1 Introduction 2. 3 Spectral informations for the single-layer potential for low frequency single scattering

Transcription

1 Spectral and Condition Number Estimates of the Acoustic Single-Layer Operator for Low-Frequency Multiple Scattering. Part I: From Single to Multiple Scattering, Dilute Media Xavier Antoine Bertrand Thierry Abstract The aim of this paper is to derive asymptotic spectral and condition number estimates of the acoustic single-layer potential for various low frequency multiple scattering problems. The obstacles are supposed to be distant (dilute media to obtain the approximation formulas. We show that an approach based on the Gershgorin disks provides limited spectral informations. We introduce an alternative approach by applying the power iteration method to the limit matrix (associated with the zero order spatial modes which results in satisfactory estimates. All these approximations are built for circular cylinders and formally extended to ellipses and rectangles and for linear boundary element methods with non uniform meshes. This study is completed in [6] by new spectral estimates for the case of close obstacles. Contents Introduction The Electric Field Integral Equation (EFIE and the single-layer operator 3. Single-layer potential and EFIE for multiple scattering Expression of the single-layer potential in the Fourier basis for the multiple scattering problem by M circular cylinders Spectral informations for the single-layer potential for low frequency single scattering 6 3. The circular cylinder case Diagonalization in the Fourier basis Low frequency spectral and condition number estimates Link with the boundary element approximation and formal extensions to other objects Boundary element approximation The circular cylinder Estimates for other geometries The authors are partially supported by the French ANR fundings under the project MicroWave NT Institut Elie Cartan Nancy, Nancy-Université, CNRS UMR 75, INRIA CORIDA Team, Boulevard des Aiguillettes B.P. 39, F-5456 Vandoeuvre-lès-Nancy, France (Xavier.Antoine@iecn.u-nancy.fr, Bertrand.Thierry@iecn.u-nancy.fr.

2 4 Condition number estimates for the single-layer potential at low frequency for distant obstacles: Gershgorin disks approach 5 4. Localization by the Gershgorin theorems and asymptotics Accuracy of the localization of the eigenvalues with smallest and largest modulus An alternative approach based on the limit of L 6 Link with the boundary element approximation and formal extensions to other objects 3 6. Boundary element approximation The circular cylinder Extensions to other geometries Conclusion 34 Introduction We propose in this (first paper some spectral and condition number estimates of the acoustic single-layer potential for low frequency multiple scattering configurations. A few studies have been developed for acoustic integral operators but for (high frequency single scattering problems. For examples, let us mention the circular cylinder case [,, 4, 5], the case of various convex and non convex single scatterers [7, 8] or also the case of open surfaces and guiding structures [, 3, 9]. Considering multiple scattering leads to new difficulties due to the presence of interactions between scatterers. Mathematically, this means that new parameters are involved into the spectral analysis: the distances between separated obstacles. We focus here on the low frequency case because the medium and high frequency cases are still out of reach and first require a better understanding of the low frequency situation that is investigated here. The single-layer potential (associated to the Electric Field Integral Equation (EFIE in Section is considered here as an example but other standard integral operators (mainly the trace of the double-layer potential and the normal derivative traces of the single- and double-layer potentials could also be studied. However, we will see that understanding the spectral properties of the single-layer potential already requires a non trivial and tricky asymptotic analysis. In particular, several regimes must be analyzed in details. We propose in this first part to examine the case of dilute media which correspond to the situation where the distances between the obstacles are larger than a few wavelengths. In a second part [6], we develop a complete analysis when the obstacles are close (dense media. The intermediate case, that is when the distances between the obstacles is of the order of the wavelength, appears to be more delicate. A first step of our approach consists in a detailed spectral and condition number analysis of the acoustic single-layer operator for single scattering in the low frequency regime (section 3. We propose to investigate the explicit case where the scatterer is a disk which results in the diagonalisation of the integral operator in the Fourier basis. We therefore obtain asymptotic expressions of the eigenvalues of the operator which are then extended to other obstacles (elliptical and rectangular cylinders and to the linear boundary element approximation of the integral operator. In Section, we give the explicit expression of the single-layer operator for multiple scattering by a collection of circular cylinders. Unlike the single scattering case, the operator is no longer diagonalizable. For this reason, a more complicate asymptotic analysis must be developed. In Section 4, we first

3 study the possibility of applying Gershgorin circle theorems to get informations on the spectrum in conjunction with asymptotic expansions for the case of distant obstacles. It unfortunately appears that this method is quite limited. For this reason, an alternative approach based on the application of the power iteration method to a small matrix associated with the zero order Fourier modes and asymptotic expansions is proposed in Section 5. We show that pretty accurate estimates of the spectrum as well as condition number of the single-layer operator can be obtained. We furthermore extend these estimates to ellipses and rectangles and to the linear boundary element approach. All these approximate formulas are validated with various numerical simulations. Finally, a conclusion is given in Section 7. The case of dense media where the obstacles are close is treated in a second paper [6] by using this last approach. The Electric Field Integral Equation (EFIE and the single-layer operator. Single-layer potential and EFIE for multiple scattering Let us consider a homogeneous acoustic medium filling the whole space R and containing a bounded open set Ω. We denote by Γ its boundary and by n its unit outward normal. Moreover, the propagation domain Ω + = R \ Ω is assumed to be connected. We consider the scattering of an incident time-harmonic acoustic wave u inc by the obstacle Ω (the time dependence is assumed to be of the form e iωt and the wavenumber k is real and positive. In other words, we seek the scattered field u, solution to the following problem u + k u = in Ω +, u = u inc ( on Γ, ( x lim x x / u x iku =, where x = (x + x /. Another formulation of system ( is based on the Electric Field Integral Equation (EFIE given through u = L ρ, ( where the single-layer integral operator L is defined by L : H / (Γ Hloc (Rd \ Γ ρ L ρ, x R d \ Γ, L ρ(x = Here, the Green s function G(, is given by Γ G(x, yρ(y dγ(y. x, y R, x y, G(x, y = i 4 H( (k x y, where H ( is the zeroth order Hankel function of the first kind. We refer to [] concerning the functional spaces. Taking the trace on Γ of the scattered field ( leads to the EFIE Lρ = u inc Γ, in H / (Γ, (3 3

