Bayliss Turkel-like Radiation Conditions on Surfaces of Arbitrary Shape

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1 Journal of Mathematical Analysis and Applications 9, 84 Ž 999. Article ID jmaa , available online at on BaylissTurkel-like Radiation Conditions on Surfaces of Arbitrary Shape X. Antoine and H. Barucq Laboratoire de Mathematiques Appliquees pour l Industrie and CNRS UPRES-A 5033, Uniersite de Pau et des Pays de l Adour, Pau, France helene.barucq@univ-pau.fr and A. Bendali Departement de Genie Mathematique, INSA, Laboratoire MIP UMR 5640, CNRS-UPS-INSA, Complexe Scientifique de Rangueil, 3077 Toulouse Cedex 4, France Submitted by Firdaus E. Udwadia Received July, 997 This paper addresses the extension of the BaylissTurkel second-order radiation condition to an arbitrarily shaped surface. The derivation is based mainly on the pseudo-differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition. The radiation condition then ranges among the most accurate of those of order two. As a by-product of the derivation, almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared. 999 Academic Press Key Words: Absorbing boundary conditions; Helmholtz equation; wave equation; pseudo-differential operators. INTRODUCTION The field of a scalar time-harmonic scattered wave is generally governed by the following model problem Žcf., e.g., in electromagnetic and 7 in. N acoustic scattering. Find u defined in an exterior domain of, with N, 3 Ž i.e., such that its complement is a compact set., satisfying the Helmholtz equation u k u 0 in, Ž X99 $30.00 Copyright 999 by Academic Press All rights of reproduction in any form reserved. 84

2 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 85 the Sommerfeld radiation condition lim x Ž N. u iku 0, x expressing that the energy propagates from the obstacle infinity, and a boundary condition x Bu g Bu inc on, toward expressed by a given boundary differential operator B. u inc stands for the incident wave and is a given solution to the Helmholtz equation in the vicinity of the obstacle; the time variation expž i t. is suppressed by linearity; 0 is the frequency; and k c is the related wave number, where c is the velocity of the propagation. Hereafter, we assume N that is a C compact manifold with boundary imbedded in. However, such a degree of smoothness is needed only for theoretical purposes. This is relative to the use of the pseudo-differential calculus and may largely be weakened at the level of the final obtained radiation conditions. It is well known that a process limiting the computational domain has to be used before any attempt to numerically solve the above problem. A broadly used approach consists of truncating the infinite domain with a terminating closed boundary enclosing the obstacle. A boundary condition is then introduced on in such a way that the resulting boundary-value problem is well posed and its solution approximates the restriction of u to the bounded domain limited by and. Two classes of such a boundary condition can be used. Exact conditions give exactly the restriction of u if no further approximation is made. The approximate Žsome- times also called radiation or absorbing. boundary conditions only yield an approximation of this restriction. They can be designed from numerous standpoints, such as low-reflecting boundary conditions, 0, one-way propagating waves 9, 0, radiation conditions obtained from a Wilcox expansion far from the obstacle 5, 8, 9, the behavior of the solution near a wave front 3, 4, etc. These boundary conditions can be seen as an exact or an approximate way to deal with the following so-called Dirichlet-to-Neumann operator. Let be the exterior domain limited by the introduced boundary. For a function defined on, consider w, the solution to Eq. Ž. in subject to Ž. and to the boundary condition w on. The operator is then defined by w n on, where n is the unit normal to inwardly directed to.

3 86 ANTOINE, BARUCQ, AND BENDALI It is well known that is a pseudo-differential operator of order Žsee 6, 3 concerning this point and for standard notation as well as functional spaces, which will be used in the sequel without further comment.. Hereafter we limit ourselves to the extreme case where exactly coincides with. The problem to be solved is then reduced to a system of two boundary equations, Bu g, u u 0 on, n in the two unknowns u and n u, classically referred to as the Cauchy data of the problem. The determination of these unknowns yields the solution entirely from the classical Helmholtz integral representation formula. The above approach is exactly what is done in a boundary integral equation method even if the effective derivation of this equation follows a different path. Unfortunately, the mapping is not a differential operator. As a result, a standard discretization of by a finite-element method leads to a full matrix. Somewhat obviously, this results in storage and computational difficulties, especially when high-frequency or three-dimensional problems are considered. Rather surprisingly, Kriegsmann et al. 6have succeeded in designing a differential approximation of the operator A in a heuristic way that yields a reasonably accurate solution in the case of a bidimensional convex-shaped obstacle. They called the resulting equation an on surface radiation condition Ž OSRC.. The essential trick of the approach is to write a radiation condition intrinsically on a large circle of radius R by putting the curvature in place of the term R and substituting the derivative with respect to the curvilinear abscissa s to the differential operator R, where is the angular variable. From numerical experiments, 3 we have observed that the most accurate second-order condition is based on the related BaylissTurkel radiation condition on the circle. Furthermore, since circular geometry has a constant curvature, a more accurate condition can be obtained by first writing the OSRC in the form of a symmetric operator Žrelative to the scalar product of L. incorporating the derivative of the curvature before formally writing it on an arbitrarily shaped obstacle. It has been known for a long time that incorporating curvature terms gives more accurate radiation conditions Žsee, for instance.. Jones 3 and, more recently, Stupfel have constructed absorbing conditions depending on the derivatives of the curvature but which do not lead to a symmetric operator. However, we have observed in 3 that Jones s conditions result in a degeneracy of the quality of the approximation when applied to problems involving polygonal scatterers through the OSRC technique.

