, bubbes, dropets, or dust partices can be assumed spherica Gumerov et a., 1988; Duraiswami and Prosperetti, We are interested in computing the soution of mutipe scattering from N spheres, with specified impedance boundary conditions at their surfaces. Numerica methods such as boundary-eement methods BEM, finite-eement methods FEM, or finite difference methods FDM are we known. Despite the reative advantages of these methods they a share a common deficiency reated to the necessity of discretization of either the boundary surfaces or of the compete space. Discretization introduces a characteristic size or ength scae * of the surface or spatia eement. For accurate and stabe computations the change of discretization ength * must not affect the resuts. This eads to a requirement that this size shoud be much smaer than the waveength, i.e., *. Practicay this condition is * B, where B is some constant smaer than 1. If computations are required for high frequencies or short waves, this eads to very fine surface or spatia meshes. For exampe, in computations of scattering of sound by human heads Kahana, 2001; Katz, 2001, for 20 khz sound in air in norma conditions, the waveength is 1.7 cm, whie the diameter of the typica human head is D17 cm. For accuracy the ength of a surface eement shoud be, say, 6 times smaer than 1.7 cm, i.e., 60 times smaer than the diameter of the head. This gives a eement discretization of the head surface in the case where trianguar eements are used in the BEM. Such discretizations require soution of very arge inear systems with dense interinfuence matrices in the case of the BEM, and even arger but sparse matrices in the case of the FEM/FDM. Soutions of these equations using direct or iterative methods are costy in terms of CPU time and memory. Another approach to sove this probem is a semianaytica method such as the T-matrix method Waterman and True, 1961; Peterson and Strom, 1974; Varadan and Varadan, 1980; Mischenko et a., 1996; Koc and Chew, In this method, the scattering properties of a scatterer and of the coection are characterized in terms of the socaed T-matrices that act on the coefficients of the oca expansion of the incoming wave, centered at the scatterer, and transform them into the coefficients of an outgoing wave from the scatterer. There are two basic chaenges reated to using the T-matrix method to sove mutipe scattering probems. The first chaenge is to obtain the soution for a singe scatterer. In the case of compex shapes and boundary conditions this soution can be obtained either numericay or via anaytica or semi-anaytica techniques, based on decomposition of the origina shape to objects of simper geometry. The fied scattered by sound-soft, sound-hard, and eastic spheres paced 2688 J. Acoust. Soc. Am. 112 (6), December /2002/112(6)/2688/14/$ Acoustica Society of America

2 in the fied of a pane wave were considered in severa papers e.g., Marnevskaya, 1969, 1970; Huang and Gaunaurd, In the present paper we consider the practicay important case of spherica objects with impedance boundary conditions in an arbitrary incident fied in particuar that generated by a source. In this case a soution for a singe scatterer in an arbitrary incident fied can be obtained anayticay. The second chaenge is reated to computation of the T-matrices or matrices of the transation coefficients for the Hemhotz equation. Severa studies that are dedicated to scattering by two spheres Marnevskaya, 1969, 1970; Gaunnaurd and Huang, 1994; Gaunaurd et a., 1995 use representations of the transation coefficients via the Cebsch Gordan coefficients or 3- j Wigner symbos. We shoud notice that whie this method provides exact computation of T-matrix eements, it is time consuming, and perhaps is practica ony for a sma number of spheres. Brunning and Lo 1971 considered the mutipe scattering of eectromagnetic waves by spheres, and pointed out that the computationa work can be reduced by severa orders of magnitude if the computation of the transation coefficients coud be performed more efficienty, e.g., recursivey. In their study they appied recursive computation of these coefficients for the case when the sphere centers are ocated on a ine. This is aways true for two spheres, and is a specia case for three or more spheres. In the present paper we refer to this as the coaxia case. A genera three-dimensiona case of acoustica scattering by N objects using the T-matrix method was considered by Koc and Chew They proposed using a numerica evauation of integra forms of the transation operators Rokhin, 1993 and the fast mutipoe method FMM appied to an iterative soution of a arge system of equations that appear in the T-matrix method. In this paper we use a variant of the T-matrix technique that is speciaized to the soution of the probem of mutipe scattering from N spheres arbitrariy ocated in threedimensiona space. The distinguishing feature of our agorithm is utiization of a fast recursive procedure for exact computation of the T-matrix eements. We shoud note that the recurrence reations for the three-dimensiona scaar addition theorem for the Hemhotz equation were obtained by Chew These reations were derived for oca-to-oca reguar-to-reguar transations of the soutions of the Hemhotz equation, but can be extended to far-to-far and far-tooca transations. This was mentioned by Chew 1992 and detais can be found in our study Gumerov and Duraiswami, 2001a where aso reations for arbitrary rotations and simpified expressions for coaxia transations are presented. The savings that woud arise from use of fast mutipoe methods can aso be reaized in our probem though they are not considered here as our focus is on the case of reativey sma numbers of scatterers. We have deveoped software impementing this technique. We compare the resuts of the computations with numerica and anaytica soutions, and demonstrate the computationa efficiency of our method in comparison with commercia BEM software. The resuts show that the deveoped method