4 where the single-layer boundary integral operator L is given by L : H / (Γ H / (Γ ρ Lρ, x Γ, Lρ(x = Γ G(x, yρ(y dγ(y. It is well known (see e.g. Theorem 3.4. of [] that the operator L defines an isomorphism from H / (Γ to H / (Γ except for the set F D (Ω of the Dirichlet irregular frequencies, that is the wavenumbers k for which the interior homogeneous Dirichlet problem { ( + k w = in Ω w = on Γ, admits non trivial solution. As a consequence, if k F D (Ω, then the single-layer potential L ρ solves the scattering problem ( if and only if ρ is the unique solution of the EFIE (3. For numerical reasons, it is classical to consider the variational formulation of the EFIE, which reads on L (Γ as { Find ρ H / (Γ such that Φ L (Γ, (Lρ, Φ L (Γ = ( u inc Γ, Φ L (Γ. (4 The inner product (, L (Γ is defined by (f, g L (Γ L (Γ, (f, g L (Γ = Γ fḡ dγ, where ḡ denotes the complex conjugate of g. The multiple scattering case consists in an obstacle Ω with M components Ω,..., Ω M with respective boundaries Γ,..., Γ M. Then, we have L (Γ = L (Γ... L (Γ M and M (f, g L (Γ = f p ḡ p dγ p, Γ p p= where f = (f,..., f M and g = (g,..., g M are L (Γ functions. Throughout this paper and as pointed out above, the density ρ is assumed to belong (at least to L (Γ. Thus, the weak formulation (4 can be written as { Find ρ L (Γ such that Φ L (Γ, (Lρ, Φ L (Γ = ( u inc Γ, Φ L (Γ. (5. Expression of the single-layer potential in the Fourier basis for the multiple scattering problem by M circular cylinders We assume that Ω is the union of M circular cylinders Ω p, p =,..., M, strictly disjoint (no sticky case. The boundary Γ p of Ω p is a circle with center O p and radius a p, the boundary of Ω being quoted Γ. The explicit expression of the single-layer potential in the case of several disks, has been obtained e.g. in [4]. Let us recall that, for p =,..., M, any point x of R is given by its polar coordinates link to the obstacle Ω p r p (x = O p x, r p (x = r p (x, θ p (x = Angle( Ox, O p x. 4

5 If O is the origin and O p is the center of the disc Ω p, we set and, for q =,..., M, p q, b p = OO p, b p = b p, α p = Angle( Ox, b p, b pq = O q O p, b pq = b pq, α pq = Angle( Ox, b pq. For each disc Ω p, p =,..., M, we introduce the Fourier basis functions (ϕ p m m Z defined by ϕ p m : Γ p C x eimθp(x πap. Each family (ϕ p m m Z forms an orthonormal basis of L (Γ p for the natural inner product. We extend to Γ the functions ϕ p m by functions Φ p m such that { Φ p ϕ p m on Γ p, m = (6 on Γ q, q p. Hence, the family B = (Φ p m p=,...,m,m Z forms an orthonormal basis of L (Γ. After projection onto the basis functions, the weak form (5 of the single-layer potential writes (Lρ, Φ p m L (Γ = ( u inc Γ, Φ p m, p =,..., M, m Z. (7 L (Γ By decomposing ρ in the basis B: ρ = M q= n Z ρq nφ q n, and by injecting this expression in formulation (7, we obtain N q= n Z ρ q n (Φ q n, Φ p m L (Γ = ( u inc Γ, Φ p m, p {,..., M}, m Z. L (Γ These equations can be written as a matrix system with L, L,... L,M L, L,... L,M L M, LM,... LM,M L ρ = Ũ, (8 ρ ρ. ρ M = Ũ Ũ. Ũ M where each infinite block L p,q, p, q M of the matrix L has coefficients, L p,q m,n = (LΦ q n, Φ p m L (Γ, m, n Z. (9 The coefficients (9 of the single-layer potential, for two objects p and q, with p, q =,..., M, for two Fourier modes m, n Z, are given by the expressions iπa p L p,q J m(ka p H m ( (ka p δ mn if p = q, m,n = iπ ( a p a q J m (ka p S nm (b pq J n (ka q otherwise. 5

6 Symbol δ mn is the Krönecker s delta function, equal to if m = n and otherwise. The quantity S nm (b pq is given by: S nm (b pq = H ( n m (kb pqe i(n mαpq, for p, q =,..., M, p q and m, n Z. A block matrix notation is more convenient iπa p L p,q J p Hp if p = q, = iπ a p a q Jp ( S p,q T Jq otherwise, where the infinite matrices J p and H p are diagonal with respective general terms J m (ka p and H ( S p,q m (ka p. The infinite matrix ( S p,q T is the transposed of the matrix ( S p,q, with coefficients m,n = S mn (b pq. Let us now consider a finite dimensional projection. To this end, we truncate system (8 by keeping, for each Fourier series (Φ p m m Z, p =,..., M, N p + modes such that: N p m N p. The resulting truncated system is denoted by Lρ = U, with L, L,... L,M L, L,... L, L = L M, L M,... L M,M ρ ρ. ρ M = where each block L p,q, of size (N p + (N q +, for p, q =,..., M, of matrix L has coefficients ( L p,q p,q m,n = L m,n, for N p m N p and N q q N q. Each block can be written in a matrix form, for p, q =,..., M, iπa p L p,q Jp H p if p = q, = iπ a p a q J p (S p,q T J q otherwise, where the diagonal matrices J p and H p of size (N p + (N p + have respective general terms J m (ka p and H m ( (ka p, and matrix (S p,q T of size (N p + (N q + is the transposed of (S p,q, with coefficients S p,q m,n = S mn (b pq. 3 Spectral informations for the single-layer potential for low frequency single scattering In the sequel, we assume that k F D (Ω to have the well-posedness of the EFIE. The goal of the present paper consists in obtaining spectral informations (eigenvalues and conditioning of the single-layer potential for multiple scattering problems. Before this, let us consider the low frequency single scattering case for a disk. In this situation, the weak formulation (5 leads to explicit coefficients of the diagonal matrix which can then be approximated by using low frequency expansions. This will provide estimates of the eigenvalues of the single-layer operator for circular obstacles. We conclude this section by formal extensions to other objects and make the link with the boundary element discretization. Let us note that this study can be seen as a first step before investigating the (more complex high frequency problem. 6 U U. U M,

