Computational method for acoustic wave focusing

Transcription

1 Int. J. Computing Science and Mathematics Vol. 1 No Computational method for acoustic wave focusing A.G. Ramm* Mathematics epartment Kansas State University Manhattan KS USA ramm@math.ksu.edu *Corresponding author S. Gutman Mathematics epartment University of Oklahoma Norman OK 7319 USA sgutman@ou.edu Abstract: Scattering properties of a material are changed when the material is injected with small acoustically soft particles. It is shown that its new scattering behaviour can be understood as a solution of a potential scattering problem with the potential q explicitly related to the density of the small particles. In this paper we examine the inverse problem of designing a material with the desired focusing properties. An algorithm for solving such a problem is examined from the theoretical as well as from the numerical points of view. Keywords: smart materials; inverse scattering; scattering of acoustic waves; computational method; focusing of waves; radiation pattern; ill-posed problems. Reference to this paper should be made as follows: Ramm A.G. and Gutman S. (7) Computational method for acoustic wave focusing Int. J. Computing Science and Mathematics Vol. 1 No. 1 pp Biographical notes: Alexander G. Ramm is the Author of 1 monographs more than 5 papers and patents. He was born in St. Petersburg Russia and came to USA in He has lectured at many Universities throughout the world. He was distinguished Foreign Professor in Mexico and Egypt in and 6 a CNRS Research Professor a Fulbright Research Professor in Israel a distinguished HKSTAM Speaker in 5 a London Mathematical Society Speaker in 5 and won the Khwarizmi Award in Mathematics in 4. r. Semion Gutman is a Professor of Mathematics at the University of Oklahoma in Norman OK. He received his Ph in Mathematics from the Hebrew University in Jerusalem Israel in He has published more than 5 papers. Copyright 7 Inderscience Enterprises Ltd.

2 A.G. Ramm and S. Gutman 1 Introduction Let 3 be a bounded connected domain with Lipschitz boundary S. enote by S the unit sphere in 3 by S the direction of the incident plane wave (incident direction) by α' S the direction of the scattered wave and by n (x) the refraction coefficient in n (x) = 1 in = 3 \. Then the scattering of a plane acoustic wave u = u (x) = e ikα x incident upon is described by the system: [ + kn( x)] ux ( ) = in (1) 3 ux ( ) = u( x) + vx ( ) () ikr e 1 x vx ( ) = A( α α) + o r: = x : = α (3) r r r where v(x) is the scattered field and α α S. The coefficient A(α α) is called the scattering amplitude k > is the wave number which is assumed to be fixed throughout the paper. For this reason the dependence of A on k is not shown. Our theory is applicable to various kinds of waves described by the equations (1) (3). Let m 1 m M be a small particle i.e. 1 ka 1 where a= max diamm k = kmax n( x). (4) 1 m M x The geometrical shape of m is arbitrary but we assume that each m has a Lipschitz boundary. Moreover the Lipschitz constant is the same for every domain m. This is a technical assumption which can be relaxed. It allows one to use the properties of the potentials of single layer used in our arguments. Let d : = min dist( ). (5) Assume that m j m j a d. (6) We do not assume that dλ that is that the distance between the particles is much larger than the wavelength. Under our assumptions it is possible that there are many small particles within the distances of the order of magnitude of the wavelength. The particles are assumed to be acoustically soft i.e. the irichlet boundary condition is assumed on the boundary of each particle: u = 1 m M. (7) S m As a result of the distribution of many small particles in one obtains a new material. We would like this smart material to have some desired properties. Specifically we want this material to scatter the incident plane wave according to an a priori given desired radiation pattern for example to focus the incident wave within a given solid angle.

