Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory

Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources IV discrete sources method in acoustic theory

IV DISCRETE SOURCES METHOD IN ACOUSTIC THEORY Within the analysis of the preceding chapters, we now are well prepared to present the mathematical justification of the discrete sources method (DSM) Only the exterior Dirichlet, Neumann and impedance boundaryvalue problems are discussed in detail since the basic concepts are fully represented in these cases We include in our analysis a description of the smoothing procedure developed by Yasuura and Ikuno [166] In addition to a strict mathematical model we also consider practical aspects of the method, such as the correct choice of the support of discrete sources, the determination of the amplitudes by a stable numerical algorithm and the estimation of the accuracy of the results In particular, we will show that the convergence and the stability of the method depends on the position of the singularities of the scattered field We then proceed to briefly analyze the acoustic scattering by axisymmetric particles We will show that the use of distributed spherical wave functions located on the particle symmetry axis reduces the scattering problem to a sequence of one-dimensional approximation problems relative to the Fourier harmonics of the fields Finally, we will present a method for correlating the position of the support of discrete sources with the singularities of the scattered field 67 68 CHAPTER IV DSM IN ACOUSTICS BASIC CONCEPTS Let us consider the boundary-value problem consisting in Helmholtz equation /HiUs -f k'^Us = in Ds, (4,1) r-T • Vn^ - jkus = ^ f FT ) ^ |x| - • oo, (4.2) the radiation condition uniformly for all directions x / |x|, and the boundary condition C{us 4- tio) = on S (4.3) Here £ is a boundary operator given by £ = for the Dirichlet problem, C = d/dn for the Neumann problem and £ =1 — ^d/dn for the impedance problem Let us construct the approximate solution to the boundary-value problem (4.1)-(4.3) as a finite linear combination of elementary sources «,A,(x) = 5;a^V^(x), (4.4) t/=l where the functions V^^ satisfy the Helmholtz equation in £)«, the radiation condition at infinity and the systems { £ ^ ^ } ^ ^ j , that is {V^^}^^i ? [d'^^y/dn]^^^ and {V^^ - 'yd\l^l/dn}^^^ are complete and linear independent in L^(5) The functions V^^ represents the localized radiating spherical wave functions u^^, the distributed spherical wave functions there exist N and N hsNix)^ J2 ^^XU^) (4.22) such that I|/i5iv4-/i||2,c /x, we construct = (f:a^i,l + «o) - j - j ^ ( E a ' ^ i > l +uo,l) t (4.24) and deduce that \\haN 4- h\\2 c ^^ LC!LI ôf)^V^^ + ^o M with c > n2fC' Thus, choosing the set {af)^}^^^, such that ^1^=1 a^^^ H- tto II < €/c we get (4.23) and the statement is proved ll2,C 72 CHAPTER IV DSM IN ACOUSTICS If we define by LQ{C) the subset of L'^{C) which is orthogonal to constants, i.e Ll{C) = {f/feL\C), if,1)2,0 = 0}, then the functions h^xt^^ — 1>2, , and hsN belongs to LQ{C) Clearly, LQ{C) is a subspace of L'^{C) To show this we firstly observe that LQ{C) is a vector subspace, because for any a, /? € C and any f,g e L§(C), {af + /3p, l>2,c = ; whence af + Pg e Ll{C) To prove that Ll{C) is complete let us choose a Cauchy sequence (/n) C Ll{C); it converges to some / € L'^{C) because JL^(C) is complete Now for any g e L?{C)^ we have (/n»y)2,c "^ if^9)2,0 as n —> 00 For ^ = we have {fnA)2C = 0, n = 1,2, ; thus {/, 1)2^^ = and s o / € L ( C ) Next, we introduce the smoothing operator {Kf){s) = I Ki{s,t)mdt, (4.25) with the kernel Ki{s,t) = 0{s - f) - (5 - t) - i (4.26) Here stands for the unit step function Direct calculations shows that ICf is an indefinite integral of the Lo'^omponent of / , ^ = /-(/,l>3,c, (4-27) and that ICf belongs to Ll{C), Thus, the operator /C acting on LQ{C) gives the indefinite integral of the function We call /C the smoothing operator since an indefinite integration usually increases the smoothness of a function A higher-order smoothing operator is defined recursively by /CP+7 = /C(/CP/), p = 1,2, (4.28) By definition, it is obviously that /C^/ belongs to LQ{C) and that l i ^ = ; C / (4.29) as Let us now integrate the right-hand side of (4.20) by parts We obtain NUMERICAL IMPLEMENTATION 73 Further integration by parts yields uM^) - «.