7 3. The circular cylinder case 3.. Diagonalization in the Fourier basis We assume here that Ω is a disk with radius a and centered at the origin O. In this case, the infinite matrix L is diagonal. Moreover, by using (, its coefficients, denoted by L mn, are given by iπa m, n Z, Lmn = δ mn J m(kah m ( (ka. ( For the finite dimensional approximation, we keep N + modes such that the indices m satisfy N m N. We denote by L = (L mn N m,n N the diagonal matrix of size (N + (N + which approximates the single-layer operator and with coefficients N m, n N, L mn = L mn = δ mn iπa J m(kah ( m (ka. Remark. For one disk, the single-layer operator is diagonal with coefficients: µ m = L mm. Furthermore, the operator L is singular if one of its eigenvalues µ m is equal to zero, which means that J m (ka =, for a certain value of m for ka fixed (let us recall that H m ( (x for any x >. This is not the case here since we assume that k F D (Ω = {k/ m N/J m (ka = }. 3.. Low frequency spectral and condition number estimates Since the matrix L is diagonal, its eigenvalues, denoted by µ m, m = N,..., N, are directly given by the diagonal terms µ m = L mm = iπa J m(kah ( m (ka, m = N,..., N. ( This means that we get the explicit expression of each eigenvalue of L. We now propose a detailed low frequency study of this spectrum, that is when ka tends towards. Let us first remark that, following [, (9..5], we have m Z, x, H ( m (x = ( m H ( m (x, J m (x = ( m J m (x. (3 These properties imply that the eigenvalues µ m of L are double: µ m = µ m, m = N,..., N, m, the eigenvalue µ being single. The eigenvalues are explicitly known. However, they are given by special functions which means that the dependence in terms of dimensionless wave number ka and truncation parameter N is not easy to analyze. To get simpler expressions we consider low frequency expansions of the eigenvalues. Let us remark that the medium/high frequency analysis is also possible [4, 5]. Nevertheless, its extension to high frequency multiple scattering is still more tricky. For this reason, we restrict ourselves to the low frequency regime. Let us consider that ka. In this particular case, the asymptotic expansions of J m and H m ( give (see relations (9.. and (9..3 in [] [ ( ] ka a ln + γ + i πa µ m + O ( (ka ln(ka for m =, a m + O ( (ka (4 for m, 7

8 where γ = is the Euler s constant. To illustrate and validate this approximation, we report on Figure the real and imaginary parts of the exact ( and approximate (4 eigenvalues by truncating the approximation to N =, for a dimensionless wave number ka =. (a = here. We observe that the approximation provides close values to the expected ones. Moreover, we remark that the eigenvalue associated with the mode m = is the only one to get a significant imaginary part. This physically translates the property that the zero order mode is the only propagating one at low frequency (which is not the case for a higher frequency. For larger ka, the considered approximations introduce more errors. Furthermore, we see that the modulus of the eigenvalues of L tend towards zero when m is large enough. These eigenvalues, which correspond to evanescent modes, have zero as limit since L is a (pseudodifferential operator of order, having therefore eigenvalues associated to high order spatial modes going to zero..8.6 Exact eigenvalues Approximate eigenvalues.6.4 Exact eigenvalues Approximate eigenvalues.4. Real part..8.6 Imaginary part Mode: m (a Real parts Mode: m (b Imaginary parts Figure : Comparison between the exact ( and approximate (4 eigenvalues of L, for ka =. and N =. Let us now introduce the condition number cond(l of matrix L defined by: cond(l = L L, for a matrix norm associated with a vectorial norm on C N+, that is If the -norm is defined by L = LX sup X C N+ \{} X. X = N m= N X m, for any vector X = (X N,..., X N T C N+, we have cond (L = L L. Since L is diagonal, it is normal and cond (L = µ max, µ min where µ max = max m N µ m and µ min = min m N µ m. From the previous low frequency analysis (4, the eigenvalue with maximal modulus is given for m = (propagating mode and 8

9 the one with minimal modulus corresponds to m = N (evanescent mode. Finally, a low frequency approximation (ka of the condition number is cond (L {a [ ln ( ka + γ] + } ( πa / N a for the single circular cylinder case of radius a with truncation Fourier parameter (N Link with the boundary element approximation and formal extensions to other objects 3.. Boundary element approximation Let us now introduce the boundary element approximation of (5. We assume that the boundary Γ and the incident wave u inc are sufficiently smooth so that the density ρ is a L (Γ function. Let Γ h be a polygonal approximation of Γ by using N h segments K j, j N h. Then we have: Γ h = N h j= K j. We designate by h j the length of segment K j and by h the maximal length: h = max j Nh h j. We choose the boundary element subspace V h of L (Γ h with continuous piecewise linear functions over Γ h (5 V h := { ρ h C (Γ h /ρ h Kj P, j N h }. (6 In this framework, the exact variational formulation (5 is approximated by the discrete form { Find ρh V h such that which leads to the linear system Φ h V h, (L h ρ h, Φ h L (Γ h = ( u inc Γh, Φ h L (Γ h (7 { Find ρh C N h such that [L h ] ρ h = [M h ]u inc, (8 where [L h ] M Nh,N h (C is the matrix associated with the discretization of the single-layer operator and [M h ] M Nh,N h (C is the mass matrix for V h, ρ h C N h is the nodal density vector and u inc the nodal incident vector. 3.. The circular cylinder We now relate the spectral Fourier approximation to the boundary element method. Let us assume that we have a uniform mesh with step h. The number of degrees of freedom N E is then: N E = N h πa h. The spectral method requires N + modes. Let us denote by µh min and µh max, respectively, the eigenvalues of [M h ] [L h ] with smallest and largest modulus, respectively. Formally replacing N by πah / in the estimates µ min and µ max, we get a µ min (a, h = πah, µ max (k, a = a [ln ( ka + γ] + 9 ( πa (9.