3 Computational method for acoustic wave focusing 3 Is this possible? If yes then how does one distribute the small particles in order to create such a material? In mathematical terms the problem is: Given an arbitrary function f() L (S ) can one distribute small particles in so that the resulting medium generates the radiation pattern A() := A( ) at a fixed k > and a fixed S such that f( β) A( β) ε (8) L ( S ) where > is an arbitrary small fixed number? The answer is yes. It is contained in the following Theorem. Theorem 1: For any f L (S ) an arbitrary small > any fixed = S any fixed k = k > and any bounded domain 3 there exists a (non-unique) potential q(x) L () such that equation (8) holds. The proof of Theorem 1 consists of the following steps: Given f we find h satisfying inequality (19). This is done in Theorem Section 3. Given h we check if condition (36) holds and if it holds then the potential q is calculated by formula (37). In our numerical experiments condition (36) always holds. If condition (36) does not hold then one can perturb h a little in the norm of L () so that condition (36) holds for the perturbed h and calculate the potential by the formula similar to equation (37) (see Lemma in Section 4). This completes the description of the steps of the proof of Theorem 1. The proof itself is finished in Section 4 with the proof of Lemma. The relation between the particle distribution density and the potential q is explained in Sections and 3. In Section 4 we give an algorithm for calculating such a potential. Numerical results are presented in Section 5. Our solution of this problem is based on our earlier results on wave scattering by small bodies of arbitrary shapes see Ramm (5b) as well as Ramm (6a 6b 7a 7b). There is an extensive literature on smart materials i.e. materials with some desired properties on cloaking as a method for creating such materials on homogenisation theory both matematical and applied to the material properties. However the theory and ideas of this paper do not have intersection with published literature and are based on the works of one of the authors. Scattering by many small bodies were also considered by a different method in the work of Marchenko and Khruslov cited in the bibliography. The solution of the 3 inverse scattering problem with fixed-energy data possibly noisy was given in Ramm () where the stability estimates were obtained for the solution of this problem (see also Ramm 5a). The statement and the solution of the inverse scattering problem with fixed-energy and fixed incident direction data were first given in Ramm (6a 6b 6c).

4 4 A.G. Ramm and S. Gutman Scattering by many small particles If many small particles m 1 < m < M are embedded in u S m = where S m is the boundary of m then the scattering problem is: M 3 + = m= 1 [ k q ( x)] u in \ (9) m u S = m = 1 M (1) m ikr e 1 x ux ( ) = u ( x) + A( βα ) + o r= x β= (11) r r r and the solution u(x) is called the scattering solution. Here n (x) is a given refraction coefficient n (x) > in 3 u is the acoustic pressure q ( x): = k [1 n ( x)] = in. Under the assumptions of Section 1 we can have d λ. We also assume that the quantity a/d 3 has a finite non-zero limit as M and a/d. More precisely if C m is the electrical capacitance of the conductor with the shape m then we assume the existence of a limiting density C(x) of the capacitance per unit volume around every point x : lim C = C( x)d x M m m where is an arbitrary subdomain of. Note that the density of the volume of the small particles per unit volume is 3 a a O as. 3 d d One can prove (see Ramm 5b p.13) that in the limit M the function u solves the equation 3 [ + k q( x)] u = in q( x) = q ( x) + C( x) (1) (13) where C(x) is defined in equation (1) and A( ) in equation (11) corresponds to the potential q(x) (see also Marchenko and Khruslov (1974) where similar homogenisation-type problems are discussed). If all the small particles are identical C is the capacitance of a conductor in the shape of a particle and N(x) is the number of small particles per unit volume around point x then up to the quantity of higher order of smallness as a/d we have: Therefore Cx ( ) = NxC ( ). qx ( ) = q( x) + NxC ( )

5 Computational method for acoustic wave focusing 5 and qx ( ) q( x) N x = (14) ( ). C Thus one has an explicit one-to-one correspondence between q(x) and the density N(x)of the embedded particles per unit volume. Remark 1: If the boundary condition on S m is of impedance type: u = ςu on S N m where N is the exterior unit normal to the boundary S m and is a complex constant the impedance then the capacitance C in formula (9) should be replaced by C Cς = 1 + ( C ς S ) where S is the surface area of S and the corresponding potential q(x) will be complex-valued see Ramm (5b p.97). 3 Scattering solutions To establish Theorem 1 recall that for a fixed k > the scattering problem equations (1) (3) is equivalent to the Schrödinger scattering problem for the potential q(x): ik x y e uq = u gxyqyu ( ) ( ) q( y)d y gxy ( ): = (15) 4 π x y for which the scattering solution u = u q is the unique solution. The corresponding scattering amplitude is 1 ikα x A( α α) = e q( x) uq ( x α)d x (16) where the dependence on k is dropped since k > is fixed. If q is known then A := A q is known. Let q L () be a potential and A q (α α) be the corresponding scattering amplitude. Fix α S and denote Then A( β ): = A q ( α α) α = β. (17) 1 ikβ x A( β ) = e h( x)d x h( x): q( x) uq ( x α). = (18)

