rical methods for pre r orrnance

D.J.W. Hardie A.B. Gallaher

Indexing terms: Boundary-element method, Finite-element method, Sonar arrays, Sonar rransducers

Abstract: Sonar systems are becoming increasingly more elaborate and complicated as greater sensitivity and noise rejection are demanded. Consequently modern sonars are expensive and difficult to construct. This is particularly so for the actual acoustic array. The importance of providing accurate performance prediction is obvious. The review considers numerical schemes for this task and assess their suitability for specific applications. Particular attention to finite-element and boundary-element methods is made. Methods suitable for active and passive arrays are considered. Possible potential improvements in accuracy and computational effort are discussed.

I

1 Introduction

There is a continuing need for improved performance in the application of sonar systems to underwater acoustics. Both active and passive sonar arrays are required to be ever more sophisticated with increased capacity for discrimination. This makes great demands on designers. With the emphasis on reducing expensive prototyping costs the need for reliable modelling of sys- tem performance is manifest. When considering com- plicated design issues with many stringent and potentially conflicting constraints, resorting to several initial trials can be prohibitive. These questions must be resolved using accurate predictive methods.

This review is generally restricted to finite-element and boundary-element methods which have shown great utility in calculating the dynamical response of general elastic structures. No attempt is made to dis- cuss in detail the governing equations pertinent to a particular method. The reader is referred to the exten- sive references cited in this text for specific mathemati- cal details. We do, however, endeavour to highlight certain aspects of a particular modelling technique for the benefit of a nonspecialist.

Attention is primarily focused on predicting quanti- ties critical for the acceptance of a sonar array, namely, 0 Crown Copyright 1996 IEE Proceedings online no. 19960424 Paper first received 23rd October 1995 and in revised form 12th February 1996 The authors are with the Sonar Systems Department, DRA, Winfrith, Dorset DT2 8XJ, UK

projector or receiver sensitivities and directivity beam patterns. This requires a fully coupled method yielding a relation between the electrical and acoustic quantities. Of course, other system parameters are important (e.g. projector depth dependence) and the methods outlined here may be of use in their prediction.

2 Theory and methods

Within the confines of steady infinitesimal linear motion the partial differential equations governing the dynamic behaviour of underwater acoustic projectors and receivers are well known [l]. There are many meth- ods for their numerical solution over the region of interest or domain. Partial wave analysis and related methods (e.g. T-matrix approaches) solve partial differ- ential equations using separation of variables. The Helmholtz equation with its associated Sommerfeld radiation condition is tractable by these means [2]. These are exact procedures provided accurate represen- tation of the eigenfunctions specific to the particular geometry under consideration is assured. Sufficient terms must be included to achieve a converged solu- tion. Coupled elastoacoustic problems can be addressed, at least for simple structures [3]. Separation of variables is restricted to specific geometries. Meth- ods exist for overcoming these restrictions at the cost of adopting more complicated numerical procedures involving nonorthogonal functions. These methods have proved to be very useful for scattering problems [4]. In the case of acoustic radiation from active arrays they have been used to a limited extent. For simplicity usually a Neuman-type boundary condition (specified surface normal velocity) is imposed, structural coupling being ignored [5]. This may be sufficient for simple directivity predictions. For general array applications these methods have been largely superseded.

A great deal of useful transducer and array design work is based on simple lumped parameter methods [6, 71. These do not attempt to solve any general field or potential problem but attempt to map the dynamics to a relatively simple equivalent circuit. To describe the behaviour of more complicated projectors over a wide operational frequency band requires an elaborate sys- tem of circuits which requires careful consideration in its fabrication [8]. More refined approaches exist. Those based on transmission line theory have proved to be accurate and straightforward in their implementa- tion [9]. When considering arrays these methods can become unwieldy but if the projectors have simple behaviour adopting a lumped parameter model can be fruitful. Lumped parameter and related methods are

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specific to known designs and their application to novel transducers is not straightforward. Nevertheless their inherent capacity to aid physical understanding will ensure that these methods will continue to be popular.