10 We then obtain the following estimate of the spectral condition number (5 with cond(k, a, Γ h := cond ([M h ] [L h ] cond app (k, a, h, ( cond app (k, a, h = {a [ ln ( ka + γ] + ( πa } / πah, ( a when ka tends towards zero. To validate (, we consider the following test cases. Let a =. We compare on Figures (a and (b the numerical condition number cond(k, a, Γ h of the single-layer potential and its estimate cond a (k, a, h for a uniform mesh. In the first case, we let ka varies for N h =. In the second case, the number of points N h varies for ka =.. In both situations, we see that our estimate is accurate. Moreover, we clearly observe that, when ka is large enough, we loose some accuracy since the low frequency approximation is no longer valid. We consider now a non uniform mesh for Figures (c and (d. To this end, we generate the sequence of segments K j = [x j, x j ], j N h, with x j = a(cos(θ j, sin(θ j, θ j = π (j + ε j (, N h N h where ε j is a random variable of uniform law on [; ] and x Nh = x. The computation of cond(k, a, Γ h is again realized numerically or approximated via formula (. In this last case, the choice of h is more delicate. Indeed, if one considers the dependence of the condition number cond(k, a, Γ h according to the geometry Γ h, this dependence is a priori global and affects µ min (a, h. In the case of a uniform discretization, we may consider an estimate of µ min (a, which is only local by taking cond a (k, a, h min or cond a (k, a, h max, with h min = min j Nh h j Nh j= h j. and h max = max j Nh h j, or global by considering cond app (k, a, h eqv, with h eqv = N h This last choice corresponds to a calculation of the conditioning based on an equivalent uniform discretization for an average step h eqv. For a uniform discretization, the definitions coincide. We finally remark that the choice of h eqv leads to the most accurate estimates Estimates for other geometries We now consider two other geometries a an elliptical scatterer of semi-axes a x along Ox and a x along Ox, b a rectangular cylinder with sidelengths a x and a x along Ox and Ox. These two objects are centered at the origin. For the first case, let us recall that the approximation of the eigenvalues with smallest and largest modulus for P boundary element are estimated respectively by µ min (a, h = a πah, µ max(k, a = a [ln ( ] ka + γ + π a 4. Then we propose to answer the following two questions: how to handle an equivalent radius, denoted by a eqv, for an ellipse, and, since the mesh is non uniform, which mesh step h can be chosen. For

11 7 6 cond(k, a, h cond app (k, a, h 8 6 cond(k, a, h cond app (k, a, h Condition number Condition number p (for N h = a. p (a Condition number vs. ka, N h = (uniform discretization.5.5 p (for N h = a. p (b Condition number vs. N h, ka =. (uniform discretization Condition number cond(k, a, h cond app (k, a, h min cond app (k, a, h max cond app (k, a, h eqv Condition number cond(k, a, h cond app (k, a, h min cond app (k, a, h max cond app (k, a, h eqv p (for N h = a. p (c Condition number vs. ka, N h = (non uniform discretization.5.5 p (for N h = a. p (d Condition number vs. N h, ka =. (non uniform discretization Figure : Validation of the approximation formula for the condition number of the single-layer potential in the circular cylinder case (uniform and non uniform meshes. the ellipse, we propose the three following equivalent radius for the approximation of the eigenvalue with largest modulus a eqv = a x + a x, a eqv = a x a x a, a 3 eqv = x + a x. (3 a x + a x The first approximation consists in considering an equivalent disk with mean radius based on a x and a x and the second one with a curvature given by the mean of the curvatures /a x and /a x. The last radius is based on taking the point (a x, a x / of the ellipse and next choosing the radius of the circle centered at the origin and passing by this point. Concerning the discretization, we propose three possibilities related to h min, h max and h eqv, for each value a j eqv, j =,, 3. Let us fix the following configuration: a x =, a x =.5 and N h = 6. We report on Figures 3(a, 3(b and 3(c respectively the smallest and largest eigenvalues (with respect to their modulus

12 and the condition number vs. k. We can remark that the eigenvalue with largest modulus is well approximated by a 3 eqv and the smallest one is not sensitive to this parameter (as expected. The choice of the equivalent discretization step is however crucial and the best approximation occurs for h min. Further test cases have been realized and always lead to the same conclusion, that is the approximations of the condition number is satisfactory. We also propose on Figure 4 a comparison between the different estimates according to the ratio between the different axes. More precisely, let us fix N h = 6, k =. and a x =. We then let a x varies between.5 and. We can again see that using a 3 eqv is the best choice for the approximation of the eigenvalue with largest modulus and µ h min is obtained for h min. We retain this choice of parameters for the estimates of the eigenvalues as well as condition number of matrix [M h ] [L h ]. Modulus of the eigenvalues h µ max µ max (k, a eqv µ max (k, a eqv µ max (k, a 3 eqv Modulus of the eigenvalues µ h min µ min (a eqv, h min µ min (a eqv, h max µ min (a eqv, h mean p (for ka x = p p (for ka x = p (a Ellipse (a x = and a x =.5: numerical and (b numerical and approximate eigenvalues with smallest modulus vs. ka x, for N h = 6. approximate eigenvalues with largest modulus vs. ka x, for N h = 6. 9 cond ([M h ] [L h ] cond app (k, a 3 eqv, h min 8 Condition number p (for ka = p x (c Ellipse (a x = and a x =.5: numerical and approximate condition numbers vs. ka x, for N h = 6. Figure 3: Validation of the formulae of the condition number for the single-layer potential in the elliptical cylinder case (non uniform mesh.

