STRUCTURAL-ACOUSTIC COUPLING G.W. Benthien and H.A. Schenck Naval Ocean Systems Center, San Diego, CA , USA

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1 STRUCTURAL-ACOUSTIC COUPLING G.W. Benthien and H.A. Schenck Naval Ocean Systems Center, San Diego, CA , USA 1 Introduction In recent years there has been much interest in problems involving acoustic radiation or scattering by an elastic structure submerged in an infinite fluid medium. The elastic structure is often very complicated: it may consist of such items as nonuniform shells, reinforcing ribs and beams, joints, and attached masses. The finite element method is widely used to analyze such complicated structures, but is not well suited to handle the exterior fluid medium which is infinite in extent. On the other hand, boundary element methods are ideally suited to handle the exterior fluid problem since they replace the infinite domain problem by an integral equation over the boundary surface of the submerged structure. In this chapter it will be shown how these two different types of methods can be combined to solve coupled structural-acoustic radiation or scattering problems. The particular boundary element method that will be used in this discussion is called CHIEF (Combined Helmholtz Integral Equation Formulation)[1,2]. This method has been implemented in a computer program by the same name. Any finite element program capable of handling the structure can be used. In addition to a description of the basic equations and how they are coupled, attention will be given to the computer implementation of these equations and to methods for improving the computational efficiency. Since the numerical method described in this chapter frequently generates a very large amount of output, several useful ways to visualize and analyze the output data are presented and discussed. 2 Basic Equations Throughout this section all pressures, displacements, and velocities will be represented by their complex, frequency dependent Fourier components corresponding to the time dependence e iωt.the elastic structure is modeled using the finite element equations [3] ( ω 2 M + K)U = F rad + F D (1) where M is the mass matrix, K is the stiffness matrix, F rad is the load vector corresponding to the acoustic loading of the fluid external to the structure, F D is the load vector corresponding to other

2 forces which drive the structure, and U is a column vector whose components are the displacement degrees-of-freedom at all the nodes in the body. The components of F rad are given by = pφ m ndγ (2) F rad m Γ where p is the acoustic pressure, n is a unit normal to the wet-surface Γ pointing into the fluid, and φ m is a finite element vector interpolation function. The displacement u(x) atapointxin the body can be expressed approximately in terms of the interpolation functions φ m as follows: u(x). = where U 1,...,U M are the components of U. M U m φ m (x) (3) m=1 The CHIEF formulation for the acoustic field is based on the Helmholtz integral relations [1,2]. The scalar acoustic wave equation for the acoustic medium is replaced by an integral equation over the wet-surface Γ, which in turn is approximated by a system of algebraic equations AP = BV + P inc, (4) where P and V are column vectors whose n-th components are the pressure and normal velocity (assumed to be constant) on the n-th subdivision Γ n of Γ; and P inc is a column vector whose n-th component is the value of the incident pressure wave at a reference point on Γ n. In a pure radiation problem, P inc = 0. The matrices A and B in Eq. (4) have components given by A mn = 1 2 δ mn Γ n G(ζ m,σ) n σ dγ(σ) (5) B mn = iωρ G(ζ m,σ)dγ(σ) Γ n (6) where δ mn is the Kronecker delta, ζ m is a reference point on Γ m,andg(ζ m,σ) is the free-space Green s function G(ζ m,σ)= e ikr(ζm,σ), (k =ω/c) (7) 4πr(ζ m,σ) where r(ζ m,σ) is the distance between ζ m and σ. The normal derivative of G(ζ m,σ) is given by the expression G(ζ m,σ) n σ = G(ζ ( ) m,σ) 1 r 2 ik + [(σ ζ m ) n(σ)]. (8) (ζ m,σ) r(ζ m,σ) It is well known [1] that the solution of the Helmholtz integral equation for p in terms of v and p inc is not unique at certain frequencies corresponding to the eigenfrequencies of the interior Dirichlet problem. In the neighborhood of these eigenfrequencies the matrix A in Eq. (4) becomes ill-conditioned. To overcome this numerical difficulty, CHIEF supplements the system of equations in Eq. (4) by additional equations of the same form in which each ζ m is a point interior to Γ. The resulting over-determined system of equations has a unique solution in the least-squares sense. The total acoustic pressure p in the exterior field can be written as the sum of the incident pressure p inc and the scattered pressure p s, i.e., p(x) =p inc (x)+p s (x). (9) The scattered pressure p s at a field point x exterior to the structure can be expressed in discretized form by p s (x) =a T (x)p+b T (x)v (10)