Finite-element methods have been applied success- fully to a number of transducer design problems [ 10-1 21. Indeed, the development of piezoelectric finite elements is as a result of demand by sonar designers [13-1.51. Using these elements a fully coupled approach is possible. Admittance loops can be predicted and more elaborate methods of assessment can be per- formed. Reliable material data are necessary to get accurate results when using piezoelectric elements. Depending on the age of the ceramic, it may be appro- priate to use degraded values [16]. The straightforward finite-element method is restricted to finite regions in space. This is not a difficulty when the structure con- tains internal fluid regions only or when the structure is immersed in a tank. Here simple acoustic fluid finite elements suffice [17]. These give rise to positive definite matrices that are frequency independent allowing for modal analysis. The matrices are banded owing to the local nature of the method; only neighbouring elements interact. These properties allow fast inversion, hence quick solution times.

To account for a surrounding fluid of almost infinite extent requires the correct acoustic radiating boundary condition to be satisfied. Simply extending the mesh ever further outward is incorrect as well as being com- putationally expensive. Work has been done to develop so-called ‘infinite finite elements’ with a radially decay- ing frequency dependent function [ 181. These are attached externally to the acoustic finite-element mesh and simulate the correct frequency dependent boundary condition at infinity. Another method is to impose external damping elements around the fluid mesh with a distribution dependant on important terms of a multipole expansion [19]. This method is very efficient, particularly in predicting near-field pressures but requires extra computation to determine the boundary pressures necessary to calculate any far-field results [20]. Establishing the correct Sommerfeld radiation condition at the boundary by employing spherical har- monics is an equivalent procedure relating the pressures and normal velocities at the mesh boundary [21]. The size and shape of the acoustic fluid mesh is critical to these methods. Generally a sphere or spheroid is con- structed. The radius of the sphere usually corresponds to the edge of the nearfield -Dlh2 (A denotes wave- length). Hence for large aperture (D) linear arrays and high frequencies the acoustic mesh rapidly becomes excessive. Careful meshing is necessary in the presence of simple sources owing to the strong l l v singularity. In their basic form these purely finite element schemes can give rise to results that are origin dependent. This is not a difficulty for radiation problems. Much develop- ment work has been done based on the above and use- ful codes are available (e.g. ATILA [22] and MAVART [23]). Their application to passive arrays has been lim- ited, however

The boundary-element method converts the associ- ated integral equation into a set of algebraic equations relating points to others on the domain surface only [24, 251. Thus a 3D problem is reduced to an equivalent 2D one. The method approximates the surface by a contiguous assembly of boundary elements or patches. The integral equation is then evaluated for each patch

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determining its influence on itself as well as on all the others. The first of these gives rise to a singular inte- gral. After a suitable choice of shape functions a sys- tem of matrix equations arises. The form of these matrices differs from those obtained from the finite-ele- ment method. They are dense, complex and frequency dependent and do not have the form to lend themselves to an efficient solution method. The procedure solves for unknown quantities at the surface only. All other results off this surface require postprocessing of the solution via a surface integration.

The integral equation describing the surrounding acoustic fluid is the Helmholtz equation. Here the Green’s function implicitly describes the influence of the surrounding infinite fluid. Unfortunately, the exter- nal Helmholtz integral equation fails at certain critical frequencies [26, 271. These frequencies are the eigenval- ues of the internal Dirichlet problem and their number increases as the frequency of interest is raised. This is a deficiency of the integral representation rather than the physics resulting in a lack of a unique solution. This gives rise to ill-conditioned matrices. This uniqueness problem can be overcome in several ways. The oldest of these methods is the CHIEF method of Shenck [28] which introduces extra interior points and yields an overdetermined system of matrix equations to be solved. The selection of the interior points is somewhat arbitrary and care must be taken, Improved CHIEF methods have been developed which try to ensure that the choice of interior points is optimal. More elegant methods exist.