13 Modulus of the eigenvalues h µ max µ max (k, a eqv µ max (k, a eqv µ max (k, a 3 eqv Modulus of the eigenvalues µ h min µ min (a eqv, h min µ min (a eqv, h max µ min (a eqv, h mean a x a x (a Ellipse (a x = and k =. : numerical and (b Ellipse (a x = and k =. : numerical and approximate eigenvalues with largest modulus vs. a x, approximate eigenvalues with smallest modulus vs. a x, for N h = 6. for N h = cond ([M h ] [L h ] cond app (k,a 3 eqv, h min Condition number a x (c Ellipse (a x = and k =. : numerical and approximate condition numbers vs. a x, for N h = 6. Figure 4: Validation of the formulae of the condition number for the single-layer potential in the elliptical cylinder case (non uniform mesh, for N h = 6, k =., a x =, a x varies between.5 and. For the rectangular cylinder with sidelengths a x and a x, we consider the following approximation a 4 eqv = ( + a x + a x. (4 To get this equivalent radius, we consider the ellipse with semi-axes a x and a x and the other one with semi-axes a x and a x along the abscissa and ordinates, respectively. In particular, the corners of the rectangle own to this last ellipse while the middle points of the sides are on the first one. Next, we consider the ellipse with semi-axes equal to the average of the semi-axes of the two ellipses, that is with semi-axes ( + a x / and ( + a x /. Finally, the disc with the equivalent radius a 3 eqv for the previous ellipse is considered leading to (4. We report on 3

14 Figures 5(a to 5(c the approximation of the largest and smallest approximate eigenvalues and the corresponding condition number. We take a x = and a x =.5 and discretize each side by using 5 points, for a total number of N h = points. As we can see, the approximation based on a 4 eqv is accurate and the choice of the minimal parameter is very satisfactory. Modulus of the eigenvalues h µ max µ max (k, a 4 eqv Modulus of the eigenvalues 7 x µ h min µ min (a 4 eqv, h min µ min (a 4 eqv, h max µ min (a 4 eqv, h mean p (for ka x = p p (for ka x = p (a Numerical and approximate eigenvalues with largest modulus. (b Numerical and approximate eigenvalues with smallest modulus cond ([M h ] [L h ] cond (k, a 4, h app eqv min Condition number p (for ka x = p (c Numerical and approximate condition numbers. Figure 5: Validation of the formulae for the eigenvalues µ h min and µh max and the condition number for the single-layer potential and a rectangular cylinder with half side lengths a x = and a x =.5, vs. ka x (N h = segments, 5 by side. As a conclusion, this first study provides some explicit estimates of the eigenvalues with minimal and maximal modulus as well as condition number estimates of the single-layer potential in the low frequency regime. Results are obtained for the circular, the elliptical and rectangular cylinders. 4

15 4 Condition number estimates for the single-layer potential at low frequency for distant obstacles: Gershgorin disks approach Unlike the single-scattering case, we do not have access directly to the eigenvalues of L for multiple scattering configurations since the matrix is not diagonal. We still assume that we are in the low frequency regime ka, setting a = max a p p=,...,m as the largest radius of the cluster of M circular obstacles. Considering multiple scattering involves some new parameters compared to single-scattering, essentially kb pq, where b pq is the distance between the centers of Ω p and Ω q. We propose to analyze the effect of this parameter on the condition number of the single-layer operator with respect to these asymptotics. We will see that one can obtain partial results for some regimes, the general case being out of reach. We essentially analyze two situations: the case of far obstacles (called dilute media in this paper and the case of close obstacles (dense media treated in [6]. 4. Localization by the Gershgorin theorems and asymptotics In this Section, we assume that the obstacles are far enough, that is kb +, where b = min p,q=,...,m,p q b pq is the smallest intercenter distance between obstacles. In this particular regime, we can expect that the coupling effects between scatterers is weak enough so that the behavior of the system is close to the single scattering case. To localize the spectrum of L, we use an approach based on the Gershgorin disks directly applied to the matrix of the truncated operator. Let us recall the first Gershgorin circles theorem [3] for L. Theorem (First Gershgorin s Theorem. For p =,..., M, and N p m N p, we introduce the lines Gershgorin disks R p m = { z C : z L p,p m,m R p m}, (5 and the row disks where the radius R p m and C p m are given by R p m = q M,q p n= N q C p m = { z C : z L p,p m,m C p m}, (6 N q Then, the spectrum σ(l of L satisfies where S R = M p= N p L p,q m,n σ(l, Cm p = (S R SC, m= N p R p m and S C = are the union of the lines and row disks, respectively. 5 N q q M,q p n= N q M p= N p m= N p C p m, L q,p n,m. (7

16 A second Theorem provides a refined localization of the eigenvalues [3]. Theorem. Let us define I as the set of admissible indices p and m I = {(p, m N Z, such that p M and N p m N p }. Let I be a subset of I and I = I \ I its complementary set. Furthermore, let S and S be the union of the Gershgorin disks associated with these sets S = Rm, p S = Rm. p (p,m I (p,m I If S S =, then S exactly contains card(i eigenvalues of L, each being counted with respect to its algebraic multiplicity, the other eigenvalues being in S. Let us first state the principal properties of the Gershgorin disks associated with L. Proposition. For p =,..., M and N p m N p, the Gershgorin disks of L fulfil i The radius of the lines and row disks are equal: R p m = C p m. ii For m, the disks are double: R p m = R p m, and C p m = C p m. Proof. We do not detail the proof [4] which is direct from some properties of the Bessel s and Hankel s functions. Proposition shows that we can restrict our study to the lines Gershgorin disks. To be more explicit, we asymptotically analyze the Gershgorin disks for the low frequency regime (ka p, p M and a dilute media (kb +. Lemma. Let p, q, m and n be such that p, q M, N p m N p and N q n N q. If ka and kb +, then the coefficient L p,p m,m of matrix L has the following asymptotic expansion ( ] kap a p [ln + γ + i πa p L p,p m,m = + O ( (ka p ln(ka p for m =, a p m + O ( (ka p (8 for m. Moreover, the coefficient L p,q m,n satisfies ( L p,q πap a q m,n = i + O m + n (ka p m (ka q n e i(kbpq π/4 (n mπ/+(n mαpq kbpq ( (ka p m (ka q n + kbpq + O ( ( (ka p m + (ka q n + O kbpq (ka p m (ka q n, (9 and its modulus is such that ( L p,q πap a q m + n (ka p m (ka q n m,n = m! n! kb pq ( ( ( (ka p m (ka q n + (ka p m + (ka q n (ka p m (ka q n + O + O + O. (3 kbpq kbpq kb pq kb pq 6