3 where a(x) andb(x) are column vectors having components given by G(x, σ) a n (x) = dγ(σ) (11) Γ n n σ b n (x) = iωρ G(x, σ) dγ(σ). (12) Γ n In the far-field of Γ, the components a n (x) andb n (x) can be approximated by. a n (x) = ike ikr (ˆr(x) n(σ))e ikˆr(x) δ(σ) dγ(σ) (13) 4πR Γ n. b n (x) = iωρe ikr e ikˆr(x) δ(σ) dγ(σ) (14) 4πR Γ n where R is the distance from an origin near the surface Γ to the field point x, ˆr(x) is a unit vector at the origin in the direction of x, andδ(σ) is the position vector of the integration point σ relative to the origin. where Eqs. (4), (9) and (10) can be combined to give p s (x) =q T (x)v +p rs (x) (15) q T (x) a T (x)a 1 B + b T (x) and p rs (x) a T (x)a 1 P inc. It can be seen from Eq. (15) that p rs (x) is the scattering that would result if the body were completely rigid (v =0). Applying the assumption that p is piecewise constant on the subdivisions of Γ to Eq. (2) gives where C mn = 1 φ m ndγ Γ n Γ n and D =diag(γ 1,Γ 2,...,Γ N ). Eqs. (1) and (16) can be combined to give F rad = CDP (16) U = ( ω 2 M + K) 1 CDP +( ω 2 M+K) 1 F D. (17) Since different interpolation schemes are used in the finite element model of the structure and the CHIEF model of the acoustic field, it is impossible to enforce exact continuity of normal displacement across Γ. However, this continuity is approximately enforced by equating the CHIEF normal velocity v n to the average of the finite element normal velocity over Γ n, i.e., v n = 1 iωu ndγ. (18) Γ n Γ n Combination of Eqs. (3), (17), and (18) gives V = iωc T U = iωc T ( ω 2 M + K) 1 CDP + iωc T ( ω 2 M + K) 1 F D. (19)

4 For computational efficiency it is useful to introduce the matrix E whose columns are the normal modes of the structure, i.e., KE = MEΩ (20) where Ω = diag(ω1,...,ω 2 M 2 ) is the diagonal matrix of in-vacuo eigenfrequencies. Since the modes are M-orthogonal, they can be normalized so that E T ME = I. (21) It is easily verified that the inverse matrix ( ω 2 M + K) 1 can be expressed in terms of the normal mode matrix E as follows: ( ω 2 M + K) 1 = E( ω 2 I +Ω) 1 E T. (22) This form is convenient since the diagonal matrix ( ω 2 I + Ω) is easily inverted at every frequency which is not one of the in-vacuo eigenfrequencies. Combination of Eqs. (19) and (22) gives V = iωc T E( ω 2 I +Ω) 1 E T CDP + iωc T E( ω 2 I +Ω) 1 E T F D. (23) Eqs. (4) and (23) can now be combined to give HP = P inc + iωbc T E( ω 2 I +Ω) 1 E T F D. (24) where H = A + iωbc T E( ω 2 I +Ω) 1 E T CD. Eq. (24) can be solved for the surface pressure vector P. OncePis determined, the velocity vector V can be obtained from Eq. (23) and the scattered pressure p s (x) can be computed at any exterior field point x using Eqs. (10) or (15). 3 Computational Techniques An outline of the computational steps involved in implementing the equations of Section 2 as well as several techniques for improving the computational efficiency of the method will be presented in this section. Obviously, there are many ways that the computations could be organized, and it is not claimed that the scheme presented here is optimal. However, this scheme has proven to be very flexible and efficient. 3.1 Outline of Computations 1. Compute the in vacuo eigenfrequencies and modes of the structure using a finite element program. Although the approach presented in Section 2 does not require the use of eigenmodes, it is more efficient to obtain the inverse ( ω 2 M + K) 1 using eigenmodes when the inverse is required at a large number of frequencies. 2. The columns of the coupling matrix C can be computed in many finite element codes as load vectors corresponding to a unit pressure on one surface subdivision with zero pressure on the rest. There is also code in the CHIEF program for computing the C matrix. 3. Compute the frequency independent product E T C occuring in Eq. (24). Note that C T E = (E T C) T.