compares favoraby in both accuracy and speed FIG. 1. The mutipe scattering probem considered in this paper. up of computations in some cases by severa orders of magnitude. II. STATEMENT OF THE PROBLEM Consider the probem of sound scattering by N spheres with radii a 1,...,a N situated in an infinite threedimensiona space as shown in Fig. 1. The coordinates of the centers of the spheres are denoted as r(x p p,y p,z), p p 1,...,N. The scattering probem in the frequency domain is reduced to soution of the Hemhotz equation for compex potentia (r), 2 k 2 0, 1 with the foowing genera impedance boundary conditions on the surface S p of the p th sphere: n i 0, p1,...,n, 2 ps p where k is the wave number and p are compex admittances, and i1. In the particuar case of sound-hard surfaces ( p 0) we have the Neumann boundary conditions, /n Sp 0, and in the case of sound-soft surfaces ( p ) we have the Dirichet boundary conditions, Sp 0. Usuay the potentia is represented in the form: r in r scat r, where scat (r) is the potentia of the scattered fied. Far from the region occupied by spheres the scattered fied shoud satisfy the Sommerfed radiation condition: im r r scat r ik scat 0. III. SOLUTION USING MULTIPOLE TRANSLATION AND REEXPANSION A. Decomposition of the scattered fied Due to the inearity of the probem the scattered fied can be represented in the form J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2689

3 N scat r p1 p r, where p (r) can oosey be thought of as the change in the scattered fied introduced by the pth sphere, though of course it contains the infuence of a the spheres. More precisey, each potentia p (r) is reguar outside the pth sphere and satisfies the Sommerfed radiation condition im r 7 r p r ik p 0, p1,...,n. 8 B. Mutipoe expansion Let us introduce N reference frames connected with the center of each sphere. In spherica poar coordinates rr p r p (r p, p, p ), the soution of the Hemhotz equation that satisfies the radiation condition 8 can be represented in the form n p r n0 A pm n S m n r p, p1,...,n. 9 mn Here A (p)m n are coefficients mutipying the S m n (r), which are mutipoes of order n and degree m, given by S m n r p h n kr p Y m n p, p, p1,...,n. 10 Here h n (kr) are spherica Hanke functions of the first kind that satisfy the Sommerfed condition, and Y m n (,) are orthonorma spherica harmonics, which aso can be represented in the form Y n m,1 m 2n1 4 nm! nm! P n m cos e im, 11 where P n m () are the associated Legendre functions. The probem now is to determine the coefficients A n (p)m so that the compete potentia N n r in r p1 A n0 pm n S m n r p 12 mn satisfies a boundary conditions on the surface of a spheres. C. Mutipoe reexpansion To sove this probem et us consider the qth sphere. Near the center of this sphere, rr q, a the potentias p (r), pq, are reguar. Accordingy the singuar soutions mutipoes S m n (r p ),pq] can be then reexpanded into a series of terms of reguar eementary soutions near the qth sphere, r q rr p q using the foowing expression: S m n r p 0 SR sm n r pq R s r q, s p,q1,...,n, pq. 13 Here R m n (r q ) are reguar eementary soutions of the Hemhotz equation in spherica coordinates connected with the qth sphere: R n m r q j n kr q Y n m q, q, p1,...,n, 14 where j n (kr) are spherica Besse functions of the first kind. The coefficients (SR) sm n (r pq ) are the transation reexpansion coefficients, and depend on the reative ocation of the pth and qth spheres, r pq. Since rr p rr p q r q, we have r p r q r pq, r pq r q rr p p r q, 15 where r pq is the vector directed from the center of the pth sphere to the center of the qth sphere. Aspects of the mutipoe reexpansion coefficients, their properties, and methods for their efficient computations are considered in Chew 1992 and Gumerov and Duraiswami 2001a. Near the center of expansion the incident fied can aso be represented using a simiar series, the radius of convergence for which is not smaer than the radius of the qth sphere: in r 0 C ins r q R s q r q. s 16 Substituting the oca expansions from Eqs. 13 and 16 into Eq. 12 we obtain the foowing representation of the fied near rr q : r 0 n0 C ins r q R s r q s 0 n mn N A n qm S n m r q p1 pq SR sm n r pq R s r q. s n0 n pm A n mn 17 Here the incident fied is aso expanded in terms of the reguar functions centered at r q, the singuar mutipoe expansion around r q is kept as is, whie the other mutipoe expansions are converted to oca expansions in terms of reguar functions around r q. Let us change the order of summation in the atter term in Eq. 17 and substitute Eqs. 10 and 14 for S n m and R n m. This expression can then be rewritten as r 0 B qs r 1,...,r N B qs j kr q A qs h kr q Y s q, q, s 18 N C ins r q p1 pq n0 D. Boundary conditions n SR sm n r pq A pm n. mn 19 From these equations we have the boundary vaues of and /n on the surface of the qth sphere as 2690 J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

4 Sq 0 B qs j ka q s Sq 1 2 ika q 0 s A qs Y s q, q kj ka q i q j ka q, 28 ns q k 0 A qs h ka q Y s q, q, B qs j ka q s A qs h ka q Y s q, q ns q q ka q 2 0 i q Sq. s A qs Y s q, q kj ka q i q j ka q 29 To obtain the coefficients, we must satisfy the boundary condition Eq. 2, which on the surface of the qth sphere eads to For the particuar case of a sound-hard sphere ( q 0) this gives 0 B qs kj ka q i q j ka q s A qs kh ka q i q h ka q Y s q, q 0. Orthogonaity of the surface harmonics yieds 22 Ss 1 ik 2 2 a q 0 ns q 0, s A qs Y s q, q, j ka q 30 B qs kj ka q i q j ka q A qs kh ka q i q h ka q, 23 whie for sound-soft spheres ( q ) we have where 0,1,..., s,...,. Note that the boundary vaues of Sq and /n Sq can be expressed in terms of the coefficients A (q)s, since B qs A qs kh ka q i q h ka q kj ka q i q j ka q, 0,1,..., s,...,, 24 and Eqs. 20 and 21 yied Sq 0 h ka q j ka q s kh ka q i q h ka q A qs Y s q, q. kj ka q i q j ka q h ka q j ka q s ns q k 0 kh ka q i q h ka q kj ka q i q j ka q A qs Y s q, q These reations can be aso rewritten in a compact form using