17 Proof. The asymptotics of the diagonal coefficients (8 directly come from relation (4 for single scattering. Let p, q =,..., M, p q, N p m N p and N q n N q. Let us recall the following asymptotic expansion of the Hankel s functions (see equations (9..7, (9..9 and (9.. in [] H ( n m (kb pq = when kb pq +, and for the Bessel s functions J m (ka p = m! π(kb pq ei(kbpq (n mπ/ π/4 + O ( kap ( kb pq, (3 m + O ( (ka p m +, (3 for ka p. The approximation (9 is then obtained by applying (3 and (3 in the expression ( of L p,q m,n. Relation (3 is derived by considering the modulus of (9. Because of the complexity of the problem and for clarity, we assume from now on that the obstacles have the same radius a p =,..., M, a p = a, (33 and that the truncation indices N p are equal to N N: p =,..., M, N p = N. assumptions simplify both the notations and proofs. Let us first study the asymptotics of the radius of the Gershgorin disks. These two Lemma. Let p =,..., M, N m N and b = min b pq. If ka and kb +, p,q=,...,m,p q then the radius Rm p of the Gershgorin disc Rm p behaves like R p m = ( ka m a m! M q=,q p π + kb pq M q=,q p O ( (ka m + kbpq + M q=,q p O ( (ka m. (34 kb pq In particular, we have R p m a(m m! ( ka m ( ( π kb + O (ka m + (ka m + O kb kb (35 Proof. Let p, q, m and n such that p, q M and N m, n N. From Lemma, for ka sufficiently small and kb large enough, we have ( ( ( L p,q π ka m a (ka m + (ka m m, = + O + O, m! kbpq kbpq kb pq and for n, L p,q m,n = O ( (ka m +. kbpq By injecting these two expressions in R p m (7 and by summing over n and q, we obtain relation (34. The inequality (35 is simply a consequence of b b pq, for any p, q =,..., M, p q. 7

18 Let us remark that, under the assumption (33, for m =,..., N, the Gershgorin disks R p m have the same center, for any p =,..., M, since p, q M, N m N, L p,p m,m = L p,p m, m = Lq,q m,m = L q,q m, m. Let L m be this center: m =,..., N, L m = L p,p m,m. From Lemma, the centers L m have the following asymptotic expansions when ka tends towards L m = { L + O ( (ka ln(ka if m = L m + O ( (ka otherwise, (36 where [ ( ] ka a ln + γ + i πa if m = L m = a (37 otherwise. m We now show that, when ka and kb +, then we can group together the Gershgorin disks into N disks R m, m =,..., N with empty intersection. This allows us to apply Theorem to explicitly localize the eigenvalues of L. Proposition. For m N, let us introduce the disks R m of center L m, defined by (36 and with radius R m = max p=,...,m R p m. Then, the following two results hold i For any p =,..., M and m =,..., N: R p m R m and R p m R m. ii There exist two constants K a > and K b > such that, if ka < K a and kb > K b, then for any m, n such that m, n N, we get: m n R m Rn =. Proof. Point i is trivial: on the one hand, for any p =,..., M and m =,..., N, the disks R m and R p m have the same center L m, and, on the other hand, by definition, the radius R m of the disc R m is larger than the radius of R p m. Concerning Point ii, let us introduce ε > defined by and let us remark that, since L m = ε = min L m L n, m n N a m for m =,..., N, we have: ε = L N L N. Parameter ε is independent of ka but depends on a. From (37, there exists one constant K a > such that, if ka < K a, then, for any m such that m N, we have: L m L ε. Hence, for any m and n such that m, n N, we get: L m L n ε, for m n. Now, let m and n be such that m, n N. The expression (36 implies that there exists a constant K a,m,n > such that, if ka < K a,m,n, then L m L m < ε 4, and L n L n < ε 4. This implies that, since L m L n = L m L m + L m L n + L n L n L m L n L m L m L n L n, then, for ka < min (K a, K a,m,n, we obtain L m L n > ε ε 4 ε 4 = ε, for m n. (38 8

19 From the inequality (35 (Lemma and for ka < min (K a, K a,m,n, we can find a strictly positive constant K b,m,n such that, if kb > K b,m,n, we have: R m < ε 4 and R n < ε 4. It comes that: R m + R n < ε < L m L n. Let us now introduce ( K a = min K a, min, K b = max K b,m,n. m,n N m,n N K a,m,n Then the following result holds: if ka < K a and kb > K b, then, for any m and n such that m, n N, we have: R m + R n < L m L n. This exactly means that R m Rn =, which is Point ii. Let us now analyze at a few examples which illustrate Proposition. We fix ka =., for five (randomly distributed unitary circular cylinders. The truncation index N is fixed to 4. We represent the Gershgorin disks and their centers (see Theorem on Figures 6 for two values of kb: kb =.5 on Figures 6(a-6(b and kb = 7.8 on Figures 6(c-6(d. In the first case (kb =.5, we remark that we are not in the situation where the obstacles are considered as far like in Proposition since the disks intersect. For the second case (kb = 7.8, the disks do not intersect. In this last situation, we see that we are close to a regime where some couplings occur and where multiple scattering is not very important, at least for the modes m. Therefore, the eigenvalues associated with the single scattering problem (centers of the disks can be considered as suitable approximations of the eigenvalues of the multiple scattering configuration. For the propagating modes m =, we remark that the five Gershgorin disks have a relatively large radius and that the eigenvalues associated with the multiple and single scattering problems are relatively different. Let us assume that ka is sufficiently small and kb large enough; then, from Point i of Proposition, the disc R exactly contains the M Gershgorin disks R p associated with L. Each disc R m, m N, contains M disks Rm p and R p m. Moreover, the disks (R m m=,...,n do not intersect because of Point ii. Theorem implies that L has M eigenvalues localized in the disc R, that we will note (µ p p=,...,m, and M other eigenvalues in each disc R m, for m, which will be designated by (µ p m p=,...,m and (µ p m p=,...,m. In the following, we assume that such a regime is reached and we estimate the eigenvalues of L with smallest and largest modulus. 4. Accuracy of the localization of the eigenvalues with smallest and largest modulus Let µ min and µ max be respectively the smallest and largest eigenvalues (in modulus of L, that is µ min = min p M, N m N µp m and µ max = max p M, N m N µp m. From the above analysis, µ min (respectively µ max is in the disc R m which is the closest (respectively the further from the origin. However, we have, on the one hand, the asymptotic behavior of the centers of the disks R m L m = L m + O ( (ka ln(ka, and, on the other hand, the ordering relation (for ka sufficiently small for the approximate centers of the disks (from relation (37 < L N < L N <... < L < R( L. 9