5 4. At each frequency of interest compute the CHIEF matrices A and B given by Eqs. (5) and (6) as well as the far-field vectors a(x) andb(x) defined by Eqs. (13) and (14) for each field point x desired. In addition, compute the incident vector P inc for each desired incidence direction. 5. Compute the products C T E( ω 2 I +Ω) 1 E T CD and C T E( ω 2 I +Ω) 1 E T F D at each frequency making use of the fact that the matrices ( ω 2 I +Ω) 1 and D are diagonal. These matrix products will be needed several times in subsequent calculations. 6. For each frequency and each incidence angle, form the matrix H and the right-hand-side of Eq. (24). 7. For each frequency and each incidence angle, solve the (possibly) overdetermined system of equations in Eq. (24) in the least-squares sense for P. The CHIEF program makes use of a Householder decomposition to solve the least-squares problem [4]. This decomposition does not have to be repeated for the multiple right-hand-sides corresponding to different incidence angles. 8. For each solution P the corresponding velocity vector V can be obtained using Eq. (23) and the matrix products previously accumulated in step The far-field scattered pressure p s (x) is obtained for each field point x using Eqs. (10), (13), and (14). Note: Although the inverse ( ω 2 M + K) 1 does not exist when ω is an in vacuo eigenfrequency, this is not a serious limitation since these resonances are usually very sharp. The problem can be eliminated entirely by adding a small amount of damping to the structural model. 3.2 Symmetry Conditions While it is important to maintain the capability of handling quite general structures, it is also important to realize that significant reductions in computation time can be achieved by taking advantage of any symmetries that might be present in the problem. In the finite element portion of the problem, planes of symmetry are handled by modelling only one-half, one-quarter, or one-eighth of the body depending on the number of symmetry planes. Appropriate boundary conditions are applied along the planes of symmetry to account for symmetric and antisymmetric modes relative to each of the planes. Circular symmetry is handled in many finite element programs by expanding all forces and displacements in Fourier series in the angular coordinate. Because of orthogonality, each circular harmonic can be computed independently. In the acoustic portion of the problem, symmetry manifests itself in the structure of the A and B matrices defined in Section 2. It has been shown by Benthien, Barach and Gillette [2] that each type of symmetry results in the A and B matrices commuting with the appropriate symmetry operator Σ, i.e., AΣ =ΣA, BΣ=ΣB. (25) It follows from these commutation relations that the eigenspaces of Σ are invariant under both A and B. Thus, A and B become block diagonal when expressed in terms of a basis consisting of eigenvectors of Σ. The sizes of the diagonal blocks are the dimensions of the eigenspaces of Σ. To see how symmetry can be used in a coupled problem, consider the special case of one plane of symmetry. The symmetry operator for one plane of symmetry is ( ) 0 I Σ= (26) I 0

