Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X

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1 Variable separation in acoustic radiation problems using Chebyshev polynomials Z. Wozniak* and C. Purcell** *ORTECH Corporation, Mississauga, Ontario, Canada, L5K1B3 **Defence Research Establishment Atlantic, Dartmouth, Nova SWz'a, Canada, B27 JZ7 Abstract A new method of computation of Boundary Element matrices for acoustic radiation problems has been developed. The method utilizes Chebyshev polynomials for separation of the frequency from space variables in the free-space Green's function. The BEM matrices resulting from discretization of the Helmholtz equation are static and can be synthesized for each required frequency using an analytical formula. Significant savings in computer time have been achieved. 1 Introduction The radiation and scattering of sound waves is a vital part of many engineering problems such as fluid-structure interactions, acoustic radiation, earthquake and vibrating foundation problems. As the Boundary Element Method (BEM) has been found to be an effective tool for solving this kind of problem, especially for unbounded domains, it is also recognized that the most time consuming part of the computational process is the formulation of the BEM matrices. Due to the oscillatory nature of the Green's

2 420 Boundary Elements function involved, the order of integration over the element surface has to increase with frequency and often the mesh itself has to be refined to achieve satisfactory accuracy. In the past few years, considerable effort has been expended in developing methods for speeding up the BEM computations. The two different approaches have been explored; series approximation and interpolation. An example of the series approximation is given by Koopman and Benner^. For small values of kr they used a truncated McLaurin series expansion for exp(ikr}. Alternatively, for larger values of kr the farfield approximation was used. The interpolation method was introduced by Schenck and Benthien^. The rapid variation of the Green's function integrands was eliminated by factoring out the exp(-ikrmn) quantity, where rmn is a distance between two reference points on the radiating surface. Modified this way integrands are slowly varying with frequency, thus allowing linear interpolation in frequency domain. Review of the range of methods used for speeding up the BEM solution is given by Kirkup and Henwood ^. They proposed an efficient interpolation method in combination with the collocation method for the Burton and Miller* formulation of BIE. The method presented in this work separates the frequency from space variables using an expansion of the Green's function in terms of the Chebyshev polynomials. This alternate formulation of the fundamental solution to the Helmholtz equation has a significant computational advantage over the standard approach since the integration over the space variables need to be performed only once and can be used subsequently for any frequency required. A similar method** has been successfully developed by the author for modeling of water surface gravity waves. Since the Chebyshev polynomials are bounded in the [-1,1] interval, a better numerical performance of the expansion has been achieved. 2 Theory The fundamental solution to the Helmholtz equation^ satisfying the Sommerfeld radiation condition is the free-space Green's function <9<1> ^K^T+^O] (2) G(P,Q) = ^- (3) where <& is the velocity flow potential, r = \P Q\ is the distance between any two points P and Q in the acoustic medium or on the body surface,

3 Boundary Elements 421 and k is the wavenumber. In order to separate the wavenumber from the space variable r in Equation (3), the following Chebyshev expansion of the exponential function is examined where Jn(p) are Bessel's functions of thefirstkind and Tn(q) are the Chebyshev polynomials. The above series converges uniformly for all p > 0 and for q within the range [-1,1]. Since Tn(q) are bounded within [-!,+!], the convergence of the series is governed mainly by the Bessel's functions Jn(p)- The error of the partial sums 5oc - &, of the Equation (4) shown in Figure (1) was computed assuming 7^ 1 for all n. n=l (4) kr/2 = 0.1 kr/2=1.0 kr/2 = 5.0 kr/2=10 kr/2= Number of terms Figure 1: Error of approximation \S^ Sn using Chebyshev expansion It can be seen that for n» p the error is rapidly decreasing due to decrease of the Bessel's terms Jn(p) as the index n increases. The magnitude of the terms truncated can be estimated using the asymptotic estimate^ applies /-(*) ~( (5) which indicates that terms of the expansion for n» k are monotonically decreasing as shown in Figure (1) The argument of the Chebyshev polynomials requires mapping of the space variable r on the [-1,1] interval. Such a mapping is always possible since practically all sound radiating bodies have a limited maximum dimension (6)