20 6 Gershgorin discs Centers of the discs.4.3 Gershgorin discs Centers of the discs Imaginary part 4 Imaginary part Real part (a kb = Real part (b kb =.5 (zoom around the origin 3 Gershgorin discs Centers of the discs..5 Gershgorin discs Centers of the discs.5. Imaginary part.5 Imaginary part Real part Real part (c kb = 7.8 (d kb = 7.8 (zoom around the origin Figure 6: Gershgorin disks of matrix L, with 5 unitary circular obstacles (ka =. and N = 4 randomly distributed in [, ] [, ] for Figures (a and (b and [, ] [, ] for Figures (c and (d. Hence, µ min is localized in R N and µ max in R if ka is sufficiently small. Since µ min is an element of R N, we have the following inequality: L N R N µ min L N + R N. Let us now compare the orders of the different parameters involved in this bound. We know that L N = and, from (34, the radius R N behaves like which clearly implies that (if N R N = O a N + O ( (ka, ( (ka N kb, µ min = a N + O ( (ka. (39

21 We then remark that the distance b between the obstacles do not appear in the estimate of µ min. Furthermore, this estimate exactly corresponds to the smallest eigenvalue obtained in the single scattering case (4. Let us recall that the M eigenvalues µ p N and µp N, for p =,..., M, of matrix L are localized in the disc R N. They also fulfill the inequality (39 and so a N is a good a estimate. Hence, N can be considered as an eigenvalue of multiplicity M, with maximal modulus, approximating the M eigenvalues (µ p N p=,...,m and (µ p N p=,...,m. Let us remark that even if the analysis is realized for m = N, we can also conclude that the eigenvalues (µ p m p=,...,m and (µ p m a p=,...,m, for m =,...N and m, can be approximated by the eigenvalue m with a multiplicity M (for each one. Since µ max is an element of R, we have the following inequality L R µ max L + R. (4 Let us precise this bound. On the one hand, following (8, we have L = L + O ( (ka ln(ka [, with L = a ln and, on the other hand, from (35, Finally (4 writes down ( ka ( ( R R ka + O + O, with R = a(m kb kb L R + O ( (ka ln(ka ( ( ka + O + O kb kb µ max L + R + O ( (ka ln(ka + O ] + γ + i πa, π(kb. (4 ( ( ka + O. (4 kb kb The term R is proportional to the number of obstacles M. In particular, for a large number of obstacles, R becomes very large and the inequality (4 is inaccurate. Coming back to the example of Figure 6(c, we have the bound.4 µ max 4.69, knowing that µ max = This bound can be satisfying for a small number of obstacles but is inaccurate when M is too large. As a conclusion, an approach based on the Gershgorin disks is limited and do not lead to an accurate estimate ot µ max. A satisfying approximation of µ min is possible but only in the case where the disks do not intersect, which is very restrictive. Indeed, when the number of obstacles is large, the radius R p, for p =,..., M, given by (34, is large and the disc Rp may then intersect with other disks Rn, q for n and q =,..., M. We propose an alternative approach based on asymptotics of the coefficients of the matrix L, obtained in Lemma. By using this approach, we show that the estimate of µ min remains valid even for a large number of obstacles and when the Gershgorin s disks intersect, like in the example of Figure 6(a. 5 An alternative approach based on the limit of L We propose an alternative approach based on Lemma giving an asymptotics of the coefficients of L. We then deduce a limit matrix of L through each block L p,q, for p, q =,..., M.

22 We begin our study with the diagonal block L p,p, for p =,..., M. Let us recall that this matrix is diagonal. Moreover, when ka, its coefficients L p,p m,m, for N m N satisfy (see Lemma { L L p,p + O ( (ka ln(ka if m = m,m = L m + O ( (ka (43 otherwise, with a L m = a m [ ln ( ] ka + γ + iπa if m = otherwise. We then build the diagonal submatrix (L p,p which only contains the dominating terms of the asymptotic expansion obtained in (43. More precisely, (L p,p has the same size as L p,p, that is (N + (N +, and is defined by L N (L p,p =..... L.... ( LN Relation (43 implies that, when ka, the following relation is fulfilled, for p =,..., M (44 L p,p = (L p,p + O ( (ka ln(ka. (46 In other words, each submatrix (L p,p is an approximation of the diagonal block L p,p of matrix L. Furthermore, from relation (45, we can see that the coefficients of (L p,p do not depend on the index p (since a p = a, for all p =,..., M, which implies that, for p, q M, we have the equality (L p,p = (L q,q. We now proceed in a similar way for the off-diagonal block L p,q, with p, q M and p q. When ka and kb pq +, the coefficients L p,q m,n of block L p,q have the following asymptotic behavior (Lemma π ia L p,q m,n = O e i(kbpq π/4 kbpq ( (ka m + n kbpq ( ( (ka + O + O kb pq kbpq if (m, n = (,, if (m, n (,. This means that, when ka and kb pq +, all the coefficients L p,q m,n of L p,q decay (at least like ( ka O kbpq except the term of indices (m, n = (,, which decays like ( O. kbpq (47