6 assuming that symmetric elements are numbered in the same order. The matrix X of eigenvectors of Σ is given by ( ) I I X =. (27) I I The inverse X 1 of X turns out to be 1 2X. Let the matrix E consisting of all in vacuo eigenmodes of the structure be arranged so that all the symmetric modes and all the antisymmetric modes are grouped together. The columns of the matrix C T E appearing in Eq. (24) are the normal components of the structural modes averaged over the CHIEF subdivisions. Thus, C T E has the form ( ) C T Es E E = a. (28) E s E a If the left-hand-side of Eq. (24) is multiplied by X 1, it can be written as X 1 HP = (X 1 HX)(X 1 P ) = [X 1 AX + iω(x 1 BX)(X 1 C T E)( ω 2 I +Ω) 1 (E T CX)(X 1 DX)] ˆP (29) where ˆP = X 1 P. It is shown in Ref. [2] that X 1 AX and X 1 BX have the form ( ) ( ) X 1 A1 + A AX = 2 0, X 0 A 1 A 1 B1 + B BX = 2 0. (30) 2 0 B 1 B 2 It follows from Eqs. (27) and (28) that ( ) X 1 C T Es 0 E = 0 E a ( and E T E T CX = 2 s 0 0 Ea T Since D has the form D = (31) ). (32) ( ) D1 0, (33) 0 D 1 it follows that X 1 DX = D. (34) It can be seen from Eqs. (29) (34) that X 1 HX is also block diagonal, and hence the system of equations in Eq. (24) decouples into two smaller systems. Since solution time goes roughly as the cube of the number of equations, it is faster to solve the two smaller systems than the one larger system. The symmetry can also be used to reduce the time needed to generate the matrices in Eq. (24). The CHIEF program can take advantage of one, two or three planes of symmetry as well as any finite order of rotational symmetry. CHIEF can also take advantage of symmetry in the forcing terms when it exists. It is shown in Ref. [5] that complete rotational symmetry can be approximated by finite order rotational symmetry in the same way that Fourier series can be approximated by discrete Fourier transforms. 3.3 Frequency Interpolation Another way to improve computational efficiency is to use frequency interpolation [6]. For coupled acoustic structural scattering problems, the spectral response often varies rapidly over a large dynamic range, requiring the response to be calculated at many closely spaced frequencies. In such problems, the computation time is often dominated by the calculation of acoustic interactions which do not vary rapidly with frequency. It will be shown that large savings in computation time can be achieved by interpolating on appropriately scaled intermediate acoustic quantities that vary slowly

7 with frequency. These scaled quantities are computed with a coarser frequency resolution than is desired in the final frequency sweep, and the intermediate frequency values are then obtained by interpolation. The only acoustic quantities involved in the system of equations (24) for P are the matrices A and B. It can be seen from Eqs. (5) to (8) that the components of A and B both involve the quantity e ikr which varies rapidly with frequency when the separation is large. This rapid variation in the integrands of A mn and B mn can be eliminated by factoring out the quantity e ikrmn where r mn is the distance between the reference points on Γ m and Γ n. Thus, the matrices  and ˆB defined by G(ζ m,σ) dγ(σ) (35) Γ n n σ ˆB mn iωρe Γ ikrmn G(ζ m,σ)dγ(σ) (36) n  mn e ikrmn are slowly varying with frequency and are simply related to A and B by A mn = e ikrmn ( 1 2 δ mn Âmn) (37) B mn = e ikrmn ˆB mn. (38) Linear frequency interpolation can be used on Â, ˆB and the matrices A, B can be recovered using Eqs. (37) and (38). The far-field calculations involve the acoustic quantities a n (x), b n (x) defined by Eqs. (13) and (14). The quantity e ikˆr(x) δ(σ) in these equations can be expanded as follows: e ikˆr(x) δ(σ) = e ikˆr(x) δn e ikˆr(x) (δ(σ) δn) (ik) m = e ikˆr(x) δn [ˆr(x) (δ(σ) δ n )] m m! m=0 (ikd = e ikˆr(x) δn n ) m [ ] m ˆr(x) (δ(σ) δn ) (39) m! m=0 where δ n is the value of δ(σ) at the reference point on Γ n,andd n is the maximum value of δ(σ) δ n over Γ n. For the small subdivisions Γ n used in practice, the m = 0 term gives a sufficiently accurate approximation. Use of this first order approximation in Eqs. (13) and (14) gives rise to a n (x) = ik e ikr eikˆr(x) δn ˆr(x) n(σ) dγ(σ) (40) 4π R Γ n b n (x) = iωρ e ikr eikˆr(x) δn Γ n. (41) 4π R For equal frequency increments the quantity e ikrmn occurring in Eqs. (37) and (38) and the quantity e ikˆr(x) δn occurring in Eqs. (40) and (41) can be computed recursively using d n e i(k+ k)r = e ikr e i kr. (42) The interpolation scheme can now be summarized as follows: 1. The matrices  and ˆB defined in Eqs. (35) and (36) are computed at a set of frequencies which have a wider spacing than the intended final frequency sweep. 2. The integrals Γ n ˆr(x) n(σ) dγ(σ) occurring in Eq. (40) are computed for each subdivision Γ n and each desired far-field point x.