4 422 Boundary Elements for both internal and external problems. Thus a new space variable p may be defined as follows P = 2^-l (7) and consequently the Green's function can now be rewritten as Substituting expansion (4) for the second term in the above formula (8) yields the alternate equivalent form of the Green's function 1 00 G(P,Q) = e-< '*-[Jo(kR/2] - 2 i"jn(kr/2)t»(p)] (9) with the wave number k separated from space variable r. It can be seen that the convergence of the above formula is governed by the argument of the Bessel's function kr/2. A better convergence exists for smaller wavenumbers as indicated by the estimate (5). The Chebyshev polynomials can be generated using well known recurrence relation To(p) = 1 Ti(p) = P (10) Tk+i(p) = 3 Numerical Implementation Following the standard procedure in BEM\ the boundary integral equation is based on the Helmholtz integral equation (ii) with boundary conditions specified on the body surface F = FI -f where the coefficient c(p) = 2?r for a node on the smooth boundary or ;(V (13) for an edge node or a corner node. The I\ and 1^ are the parts of the body surface with specified potential and the normal gradient of the potential, respectively.

5 Boundary Elements 423 In the course of discretization, the surface integral (11) is replaced by a sum of integrals over a number of surface elements nodes m, C,;$,; = (Aifr - B^) J = l elem,. -/-. AH = Is^N^dS^ (14) m=l Bij = *E' Is^NjGjdSm m=l where i and j are the node numbers. Here, the mapping of the global-to-local coordinates is accomplished by using the shape functions Nj(r, s) X(r,s) = ^XjNj(r,s) (15) j For convenience, the same shape functions are being used for interpolation of thefluidpotential 0 together with the gradient d$/dn on the boundary where j is the surface element node number. The discretized version of the surface integral (11) is given in a form J 00 (A]{*} = [B}{^} (17) where complex matrices A and B are assembled row- by-row for each frequency from surface element integrals. The advantage of the proposed series formulation of the Green's function is that it makes possible to synthesize these matrices using formulas derived from the Chebyshev expansion (4). Substituting Equation (4) into the formulas for element vectors A^ and B^ (14), and grouping terms, one can obtain alternate expressions for the element vectors oc"=0 (18) B% = E#k&uk 6=0 where the coefficients ak(kr/2) depend on frequency only

6 424 Boundary Elements and the coefficients a^, CLijkt &^d 6^-& depend on the space variable only, and may be obtained by integration over the element surfaces Sm elem. at* = Js^NftgdSn ra=l at* = "izfs^n&sidsn (20) 771=1 elem. bijk = E Js^NftdSm 771=1 Since integrands of the coefficients a^ and a^ (20) are well behaved in r, they can be integrated by simple Gaussian quadrature rule. For the coefficients bijk however, the problem of an integrable singularity arises when the "root" node is in the element under consideration. All the Chebyshev polynomials have a value +1 or -1 at the root node and consequently all the integrals b^k (20) are singular. The integration method used in this work is a adaptation of the method given by Lachat and Watson^. The quadrilateral element is divided into two triangles with one vertex located at the point of singularity. As shown elsewhere^, that singularity can be effectively removed by replacing triangles with quadrilaterals collapsed at the "root" node. Assembling of the BEM matrices A and B of Equation (17) for any desired frequency proceeds in a usual manner (Equation 14) with an addition of one inner loop computing element vectors (18) from the Chebyshev expansion. The process of assembling is very fast even if the coefficients of expansion are stored on disk. The most time consuming part is a computation of the coefficients of expansion, but it occurs only once at the initial stage of computation. The time gains of fast assembling of the matrices A and B clearly outpace this initial expense especially if the restart option is implemented and stored coefficients can be used over and over again for analysis involving different frequency ranges. Another important advantage of this method is that the matrices A and B are the continuous functions of the frequency. This feature effectively renders an interpolation unnecessary when A and B are to be evaluated only at certain frequencies. 4 Examples The proposed method of solution to the Helmholtz equation has been implemented into a Boundary Element analysis program. The purpose of the examples presented is to examine performance of the method and compare the results to known analytical and numerical solutions. Both internal and external problems are considered. The first example of an internal problem is an acoustic wave inside a square tube. The piston at x 0 vibrates with uniform velocity V(0) = V exp(-iujt), and at the other end of the tube, at x=l, there is a prescribed