The method of Burton and Miller [29] has received much attention. They proposed solving a linear combi- nation, with arbitrary complex coefficients, of the acoustic integral equation and its normal derivative form. This involves evaluating integrals with a hyper- singularity. Various ways for evaluating these integrals have been reported [30-321. In its original form the Burton and Miller method is unsuitable for use with (the commonly used) quadratic elements or patches and resort to constant pressure patches is the common recourse. For quadratic patches the nodes are located at the element boundary and the surface normal may be discontinuous across element boundaries. Formula- tions which resolve this difficulty have been proposed [33, 341. The results should not be dependent on the choice of the complex coefficients required for the lin- ear combination. However, the user must be aware of any possible sensitivity of his results to changes in these values. Some recommendations for a suitable choice are available [35].

The boundary-element method is applicable to uncoupled external acoustics, both radiation and scat- tering. It has been applied to structural dynamic prob- lems and a coupled elastoacoustic formulation based purely on boundary elements has been proposed [36]. However a more attractive scheme is to adopt a finite element description of the structure and a boundary element formulation for the surrounding infinite acous- tic fluid. The two methods are coupled at the fluid- structure surface via the continuity relations. Detailed descriptions of this approach are given in [37], suffice to say that the fluid-variable methodology as recom- mended by Mathews [38] and independently by Wilton [39] is the most suitable from an acoustic array per- spective.

These procedures are exact, within the terms of the numerical impllementation. They are low to moderately

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high fi-equency methods. It is important to realise that they require considerable computational effort which increases markedly as the wavelength becomes much shorter when compared to a typical surface length scale L. An important factor is the degree of fineness of the surface mesh. As a guide the authors recommend that the minimum number of boundary element degrees of freedom over a surface (of area A ) be k2A, while along a length there should be kL collocation points. This is applicable to both boundary-element and finite-element meshes. Here k is the acoustic wavenumber. A similar criterion exists for the minimum number of terms in a partial wave expansion [40]. The size of the resulting matrices from a coupled boundary-element and finite- element approach is a crticial consideration when solv- ing a practical problem numerically. The time for their construction is of the order of the number of surface degrees of freedom squared and their inversion time during solution is proportional to that number cubed. For moderate jobs the matrix construction time is dominant. This must be done at each frequency consid- ered. Interpolation of the frequency dependent matrices should be beneficial when considering problems with wide bandwidths.

Provision for direct coupling between acoustic boundary elements and acoustic finite elements can reduce the critical surface area by filling cavities and smoothing out irregularities that may be present in the actual surface of the structure. Hardie [41] has pro- posed a combined finite-element and boundary-element method exploiting an approximate formulation of the Helmholtz integral equation, the doubly asymptotic approximation (DAA2c) of Geers [42]. The idea is to use the DAA2c boundary element as a general bound- ary condition for the finite element mesh akin to the pure finite-element approaches. Here no specific shape of finite-element mesh is needed, except that it be con- vex. The requirement to describe the whole of the array’s nearfield is also relaxed. Only a small acoustic finite element mesh is needed. For an active array of test rings no distinction is seen with an exact treatment [43]. Further advantages over conventional boundary element approaches are that the resulting matrices have a simple frequency dependence requiring to be con- structed only once and that no critical frequencies exist. Thus a wideband analysis can be performed very quickly. This is of particular interest when considering fine frequency steps as in predicting admittance loops. This method needs further investigation and possible refinement to put it on a rigorous foundation [44].

We now consider the application of these methods to sonar array design problems. There are different prob- lems associated whether the array is active or passive. Some numerical methods are more appropriate for one or other array type although most have general appli- cability.