6 6 A.G. Ramm and S. Gutman Our goal in Theorem 1 is to find a potential q for which equation (8) is satisfied. First we find an h(x) that satisfies 1 f( β ) + e h( x)d x <. The existence of such an h follows from Theorem. ikβ x ε (19) L ( S ) Theorem : Let f() L (S ) be arbitrary. Then 1 ikβ x inf f( β ) + e h( x)dx. = () h L ( ) L ( S ) Proof of Theorem : If equation () fails then there is a function f() L (S ) f such that d ( ) ik β x β f β e h( x)dx = h L ( ). S (1) This implies ik x β () S ϕ( x): = d β f( β) e = x. The function ϕ(x) is an entire function of x. Therefore equation () implies 3 ϕ ( x) = x. (3) This and the injectivity of the Fourier transform imply f() =. Note that ϕ(x) is the Fourier transform of the distribution f()(k λ)λ where (k ) is the delta-function and is the Fourier transform variable. The injectivity of the Fourier transform implies f(β)λ (k ) = so f() =. Theorem is proved. Finding h from a given f is an ill-posed problem which is similar to solving first-kind Fredholm linear integral equation with the kernel (1/)e ikβ x. It can also be considered as the problem of approximation of a given f L (S ) by an entire function of exponential type. If inequality equation (19) holds with some > and c > is a constant then the function cf is approximated with the accuracy c by the integral in equation (19) with ch replacing h. Therefore although for a fixed and a fixed f the norm of h may be very large one can take sufficiently small c > such that ch is small for instance ch < 1 and the integral in equation (19) with ch replacing h will approximate the function cf with accuracy c. The function cf describes the same radiation pattern as f but it is normalised so that its maximum is c max f for a bounded f. Multiplication by c α S does not change the dependence of f on it only changes the amplitude of f. Therefore one may say that f and h are proportional in the sense that multiplication of f by a positive constant results in multiplication of h and by the same constant. From the applications point of view this is important: One may find a small h such that equation (19) holds for cf and then arrange for a linear amplifier with the amplification coefficient 1/c to get from cf the desired f.

7 Computational method for acoustic wave focusing 7 To find an h that satisfies equation (19) one can proceed as follows. Let { Yl( β )} l= Yl = Yl m l m l be the orthonormal in L (S ) spherical harmonics Y ( β) = ( 1) Y ( β) Y ( β) = ( 1) Y ( β) (4) l l + m lm lm lm l m 1/ π jl(): r = Jl+ (1/ ) () r (5) r where J l are the Bessel functions and the overbar stands for the complex conjugate. It is known that ikβ x l = 4 π( ) l ( ) lm ( ) lm ( β) : =. (6) l= l m l x e i j kr Y x Y x Let us expand f into the Fourier series with respect to spherical harmonics: f( β) = f Y ( β). (7) l= l m l lm lm Choose L = L() such that flm ε. (8) l> L With so fixed L take h lm (r) l L l m l such that 1/ b l π 3/ lm = l+ (1/) lm f ( i) r J ( kr) h ( r)d r k (9) where b > the origin O is inside the ball centred at the origin and of radius b belongs to and h lm (r) = for r > b. There are many choices of h lm (r) which satisfy equation (9). If equations (8) and (9) hold then the norm on the left-hand side of equation () is smaller than. A possible explicit choice of h lm (r) is x where f i l L h = k g k l > L l lm ( ) lm π ( 1 l + (1/ ) ( )) (3) 1 µ + (1/ ) µ v( ): v( )d. g k = x J kx x This integral can be calculated analytically see Bateman and Erdelyi (1954) formula We have assumed that h(x) = for x > 1 and b = 1 in equation (9). Finally let L hx ( ) = hlm ( ry ) lm ( α ). l = This function satisfies inequality equation (19) by the construction.