4 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 87 The above heuristic derivation of the OSRC from the radiation condition of Bayliss and Turkel may also be questionable. Is it really the most accurate OSRC of the second order that can be designed? Since the curvature operator is scalar for an arbitrarily shaped boundary in the two-dimensional case, the heuristic transposition of the BaylissTurkel radiation condition can be made in a straightforward way. But what about such a transposition for three-dimensional problems where this operator is scalar only for spherical geometries? An attempt to answer these questions can be made by using the formalist background of microlocal analysis Žthe calculus of pseudo-differential operators.. Such an approach has been initiated by Engquist and Majda 9. In this paper, we develop a clear and precise handling of the construction process that explains why the above modified BaylissTurkel condition is the most accurate and results in a condition of the same accuracy in three dimensions. Let us briefly describe this approach. Using Nirenberg s decomposition theorem and symbolic calculus, one easily derives an asymptotic expansion of the operator in the form Ý, Ž 3. j where the sign has the following meaning: for every nonnegative integer m, the operator Ý j mj is a pseudo-differential operator of order at most Ž m.. In this way, the expansion Ž 3. gives an explicit expression of the operator, since the symbol of j can be obtained from an explicit recursive formula. Apparently two steps are necessary for the construction of a differential operator approximating. First, an approximation of the operator by a finite sum Ý j mj has to be made. Strictly speaking, this process results in an approximation of the operator such that the remaining correction is a regularizing operator producing Ž m. supplemental orders of derivations every time it is applied to a function. The question is: How many terms in this expansion have to be retained for a fixed accuracy? Second, the explicit operators j have themselves to be approximated by a differential operator. How can this approximation be made without discarding any nonnegligible term? The following property gives the precise criterion for the construction. Essentially every operator j has a symbol j that behaves like k j for large k. The symbol of Ý j j is expanded by a Taylor expansion in powers of frequency up to the term k. The resulting expression differs only by terms in k 3 with the symbol relative to the accurate condition, heuristically designed in the two-dimensional case 3. Clearly, the same procedure can be used in dimension 3 to design an extension of the BaylissTurkel boundary condition for an arbitrarily j

5 88 ANTOINE, BARUCQ, AND BENDALI shaped surface. Furthermore, we shall see that this approach gives a common framework from which almost all radiation conditions of order less than or equal to can be recovered. Here we choose not to complicate the exposition by considering a general frame for a space of dimension N equal to either or 3. We concentrate on the three-dimensional case since the heuristic procedure cannot be used there to design the most accurate second-order condition. The radiation condition for N can then be obtained as a special form of the three-dimensional one. This paper is organized as follows. We begin in Section by reviewing the description of a coordinate system that is well adapted to the description of a thin layer in the vicinity of a surface. Then we give the expression of the Laplace operator in this coordinate system. The coordinate change reduces the study to the case where the problem is set in a half-plane but involves a variable coefficient wave equation. In Section 3, we define the Dirichlet-to-Neumann nonlocal pseudo-differential operator as the outgoing part of the solution to the wave equation. Several properties of this operator that are crucial to the method are briefly noted, as is the application of the Nirenberg factorization theorem, which yields a formula that permits recursive determination of the entire sequence 4 j j. Moreover, we prove that the leading term of a symbol j associated with an operator j behaves like k j as k tends to infinity. In Section 4, we introduce a precise criterion based upon the combination of a symbolical truncation and a Taylor s expansion in k of symbols, providing a hierarchy of surface radiation conditions of different orders. We then show how most of the well-known radiation conditions can be recovered. Finally, we design a second-order BaylissTurkel-like radiation condition for an arbitrarily shaped surface.. REVIEW OF DIFFERENTIAL GEOMETRY In this section we intend to write the Laplace operator in a system of local coordinates defined in a so-called tubular neighborhood of the surface. For 0, this type of neighborhood U is defined by 4 U x 3 ; dž x., where the function d stands for the distance from x to. When is chosen small enough, there exists a unique p in such that x p dž x. for any x in U Žsee, for instance,. 8. The vector p is the orthogonal projection of x onto. The unit normal vector to is pointing into, that is, defining nž p. dž x. if x,