8 3. The exponentials e ikrmn and e ikˆr(x) δn are computed for a starting wavenumber k 0 and a wavenumber increment k corresponding to the final frequency resolution desired. 4. At each wavenumber k n = k 0 +(n 1) k, the matrices  and ˆB are computed using linear frequency interpolation. 5. The quantities e ikrmn and e ikˆr(x) δn are updated using the recursion relation (42). 6. The matrices A and B corresponding to k n are computed using Eqs. (37) and (38). 7. Eq. (24) is solved for the surface pressures P. 8. The surface normal velocities V are computed using Eq. (23). 9. The far-field quantities a n (x) andb n (x) are computed using Eqs. (40) and (41). 10. The far-field pressure p s (x) is computed for each field point x using Eq. (10). Examples will now be shown which illustrate the accuracy and improvement in computational efficiency that can be obtained using this scheme. The first example involves the backscattering from a spherical shell in water. The ratio of the thickness to the mean radius of the shell is The material parameters of the shell are Density = 7669 Kg/m 3 Young s modulus = N/m 2 Poisson s ratio = 0.3. The density of the fluid medium was taken to be 998 Kg/m 3 and the sound speed was taken to be 1486 m/s. Fig. 1 shows the backscattered form function for this spherical shell versus the normalized frequency for increasingly coarse interpolation in. The normalized frequency is defined by =2πfa/c where f is the frequency, a is the mean radius, and c is the sound speed in water. The increment used in the acoustic calculations (CHIEF) was taken successively to be 0.01, 0.1, 0.2, 0.3. The solid curve was obtained from an analytic series solution [7]. It can be seen from Fig. 1 that there was only a modest decrease in accuracy as was increased. Table 1 shows the number of frequencies at which the CHIEF calculations needed to be made for each value of. Table 1. Number of CHIEF frequencies required for sphere backscattering problem. CHIEF Frequencies As would be expected, the decrease in the number of CHIEF calculations results in a large reduction in overall computation time. Fig. 2 shows the central processing unit (CPU) times required on a CONVEX CS-1 minisupercomputer for 100 frequencies and 91 far-field angles as a function of maximum. The top curve shows the CPU time required for the acoustic calculations with no frequency interpolation ( =.01). The next lower curve gives the total (acoustic, finite-element,

9 ACCURACY OF FREQUENCY INTERPOLATION J - F MAG (FORM FUNCTION) Figure 1: Backscattering from spherical shell utilizing frequency interpolation ACCURACY OF FREQUENCY INTERPOLATION J - F MAG (FORM FUNCTION) Figure 2: CPU time versus with and without frequency interpolation

10 and coupling) CPU time required when =0.2. The remaining curves show the CPU times for various parts of the calculation. With the new interpolation scheme, the time is dominated at low by the portion of the program which couples the CHIEF and finite element results. At higher the finite element computations dominate. In general, the frequency interpolation scheme has resulted in an overall reduction in computation time of about 15:1. The second example involves the backscattering off the end of a capped cylindrical shell in water. The ratio of the mean radius a to the length L of the shell is 0.125, and the ratio of the thickness h to the mean radius a of the shell is The ends of the cylinder are capped with circular disks having the same thickness as the shell. The material properties of both the shell and endcaps are Density = 8977 Kg/m 3 Young s modulus = N/m 2 Poisson s ratio = Again the density of the fluid medium was taken to be 998 Kg/m 3 and the sound speed was taken to be 1486 m/s. Fig. 3 shows the backscattered normalized pressure for an incident plane wave travelling along the axis of the cylinder. The far-field scattered pressure is normalized by the high frequency plane-wave approximation to the rigid backscattered pressure. The plane-wave approximation involves setting p s = ρcv s on the surface where p s and v s are the scattered pressure and normal velocity respectively. For a rigid body the scattered and incident normal velocities on the surface are related by v s = v inc. Thus, the plane-wave approximation reduces to setting p s = ρcv inc on the surface. The far-field scattered pressure can be obtained from the approximate values of p s and v s on the surface. The solid curve represents computed values at increments of with no interpolation. The dotted curve (which is almost indistinguishable from the solid curve) corresponds to CHIEF computations at increments of 0.4 with interpolation to increments. This curve represents CHIEF computations at only 6 values out of a total of 71. There is essentially no loss of accuracy in this frequency range for this degree of interpolation. It should be noted that the final frequency resolution is not limited to the value = used in this example. Interpolation can be performed to any desired frequency resolution without any further CHIEF computations. Fig. 4 illustrates the importance of scaling the components of the CHIEF A and B matrices by e ikrmn. The solid curve again represents calculations at increments of with no interpolation. The dotted curve represents increments of 0.4 for the CHIEF computations with interpolation performed on the A and B matrices directly without any scaling. It is clear that the agreement is not nearly as good as it was for the scaled interpolation case shown in Fig Analysis of Output The computational technique described in Section 3 usually produces a large amount of output data. Therefore, it is very important to develop methods which simplify the analysis of these data. In this section, several ways of visualizing and analyzing the output data to aid in the interpretation of the results will be described.