8 8 A.G. Ramm and S. Gutman 4 Reconstruction of the potential In the previous section we have shown how to find a function h L () that satisfies equation (19). In this section a potential q satisfying the conditions of Theorem 1 is constructed from such an h. The possibility of such a reconstruction follows from Theorem 3. Theorem 3: Let h L () be arbitrary. Then inf h qu ( x α) =. (3) q L ( ) q Here S and k > are arbitrary fixed. Moreover if h L ( is sufficiently small ) then there exists a potential q such that hx ( ) = qxu ( ) ( xα ). (33) q This theorem follows from Lemmas 1 and stated and proved below. For convenience let us summarise the method for finding a potential q satisfying the conditions of Theorem 1. Method for potential reconstruction Let >. Step 1: Given an arbitrary function f() L (S ) find h L () such that equation (19) holds. This can be done using equation (3). Let L hx ( ) = h ( ry ) ( α ) (34) l = where L = L(). lm lm Step : Use h obtained in Step 1 to find a potential q L () satisfying h qu ( x α) < ε. q For f with a sufficiently small norm f ( β ) so that for the corresponding h L ( S ) condition (36) holds such a potential q can be found using formula (37) see below. Formula (37) can be used for any f for which condition (36) holds. In this case Step is done analytically. In our numerical experiments condition (36) always held. If equation (36) does not hold then one can perturb h a little in L ()-norm so that condition (36) is satisfied and the corresponding potential calculated by formula (37) is bounded as follows from Lemma stated below. Therefore theoretically the smallness restriction on f (or on h) is not essential. If it holds then there is a potential q L () given by formula (37) analytically and the infimum in formula (3) is attained at this potential. See also Remark 3 below.

9 Computational method for acoustic wave focusing 9 Step 3: This potential q generates the scattering amplitude A() at fixed and k such that f( β) A ( β) Cε q L ( S ) holds for some constant C independent of. Indeed let Then : =. L ( S ) 1 ikβ x f( β) Aq ( β) = f( β) + e q( x) u( x)dx 1 ikβ x ε f( β ) + e hdx + (35) ε ε + = meas. This concludes the description of the reconstruction method. Numerically this method is applicable with the additional regularisation Step 1(b) described in Section 5. The role of this additional step is to bound the norm of the function h which is calculated in Step 1. Lemma 1: Assume that supx ghdy < 1 or more generally that ik x y e inf u ( x) gxyhydy ( ) ( ) g gxy ( ) :. x > = = (36) 4 π x y Then equation (33) has a unique solution: hx ( ) qx u x e q L u( x) gxyhydy ( ) ( ) ikα x ( ) = ( ) = ( ). (37) Remark : It follows from Theorem and the discussion afterward that f and h are proportional so that if f is sufficiently small then L ( S h ) L ( is small and then ) condition (36) is satisfied. Proof of Lemma 1: The scattering solution corresponding to a potential q solves the equation ik x u = u g( x y) q( y)d y u : = e α. (38) If h(x) = q(x)u q (x ) holds i.e. if h corresponds to a q L () then u u ghd. y Multiply this equation by q and get qxux ( ) ( ) qxu ( ) ( x) qx ( ) gxyhy ( ) ( )d y. = =

10 1 A.G. Ramm and S. Gutman Using equation (33) and solving for q one gets equation (37) provided that Condition (36) holds. Condition (36) holds if h L ( is sufficiently small. One has ) and 1 1 gxyhy ( ) d sup h (39) L ( ) 4 π x x y L ( ) 1 dy 1/ 4 π x y 4 a π where a =.5 diam. If for example a h < 1 L ( ) then condition (36) holds and formula (37) yields the corresponding potential. This explains the role of the smallness assumption. Remark 3: If condition (36) fails then formula (37) may yield a q L (). As long as formula (37) yields a potential q L p () p 1 our arguments essentially remain valid. In our presentation we have used p = because the numerical minimisation in L -norm is simpler. The difficulty arises when formula (37) yields a potential which is not locally integrable. Numerical experiments showed that this case did not occur in practice in several test examples in which the smallness condition was not satisfied. We prove that a suitable small perturbation h of h in L ()-norm yields by formula (37) a bounded potential q. This means that the smallness restriction on the norm of f is not essential. Lemma : Assume that h is analytic in and bounded in the closure of. There exists a small perturbation h of h h h < such that the function L ( ) is bounded. h ( x) q : = u( x) gxyh ( ) ( ydy ) Outline of proof: Suppose that for a given h L () condition (36) is not satisfied. Let us approximate h by an analytic function h 1 in for example by a polynomial so that 1 ikβ x f( β ) + e h1 ( x)d x ε. <