6 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 89 we get an extension of the field n to all of U. Therefore U natural parametrization by U x p rnž p.; r, p 4. admits a More precisely, for sufficiently small 0, the mapping Ž p, r. x p rnž p. Ž 4. is a C -diffeomorphism of manifolds between Ž,. r, the surface r parallel to can be defined by 4 and U. For x 3 ; x p rnž p.. Ž 5. r The unit normal to r at a point x is then given by the previous extension of the unit normal nž x.. According to Ž 5., is simply the surface 0. Now consider a coordinate patch V, of where : V. Each point p of Ž V. is then given by p Ž s., and the normal vector to can be defined by the following rule. The coordinate patch gives rise to a basis Ž,. of the tangent plane T Ž.: p, i,, i s i where s and s are the coordinates of s. We assume that this coordinate patch is compatible with the orientation of the unit normal n, that is, nž p., where stands for the usual vector product. Now consider a function depending on x and define Ž s, r. Ž x..to write the Laplacian in the coordinates Ž s, r., we first use the two following relations given by the chain rule: 3 s Ž s, r. Ý x s xi s x, j i j j i 3 Ý Ž s, r. x x, j,. r xi r i r i w w denotes the usual scalar product of two vectors w and w with three Ž possibly complex. components. Thus, since x Ž s. rnžž s.., we get x n Ž s., x r n, j,. r sj j sj

7 90 ANTOINE, BARUCQ, AND BENDALI The derivatives n R, j, sj j can be expressed using a self-adjoint operator R of the tangent plane, thereby giving the curvature of at the point p Ž s. Žcf., e.g.,. 8. If we denote by the identity of the tangent plane, we then get x Ž r R., j,. sj Since R is self-adjoint, the relation Rj R j yields Ž s, r. Ž r R. Ž r R., j,, sj j j where is the projection onto the tangent plane. The above equation can be written intrinsically as j r R, where is the surface gradient of the partial mapping p Ž p, r.. For the sake of brevity, we do not distinguish in the sequel between Ž s, r. and the function obtained from by the variable change Ž. 4. For a regular function compactly supported in U, we can write H H 4 du Ž. Ž n.ž n. du. U U Thus, since the volume element on U we get H U is given by du detž r R. d dr, H Ž r r. Ž,. du r R Integrating by parts, we obtain detž r R. d dr. H H ž Ž. dx div r R det r R Ž,. detž r R. d dr, Ž 6. r r /

8 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 9 where div is the surface divergence of a tangent vector field on. For any function compactly supported in U, the following Green formula: H H dx dx, U U Ž. according to 6, directly yields the expression of the Laplacian in variables s and r: ž det r R div det r R r R rž r./ det r R. Now to get a more tractable form for this expression, we make a particular choice for the coordinate patch. In fact we can choose a particular coordinate patch so that the coordinate system Ž s, r. may be an orthonormal system Ži.e., Ž,. is an orthonormal basis of the tangent plane. and, furthermore, such that j is an eigenvector of R. Such a basis is called the principal basis of T Ž. p, whereas the vectors and are known as the principal curvature directions of and satisfy R C, j j j where C and C are the principal curvatures of. The Gauss curvature and the mean curvature can now be respectively defined by K CC and H Ž C C.. Defining h j r C j, j,, we can write detž r R. hh and Ž r R. j h j j, j,. Hence, in the orthonormal coordinates system, the Laplace operator takes the following expression: ž / r Ž hh. h h r hh r hh s s s s. h h Hereafter, we denote by Hr the mean curvature of the parallel surface r given by r Ž hh. Hr, hh and we drop the symbol to simplify the notation.

9 9 ANTOINE, BARUCQ, AND BENDALI 3. APPROXIMATION OF THE DIRICHLET-TO-NEUMANN OPERATOR 3.. The Dirichlet-to-Neumann Operator To keep the convention on the time variation and use the formalism of pseudo-differential operators, we write the Helmholtz equation as a wave equation, hence depending explicitly on the time t: Lu u u 0, t where už x, t. Ž x. expž ikt.. Consequently, we do not consider the multiplication by k as a zero-order operator. In the local coordinate system, the above equation can be rewritten as ž / h h Lu r u Hr ru hh s s u s s u t u 0. h h Ž 7. The symbol L of L, with respect to the variables s and t and their respective dual variables and, depends smoothly on the variable r, which will hereafter play the role of a parameter. Thanks to the expression of Ž. 7 for the Helmholtz operator L in the variables s and r, this symbol has the following explicit expression: L H h h r r r i h i h s s. hh h hh h The selection of waves propagating in the normal direction is based on the following decomposition resulting from Nirenberg s factorization theorem : There exist two classical pseudo-differential operators and of order, depending smoothly on r, such that Lu i i u. Ž 8. r In addition, the uniqueness of the decomposition is ensured by the following characterization. Let and be the respective symbols of and. From the definition of pseudo-differential operators of order, the symbols and belong to the symbol class S Ž, 0 cf., e.g., Chazarain and Piriou 6 for the definition of this class of symbols.. By classical pseudo-differential operators, we mean that the symbols r