the Wronskian for the spherica Besse functions, as W j ka,h ka j kah ka j kah ka ika 2 27 Sq 0, ns q E. Matrix representation i ka q 2 0 s A qs Y s q, q. j ka q 31 To determine the boundary vaues of the potentia and/or its norma derivative and obtain a spatia distribution according to Eq. 12 we need to determine the coefficients A (q)s in Eqs. 23 and 19, which are vaid for any sphere, q 1,...,N. These equations form a inear system that may be represented in standard matrix-vector form. First, we note that coefficients of expansions to spherica harmonics, such as A n m, n0,1,2,...; mn,...,n, can be stacked into one coumn vector, e.g., AA 0 0,A 1 1,A 1 0,A 1 1,A 2 2,A 2 1,A 2 0,A 2 1,A 2 2,... T, 32 where the superscript T denotes the transpose. In this representation the eements of the vector A are reated to coefficients A n m by A t A n m, tn1 2 nm, n0,1,2,...; mn,...,n; t1,2, The same reduction in dimension can be appied to the reexpansion coefficients (SR) sm n. The coefficients can then be stacked in a two-dimensiona matrix as J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2691

5 SR 00 SR00 10 SR SR SR SR SR SR SR SR SR SR SR SR SR SR SR 10 SR 11 SR 11 SR 11 SR SR 21 SR 21 SR 21 SR 22, 20 SR 20 with the foowing correspondence of the matrix eements and coefficients: SR rt SR sm n, r1 2 s, tn1 2 nm,,n0,1,2,...; mn,...,n; s,...,. 35 Using this representation we introduce the foowing vectors and matrices where A q A q t T, D q D q t T, L qp L qp rt, A t q A n qm, q1,...,n, p1,...,n, 36 D t q kj nka q i q j n ka q kh n ka q i q h n ka q C n inm r q, L qp rt kj ka q i q j ka q kh ka q i q h ka q SR n sm r pq, for pq, L qq rt rt, tn1 2 nm, n0,1,2,...; r1 2 s, mn,...,n; 0,1,2,...; s,...,; q1,...,n, p1,...,n, 37 where rt is the Kronecker deta; rt 0 for rt and rr 1. Equations 23 and 19 then can be represented in the form N p1 L qp A p D q, q1,...,n, 38 or as a singe equation LAD, where the matrices and vectors are 39 L L11 L12 L1N L 21 L 22 L 2N NN, L N1 L N2 L A A1 A 2 A N, D D1 D 2 D N. 40 This inear system can be soved numericay using standard routines, such as LU-decomposition. The bock-structure can aso be expoited using bock-oriented sovers, though we do not pursue this here. An important issue is the truncation of the infinite series and corresponding truncation of the associated matrices. A first way to truncate the series is to truncate the outer sum at a fixed number M in each expansion, and is the approach taken here. This number is seected via a heuristic based on the magnitude of the smaest retained term. In this case n 0,1,...,M; mn,...,n, then the ength of each vector A (p) and D (p) wi be (M1) 2, and the size of each sub-matrix L (qp) wi be (M1) 2 (M1) 2, the size of the tota vectors A and D wi be N(M1) 2 and the size of the tota matrix L wi be N(M1) 2 N(M1) 2. Severa ideas coud be empoyed to improve this heuristic. For exampe, the use of fast mutipoe methods coud ead to acceerated schemes to sove these equations iterativey Koc and Chew, Further, the expressions used to seect the truncation of the series coud be seected in a more rigorous manner. However, as wi be seen beow, even with a fixed size of M we are abe to sove arge probems using conventiona techniques. F. Computation of mutipoe reexpansion coefficients The (SR)-mutipoe transation coefficients can be computed in different ways incuding via numerica evauation of integra representations, or using the Cebsch Gordan or Wigner 3- j symbos e.g., see Epton and Dembart, 1995; Koc et a., For fast, stabe, exact and efficient computations of the entire truncated matrix of the reexpansion coefficients we used a method based on Chew 1992 recurrence reations and symmetries, who derived them first for the (RR)-transation coefficients transation of reguar-toreguar eementary soution, see Eq. 14. Fortunatey since the singuar and reguar eementary soutions of the Hemhotz equation satisfy the same recurrence reations these reations hod for (SR)-mutipoe transation coefficients 2692 J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

6 aso, but with different initia vaues for recursive computations 44. This was aso pointed out by Chew Note that for other equations e.g., for the Lapace equation the singuar and reguar eementary soutions may not satisfy the same recursions. In Gumerov and Duraiswami 2001a proofs and detais may be found. We ony provide necessary reations and initia vaues for using the recurrence procedures here. A the (SR) sm n (r pq ) transation reexpansion coefficients can be computed in the foowing way. First, we compute the sm so-caed sectoria coefficients of type (SR) m and (SR) sm sn using the foowing reations: b m1 m1 SR s,m1,m1 b s s1,m SR 1,m where b s1 1 SR s1,m 1,m, b m1 m1 SR s,m1,m1 b s s1,m SR 1,m b n m b s1 1 SR s1,m 1,m, 0,1,..., m0,1,2,..., nm1nm 2n12n1, nm1nm 2n12n1, 0, mn, and the recurrence process starts with SR s0 0 r pq 41 S s r pq, s,...,, 0mn, nm0, SR 0m 0n r pq 4S m n r pq. Due to the symmetry reation SR m,s m 1 m SR sm m, 0,1,2,..., s,...,, mn,...,n, 45 a of the sectoria coefficients (SR) sm sn can be obtained from the coefficients (SR) sm m. Once the sectoria coefficients are computed, a other coefficients can be derived from them using the foowing recurrence reation, which does not change the degrees s,m of the reexpansion coefficients: a m n1 where sm SR,n1 a s SR sm 1,n a n m a m sm n SR,n1 s a 1 sm SR 1,n,n0,1,..., s,...,, mn,...,n, 46 n1mn1m, nm, 2n12n3 0, mn. Due to the symmetry, 47 SR sm n 1 n SR m,s n,,n0,1,..., s,...,, mn,...,n, 48 ony those coefficients with n need be computed using recurrence reations. In addition, the SR coefficients for any pair of spheres p and q need be computed ony for the vector r pq, since for the opposite directed vector we have the symmetry SR sm n r pq 1 n SR sm n r pq,,n0,1,..., mn,...,n. 49 Software based on this agorithm was deveoped and entited MutisphereHemhotz. Resuts of tests using this software are presented and discussed in Sec. V. IV. COAXIAL SPHERES The case