11 BACKSCATTERING FROM END OF CYLINDER 10 0 No Int 0.4 Int 8 7 Normalized Pressure Mag Figure 3: Backscattering from capped cylinder, CHIEF increment of 0.4 BACKSCATTERING FROM END OF CYLINDER 10 0 No Int No Scale 8 7 Normalized Pressure Mag Figure 4: Effect of scaling on frequency interpolation

12 4.1 Spectral and Spatial Descriptions of Far-Field Pressure The output of greatest interest in structural-acoustic scattering problems is usually the far-field scattered pressure or some quantity directly related to it such as the form function or target strength. The far-field pressure is computed using Eqs. (10), (13), and (14). It is, of course, a function of the material, fluid, and dimensional parameters of the problem, but usually we are most concerned with a spatial and spectral description of this result. Thus, it is sufficient to write the functional dependency of the normalized far-field pressure as p s = p s (x inc,x obs,ω) (43) Furthermore, for plane-wave incidence and far-field observation, both x inc and x obs is independent of distance from the scatterer and can each be defined by the spherical angles θ and φ. Thus, we are usually interested in plotting the scattered pressure as a function of some angle at a fixed frequency (a spatial description, or directivity pattern) or as a function of frequency at a fixed angle (a spectral description). Typical spectral and spatial plots for the previously described spherical shell example are shown in Figs. 5(a) and 5(b). It should be noted, however, that such plots are very sparse descriptions of the far-field pressure which can vary rapidly with both spatial and spectral parameters. When the computational effort to obtain a high resolution description of the scattering has been expended, it is essential to try to visualize the complete output data set. Fig. 5(c) shows a surface which is a combined spatial and spectral description of the variation of target strength for a fixed incidence angle. This plot contains nearly 50,000 calculated points (331 frequencies by 145 angles). Fig. 5(a) represents just one horizontal cut or slice through this surface, and Fig. 5(b) represents one of many possible vertical cuts in the surface. Surfaces such as shown in Fig. 5(c) have proved to be an excellent tool for visualizing the global features of the scattering response and for comparing the response of one object with the (calculated or measured) response of another object 1. Once the important features have been identified, then specific slices can be produced as line plots to illustrate the quantitative variations. Other graphical variations of this technique are shown in Fig Surface Pressures and Normal Velocities In addition to the far-field description of scattering, it is often useful to know the variations of acoustic pressure and velocity on the surface of the scatterer. In the method described in this chapter these surface quantities are determined as part of the process of generating the far-field description (see steps 7 and 8 of Section 3.1). In principle, the (total or scattered) pressure and the complete vector velocity is known at all points on the surface of the scatterer. In practice we usually seek to plot the spatial and spectral variations of these quantities to better understand the physical mechanisms and parametric sensitivity associated with the pressure and velocity. Once again we have a challenging visualization problem to present all the data that are generated. For the same spherical shell as described previously, two different views of the total complex surface pressure are presented in Fig. 7. In Fig. 7(a), we see the variation of the real and imaginary parts of the surface pressure as a function of the projected distance along the axis of the sphere which is aligned with the incident direction, for a particular frequency or near the first resonance of the fluid-loaded shell. The frequencies of maximum response are illustrated in Fig. 7(b) which is a plot of the total complex surface pressure as a function of frequency () at a specific location on the sphere, namely the pole where the incident wave first impinges on the sphere. Many such plots would be needed to characterize completely the spatial and spectral variations of these or other surfacerelated variables. As in the case of the far-field quantities, we can capture the global variations in 1 In this book all surface images are rendered using a gray scale; however, the interpretation of features can be enhanced by using a color scale.