11 Computational method for acoustic wave focusing 11 enoting h 1 by h again we may assume that h is analytic in and in a domain which contains. We prove that it is possible to perturb h slightly so that for the perturbed h denoted h condition (36) is satisfied and formula (37) yields a potential q L () for which inequality equation (8) holds. For convenience of the reader we include the proof from Ramm (6c). Since the set of analytic functions is complete (total) in L () we can assume without loss of generality that h is analytic. Moreover by the same reason we can assume that h is a polynomial. Let where N : = { x: ( x) = x } : u ( x) g( x y) h( y)d y. = This set is generically a line defined by two simultaneous equations j = j = 1 where 1 := Re and := Im. Let N : = { x: ( x) < x } and := \N. Generically c > on N and therefore by continuity in N. A small perturbation of h will lead to these generic assumptions. Consider the new coordinates s = s = s = x s = ( s s s ) Choose the origin on N. The Jacobian ( s1 s s3) J : = ( x x x ) 1 3 is non-singular in N 1 sup x N( J + J ) < c. By c we denote various positive constants. The vectors j = 1 are linearly independent in N. efine h = h in and h = in N. Let where h q : = in q : = in N : u ( x) g( x y) h ( y)d y. =

12 1 A.G. Ramm and S. Gutman We wish to prove that the function q is bounded. It is sufficient to check that > c > in N. By c we denote various positive constants independent of. One has where I( ) I( ) M dy I( ): = x M max. 4 π = N x y x The proof will be completed if the estimate I() = o() is established. We prove a stronger result: I ( ) = O( ln( ) ). Let us derive this estimate. It is sufficient to check this estimate for the integral dy ds I : = πc d ρρ N y ρ 1 3 s3 + ρ where ρ = s1 + s we have changed the variables y to s used the estimate J 1 < c and took into account that the region N is described by the inequalities s 3 1. A direct calculation of the integral I yields the desired estimate: I = O( ln() ). The proof of Lemma is completed. Finally we make some remarks about ill-posedness of our algorithm for finding q given f. This problem is ill-posed because an arbitrary f L (S ) can not be the scattering amplitude A q () corresponding to a compactly supported potential q. Indeed it is proved in Ramm (199 ) that A() is infinitely differentiable on S and is a restriction to S of a function analytic on the algebraic variety in 3 defined by the equation = k. Finding h satisfying equation (19) is an ill-posed problem if is small. It is similar to solving the first-kind Fredholm integral equation 1 e ikβ x h ( x )d x = f ( β ) whose kernel is infinitely smooth. Our solution (3) shows the ill-posedness of the problem because the denominator in equation (3) tends to zero as l grows. Methods for stable solutions of ill-posed problems (see Ramm 5a) should be applied to finding h. If h is found then q is found by formula (37) provided that equation (36) holds. If equation (36) does not hold one perturbs slightly h according to Lemma and get a potential q by formula (37) with h in place of h. 5 Numerical results In this section we present results of numerical experiments for a design of the material capable of focusing the incoming plane wave into a desired solid angle. First let us note that a direct implementation of the algorithm presented in the previous sections produces potentials q with large magnitudes (of the order of 1 5 ) in our examples. This happens