of two spheres is interesting, since on the one hand the scattered fieds due to the spheres interact with each other mutipe scattering, whie on the other, the interaction is sti simpe enough that it can be investigated in detai and understood more intuitivey than the genera case of N spheres. Additionay, in this case, the computation of the reexpansion matrices can be simpified via a proper seection of the reference frames. Indeed, for two spheres we can introduce a reference frame which has its z axis directed from the center of one sphere to the center of the other sphere. Since the reexpansion coefficients depend ony on the reative ocation of the spheres, for this frame orientation, there wi be no anguar dependence for the reexpansion coefficients. This does not require the incident fieds to be symmetric with respect to this axis. The same statement hods for the case when there are N spheres arranged aong a ine, taken to be the z axis. In these particuar cases, the genera reexpansion formua Eq. 13 simpifies to S m n r p m SR mm n r pq R m r q, p1,...,n, pq. 50 The coefficients SR m n r pq SR mm n r pq,,n0,1,..., mn,...,n, 51 satisfy genera recurrence reations and can be computed using the genera agorithm we have deveoped. However, there are simper reations that take advantage of the coaxiaity of the spheres, whie resuting in substantiay ower dimensiona matrices. Note that the sign of coefficients (SR) m n (r pq ) depends on the direction of the vector r pq. To be definite, we use the convention that r pq corresponds to r pq and r qp corresponds to r qp r pq. Since (SR) mm n (r pq )(1) n (SR) mm n (r pq )(1) n (SR) mm n (r qp ), we wi have SR m n r pq 1 n SR m n r qp,,n0,1,..., mn,...,n. 52 J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2693

7 A. Matrix representation According to Eq. 50 harmonics of each order m can be considered independenty. Equations 19 and 23 can be rewritten in the form N kj ka q i q j ka q SR m pm n r pq A n kh ka q i q h ka q p1 nm pq where A qm D qm, m0,1,2,..., m,m1,..., q1,...,n, 53 D qm kj ka q i q j ka q kh ka q i q h ka q C inm r q. 54 This inear system can be represented in the foowing form N p1 L qpm A pm D qm, m0,1,2,..., q1,...,n, 55 where the vectors A (q)m and D (q)m and matrices L (qp)m are stacked as foows: A qm A n qm T, D qm D n qm T, L qpm L qpm n, q1,...,n, p1,...,n, m0,1,2,...,,nm,m1,..., with the individua matrix eements given by L qpm n kj ka q i q j ka q kh ka q i q h ka q SR n m r pq, 56 for pq, L n n. 57 Since a equations can be considered separatey for each m, the inear system Eq. 55 can be written as L m A m D m, m0,1,2,..., 58 where L L11m L12m L1Nm L 21m L 22m L 2Nm m NNm, L N1m L N2m L A m A1m A 2m Nm, A Dm D1m D 2m Nm. D 59 As in the genera case considered above, the infinite series and matrices need to be truncated for numerica computation. If we imit ourseves to the first M modes for each expansion of spherica harmonics, so that m0,1,...,m, n m, m1,...,m, then the ength of each vector A (p)m and D (p)m is M1m, the dimensions of each matrix L (qp)m is (M1m)(M1m), the size of the tota vectors A m and D m are N(M1m), and the size of the tota matrix L m is N(M1m)N(M1m). The probem then is reduced to soution of 2M1 independent inear systems for each m. Note that the coaxia, or diagona, transation coefficients (SR) m n (r pq ) are symmetrica with respect to the sign of the degree m, (SR) m n (r pq ) (SR) m n (r pq ) see Gumerov and Duraiswami, 2001a. Therefore the matrices L m are aso symmetrica see Eq. 57 L m L m, m0,1,2,..., 60 and can be computed ony for non-negative m. At the same time the right-hand side vector D m, generay speaking, does not coincide with D m, so that the soution A m can be different from A m. Let us compare the number of operations required for determination of a expansion coefficients A (q)m n, m0, 1,...,M, nm, m1,...,m, q1,...,n, using the genera agorithm and using the agorithm for coaxia spheres. Assuming that a standard sover requires CK 3 operations to sove a inear system with matrix of dimension KK, where C is some constant, we can find soution using genera agorithm with N genera operations CN 3 M operations. Using the agorithm for coaxia spheres we wi spend CN 3 (M1m) 3 operations to obtain A (q)m n for each m0,1,...,m. The tota number of operations wi be therefore N coaxia operations CN 3 M mm M1m M CN M1 3 2 m1 m CN 3 M M 2 M CN 3 M1 4 M1 2. Therefore, for M1, we have genera N operations coaxia N operations 2M which shows much higher efficiency of the agorithm for coaxia spheres. Note aso that in the case of coaxia spheres the number of the mutipoe reexpansion coefficients that needs to be computed for each pair pq requires O(M 3 ) operations, whie in the genera case such computations can be performed in O(M 4 ) operations. These numbers are smaer than the eading order term in the compexity, that are required for soution of inear equations, and thus Eqs provide a comparison between the two methods. The above expressions for coaxia spheres assume that the z axis coincides with the direction from the center of one 2694 J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