13 Polar angle Target strength 40 db 20 (a) Spectral cut (c) Combined spatial and spectral description 40 db max, 10 db/ring (b) Spatial cut Figure 5: Spatial and spectral descriptions of the target strength of a spherical shell (a) (b) Figure 6: Alternative graphical representations of target strength.

14 a semi-quantitative way by plotting these complex quantities as a surface which represents both the spatial variations (along the abscissa) and the spectral variations (along the ordinate). Fig. 8 shows the real and imaginary parts (magnitude and phase would of course be an alternative representation) of the total normal velocity. Both plots are scaled to the same maximum and minimum values. One can see that the variations are relatively mild over much of the spatial and spectral ranges, but the variations are large and rapid near the resonance frequencies. 4.3 Acoustic Intensity and the Effect of Structural Damping Since we have representations of the complex surface pressures and normal surface velocities at each point on the scatterer, we can compute the ratio of the acoustic intensity I(x, ω) to the incident plane-wave intensity I inc as a function of frequency in the following manner: I(x, ω) = ρc Real[p(x, ω)v (x, ω)], (44) I inc where the superscript v denotes the complex conjugate of v. At any particular location or frequency, the intensity ratio may be positive or negative, indicating whether power is flowing out of or into the scatterer. However, in the absence of structural losses, the total power flowing into or out of the object must vanish. In Fig. 9(a) we compare the average intensity ratio (total power divided by the surface area of the scatterer and by the incident intensity) to the maxima and minima of the intensity ratios over all locations on a spherical shell under the assumption that there are no material losses in the shell. One can see that, in spite of very large spatial and spectral fluctuations, this average value is essentially zero (it is actually 4-5 orders of magnitude smaller than the peak intensity in this specific numerical example). This is a good test of the accuracy of the numerical results in the sense that, unlike the far-field scattered pressure, the average intensity is sensitive to the local evanescent variations of surface pressure and velocity. 2 When a small damping coefficient of 2% is attributed to the shell material, the results are as shown in Fig. 9(b). Note that the local variability of the intensity ratio is reduced, and that the average intensity ratio is no longer zero at all frequencies, but takes on a negative value proportional to the damping coefficient near the resonance frequencies. These negative values indicate that power is being dissipated in the shell. The global effect of adding material damping to this coupled structure-acoustic problem is illustrated in Fig. 10. The spatial and spectral variation of intensity is displayed in Fig. 10(a) for an undamped spherical shell whose vibrational amplitude is solely limited by acoustic radiation at its resonance frequencies. When a small amount (2%) of damping is introduced in the shell material, the result is shown in Fig. 10(b). Note that the higher frequency resonances have completely disappeared and that, although their amplitude is slightly diminished, the first few resonances are still strong indicating that the acoustic radiation damping is still dominant. In a material with high losses, even these peaks would be obscured. 4.4 Mode Identification The frequency behavior of the scattered acoustic field produced by a plane wave impinging on an elastic body immersed in an infinite fluid medium is dominated in certain frequency ranges by large peaks arising from resonant modes in the elastic structure. A technique for computing the scattered acoustic field was described in the previous sections. It will now be shown how the above technique canbeusedtoidentifywhichin vacuo structural modes are the main contributors to each peak in the 2 The idea of using this relation as a check on the accuracy of the numerical computation is due to Earl Williams, of the Naval Research Laboratory.

15 Solid line = real part Dashed line = imaginary part Solid line = real part Dashed line = imaginary part Surface pressure Surface pressure Normalized axial distance (a) Surface pressure vs. axial distance (b) Surface pressure vs. frequency Figure 7: Surface pressure on a spherical shell. Axial distance (a) Real part of surface velocity Axial distance (b) Imaginary part of surface velocity Figure 8: Spatial and spectral variations of surface velocity on a spherical shell.

16 Intensity Intensity (a) No damping (b) Damping coefficient = 0.02 Figure 9: Comparison of local and average acoustic intensity on the surface of a spherical shell. Axial distance (a) No damping Axial distance (b) Damping coefficient = 0.02 Figure 10: Spatial and spectral variation of acoustic intensity on the surface of a spherical shell.

