Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory

Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory

V NULL-FIELD METHOD IN ACOUSTIC THEORY In this chapter we will present the fundaments of the null-field method (NFM) for solving the Dirichlet and Neumann boundary-value problems We begin by showing that the scattering problem reduces to the approximation problem of the surface densities by convergent sequences We then present convergent projection methods for the general null-field equations Next we will investigate the conventional null-field method with discrete sources The foundations of the method include convergence analysis following Ramm's treatment [128] and derivation of sufficient conditions which guarantee the convergence of the approximate solution The conclusion of this analysis is that the null-field method converges if the systems of expansion and testing functions form a Riesz basis in L'^{S) Finally, we will present the equivalence between the null-field method and the auxiliary current method BASIC CONCEPTS Let hg solve the null-field equation for the exterior Dirichlet problem and let hsN be an approximation of h Define the approximate scattered field 93 94 CHAPTER V NFM IN ACOUSTICS by «,w(x) = - kN(y)5(x,y,fc) + t x o ( y ) ^ ^ ^ ] d5(y), X e £>„ (5.1) and the residual field by SuNix) =J L;v(y)5(x,y,fc) + «o(y)^^^j^]d5(y),xeA (5.2) s Then, the estimates 11^5 - ^SNWOO.GS ^ ^ W^^ - f^sNh^S (5-3) and ¥uN\UG, |x|, 00 n "W = E E «mn"mn(x), n=Om=—n where amn = ^T>mnUlrnn{y) and a'^^ = ^ P m n u l m n ( y ) - (5.29) 100 CHAPTER V NFM IN ACOUSTICS Thus, assuming that the representations (5.28) and (5.29) are vahd on the entire surface S we will contradict the spherical wave expansion of the Green function In the single spherical coordinate-based null-field method, the infinite set of null-field equations guarantees that the total field will be zero inside the maximal inscribed sphere Because of its analyticity, the total field vanishes throughout the entire interior volume If we consider a finite sequence of null-field equations, we guarantee that the residual field tends to zero inside the maximal inscribed sphere But in general this result does not imply that the residual field converges to zero within the entire interior volume If instead of localized multipoles we use distributed sources (spherical multipoles and point sources) it is possible to overcome the numerical instabilities associated with the single spherical coordinate-based null-field method The explanation is that the null-field conditions will be satisfied in the interior of the support of discrete sources, whose form and position can be correlated with the boundary geometry A similar technique was used by Bates and Wall [11] Using the bilinear expansion of the Green function in the spheroidal coordinate system Bates and Wall imposed the null-field condition inside the inscribed spheroid In this way it was possible to reduce numerical instabilities by decreasing the part of the null-field region not included in D^ Although this method enables many bodies to be analyzed satisfactorily, one can devise shapes for which this method is not particularly suitable In this context, the method based on discrete sources appears to be more flexible We conclude this section by presenting the equivalence between the null-field method and the auxiliary current method Let us consider the Dirichlet boundary-value problem and let S~ be a surface of class C^^ enclosed in Di Define the operator Ti : L^{S) —> L^{S~) by (W/i)(x )=Jh{y)g{x,y,k)dS(y) (5.30) T H E O R E M 2.1: Consider Di a bounded domain of class C^; let the surface S~ be enclosed in Di and assume k ^ p{D~), where D~ is the interior of S~ Let h solve the integral equation of the first kind Hh = UQ (5.31) Then h solves the general null-field equation (2.82) and conversely Proof: Let ii(x) = u o ( x ) - /"/i(y)^(x,y,fc)d5(y), x € R^ - (5.32) CONVENTIONAL NULL-FIELD METHOD 101 Since h solves (5.31) we see that u = on S~ From k ^ p[D~) we find that w = in D^\ whence, by the analyticity of iz, i/ = in Di follows Hence, h satisfies the general null-field equation The converse theorem is immediate Because of this equivalence the integral equation (5.31) has precisely one solution and this solution belongs to C^'^{S) The following theorem is the analog of theorem 2.3 given in Chapter for the operator H T H E O R E M 2.2: Consider Di a hounded domain of class C^ and let the surface S~ he enclosed in Di The operator H defined hy (5.30) is injective and has a dense range provided that k is not an eigenvalue for the interior ofS- Proof: The injectivity of H follows from the assumption k ^ p{D~) and theorem 2.2 of Chapter For proving the second part of the theorem we have to show that N{V)) = {0}, where H^ is the adjoint operator of H and N{H^) is the null space of H ^ Since the adjoint operator H^ : L^{S-) -^ L'^(S) is given hy {n^a) (y) = J a(x)p*(x,y,fc)d5(x), (5.33) swe may proceed as in theorem 2.3 of the precedent chapter to conclude The operator 7i has an analytic kernel and therefore the integral equation is severely ill-posed Actually, the integral operator H acting from L^{S) into 1/^(5"") is a compact operator with an open range