8 of the spheres to the center of some other sphere. If coordinates of spheres are specified in the origina reference frame, with the axis z oriented arbitrariy with respect to the ine connecting the sphere centers, we can rotate them so that the new reference frame is convenient for use with the coaxia agorithm. B. Computation of coaxia reexpansion coefficients Due to the symmetry reations see Eqs. 49 and 51 SR n m SR n m, m0,1,2,..., 64 the coaxia coefficients (SR) m n (r pq ) can be computed ony for nm0. The process of fiing the matrix (SR) m n can be performed efficienty using recurrence reations that first fi the ayers with respect to the orders and n foowed by advancement with respect to the degree m. If such a procedure is seected, then the first step is fiing of the ayer m0. The initia vaue depends on the orientation of r pq vector reative to the axis i z or i ẑ if rotation is performed, and is given by where SR 0 0 r pq SR 00 0 r pq 41 S 0 r pq pq pq 21h kr pq, r pq i z r pq 1, for r pq i z r pq, 1, for r pq i z r pq To advance with respect to the order n at fixed m0 we can use Eq. 46 for sm, a m m n SR,n1 a m n1 a m 1 m SR,n1 m SR 1,n a m m SR 1,n, nm,m1,..., 67 with the a s given by Eq. 47. For advancement with respect to m it is convenient to use Eq. 41 for sm1, b m1 m1 m1 SR,m1 b m1 m SR 1,m m1,m2,..., b m 1 m SR 1,m, 68 with the b s given by Eq. 43 and then obtain the other (SR) n m1 using Eq. 67. V. COMPUTATIONAL RESULTS We appy our method to severa exampe probems. First, we compare the present technique with the BEM for probems invoving two and three spheres, and show that the present method is accurate, and much faster. Next, we consider probems that, due to their size, woud be impractica to hande with the BEM, and demonstrate that the present technique can dea with them. Convergence for these cases is demonstrated as the truncation number M is increased. FIG. 2. Notation denoting the different reference frames used in the mutipoe reexpansion technique. A. Singe sphere For the case of a singe sphere there are no mutipe scattering effects. Despite this, the case is interesting from a practica viewpoint. We were unabe to find this soution in the standard handbooks, and we incude this imiting case to provide an anaytica soution for both an arbitrary incident fied and for a monopoe source. From Eqs we have L (1) I, A (1) D (1),or A 1m n kj nkaij n ka kh n kaih n ka C inm n r Here we wi drop subscript 1 for and a, whie keeping them for the coordinates. Substituting this expression into Eqs. 28 and 29, we obtain expressions for the potentia and its norma derivative on the sphere surface as S1 1 ika 2 0 ns 1 ka 2 0 i S1. s s C 1 ins r 1 Y s 1, 1 kh kaih ka, C ins r 1 Y s 1, 1 kh kaih ka The coefficients C n (in)m (r 1 ) are determined using Eq. 16 and depend on the incident fied. In the case of a monopoe source ocated at some point rr source, the incident fied corresponds to the fundamenta soution of the Hemhotz equation eikrr source in rqg k rr source Q 4rr source, 72 where Q is the source intensity compex, if the phase is not zero, QQe i. 73 For mutipe spheres, expansion of this function near the center of the qth sphere rr q see Fig. 2 can be found ese- J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2695

9 FIG. 3. The normaized surface transfer function H, Eq. 81, for a singe sphere of radius a8.25cmanda monopoe source, ocated at a distance d/a10 from the center of the sphere for spheres of different vaues of /k shown near the curves. The continuous and dashed ines show resuts of computations using the present method with truncation number M8. Circes and diamonds respectivey show the resuts of Duda and Martens 1998 and of BEM computations for 0. In the BEM the sphere surface was discretized using 2700 inear eements. where e.g., Morse and Feshbach, 1953, in rqik n0 n S m n r source r q R m n r q, mn r q r source r q. Comparing Eqs. 16 and 74 we obtain C n inm r q QikS n m r source r q A simpification of the genera formua for C n (in)m, Eq. 75, is possibe, since the probem for monopoe source and a singe sphere is axisymmetric reative to the axis connecting the sphere center and the source ocation. Setting the axis z 1 to be this axis, we have C n inm Qik m0 h n kr source r 1 Y n 0 0,0 Qik m0 2n1 4 h nkd, 76 where d is the distance between the source and the sphere center. The expression for the surface vaues of the potentia and its derivative Eqs. 70 and 71 become S1 Q 4a 2 0 iq ns 1 4a 2 0 i S1. 21h kdp cos 1, 77 kh kaih ka 21h kdp cos 1 kh kaih ka 78 For the particuar case of a sound-hard (0) surface this gives S1 Q 4ka 2 0 ns q 0, 21 h kd h ka P cos 1, whie for sound-soft () surfaces we have 79 S1 0, 80 Q 21 h kd ns 1 4a 2 0 h ka P cos 1, These imiting cases are cassica and their expressions can be found esewhere e.g., see Hanish, We compared resuts of computations of the potentia on the surface with those provided by Eq. 79 and that obtained using the BEM, as reaized in the software package COMET. The figures beow iustrate comparisons of the foowing transfer function H measured in db, which represents the ratio of the ampitude of the acoustic fied at a specified ocation of the surface to the ampitude of the incident fied at the center of the sphere: H20 g S 1 81 in r 1. The soution, Eq. 79, for a singe sphere was used by Duda and Martens 1998, for an investigation of scattering cues in audition. Figure 3 shows good agreement between computations using a methods. In this exampe we incude aso computations for different impedances of the sphere. We aso tested the resuts of computations obtained using different truncation numbers. Duda and Martens 1998 used truncation based on comparison of subsequent terms in the series in this particuar case the series was truncated when the ratio of such terms is smaer than 10 6 ). Experiments showed that exceent agreement with these resuts is achieved, if the truncation number is seected using the heuristic Meka, e , 82 which shows that it increases with the wave number. For arge ka good agreement was observed for M1/2eka, whie differences were visibe for smaer M. For moderate ka such differences appeared for truncation numbers beow the vaue provided in Eq. 82. B. Two spheres Since there is no cosed-form anaytica soution for two spheres in a simpe form, we compared the numerica resuts obtained using our code, and by using the BEM. As an exampe we considered computation of the function Eq J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