(x) = (-1)" I ^ c ( ^ ^ ) C^" ihsN + h)) {s)ds (4.31) for p = 1,2, Thus, we can construct an approximate solution by minimizing the boundary residual WlC^hgN -I- f^^h\\2Q , i.e a^ = arg WlC^hsN + IC^hWlc^ (4.32) We note that the systems {fC^xli 2^ = 1,2, , i/^^fi}, p = 1,2, , are complete in LQ{C) TO prove this we use the identity (/C5,/)2.c + (ff.'C/)2,c = 0, (4.33) which holds for any / , ^ € L^{C) In the case p = we start with (iCxli 7)2 c = for 1/ = 1,2, , t/ 7^ A^, and / E LĐ(C) We get (x^^/>2,c = ôằ ^ = 1,2, , V ^ ii\ whence, by the completeness of the system {xj, 1/ = 1,2, , u ^ yi)^ we arrive at /C/ = on C Since d(/C/) /ds = and / G L'Q{C) we deduce that / = and the proof is complete For p > we proceed by induction Since \\Kr>{KN + h)\\^^c < •^W-'ih.N+ h)\\^ ,.< we see that the mean-square error of the pth order solution decreases faster than the mean-square error of the (p — l)th order solution as the number of truncation index N increases In fact, the convergence of solution becomes faster as we increase p However, it is noted that we should not expect to have good solutions by this method with p greater than some po, which is decided by the nature of the problem and the accuracy of the numerical computation This is because a higher-order smoothing procedure may neglect an important contribution of some space harmonics which closely relates a higher-order Fourier component The practical range of p may be 12,5» rn,n = 1,2, , AT, is positive definite It is worthwhile to note that when N increases the condition number of G^v increases This is due to the fact that the points x;;^ are more condensed on S~; thus the linear independence of the system of discrete sources is *altered\ In other words, some basis functions fit the data about equally well so that the Gramm matrix, unable to distinguish between them, becomes singular This fact is also reflected by the asymptotic behavior of the determinant g^ as AT —• cx) THEOREM 2.5: Consider Di a bounded domain of R^ and let the smooth surface S" be enclosed in Di, Choose a mesh on S"and let x~ be located at the center of each cell Assume for simplicity that |x^ — x ^ | = h for any neighborhood cells n and m Then, for sufficiently large N we have gN=0 [inhf) (4.53) Proof: Without loss of generality we assume that N = 2L and use the well-known estimate for the Grammes determinant (cf., e.g Aleksidze [2]) to get 9N{^T^^2^-">^2L) < 92{^T ^^2)"'92{^2L-V^2L)- (4-54) Here (^~ and ffn-^-i correspond to the neighboring cells n and (n -f 1) Let us evaluate the second-order determinant Prom the mean value theorem fZ+i (x) = 'Pn (x) - hn,-^-^^ • Vip- (x) + 0(/i2) (4.56) 81 NUMERICAL IMPLEMENTATION we obtain 92{'Pn+l) = / l ' f | | v ; | | , S ll^nll'.S " | ( v ; , * „ > , s f + ^ih) , (4.57) where nx;:*;^., = (x„+i-x„)/ |x„+i - x„ | and *„(x) = n^-x;^, •^'fin (x)Retaining the terms in h^ we receive < < ifih)'^ as N -*oo, (4.58) with u= max _ , \/\kn\\ls\\'^jls-\{fn,^n),^s\'n=l,3, ,2L- (4-59) The theorem is proved As a consequence of the above theorem we observe that for any e > and arbitrary surfaces S" and S , there exists an integer iVo, depending on S" and 5, such that QN < £ for all N > NQ, In order to reduce the computational effort, a discrete least-squares problem is formulated Let us choose a mesh on S with y^, j = 1,2, , J, located at the center of e€u:h cell Assume that the area of each cell on S is proportional to /IQ Then, using the rectangular integration rule N / wo(y) + X^anV?n(y) d5(y) n=l N (4.60) ^o(yj)-hX^an 0, z £ R } and define the complex plane E = {z = (Rez, Imz) / Rez, Imz € R } in such a way that the real axis Rez coincides with the 2-axis The spherical wave functions can be expressed in terms of the coordinates of the source point z € E and the observation point r; G E by using the analytic continuation procedure, that is «L(x) = h(^\kR)P}r\cosd)ei"^^, (4.89) where R^=p^ + {z- z)2, sin? = -^, cos? = ^ ^ (4.90) We choose the sign of R such th^t Re i? > By definition the point z € E is called the image of the point r/ € E if ^Tjz ~ 0* Taking into account that i?2^ = {p- Imz)(p-f- Imz) - j l m z ( ^ - Rez) + {z - Rez)^ (4.91) we see that for each point 7/ there exist two images z^'^ with Re z^'2 ^ ^^ ijn 2i'2 ^ ^ ^ ^4 92) MODEL WITH DISTRIBUTED SPHERICAL WAVE FUNCTIONS 89 Imz FIGURE 4.3 Illustration of the complex plane The symmetry axis coincides with the Rez-axis The curve L is the image of the curve L Consequently, the^ region of analytic continuation is the domain JD C E, whose boundary L coincides with the image of Jb^he generator of revolution L C S as pictured in Figure 4.3 A point z € E generates the singularity point 77 € S of coordinates p = |Imz|, z = Rez (4.93) and evidently, the singularities are distributed in the real space at the ring of radius p From (4.93) we see that the original complex source z = (Re z, Imz) and the conjugate source % = (Re z, —Imz) are indistinguishable in the sense that they produce the same singularity point in the real space We now pay attention to the method of solution T H E O R E M 3.7: Assume S is a surface of class C^ and let (z^) C D be a sequence of points distributed symmetrical with respect to the real axis Re z and having at least two limit points in D, Then the system {(TL ^ = 1.2, , m € Z } is complete in L^{S) Proof: To prove this, we use the same technique as in theorem 1.4 of Chapter and the uniqueness theorem of analytic functions Relying on the above theorem we conclude that the approximate solution to the boundary-value problem (4.1)-(4.3) can be constructed as a finite linear combination of spherical wave functions distributed in the complex plane This technique allows us to correlate the position of the support of discrete sources with the singularities of the analytic continuation of the scattered field into Dj 90 CHAPTER IV DSM IN ACOUSTICS TABLE 4.1 The residual field for different dimensions of the algebraic system of equations residual field = 0° = 30° = 60° ^max ^max Jmax 11 13 15 17 19 12 14 20 24 28 32 36 0.1446 0.0280 0.0057 0.0012 0.0003 0.0001 0.0001 0.1114 0.0325 0.0094 0.0033 0.0020 0.0020 0.0019 0.0722 0.0274 0.0176 0.0163 0.0161 0.0161 0.0161 15 17 28 32 0.0003 0.0001 0.0008 0.0003 0.0028 0.0027 Finally, we present some numerical results computed with the above method The scattering problem is that of a spheroid with Dirichlet boundary conditions The size parameters are fca = and kb = The ratio of the generator length to the wavelength is of about 2.13 The poles and the collocation points are chosen in the half-plane E accordingly to Zn = acosdn (4.94) Zj = acosT?j, Pj = 6sint?j, (4.95) and respectively, where 'dn = -x ^^max + (^-1) , n = l,2, ,nmax, (4.96) + (n - ) T ^ , j = 1,2, , j„,ax (4.97) ^max and ^j = :Tf^Jmax Jmax In Table 4.1, we present the residual field for different dimensions of the algebraic system of equations It can be seen that for a 60° incidence and ^max = the residual field is of about 0.016, while in the case mmax = the residual field is smaller than 0.0028 These values correspond to a number of 15 sources and 28 collocation points In Figure 4.4 we present the absolute value of the far-field pattern in the azimuthal plane (/? = 0°, for normal and oblique incidences (7 = 60°) The data correspond to nmax = 15 and jm&x = 28 In Chapter we will extend this technique to the electromagnetic case NOTES AND COMMENTS 91 lO-M FIGURE 4.4 incidences 60 120 180 240 300 360 Scattering angle (deg) Far-field pattern in the azimuthal plane y? = 0°, for normal and oblique NOTES A N D C O M M E