3 Active arrays

Active arrays radiate acoustic energy and the quality of the sound field is characterised by its directivity and source level. Their operational bandwidth is usually limited. Generally only a few acoustic projectors are involved as individual sources are expensive and of appreciable size. The small number of structures and low relative frequency of operation (typically, kL < 5 ) makes the modelling of complete active arrays numeri- cally feasible. Whether there are strong interaction effects present is of fundamental importance to the

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function of an active system. This is increasingly an issue of importance as greater emphasis is made towards low frequency and therefore long-range sys- tems. Sonar arrays are constrained to be attached to or be deployed from the platform vessel. The arrays are made to be as compact as possible. Consequently the projector separation within the array may be signifi- cantly less than a wavelength at the lowest part of the operating frequency band.

Piston stack Tonpilz projectors (Fig. 1) are com- monly used acoustic sources [6, 451. Their simple oper- ation enables successful modelling using a variety of techniques. Much work has been done using finite-ele- ment techniques to describe the behaviour of Tonpilz transducers. The compliance of the stack dominates the resonant character of a single device and care must be made to include the effects of the joints between the active ceramic pieces [46]. When deployed in an array, these devices normally require a baffle to perform effi- ciently. This increases the computational cost. If the baffle is rigid, planar and of large extent this can be reduced by using a combined finite element and bound- ary element approach with appropriate symmetry con- ditions. Good agreement with an exact result is seen for a single piston in an infinite rigid plane [47]. The structural part of the problem can be reduced further by employing a mixed method [48] adopting a trans- mission line approximation for the Tonpilz stack. Audoly [49] used an coupled boundary-element approach to quantify the interaction between adjacent Tonpilz transducers in a planar rigid baffle of finite size. A full finite-element treatment of the transducers allowed for consideration of the piston head velocities when a constant voltage is applied to the array. The nearfield is strongly affected by mutual interaction for closely packed arrays with spacing -114, but the direc- tivity of the beam appears to be less sensitive. Only by adopting a fully coupled method can these issues be addressed adequately.

e n d brass cap

Fig. 2 Barrel slave Jextensionul acoustic projector

IEE !‘roc.-Radar, Sonar Navig , Vol 143. No. 3, June 1996

Fig. 3 Cluss IV flextensionul ucou,rtic projector

A novel multihead array of Tonpilz projectors arranged in a cylindrical fashion has been studied by Bernard et al. [SO]. The array is self baffling with the projectors in free flood. Different configurations based on different spacings and with or without air-filled compliant tubes The pure finite element code ATILA was used to predict the array’s performance when amplitude or phase shading was imposed The resulting mesh is very complicated even when exploiting symme- try to the full. A large radius part-sphere was con- structed of acoustic finite elements. Calculations show good agreement with measurement with recommenda- tions for maximising the source level.

Flextensional projectors are popular sources for active arrays. Many successful types of flextensional exist such as the barrel-stave designs and the more common class IV configurations (Figs. 2 and 3) [51, 521. The operation of flextensionals is based on the flexing of a thin shell of simple form. Adopting a finite- element approach is therefore attractive. Indeed most theoretical work on flextensional projectors has been based on this technique. Hardie [S3] considered the action of hydrostatic pressure on the in-water shell dynamics of a single flextensional using the finite- element code ABAQUS [S4]. Good agreement with observation was seen. However success has been achieved using other techniques. Nelson and Royster [5S] use the finite-difference method in their analysis of in-air shell response. Lipscome has developed a lumped-parameter approach for investigation of generic issues which can give useful design information with very low computational effort [S6]. Brind [57] accurately predicted the in-water resona,nce frequencies of a single class IV device using a combination of the finite-element method to describe the transducer and a boundary-element formulation based on a form of the DAA approximation for the water loatding. Here the fluid is assumed to be effectively incompressible or equivalently that the structure is much less than an acoustic wavelength. This is the case for this type of flextensional device which offers a high power acoustic source that is compact and self contained.