10 FIG. 4. An exampe of BEM COMET 4.0 computations of potentia distribution over the surface of two spheres generated by a monopoe source. Each sphere surface is discretized to 2700 trianguar eements. The ratio of sphere radii is and they touch in one point. The mode consists of two spheres, which are touching at one point. In the computations the ratio of sphere radii was taken to be The origin of the reference frame was ocated at the center of the smaer sphere and the direction of the y axis was from the arger sphere to the smaer one. The z axis was directed towards a monopoe source, generating the incident fied, which was ocated at the distance of 10 radii from the smaer sphere. The frequency of the incident wave nondimensionaized with the radius of the smaer sphere corresponded to ka The mesh for computations using the BEM contained 5400 trianguar eements 2700 eements for each sphere. A picture of this two-sphere configuration with computationa mesh and distribution of the acoustic pressure is shown in Fig. 4. In the computations the admittances of both spheres were set to zero. For computations using MutisphereHemhotz the truncation number was automaticay set to M 1 2ekr For the above given vaues of a 2 /a 1 and ka 1, this provided M9, so that the tota number of modes n in the mutipoe expansion was 10 for each sphere or 100 A n m coefficients for each sphere. Figure 5 shows a comparison between the BEM and the MutisphereHemhotz computationa resuts for function H cacuated for sphere 1 according to Eq. 81. Each curve corresponds to a fixed vaue of the spherica poar ange 1 and demonstrates dependence on the ange 1. Note that dependence on the ange 1 is ony due to the presence of the sphere 2. The comparison shows a good agreement between the resuts obtained by different methods. Some sma dispersion of the points obtained using BEM is due to the mesh discretization of the sphere surface, which normay can be avoided by additiona smoothing/interpoation procedures we did not appy such smoothing in the resuts potted. Our code far outperformed BEM computationay, both in achieving much faster computations seconds as opposed to tens of minutes on the same computer and memory usage. Figure 6 demonstrates computations of H for the two sphere geometry with the parameters described above, but for a higher wave number, and with different impedances for the arger sphere. For the given geometry, the automaticay seected truncation number was M31. This number is arge enough to observe a substantia difference in speed of computations and memory usage between the genera agorithm and the coaxia one see Eq. 63. Proper seection of the truncation number is an important issue for appications of mutipoe transation techniques. Figure 7 shows convergence of the computations with increasing truncation numbers for H at a specified point on the surface ( 1 60 degrees and 1 0 degrees in the case shown in the figure. Computations with ow truncation numbers may provide poor accuracy. At some particuar truncation number depending on the nondimensiona wave number ka) the computationa resuts stabiize note that since the soution H depends on the wave number, and so for each ka the soution asymptotes to the corresponding vaue. Further increase in the truncation number increases both the accuracy of the resuts and computationa time/memory, since the matrix size grows in proportion to M 4 in the genera case, and to M 3 for coaxia spheres. However, at some FIG. 5. Comparison of the anguar dependence of function H, Eq.81, over the surface of a smaer sphere computed for the two sphere geometry shown in Fig. 4 using the BEM and the mutipoe reexpansion technique. Both spheres are sound-hard. J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2697

11 FIG. 6. Anguar dependence of H in Eq. 81 over the surface of the sound-hard smaer sphere for the geometry shown in Fig. 4 for different admittances of the arger sphere. Resuts computed using the present method with automaticay seected truncation number M31. truncation numbers, which sighty exceed the vaue provided by Eq. 83, the computations can encounter difficuties that arise from the exponentia growth of portions of the terms in the expansions, eading to overfow reated errors. These are due to the spherica Hanke functions of arge order, h n (kr pq ), entering the reexpansion system matrix. Asymptotic expansion of the Hanke function at arge n and fixed kr 12 shows that the growth starts at n1/2ekr 12. This is used as the basis for automatic seection of the truncation number Eq. 83. Of course, this imitation is purey based on the order of computation of the eements, and the product of terms remains finite, and cacuations can be performed for much arger M than given by Eq. 83. However, we have not yet modified our software to use this order of computation. The computations presented in Fig. 7 show that the actua stabiization occurs at smaer M than those given by Eq. 83 where we have for ka 1 1,5,10,20,30 the foowing vaues: M3,15,31,63,94, respectivey. Resuts of our numerica experiments show that for arge kr 12 reasonabe accuracy can be achieved at M*1/2M max, where M max is provided by Eq. 83. At the same time, for ower kr 12 formua 83 provides vaues which cannot be reduced, and accurate computations can be achieved with M sighty arger than M min, with M min provided by Eq. 83. In a recent paper Gumerov and Duraiswami, 2001b we presented resuts of the computation of the surface potentia for a sphere near a rigid wa. In this case the rigid wa coud be repaced by an image sphere and an image source and the coaxia mutipoe reexpansion can be used. The probem of sound scattering by a sphere near a rigid wa is in some sense a simpification of the genera probem for two spheres, since both the rea and the image spheres in this case have the same radius and impedance. For this case, the coefficients of the mutipoe expansions near each sphere A n (1)m and A n (2)m are symmetrica and the dimension of the system can be reduced using this symmetry by a factor of 2. In that paper, we aso provided a study and discussion of the infuence of the distance between the sphere and the wa and frequency of the fied on the surface potentia. C. Three arbitrariy ocated spheres If the cases of one and two spheres can be covered using simpified codes, the case of three non-coaxia spheres requires the genera three-dimensiona mutipoe transation. As in the case of two spheres discussed above, we compared resuts of computations for three spheres using Mutisphere- Hemhotz and the COMET BEM software. For this computationa exampe we paced an additiona sphere 3 to the case described above. The distance between the centers of spheres 1 and 3 was the same as the distance between the centers of spheres 1 and 2. The parameters of the incident fied were the same as for the case of two spheres. The mesh for computations using the BEM contained 5184 trianguar eements, 1728 eements for each sphere. A picture of this configuration with the computationa FIG. 7. Dependence of the computations of H, Eq.81, for the two sphere geometry see Fig. 4 on the truncation number M for different nondimensiona wave numbers J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