N T S The mathematical background of the conventional method of auxiliary sources was given by the Georgian mathematicians I N Vekua, V D Kupradze, and M A Aleksidze In [145]-[148], Vekua formulated the 'method of expansion in terms of metaharmonic functions' as a method of solving a boundary-value problem by using expansions in terms of singular solutions to the corresponding differential equation Kupradze [85] and Kupradze and Aleksidze [86] used representations in terms of fundamental solutions to the differential equation in question and elaborated the 'method of generalized Fourier series' This technique required an orthogonalization of the sets of fundamental solutions beforehand Numerical simulations performed by Aleksidze [2] and Zaridze and Tsverikmazashvili [170] showed the nonoptimally of the conventional orthogonalization procedure In this context, Bakhvalov [7] showed that the optimal solution can be achieved by using expansions in terms of nonorthogonal functions and the collocation method Independently, from the Georgian mathematicians the basic concepts of the discrete source method for two-dimensional problems was given by Yasuura and Itakura [167] The authors introduced the 'set of modal functions' (the functions X/JI) and established a method of solution called the conventional Yasuura method Later Yasuura and Ikuno [166] equipped a smoothing procedure with the original method to accelerate the convergence of the approximate solution Yasuura and Okuno 92 CHAPTER IV DSM IN ACOUSTICS [168] have made some modifications of the smoothing procedure for problems with edged boundaries This method is known as the Yasuura method with singular-smoothing procedure In the context of the so-called discrete singularity method Nishimura and Shigesawa [118] used distributed two-dimensional point sources or line singularities to compute scattering by cylindrical structures The same technique (otherwise known as the multifilament technique) has been used by Leviatan and Boag [92] Distributed spherical vector wave functions (lowest order multipoles) have been introduced by Sveshnikov and Eremin [141] for solving threedimensional axisymmetric problems The procedure of analytic continuation of the fields onto the complex plane has been done by Eremin [47] It has been shown that for oblate spheroids the distribution of discrete sources in the complex plane increases the stability of the computational scheme and reduces the residual fields Various numerical implementations of the discrete sources method with distributed spherical wave functions have been considered by Eremin and Sveshnikov [55] The first 2D program for calculating guided waves on cylindrical structures using multiple two-dimensional cylindrical wave functions was written in 1980 by Hafner [65] Later, this program was extended to solve boundary-value problems in 2D electrostatics by including a large class of expansion functions (see Hafner [66]) A program for the automatic choice of the expansion functions was given by Leuchtmann [91] Further improvements on the theory of the discrete sources method has been done by Leviatan and his coworkers In this work it has been shown that the discrete sources method can be more effective when specially devised sources are used The idea is to arrive at a localized impedance matrix, which can in turn be rendered sparse by a thresholding procedure The matrix localization may be effected by using array of multipole sources arranged in such a manner as to produce a field focus on the surface (see Pogorzelski [126]) The use of dipoles and multipoles located in complex space has been discussed by Erez and Leviatan [61], [62] The localization may also be attained by applying a discrete Fourier transformation so as to transform the individual sources into directional arrays (see Leviatan et al [93] for details) A multiresolution analysis based on wavelet transformations has been discussed by Baharav and Leviatan [6]