10 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 93 admit the following asymptotic expansion: Ý j, 9 j where are homogeneous functions of degree j in Ž,. j. The expansion holds in the following sense: m Ý j Ž m. j m, S. m Hereafter, as in 3, we more concisely denote by S the symbol class S, m 0. Hence the corresponding classical pseudo-differential operators and are in OPS. According to 7 we define the cone of propagation Ž as the set of dual variables, satisfying h h. 0. The decomposition is then made unique by requiring that be positive in the cone of propagation. The waves propagating along the normal direction Ž the so-called physical outgoing waves. satisfy the equation Ž r i. u 0, in the microlocal sense Žcf. Taylor 3.. Therefore, we explicitly obtain the Dirichlet-to-Neumann operator as the nonlocal operator such that. The exact radiation condition on is given by r0 i u i r r0 n u 0. Thus, the problem is to determine. To this end, we first need to Ž. develop the right-hand side of 8 : Lu r u i ru i r u u 0. Ž 0. On the basic of the integral representation of pseudo-differential opera- Ž. tors from their symbol cf., e.g., 6, 3, an immediate calculation gives už r, s, t. e iž t s. Ž r, s, t,,. uˆ Ž r,,. d d r rh H r r e iž t s. uˆ uˆ d d. Then we introduce the following convenient notation. 4 is the pseudodifferential operator with the symbol, and we get the following expression: už r, s, t. 4u u. r r r

11 94 ANTOINE, BARUCQ, AND BENDALI Thus relation Ž 0. can be equivalently written as Lu u iž. u i 4u u. Ž. r r r Ž. Identifying in 7 and the coefficients of the derivatives with respect to r, we obtain the system ih r, h h Ž. i 4 r t s s s s. hh h h ž / Thanks to the integral representation formula of pseudo-differential operators, the operators and are explicitly known from the determination of their symbols. Using the calculus of pseudo-differential operators Žsee 3., system Ž. can be written at the symbol level: i H r i Ý s r 0! i Ž 3. h h Ž. i h i h s s. hh h hh h To determine the symbols and, we then use asymptotic expansion Ž. 9. The first equation of system 3 yields j j 0 if j 0 and 0 0 ih r. Ž 4. Since we only need to compute symbol, relations Ž 4. will be used for the elimination of. Thus, the combination of the second equation of 3 and 4 completely determines the operator. Finally, the Dirich- is obtained from the relation r0. let-to-neumann operator Obviously, the computation of all of the symbols j is not practicable. Furthermore, our aim is to construct local boundary conditions, whereas is a nonlocal operator. Fortunately, as will be shown, only the first few symbols j, j, are needed for the derivation of radiation conditions of order. 3.. Recursie Determination of the Asymptotic Expansion of the Symbol of the Dirichlet-to-Neumann Operator Based upon an identification of symbols with the same degree of homogeneity, a recursive formula permits us to express all of the symbols

12 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 95 from the first one,. More precisely, we remark that Eq. Ž 4. j enables us to readily express from and then to solve system Ž 3. j j relative to the symbols j. The computation is done in three steps. In the first two, and 0 are obtained by a straightforward calculation. The last step is devoted to the derivation of a recursive formula that expresses m, for m 0, in terms of previously determined symbols as well as their derivatives. Starting with an identification of homogeneous symbols of the highest degree, we get Ž h h.. From 4 and the condition 0, we directly obtain h h. Going further with an identification of the homogeneous symbols of the next higher degree, we find a relation between the unknowns and : 0 0 i i i 0 0 s s r Ž. ih h h h h h. Eliminating and, we get 0 s s i 0 ž Hr s s Ž./ h h h h hh. r s s The recursive formula derived from similar relations for lower degrees of homogeneity. At first, the left-hand side of the second equation in Ž 3. can be rewritten as Ž i. j sk i r l. Ž 5.! Ý Ý Ý Ý 0 j k l Ž. Next, we make use of the following properties see 3 : This, in turn, gives S j, S k. j s k S Ž jk.. j s k

13 96 ANTOINE, BARUCQ, AND BENDALI Thus, in 5 for any nonnegative integer m, the homogeneous part of degree m is m Ž i. i.! Ý Ý j s k r m 0 jkm j, k Since there is no homogeneous symbol of degree m on the right-hand side of the second equation of system Ž 3., an identification of symbols of the same homogeneity leads to the equation m Ž i. i 0, for m 0.! Ý Ý j s k r m 0 jkm j, k Moreover, we have m m. Hence, the previous equation, symbol m, m 0, can be recursively expressed from homogeneous symbols of higher order by m j k jkm j0, k0 Ý m Ž i. 0 Ý Ý j s k r m! jkm j, k i, Ž 6. Now we intend to establish that the design of radiation conditions up to the second order needs the explicit computation of only the first few symbols 4 j j. This property will be obtained basically from the following result. LEMMA 3.. following form: Each homogeneous symbol of order m, for m, has the m m Ž. P s, r ; X, 7 where X and PŽ s, r; X. is a polynomial in ariable X with smooth functions of Ž s, r. as coefficients. More precisely, polynomial PŽ s, r; X. is of the form PŽ s, r ; X. a Ž s, r. X. Ž 8. Ý d