12 FIG. 8. BEM COMET 4.0 computations of the potentia distribution over the surface of three spheres generated by a monopoe source. Each sphere surface is discretized using 1728 trianguar eements. Two spheres of nondimensiona radii 1 sphere 1 and sphere 2 touch at one point. Sphere 3, with nondimensiona radius 1, is ocated at a distance from the center of sphere 1, on the ine connecting the source and the center of sphere 1. mesh and distribution of the acoustic pressure is shown in Fig. 8. In the computations the three spheres were a taken to be sound-hard. Resuts of comparisons between BEM and Mutisphere- Hemhotz computations with M9 are shown in Fig. 9. The comparison is as good as in the case of two spheres. Since Figs. 5 and 9 represent simiar dependence, we can notice that the presence of the third sphere reduced at some points by 3 4 db the ampitude of the sound fied on sphere 1. D. Many spheres FIG. 10. A sketch of the probem geometry for sound propagation through a screen of spherica partices. The case of sound and eectromagnetic wave scattering by many arbitrariy ocated spheres has numerous practica and theoretica appications, incuding acoustics and hydrodynamics of mutiphase fows, sound propagation in composite materias, eectromagnetic waves in couds, and inverse probems, such as the detection of buried objects, medica tomography, etc. We considered the foowing hypothetica probem of sound scattering by a screen of spherica partices. The geometry of the probem is shown in Fig. 10. Here the incident fied is generated by a monopoe source ocated at distance d from a fat partice screen of thickness. A sensor e.g., a microphone measuring the acoustic pressure is ocated behind the screen at the same distance as the source. N spheres with the same acoustic impedance in the exampes beow we took the spheres as sound-hard, but with possiby different sizes, are distributed according to some distribution density over their radii, and have ocations of their centers within a box hh representing the screen. The objective is to evauate the effect of the screen on the sound propagation. This effect can be evauated by defining a screen transfer function STF, which we define as STF20 g r mic / in r mic, 84 FIG. 9. Comparison of the anguar dependence of H 81 over the surface of smaer sphere computed for the three sphere geometry shown in Fig. 8 using the BEM and the mutipoe reexpansion technique. A three spheres have sound-hard surfaces. J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2699

13 FIG. 11. A scattering screen of 16 spheres with random sizes and random ocations of their centers. The centers are uniformy distributed inside a box 10x15, 25y25, 25z25. The radii are uniformy distributed over 0.5a1.5. FIG. 13. The view of a scattering screen of 121 spheres of the same size a1 and ocation of their centers at the nodes of a square grid 25y 25, 25z25 with the grid size yz5. where r mic is the radius-vector of the microphone ocation and which is measured in decibes. Two exampes of computations with many spheres are presented beow. In the first exampe we paced 16 spheres with uniform random distribution of their dimensioness radii from a min 0.5 to a max 1.5. Dimensioness parameters of the screen were d10, 5, and h50. The sphere centers were distributed uniformy within the screen. The view of this screen in the yz pane is shown in Fig. 11. In the computations we chose three different wave numbers, ka1, ka3, and ka5, where a is the ength scae the characteristic sphere radius. Computations were performed using the genera software with increasing truncation number M. Dependences of the STF on M at various ka are shown in Fig. 12. It is seen that resuts converge to some vaue depending on ka. This type of check shows that for the given geometry reativey ow truncation numbers can be used, which are much smaer than given by Eq. 83, where instead of r 12 some representative intersphere distance r pq is seected. We noticed, however, that in some cases when we have many spheres with very different intersphere distances from touching spheres, to spheres ocated at arge kr pq ), stabiization of computation ony occurs at higher truncation numbers than prescribed by Eq. 83. Thus, at this point the procedure we recommend is an experimenta one to check the resuts of severa cacuations and find the truncation number at which resuts converge. In the second exampe we put a reguar monoayer screen (d10, 0, and h50) of 121 spheres of the same radii a1, which centers form a reguar grid in pane xd see Fig. 13. Again we compared resuts obtained with the aid of MutisphereHemhotz at increasing M and different wave numbers and found fast convergence see Fig. 14. VI. CONCLUSIONS FIG. 12. Convergence test for the probem of sound scattering by a screen of 16 randomy sized spheres with random ocation of their centers as shown in Fig. 11. Three different curves computed at different ka, where a is the mean of the sphere radii distribution. We have deveoped a procedure for soution of the Hemhotz equation for the case of N spheres of various radii and impedances arbitrariy