13 Computational method for acoustic wave focusing 13 because of the ill-posedness of the inverse scattering problem. To remedy this situation we have introduced an additional step in the potential reconstruction algorithm stated in Section 4. Step 1(b): Let h lm be the coefficients of h obtained according to equation (3). Bound the magnitudes of the coefficients by a predetermined constant T > that is Let hlm if hlm T h lm = hlm T if hlm T. hlm L hx ( ) = h ( ry ) ( α ). (4) l = lm lm The bound on the function h has the effect of bounding the potential q. This procedure regularises the ill-posedness of the reconstruction process as discussed at the end of Section 4. The ill-posedness manifests itself in the divergence of the series (34) with L = when the regularisation we have used by introducing the bounding constant is not applied. However from the numerical observations the series equation (4) practically did not change with the increase in L for L > 5. As expected an increase in the value of T improves the precision of the approximation of the desired scattering amplitude f but it also increases the magnitude of the potential q. A reduction in the value of T leads to a deteriorating approximation. In all the experiments the incident direction = ( 1) k = 1. and L = 6. The domain is the ball of radius 1 centered in the origin. In our first numerical experiment the goal was to focus the incoming plane wave into the solid angle π/4 where is the polar angle measured from the incident direction = ( 1). Figure 1 shows the cross-section through the incident direction of the desired (dotted line) and the attained absolute value (solid line) of the scattering amplitude. Figure 1 Attained (solid line) and targeted (dotted line) scattering amplitude f() in experiment 1 Figure shows the contour plot of the absolute value of the recovered potential q in a cross-section through the z-axis. The darker colors correspond to the larger values of q. In this experiment the maximum of the absolute value of the potential q was about 19 corresponding to the bounding constant T = 1. This value of T was found by examining numerical results with larger and smaller values for this constant. For smaller

14 14 A.G. Ramm and S. Gutman T the resulting radiation pattern has smaller magnitudes i.e. the plot of its absolute value is located closer to the origin. For larger values of T the maximal value q of the potential approaches the order of 1 5. Figure Contour plot of the potential q in experiment 1 Similarly Figures 3 and 4 show the results of the numerical experiment aimed at focusing the same incident plane wave into the solid angle.π.5π. The maximum of q was about 184 in this case corresponding to the bounding constant T = 8. This value of T was found experimentally as above. For smaller values of T the resulting radiation pattern has a significant component in the region.π i.e. it produces a poor approximation for the desired scattering amplitude. Figure 3 Attained (solid line) and targeted (dotted line) scattering amplitude f() in experiment Figure 4 Contour plot of the potential q in experiment

15 Computational method for acoustic wave focusing 15 6 Conclusions A method is developed for finding the number N(x) of small acoustically soft particles to be embedded per unit volume around every point x in a bounded domain filled with a known material in order that the resulting new material has the desired radiation pattern. Any wave field not necessarily acoustic wave field which satisfies equations (9) (11) is covered by our theory. On the boundary of each acoustically soft particle the irichlet condition holds. The method is justified theoretically. Numerical examples of its application are presented. The ill-posedness of our problem is discussed and a regularisation method for its stable solution is proposed and successfully tested numerically. The direct application of the derived formula (3) may lead to large values of q. To remedy this situation the coefficients h lm are bounded. This is a way to handle the ill-posedness of the inverse problem. The resulting algorithm exhibits a stable behaviour. It serves as a regularising algorithm for solving the original ill-posed problem. Numerical results show that the method can produce materials with the desired focusing properties under the limitation that the desired radiation pattern f() is well approximated by a sum of spherical harmonics with not large L. References Bateman H. and Erdelyi A. (1954) Tables of Integral Transforms McGraw-Hill New York. Marchenko V. and Khruslov E. (1974) Boundary-value Problems in omains with Fine-grained Boundary Naukova umka Kiev (in Russian). Ramm A.G. (199) Multidimensional Inverse Scattering Problems Longman/Wiley New York. Ramm A.G. () Stability of solutions to inverse scattering problems with fixed-energy data Milan Journ. of Math. Vol. 7 pp Ramm A.G. (5a) Inverse Problems Springer New York. Ramm A.G. (5b) Wave Scattering by Small Bodies of Arbitrary Shapes World Sci. Publishers Singapore. Ramm A.G. (6a) istribution of Particles which Produces a Smart Material Paper org/abs/math-ph/663. Ramm A.G. (6b) istribution of particles which produces a desired radiation pattern Communic. in Nonlinear Sci. and Numer. Simulation (to appear). Ramm A.G. (6c) Inverse Scattering Problem with ata at Fixed Energy and Fixed Incident irection submitted. Ramm A.G. (7a) istribution of particles which produces a desired radiation pattern Physica B (to appear). Ramm A.G. (7b) Many-body scattering by small bodies J. Math. Phys. (to appear).