14 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 97 Proof. Let PŽ s, r; X. and Qs, Ž r; X. designate various functions Žnot the same in all instances. of the form Ž 8.. The chain rule gives directly that PŽ s, r ; X., PŽ s, r ; X., PŽ s, r ; X.. j sj r Then an inductive argument on the order of derivation easily shows that the derivatives of a function of the type Ž. PŽ s, r; X. have the same form given by Ž. PŽ s, r ; X. Ž. QŽ s, r ; X., s Ž. PŽ s, r ; X. Ž. QŽ s, r ; X., r Ž. PŽ s, r ; X. Ž. QŽ s, r ; X.. The end of the proof is then easily obtained by an inductive argument due to the fact that relation 4 implies that m and m, are both of the form Ž 7. whenever it holds for one of them. We can then link up the approximation of the symbol of the operator through its asymptotic expansion Ž. 9 in a precise manner. THEOREM 3.. Let m be an integer. Any symbol gien by Ý jp Ž 9. for p m differs from Ý j m j by a term at most of order Ž m.. Proof. j Immediate from the above lemma. 4. SURFACE RADIATION BOUNDARY CONDITIONS 4.. The Approximation Criterion We are now in a position to give a precise criterion allowing the derivation of various radiation conditions and to compare their respective accuracy. We limit ourselves to radiation conditions that can be built through the following process. Let l be a positive integer and be the symbol of a pseudo-differential operator 4 defined on Ž,. such that Ý is of order Ž. l j p j for all sufficiently large p. Denoting by the symbol defined on by r0, we write out the

15 98 ANTOINE, BARUCQ, AND BENDALI radiation condition u iu 4u 0 n on, which we call a complete radiation condition of order l. Such a condition is considered to agree with standard definitions of the order of classical radiation conditions Žcf., for instance,.. Hence, in view of Theorem 3., the definition of a complete radiation condition is consistent, and the most straightforward one of order l is obtained as ½ Ý j 5 jl u i u 0 on. n Unfortunately, the approximate operator has the undesirable property of being nonlocal, just like the exact one. To obtain local radiation conditions Ž i.e., expressed through a local operator., we have to consider some approximation process of the symbols j. With this end in view, we follow an approach that is similar although slightly different from that considered by Engquist and Majda 9. From the retained definition of a complete radiation condition, it is quite obvious to approximate each j by its Taylor expansion in powers of up to the term Ž. l denoted Ž. j l. A complete radiation condition of order l can then be obtained as well by ½ Ý ž j / l5 jl u i u 0 on. n Surprisingly enough, as seen below, this construction always leads to a local radiation condition for l 4 Ž i.e., for conditions of order.. Actually, these conditions involve operators with a symbol polynomial in,, and, with coefficients smoothly depending on s and r. Since the symbol corresponds to an integration in t, from the assumed time dependence of the solution, this symbol only corresponds to the multiplication by k. Engquist and Majda have considered the same kind of approximation Ž Taylor expansion. but have used the angle of incidence rather than the wavelength as a small parameter. The same standpoint has been adopted by Hanouzet and Sesques `, who have kept Žh h. as a small parameter for the expansion. The approach that has been retained here can mainly be seen as a high-frequency approximation, as opposed to that of Engquist and Majda.

16 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 99 Incomplete radiation conditions can also be created by making a Taylor expansion at a given order l but by retaining fewer terms in the asymptotic expansion of the symbol Ž 9., i.e., m ½ Ý ž j/ l5 j u i u 0 on, n with m l. In this way, we will be able to recover almost all known radiation conditions of order. Finally, approximating the symbol of the complete radiation condition of order in a suitable manner, we derive another one that coincides with the BaylissTurkel second-order radiation condition when is a sphere. 4.. Computation of Symbols Since we are planning to deal with radiation conditions of order, from the above discussion we see that we have to calculate j only for j and to expand them in powers of until the term. For symbol, we get h h terms in. 3 Hence, the approximate symbol involved in the construction of radiation conditions up to the order is given by Ž.. Ž 0. Ž. Note that lower-order approximations l of with l are ob- i Ž tained simply by dropping terms in with l i from.. The second function is determined recursively from the first and has the following expression: ir s C s C s C s C 0 ihr h h h h i 3 3 Ch Ch ir s s Ž. C h C h h ir s s Ž. C h C h. h

17 00 ANTOINE, BARUCQ, AND BENDALI Making use of the previous expansion of, we get ir Ž s C s C. Ž s C s C. 0 ihr h h h h ir 3 3 ž h s C h s C h 3 Ž.Ž s s./ h C h 3 C h 3 i 3 3 Ž Ch Ch. terms in 3 and then deduce the approximate symbol of index zero, iž C C. 0 ih. For the third function, we remark that since all of the terms of order j Ž. j j are dropped when j and behaves like, only terms of order Ž. j with 0 j, occur in the expression of. Hence we write in the following form: H H s C s C s C s C r r r hh 4 hh We then deduce that s Hr s Hr terms in. 3 hh hh H H s C s C s C s C r r r hh 4 hh h s Hr h s Hr terms in, 3 which leads, once more taking 0 into account, to the following symbol approximation: Ž K H. Ž s C s C..