23 Let (L p,q be the submatrix of size (N + (N + defined by (L p,q =.... (L p,q...,. with (L p,q π, = ia e i(kbpq π/4. ( kbpq... By using (47, when ka and kb +, the following relation holds, for p, q =,..., M and p q ( ( L p,q = (L p,q ka + O + O. (49 kbpq Hence, each block (L p,q is constituted of an approximation of the block L p,q of L. Let us introduce the block matrix L which contains each submatrix (L p,q (L, (L,... (L,M L (L, (L,... (L,M = (L M, (L M,... (L M,M Let us summarize these results in the next Lemma. Lemma 3. When ka and kb +, we have L = L + O ( (ka ln(ka + O kb pq ( ( ka + O. (5 kb kb From now on, we respectively denote by (µ p m p M, N m N and ((µ p m p M, N m N the eigenvalues of L and L. For p =,..., M and m = N,..., N, we assume that: µ p m (µ p m, which is coherent with (5. To motivate our approach and to understand how the eigenvalues of L are distributed, we compare on Figure 7 the eigenvalues of L and L, for 5 unitary disks randomly distributed in [, ] with ka =., kb = and the truncation index N = 4. These eigenvalues are computed numerically with the help of the eig Matlab function. Globally, the approximation of the eigenvalues (µ p m p=,...,m,m= N,...,N of L by the eigenvalues ((µ p m p=,...,m,m= N,...,N of L are satisfying. Let us begin by the spectral study of L by noticing that the matrix is almost diagonal since all the off-diagonal elements are zero except the ones with indices (m, n = (, (see (45 and (48. This implies that each coefficient (L p,p m,m of the diagonal of L with index m is an eigenvalue of L. Moreover, for p, q =,..., M and m, we have the following sequence of equalities (L p,p m, m = (L p,p m,m = (L q,q m,m = (L q,q m, m = L m. Hence, L m is an eigenvalue of L with multiplicity M, for m =,..., N. We fix the following ordering, for p =,..., M, m and N m N (µ p m = L m = a m. (5 3

24 3.5 3 Eigenvalues of L Eigenvalues of L.6.4 Eigenvalues of L Eigenvalues of L.5. Imaginary part.5 Imaginary part..4.6 µ min and µ min Real part (a All the eigenvalues Real part (b Zoom around the eigenvalues associated with the modes m. Figure 7: Eigenvalues of the matrices L and L, for M = 5 unit disks (ka =. and kb =. Then we know explicitly NM eigenvalues of L. Therefore, the computation of the M last eigenvalues ((µ p p=,...,m can be made by using the smaller matrix built from the zero order modes (as done later. Let us now come back to the previous example. The approximation of the eigenvalues of L by the ones of L is reasonable for the eigenvalues related to the modes of order m = (figure 7(a and for the higher order modes, m (figure 7(b. In particuliar, we remark that the approximation of the eigenvalues (µ p p=,...,m associated with the first order modes are less accurate. This is probably due to the coupling between the propagative modes m = and the modes of order m =. If the obstacles would be further, then this coupling can be expected weaker and the approximation better. We now try to localize the eigenvalues of L with smallest and largest modulus. To begin, we have the following ordering relations from (43 < L N < L N <... < L < L. (5 Since L m, for m =,..., N, is an eigenvalue of L with multiplicity M, we have ( µ min = min L N, min p=,...,m (µ p. However, Figure 7 suggests that the M eigenvalues ((µ p p=,...,m satisfy the inequality (µ p > L N. (53 When kb pq +, this relation can be proved by using the Gershgorin disks for the matrix L as in the preceeding section. Indeed, the off-diagonal coefficients (L p,q, tend towards when kb pq + (see Equation (48. Practically, this relation seems to be satisfied even when kb pq is not sufficiently large to apply the approach based on the Gershgorin disks. We therefore assume in the sequel that relation (53 holds. We estimate then µ min by L N, i.e. µ min = L N = a N. 4

25 Since L N is an eigenvalue with multiplicity M, then this is also the case for µ min. This conclusion confirms our study based on the Gershgorin disks realized in Section 4.. Now let us focus on the eigenvalue of L with largest modulus, denoted by µ max. Figure 7 suggests that µ max is an eigenvalue related to the propagative modes and that its modulus is larger than the one which would come from the simple scattering case. To prove these results and study more easily the M eigenvalues ((µ p p=,...,m, we build the smaller matrix L for the modes of order of L. More precisely, L = (L p,q p,q M has size M M and for all p, q M L p,q = (L p,q, = [ ( ka L = a ln ia π e i(kbpq π/4 kbpq + γ ] + i πa if p = q, otherwise. (54 The eigenvalues of L are then exactly ((µ p p=,...,m, as it is proved in the following lemma. Lemma 4. The M eigenvalues of L are exactly the M eigenvalues (µ p, for p =,..., M, of the matrix L. Proof. This lemma can be proved because of the particular structure of the matrix L, defined by relations (45 and (48. Indeed, we can build two permutation matrices P and Q (that we do not explicit such that PL Q = ( L L, (55 where the matrix L is defined by (54 and where the smaller matrix L := ((L p,q p,q M, of size NM NM, contains the diagonal of L without the modes of order. More precisely, for p =,..., M, each diagonal block (L p,p has for coefficient, for m = N,..., N, m, (L p,p m,m = (L p,p m,m = L m = (µ p m, (56 where we used the equality (5. Let us remark now that, since the matrix L is diagonal, its eigenvalues are also the diagonal coefficients, that is, from (56, (µ p m, for p =,..., M, m = N,..., N with m. This implies that the eigenvalues of L must be (µ p, for p =,..., M. We now have the two following results. Lemma 5. The following inequality holds Proof. The trace tr(l of L writes max p=,...,m (µ p L. (57 tr(l = M L = M L. Moreover, it is also equal to the sum of the eigenvalues L, leading to p= M L = M (µ p. p= 5