7 Boundary Elements 425 velocity potential $(!/) = 0. The analytical solution for this simple internal problem is $W = V^-^ kcos(kl) (21) The numerical solution to this problem was obtained using a mesh with 88 quadrilateral elements and 116 nodes. Tube dimensions were 1 m x 1 m x 5 m and the velocity V = 1 m/s. _O CD O Q_ BEM tf exact <f> ct d<f/dn Wave number k Figure 2: Transmission of the acoustic wave inside the tube Summary of the results is shown in Figure (2). A second example, of an external problem is calculation of the acoustic radiation from a pulsating sphere. The sphere is centered at r=0 and has radius r=r. There is a prescribed uniform radial velocity V(R) = Ve-xp(-iujt), and therefore complete spherical symmetry exists. The velocity potential for this case is given by Junger and Feit l + ikj? On the surface of the sphere the potential is r (22) (23) A sphere of radius R = 1 m, with the velocity V = 1 m/s was discretized with 384 quadrilateral elements and 386 nodes.

8 426 Boundary Elements Wave number k Figure 3: Velocity potential on the surface of the pulsating sphere The resulting velocity potential on the surface of the sphere is shown in Figure (3). The noticeable discrepancies between BEM and theoretical values at the wave numbers k = TT, 2?r,... are due to nonuniqueness of the BEM solution^ at the frequencies coinciding with the eigenfrequencies of the corresponding inner problem. For both problems, the results obtained using the proposed method based on the Chebyshev expansion are practically undistinguishable (accuracy e 10~^ was imposed) from those obtained using standard BEM approach. This should be expected because both methods are using the same Green's function and the same order of integration was applied during computation. The only difference is in the way the Green's function is represented. It is interesting to compare in Table (1), the execution time for these problems Table 1: Comparison of the execution time, sec Problem Tube Sphere Chebyshev expansion coefficients assembling Standard method assembling All computations were done on an AST Bravo 486/25 PC with a hard disk of 19 ms average access time. The compiler used was Z or tech C+H- version 3.0.

9 Boundary Elements Conclusions A highly efficient method of computation of acoustic problems has been developed which is suitable for both internal and external problems. When using the traditional BEM the formulation of the matrices takes n^tr CPU time, where n^ is the number of wave numbers and i? is the time taken for the computation at each wave number. The new method takes ts + n^n where {3 is the set-up time and t^ is the time required to assemble the BEM matrices for each individual wave number. For the two examples shown, the new method is about 4 to 6 times faster in assembling than the standard BEM procedure. The method is particularly attractive for problems which require computation of a large number of frequency steps or extraction of the natural frequencies of complex fluid-solid systems. Bibliography 1. Burton, A.J., Miller, G.F. The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary Value Problems, Proc. o/zae Ao?/a/ 5ocze%/ Z/oWon, Vol. A323 (1971), Ciskowski, R.D., Brebbia, C. A., Boundary Element Methods in Acoustics, Comp. Mech. Publ. and Elsevier Appl. Science (1992). 3. Jeans, R.A., Mathews, I. C., Solution of fluid-structure interaction problems using coupled finite element and variational boundary element oc./w. 88, (November 1990), Koopman, G.H., Benner, H. Method for computing the sound power of machines based on the Helmholtz integral, J.Acoust.Soc.Am. 71(1) (Jan. 1992), Kirkup, S.M., Kenwood, D.J. Methods for Speeding Up the Boundary Element Solution of Acoustic Radiation Problems, Journal of Vibraand /Icows^cs, 114 (1992), Lachat, J.C., Watson, J.O., Effective numerical treatment of boundary integral equations, Int. J.N urn. Methods Eng. 10, (1976), Lamb, ^..Hydrodynamics, Dover, New York, Paszkowski, S., Numerical Applications of the Chebyshew Polynomials and Chebyshev Series, Pans twowe Wydawnictwo Naukowe, Warsaw (1975).

10 428 Boundary Elements 9. Press, W.H, at al,numerical Recipes in C, Cambridge University Press (1988). 10. Schenck, H.A., Benthien, G.W. The Application of a Coupled Finite- Element Technique to Large-Scale Structural Acoustic Problems, Advances in Boundary Elements, Proceedings of the Eleventh International Conference on BEM, Cambridge, Massachusetts, Brebbia, C.A., and Connor, J.J.,eds., Vol Wozniak, Z.Efficient Algorithm for Computation of the Green's Function in Frequency Domain, Proc. llth Int'l Conf. of the Canadian Applied Mathematics Society, (1990).