18 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 0 Following the same approach again, we finally write symbol form ih H ih s r s Hr i HrŽ r Hr. i r H r 4Ž. 4Ž. Ž. 4Ž. terms in. 3 Ž. in the The approximate symbol Ž. is then given by i i H H H K H s s Sommerfeld Radiation Condition The first approximation consists only of keeping terms of order. We then consider the complete radiation condition of order given by Ž. 4 Ž u i u 0. Making the operator. 4 n explicit and returning to a function u with a time dependence in e ikt yields the condition u iku 0 n on. This well-known condition, called the Sommerfeld radiation condition, only specifies that wave u is outgoing to infinity and may be seen as the crudest radiation condition that can be designated First-Order Radiation Condition Going further with the same procedure, we derive a complete radiation condition of order. Dropping any term of order j with j 0, we are 0 Ž led to u i Ý. 4u 0, and then to n j j 0 n u iku Hu 0. Ž. This is again a classical condition that has already been obtained by many authors 5,, 3, following very different paths Radiation Condition of Order 3 Now keeping the terms of order j with j, i.e., ½ Ý ž j/ 5 j u i u 0, n

19 0 ANTOINE, BARUCQ, AND BENDALI we obtain the complete radiation condition of order 3: i n u iku Hu Ž K H. u u 0, k ik where stands for the LaplaceBeltrami operator on. It is the same as Jones s first-order condition, which has been obtained in a different way. Note that this complete condition of order 3 does not correspond to another condition designed by Jones in 4 Žfrom an approach based upon successive approximations.: n u iku Hu u 0. ik Nonetheless, from our point of view, the latter condition can be con- 0 Ž structed from the approximation u i Ý. 4 n j j u 0 and can be seen as an incomplete condition of order 3. Note that incomplete conditions coincide with complete conditions for a flat boundary Radiation Conditions of Order The most accurate radiation condition that can be considered in the Ž above framework is given as u i Ý. 4 n j j u 0. Proceeding as for the previous conditions, we get the following result. THEOREM 4.. A complete radiation condition of order is gien by ž / i H H n u iku Hu i Ž K H. u u k k 4k i R div u 0. Ž 3. ik k The above complete second-order condition is different from the condition established by Engquist and Majda, who have used the following 0 Ž approximation, corresponding in our notation to u i Ý. 4 n j j u 0. Making the above relation explicit, one obtains the incomplete condition of order : i R n u iku Hu u 0, Ž 4. ik k where terms of order coming from j, with j or, are not considered. It is worth noting that condition Ž 4., except in the case in which is a sphere, does not lead to symmetric variational formulations

20 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 03 when it is used either to terminate a finite element formulation or as an OSRC condition, although it is an important advantage with condition Ž 3.. Remark. Following the same path, we can construct an incomplete condition by retaining only, 0, and in the asymptotic development and making a Taylor approximation up to the terms of order of the involved symbols. We get another incomplete second-order radiation condition, ž / i i R n u iku Hu Ž K H. u div u 0, k ik k Ž which is derived from the approximation u i. 4 n 0 u 0. At first sight, condition Ž 3. seems to be different from Jones s secondorder condition. In fact, it is exactly the same. But it is necessary to write out some canceling terms explicitly to see that the two expressions correspond exactly. Indeed, in our notation, Jones s second-order condition is defined on the parallel surface r by n u Ž ik Hr. u Ž Ž Hr Kr. u div u. ik H H K u H div u div Ž H u ž rž r r. r / r. k 4 4 h s Ž C C. s u 4h h h ž h / h s C C s u 0. 4h h To find condition 3 on the surface again, it is sufficient to develop the expressions involving derivatives of u and H and to remark that r hh h h 0, for j,. sj r0 sj r Second-Order BaylissTurkel-like Radiation Conditions Now we come to one of the main motivations of this work. As we said before, it consists of the justification of our heuristic derivation of an accurate second-order radiation condition from the BaylissTurkel condition in the two-dimensional case 3 and its extension to a surface of

21 04 ANTOINE, BARUCQ, AND BENDALI arbitrary shape in dimension 3. The result is embodied in the following theorem. THEOREM 4.. A complete second-order radiation condition is gien by k i H H n u iku Hu i Ž K H. u u k k 4k i R div u 0. Ž 5. ik 3 When is the sphere S x ; x r 4 r, this condition coincides with the corresponding BaylissTurkel second-order radiation condition. Proof. At first, we recall a suitable form for the BaylissTurkel second-order radiation condition, which is really due in the present case to Bayliss et al. 5. In its basic form, this condition is written in spherical coordinates Ž r,,. as 3 r ik r ik u 0 on S r. r r Using the Helmholtz equation expressed in the coordinates r,, to eliminate, we can write the above equation in an equivalent form as r u r u iku Ž Ž sin u. u. 0. r ikž ikr. r sin Ž 6. Now, observing that l l l ž / icl icl terms in, 3 ic s icl s terms in, 3 H H i i terms in, 3