of values The integral equation (5.31) may be solved by using the Tikhonov regularization, that is by solving Xhx-hH^nhx = n^uo (5.34) with the regularization parameter A > From the classical theory of the Tikhonov regularization scheme, we know that the operator AJ 4- H^H : L^{S) —• L^{S) is bijective and has a bounded inverse Furthermore, since H is injective TZx = {XI4- H^H)~^H^ defines a regularization scheme with ||72,;^|| < / \ / A (cf Colton and Kress [35] for a detailed discussion) Note that the Tikhonov regularization can be interpreted as a penalized residual minimization since hx solving (5.34) minimizes the Tikhonov functional /iA = argmin{||W/i-uo|l2,5- + A||ft||2,5} (5.35) Projection methods for the integral equation (5.34) are given by theorem 3.4 of Chapter with A = XI and B = V)H The approximate solution hxN is sought in the form of a linear combination of regular fields hxN = X^^=i ^^i^^, a-nd assumed to satisfy the projection relations ({XI + V)U) hxN - W^uo, i^D^s = 0, 1/ = 1,2, , N (5.36) 102 CHAPTER V NFM IN ACOUSTICS In the electromagnetic case, we will analyze the Tikhonov regularization from a computational point of view A projection scheme for the integral equation (5.31) is 2 5- = , ^ = l,2, ,iV, (5.37) where h^ = Y^^z^i ^^V'^- Note that the system {'Hipl}^_^ is complete in 1/^(5"") provided the system {i^l}^^^ is complete in L^{S) and k ^ p{D^), The projection scheme (5.37) is equivalent to the minimization problem a = argmin \\HhN — tio||2 Il2,5s- » (5.38) iT with a = [a^y ,ti = 1,2, ,N Now, let /ijv(y) = E n = i « n % - y n ) j y €5, where { y n } ^ i is a dense set of points on Choose a mesh on S with Xj, j = 1,2, , J, located at the center of each cell Then, we may compute a by solving the discrete version of the minimization problem (5.38), i.e a = arg || A x - f Ufa, (5.39) where A = [ajn]y ajn = ^(xj,y„,A:), j = 1,2, , J, n = 1,2, , AT, is a design matrix and f = [fj] , fj = uo(xj), j = 1,2, , J The above leastsquares problem is similar to that obtained in the auxiliary sources method but with the collocation points replaced by the source points and conversely In other words, the auxiliary sources are kept on the surface and the collocation surface is shifted inside the body Numerical simulations performed by Zaridze et al [171] demonstrate that the computation of the surface fields by the above scheme does not avoid the problem of scattered field singularities In Figures 5.2 and 5.3, the dependence of the far-field pattern on the geometry of the auxiliary surface is shown The plotted data show that the convergence occurs when the auxiliary and extended boundary surfaces contain the scattered field singularities Conversely, when the singularities are outside of these surfaces the solution diverges 2.3 Transition matrix The transition matrix relates the expansion coeflScients of the incident and the scattered fields and plays an important role in multiple scattering and orientation averaging problems Let us assume that the systems {^J,} _i and {^V^i/5n}^_j form a Schauder basis in L^{S) Then, h can be represented as CONVENTIONAL NULL-FIELD METHOD 103 Far-field pattern 1.15 1.10 1.05^ 1.00 0.95 10 14 18 Number of sources 22 FIGURE 5.2 Convergence of the far-field pattern at ^ = 0° versus the number of discrete sources The scatterer is a perfectly conducting cylinder with an aspect ratio of h/a = 0.6 The auxiliary curve C " is elliptic with 6i/o = 0.55 and: (a) a i / o = 0.55, (b) ai/a = 0.70 and (c) ai/a = 0.85 The data are computed with the auxiliary sources method Far-field pattern 1.15 1.10 1.05 1.00 0.95 10 14 18 Number of s o u r c e s FIGURE 5.3 method The same as in Figure 5.2 but the data are computed with the null-field 104 CHAPTER V NFM IN ACOUSTICS for the Dirichlet problem and as oo h^Y.a^^l (5.41) for the Neumann problem Let us express the scattered and the incident fields as oo oo Us = ^VJ^ul, uo = ^V^alul, (5.42) where the first series converges uniformly outside a sphere enclosing , while, by assumption, the second series converges uniformly inside this sphere Here, i/ is a complex index containing m and n, that is i/ = (m, n) and V = 1,2, , when n = 0,1, , and m = —n, ,n The transition matrix is defined as the connecting matrix which generates the coefficients of the scattered wave by premultiplication on the coefficients of the incident wave Thus, we have [M = T [ a ° ] , (5.43) T = BA-^Ao (5.44) where Explicit expressions for the matrices A, AQ and B are given below: < = /^^^^^•< = / ^ i ^ ^ ^ ' s AZ, ^^^ s = ^ / ( v ^ ' ^ ^ - < ^ ) d = -^oV TT y dn "^"^^^ ^^^'^ s (5.45) IT J "^^ dn ^^' s where V = {—m^n) Representing the transition matrix in the form (5.44) we tacitly assumed that A~^ exists The following result which is due to Kleinman et al [82] shows that the matrix A is indeed invertible 2.3: Let { ^ ^ } ^ j and { ^ } ^ ^ be any two bases of a Hubert space H If a matrix A is defined to have elements v4^^ = (-0^, 0i,) „ , I/,/x = 1,2, , then A~^ exists THEOREM CONVENTIONAL NULL-FIELD METHOD 105 Proof: To show that A~^ exist we must estabhsh that there exist ^v^^ ^,/^ = I5 •••? with the property that 00 oc in which case A~^ = [V^/UT/I^ ^'/^ = l^ - Let w^i^ r ^^^ be biorthogonal to the systems {^^/}^i Q^nd {