ocated in three-dimensiona space. This soution uses a T-matrix method where the matrices are computed using a mutipoe reexpansion technique. We presented computationa resuts for two and three spheres obtained both using the boundary eement method with fine discretization of the surfaces thousands of eements, and showed that our soutions are correct. For the case of arger numbers of spheres, we demonstrated that our resuts are consistent, by showing that they converge as the truncation number increases. An open probem that remains is the proper choice of the truncation number as a function of the wave number, sphere sizes, and intersphere distances. In the case where the truncation number is propery seected, the soution showed high accuracy, and substantia speed-up in comparison to the boundary eement method. In cases when the centers of the spheres are ocated on a ine which is aways true for two 2700 J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres

14 FIG. 14. Convergence test for the probem of sound scattering by a screen of 121 equa sized spheres ocated at the nodes of a reguar grid as shown in Fig. 13. Three different curves are computed at different ka indicated near the curves. spheres, and can be reaized in particuar probems for N spheres computations using a coaxia mutipoe reexpansion can be performed with much higher efficiency than in the genera case. Future improvements to the method proposed here may incude use of spatia grouping of nearby spheres into custers represented by one mutipoe expansion, variabe truncation numbers depending on the particuar distribution of spheres, impementation of a fast mutipoe method based evauation of the matrix-vector product in Eq. 39, and deveopment of iterative agorithms for soution of that equation. ACKNOWLEDGMENTS We woud ike to gratefuy acknowedge the support of NSF Award No We woud aso ike to thank Dr. S. T. Raveendra of Coins and Aikman Automotive Interior Systems, Pymouth, MI for providing us with the COMET BEM software. We aso thank the reviewers of this paper for pointing out missing references, prior work, and for their hepfu comments. Agazi, V. R., Duda, R. O., Duraiswami, R., Gumerov, N. A., and Tang, Z Approximating the head-reated transfer function using simpe geometric modes of the head and torso, J. Acoust. Soc. Am. 112, Brunning, J. H., and Lo, Y. T Mutipe scattering of EM waves by spheres, parts I and II, IEEE Trans. Antennas Propag. AP-193, Chew, W. C Recurrence reations for three-dimensiona scaar addition theorem, J. Eectromagn. Waves App. 62, Duda, R. O., and Martens, W. L Range dependence of the response of a spherica head mode, J. Acoust. Soc. Am. 104, Duraiswami, R., and Prosperetti, R Linear pressure waves in fogs, J. Fuid Mech. 299, Duraiswami, R., Gumerov, N. A., Davis, L., Shamma, S. A., Eman, H. C., Duda, R. O., Agazi, V. R., Liu, Q-H., and Raveendra, S. T Individuaized HRTFs using computer vision and computationa acoustics, J. Acoust. Soc. Am. 108, 2597A. Aso avaiabe on the web at Epton, M. A., and Dembart, B Mutipoe transation theory for the three-dimensiona Lapace and Hemhotz equations, SIAM J. Sci. Comput. 4, Gaunaurd, G. C., and Huang, H Acoustic scattering by a spherica body near a pane boundary, J. Acoust. Soc. Am. 96, Gaunaurd, G. C., Huang, H., and Strifors, H Acoustic scattering by a pair of spheres, J. Acoust. Soc. Am. 981, Gumerov, N. A., and Duraiswami, R. 2001a. Fast, Exact, and Stabe Computation of Mutipoe Transation and Rotation Coefficients for the 3-D Hemhotz Equation, University of Maryand Institute for Advanced Computer Studies Technica Report UMIACS-TR-# Aso CS-TR- #4264. Avaiabe at CS-TR-4264.pdf. Gumerov, N. A., and Duraiswami, R. 2001b. Modeing the effect of a nearby boundary on the HRTF, Proc. Internationa Conference on Acoustics, Speech, and Signa Processing 2001, Sat Lake City, UT, 7 11 May 2001, Vo. 5, pp Gumerov, N. A., Duraiswami, R., and Tang, Z Numerica study of the infuence of the torso on the HRTF, to appear in Proc. of the Internationa Conference on Acoustics, Speech and Signa Processing, Orando, FL, Gumerov, N. A., Ivandaev, A. I., and Nigmatuin, R. I Sound waves in monodisperse gas-partice or vapour-dropet mixtures, J. Fuid Mech. 193, Hanish, S A Treatise on Acoustica Radiation, Nava Research Laboratory, Washington, DC, pp Huang, H., and Gaunnaurd, G. C Acoustic scattering of a pane wave by two spherica eastic shes, J. Acoust. Soc. Am. 98, Kahana, Y Numerica modeing of the head-reated transfer function, Ph.D. thesis, University of Southampton. Katz, B. F. G Boundary eement method cacuation of individua head-reated transfer function. I. Rigid mode cacuations, J. Acoust. Soc. Am. 110, Koc, S., and Chew, W. C Cacuation of acoustica scattering from a custer of scatterers, J. Acoust. Soc. Am. 103, Koc, S., Song, J., and Chew, W. C Error anaysis for the numerica evauation of the diagona forms of the scaar spherica addition theorem, SIAM Soc. Ind. App. Math. J. Numer. Ana. 36, Marnevskaya, L. A Diffraction of a pane scaar wave by two spheres, Sov. Phys. Acoust. 143, Marnevskaya, L. A Pane wave scattering by two acousticayrigid spheres, Sov. Phys. Acoust. 154, Mishchenko, M. I., Travis, L. D., and Mackowski, D. W T-matrix computations of ight scattering by nonspherica partices: a review, J. Quant. Spectrosc. Radiat. Transf. 55, Morse, P. M., and Feshbach, H Methods of Theoretica Physics I McGraw Hi, New York. Peterson, B., and Strom, A Matrix formuation of acoustic scattering from an arbitrary number of scatterers, J. Acoust. Soc. Am. 56, Rokhin, V Diagona forms of transation operators for the Hemhotz equation in three dimensions, App. Comp. Harmonic Anaysis 1, Varadan, V. K., and Varadan, V. V Acoustic, Eectromagnetic and Eastic Wave Scattering: Focus on the T-Matrix Approach Pergamon, New York. Waterman, P. C., and True, R Mutipe scattering of waves, J. Math. Phys. 2, J. Acoust. Soc. Am., Vo. 112, No. 6, December 2002 Nai A. Gumerov and Ramani Duraiswami: Mutipe scattering from N spheres 2701