22 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 05 Ž the above expression of Ý. j j may be rewritten as i H i j i H Ž K H. H 4 Ý j l Ý s l l l Ž l. Ž l. Ž i. ic ic terms in. 3 Dropping the terms in 3 and reusing the previous procedure, the above relation establishes that condition Ž 5. is another complete radiation condition of order on a general surface. Since for Sr the operator Ž i Rk. is scalar and is given by Ž i Rk. Ž ikr. and Ž K H. 0, conditions Ž 5. and Ž 6. are clearly identical. Remark. Using a suitable notation that permits us to particularize the above formula to the two-dimensional case, we readily obtain the following complete second-order radiation condition: C ic s C n u iku u u u 8kŽ ick. 8k s s u 0. Ž 7. ikž ick. In the above formula, is a curve of the plane, C is its curvature, and s is its curvilinear abscissa increasing in the counterclockwise direction. The latter formula exactly coincides with the following form of the usual BaylissTurkel radiation condition of order in the two-dimensional case: r u iku u u 0 r 8ikr Ž ikr. ikr Ž ikr. r for x r. Ž 8. Note that condition Ž 5. contains the two special forms of the Bayliss Turkel radiation conditions Ž 6. and Ž 8.; this feature is not obvious from condition Ž 6. only. In fact, the two-dimensional case has already been considered in a previous work 4. Radiation conditions were derived there, following the same approach as in this paper. An EngquistMajda-like condition can be

23 06 ANTOINE, BARUCQ, AND BENDALI obtained as an incomplete second-order condition given by ž / C ic ic n u iku u u s s u 0. 8k ik k Numerical experiments have shown Žsee 3 and the following numerical examples. that this condition, considered as an on-surface radiation condition, is less accurate than Ž 7.. Therefore, from this point of view, the theoretical study of this paper finds one of its main justifications. Remark. A high degree of smoothness of the surface is needed for the derivation of such conditions by the pseudo-differential analysis. However, some geometrical singularities can be considered as limiting cases of more regular geometries using a suitable scheme,. This is, for example, the case for the treatment of the two-dimensional scattering problem by a polygonal convex obstacle in the OSRC context. To conclude, we design another complete radiation condition of order that is close to a condition recently obtained by Stupfel, yet with the advantage of leading to symmetric variational formulations. The result is summarized in the following theorem. THEOREM 4.3. A complete second-order radiation condition is gien by ž / i H H n u iku Hu i Ž K H. u u k k 4k div i H n i R Ž n. u 0. Ž 9. ik k k Proof. The only difference the previous radiation condition lies in writing the terms Ž ic. and i C in the following form: l s l l icl ih icl ž / ž / ž / terms in, 3 icl ih icl s l s l 3 terms in, where l stands for when l. Dropping the terms in 3,we first obtain the following expression of the radiation condition: i H s l H r Ý k k l 4k u iku Hu i K H u u Ý s Ž l. l s l ik l ihk ic k u 0.

24 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 07 Observing now that u Ý u, Ž i Rk. Ž ic k. l s l l l l, l and n Ž. l we readily get Ž 9.. l l Remark. Following the criterion used in this paper, the radiation conditions Ž 3., Ž 5., and Ž 9. have the same accuracy. Numerical experiments Žcf., 3. have shown that, even with incomplete conditions, all of the radiation conditions of order give the same accuracy as long as the frequency is taken to be sufficiently high and symmetric formulations involving the derivatives of the curvature are used. This does not remain true for moderate or low frequencies. Therefore, other criteria have to be satisfied to distinguish between complete radiation conditions of the same order. This can be accomplished either by numerical experiments or by examining particular examples. In our opinion, a theoretical treatment of this problem is a harder task Surface Radiation Conditions for an Ellipsoidal Scatterer This part is devoted to the explicit expression of the previous surface radiation conditions for the special case of an ellipsoid centered at the origin. We intend to compute the OSRC solution to a scattering problem with a Neumann boundary condition: n u g on. To this end, we use the weak OSRC formulation, : find u H such that H A u u d g d. 30 The test functions are defined in the H space. The operator A is a zeroth-order operator of the tangent plane, and is a complex valued function of the surface. They are both fixed by the chosen surface radiation condition. The different geometrical parameters involved in such a problem are the following. The chosen coordinate system is : Ž,. 0; 0; Ž a cos sin, b sin sin, c cos.. We designate by a, b, and c Ž 0., respectively, the semiaxis of the obstacle along the axes Ž Ox., Ž Ox., and Ž Ox. 3 of the Cartesian coordinate system. We obtain the basis vectors Ž,. of the tangent plane as H Ž a sin sin, b sin cos,0., a cos cos, b cos sin, c sin. Ž 3.

25 08 ANTOINE, BARUCQ, AND BENDALI If the function is set as Ž. ab cos c sin a sin b cos, it follows that the mean and Gauss curvatures are easily computed as abc H a cos cos sin 3 b Ž sin cos cos. c sin, a b c K. 4 Let us introduce G as the metric tensor operating from the cotangent plane onto the tangent plane and defined by E F G Ž.. F G i j It can be shown from 3 that the first fundamental form coefficients E, F, and G are given by E Ž a sin b cos. sin, F Ž b a. cos sin cos sin, G Ž a cos b sin. cos c sin. Since the vector bases Ž,. of the cotangent plane are related to Ž,. by we obtain with Ž,. G Ž,., u u u, Ž G F. and Ž F E.. sin sin

26 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 09 The LaplaceBeltrami operator applied to the mean curvature can be computed using abc H Ž,. G Ž H, H.. abc Finally, the approximation of the solution to Ž 30. is based upon the use of a surface finite-element method, Some Numerical Results We present some radar cross section Ž RCS. calculations in the OSRC context. We consider the scattering problem for a Neumann boundary condition. The problem of the approximation is treated in,. The reference solution is given by a boundary integral equation method. The first example consists of the two-dimensional scattering problem with an incident plane wave illuminating an elliptical cylinder. The obstacle is characterized by its semiaxis a and b along the axis Ž 0 x. and Ž 0 x..as can be seen in Fig., the use of the BaylissTurkel-like second-order radiation condition improves the accuracy of the method compared with the EngquistMajda-like second-order radiation condition. A second example Ž cf. Fig.. is the three-dimensional scattering problem of a plane FIG.. Bistatic RCS of an elliptical cylinder.