17 scattered field frequency response [8]. This will be accomplished by developing a modal expansion for p s (x). Each column of E represents an in vacuo mode of the structure. If e n is the nth column of E, then Eq. (22) can be written in the alternate form ( ω 2 M + K) 1 = Combination of Eqs. (17) and (45) gives U = M ( e T m CDP m=1 In view of Eq. (19), it follows that V = iω ω 2 ω 2 m M ( e T m CDP m=1 ω 2 ω 2 m Substitution of Eq. (46) into Eq. (15) gives p s (x) p rs (x) =iω M m=1 ) e m 1 ω 2 m ω2 e me T m. (45) M ( e T m F D m=1 ) C T e m iω m=1 ω 2 ω 2 m M ( e T m F D m=1 ω 2 ω 2 m ) e m. (46) ) C T e m. (47) M [ e T m (CDP F D ] ) ω 2 ωm 2 q T (x)c T e m. (48) The nth term of the sum in Eq. (48) is the contribution of the in vacuo mode e n to the scattered field. Once Eq. (24) is solved for P, the contributions of various modes can be determined by plotting individual terms or partial sums of Eq. (48) versus frequency and comparing the results with the plots of p s (x) p rs (x) versus frequency. The mode identification technique will now be applied to the same sphere problem described in Section 3.3. Fig. 11 shows the contribution of the second mode of the sphere to the magnitude of the form function. The dotted curve is the sum of the rigid scattering form function and the form function obtained by using the m = 2 term of Eq. (19). It should be noted that the eigenfrequency of mode 2 in vacuo occurs at a of 2.55, whereas its maximum contribution to the overall response occurs at a of about 1.58 due to the mass-like loading of the acoustic field. Fig. 12 shows a similar result for the fourth structural mode. In fact, each of the peaks shown in the overall response is primarily due to a single (higher frequency) in vacuo mode of the spherical shell. This simple situation is not usually observed in the scattered field produced by other shapes where peaks are often due to the interaction of several structural modes. A more complete discussion of this technique can be found in reference [8].

18 BACKSCATTERING FROM SPHERICAL SHELL all modes + RS mode 2 + RS 10 MAG (FORM FUNCTION) Figure 11: Contribution of mode 2 to backscattering from spherical shell BACKSCATTERING FROM SPHERICAL SHELL all modes + RS mode 4 + RS 10 MAG (FORM FUNCTION) Figure 12: Contribution of mode 4 to backscattering from spherical shell

19 5 Conclusions A numerical method for solving coupled structural-acoustic radiation and scattering problems has been presented. The accuracy of this method has been validated for a spherical shell by comparison with an analytic series solution. It has been shown that computational efficiency can be greatly increased by taking account of any symmetries present in the problem and by using frequency interpolation on certain intermediate acoustic quantities. Several useful ways for displaying the large amount of output data were presented. In addition, a mode identification scheme was presented which can aid in the interpretation of the results. References 1. Schenck, H.A., Improved Integral Equation Formulation for Acoustic Radiation Problems, J. Acoust.Soc.Am.,44, pp , Benthien, G., Barach, D., and Gillette, D., CHIEF Users Manual, NOSC Technical Document 970, revision 1, September Zienkiewicz, O.C., The Finite Element Method, Third Edition, McGraw-Hill, London, Chapter 3, Businger, P., and Golub, G.H., Linear Least Squares Solutions by Householder Transformations, Numerische Mathematik, 7, pp (1965). 5. Benthien, G., Numerical Computation of Asymmetric Scattering From Axisymmetric Bodies Using CHIEF, NOSC Technical Note 1602, November Benthien, G., Application of Frequency Interpolation To Acoustic-Structure Interaction Problems, NOSC Technical Report 1323, November Junger, M.C., and Feit, D., Sound, Structures, and Their Interaction, Second Edition, The MIT Press, Cambridge, p.354, Benthien, G., Contribution of Individual Structural Modes to the Scattered Acoustic Field, NOSC Technical Report 1329, November, Acknowledgements The authors would like to thank Don Barach and David Gillette of the Naval Ocean Systems Center for converting the techniques described in this paper into efficient computer programs and for obtaining the included numerical results. This work was sponsored by the Office of Naval Technology through the RL3A/01 Block Program on ASW Support Technologies at the Naval Research Laboratory.