27 0 ANTOINE, BARUCQ, AND BENDALI FIG.. Bistatic RCS of an ellipsoidal scatterer: complete radiation conditions of order, 3, and. wave by an ellipsoidal scatterer. If an improvement is obtained by using the complete radiation condition of order 3 compared with the first-order condition Ž 3., we can conclude that the BaylissTurkel-like radiation condition Ž 5. is the most accurate second-order condition. ACKNOWLEDGMENTS The authors are very thankful to the referee for his suggestions and remarks. REFERENCES. X. Antoine, Conditions de radiation sur le bord, Ph.D. thesis, UFR Sciences, Universite de Pau et des Pays de l Adour, Pau, France, X. Antoine, Fast Computation of time-harmonic scattered field using the on-surface radiation condition method, submitted to J. Comput. Phys. and available as preprint 984, Analyse Numerique des EDP, Laboratoire de Mathematiques Appliquees pour l Industrie and UPRES-A 5033, Universite de Pau et des Pays de l Adour, France.

28 BAYLISSTURKEL-LIKE RADIATION CONDITIONS 3. X. Antoine, H. Barucq, and A. Bendali, Formulations symetriques dans le calcul de diffraction d ondes par des conditions de radiation sur le bord, preprint 95, Analyse Numerique des EDP, Laboratoire de Mathematiques Appliquees pour l Industrie and UPRES-A 5033, Univerisite de Pau et des Pays de l Adour, France. 4. X. Antoine, H. Barucq, and A. Bendali, New absorbing boundary conditions including derivatives of the curvature for the two-dimensional Helmholtz equation, preprint 9607, Analyse Numerique des EDP, Laboratoire de Mathematiques Appliquees pour l Industrie and UPRES-A 5033, Universite de Pau et des Pays de l Adour, France. 5. A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math. 4 Ž 98., J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland, AmsterdamNew York, G. Chen and J. Zhou, Boundary Element Methods, Academic Press, Harcourt Brace Jovanovitch, San Diego, M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall International, Englewood Cliffs, NJ, B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 3 Ž 977., L. Halpern and L. N. Trefethen, Wide-angle one-way wave equation, numerical analysis report 86-5, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, July B. Hanouzet and M. Sesques, ` Absorbing boundary conditions for Maxwell s equations, in Non-Linear Hyperbolic Problems: Theoretical, Applied and Computational Aspects, Ž A. Donato and F. Olivieri, Eds.., Notes Numer. Fluid Dynamics 43 Ž 99., J. Jin, The Finite Element Method in Electromagnetics, Wiley, New York, D. S. Jones, Surface radiation conditions, IMA J. Appl. Math. 4 Ž 988., D. S. Jones, An approximate boundary condition in acoustics, J. Sound Vibration Ž. Ž 988., J. J. Kohn and L. Nirenberg, An algebra of pseudodifferential operators, Comm. Pure Appl. Math. 8 Ž 965., G. A. Kriegsmann, A. Taflove, and K. R. Umashankar, A new formulation of electromagnetic wave scattering using the on-surface radiation condition approach, IEEE Trans. Antennas and Propagation 35 Ž 987., A. Majda and S. Osher, Reflection of singularities at the boundary, Comm. Pure Appl. Math. 8 Ž 975., D. B. Meade, A. F. Peterson, and C. Piellusch-Castle, Derivation and comparison of radiation boundary conditions for the two-dimensional Helmholtz equation with non-circular artificial boundaries, in Proceedings of the Third International Conference on Mathematical and Numerical Aspects of Wave Propagation, Mandelieu-La-Napoule, 995 Ž Gary Cohen, Ed.., R. Mittra and O. Ramahi, Absorbing boundary conditions for the direct solution of partial differential equations arising in electromagnetic scattering problems, in Progress in Electromagnetics Research: Finite Element and Finite Difference Methods in Electromagnetic Scattering Ž M. A. Morgan, Ed.., Elsevier, Newark, NJ, G. Mur, Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic field equations, IEEE Trans. Electromagnet. Compat. 3 Ž 98., L. Nirenberg, Pseudodifferential operators and some applications, Lectures on Linear Partial Differential Equations, CBMS Regional Conf. Ser. in Math. 7 Ž 973., B. Stupfel, Absorbing boundary conditions on arbitrary boundaries for the scalar and vector wave equations, IEEE Trans. Antennas and Propagation 4Ž.Ž , M. E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, NJ, 98.