Arrays of flextensional projectors have been mod- elled. The compact nature of the projectors makes them behave as almost simple sources, at least around the first resonance. For certain array design questions

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treating the devices as acoustic monopoles may be suf- ficient. However for detailed analyses (e.g. quantifying interaction effects) recourse to fully coupled numerical methods must be made. Rynne and Gillete [58] con- sider an array of flextensionals for an application requiring low power but wide bandwidth. The rela- tively high frequencies of interest (kL -1 0) demand that the directional nature of the higher acoustic modes be accounted for. Including interaction implicitly in the method gave rise to a significant computational task which was reduced by adopting a simplified 2D model. Densely packed flextensional arrays have been investi- gated [S9]. Here the objective was to assess the strength of the mutual interaction between class IV projectors in close proximity and in different degrees of alignment. The flextensional transducers were fitted with relatively light glass fibre wound shells. A study of the effect of shell damping due to the rubber seals was necessary to achieve representative results. For other classes of flex- tensional device, the coupling between the shell and the rest of the structure needs careful attention to correctly describe the dynamics [60].

Fig. 4 &gmented,fiee$ooded ring projectov

Free-flooding rings are attractive acoustic sources for high-power active systems (Fig. 4). Simple theoretical descriptions of their dynamical behaviour are available, at least with regard to classifying the various acoustic resonances 1611. Sherman and Parke [62] used a partial wave approach with Neuman boundary conditions to derive directivity beam patterns for line arrays. Despite these successes it is necessary to resort to numerical means to predict completely the performance of even a single ring [63]. Gallaher [64] predicted the acoustic field radiating from a line array of radially poled piezo- electric rings using a combined boundary-element and finite-element method incorporated in the PAFEC code 1651.

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Comparing with the series expansion of Sherman and Parke for the directivities revealed great disparities away from resonance. Experiment showed somewhat better accord with the calculations of Sherman and Parke for compact arrays with less than XI2 spacing. This is surprising as Sherman and Parke ignore any mutual interaction between the rings. The actual rings are waterproofed with a polymer coating and this appears to dampen the motion and hence reduce mutual interaction. Accounting for structural damping by a simple term in the PAFEC model resulted in excellent accord with measurement both for source lev- els and beam patterns. The structural damping can be introduced explicitly by using viscoelastic finite ele- ments to describe the polymer coating [66]. The undamped PAFEC results indicate that, unlike a baf- fled Tonpilz array the directivity of a compact free- flooding ring-line array can be strongly affected by mutual interaction effects.

steel insert

electrodes

Fig. 5 Mesh fur single segmentedfree-flooded ring projector

Investigating the acoustic behaviour of segmented rings continues. Their large size requires an elaborate structural design which gives rise to potentially compu- tationally intense models. Bonin et al. [21] calculated the projector sensitivity of a single large segmented free-flooding ring (Fig. 5) . They compare numerical schemes based on a pure finite-element method (MAVART) and the combined boundary-element and finite-element method (PAFEC). The MAVART model is axisymmetric using generalised piezoelectric elements and effective properties while the PAFEC mesh is 3-D exploiting full symmetry. Little difference between the sets of results is apparent over the main frequency band. Similar levels of agreement are seen in compar- ing different methods on other segmented ring designs [67]. To our knowledge, full 3-D analyses of arrays of rings exceeding two or more have been attempted only by a combined boundary element and finite element method.

By way of illustration, we present calculations of the projector sensitivity of a four-ring line array of large segmented projectors using the PAFEC code. A fully coupled finite-element and boundary-element method was used. Owing to the relative thinness of the rings no

critical frequencies were to be expected over the band of interest (0.5-3.OkHz). A full 3-13 mesh based on a single 9" segment was constructed for each ring in the line array (Fig. 5) . Only the crucial part of each projec- tor was described, all unnecessary features being ignored. Constant-pressure boundary elements sur- round the finite-element structural mesh comprising piezoelectric elements. The acoustic mesh is over-fine in the circumferential direction owing to the requirement of a one-to-one coupling between the boundary-ele- ment patches and the finite-element surfaces. This con- straint can be relaxed using a more general coupling scheme allowing for different levels of discretisation in the fluid and in the structure. The acoustic field was evaluated in the broadside and endfire directions for a constant-voltage input yielding the radial and axial projector sensitivities. Good agreement with measure- ment is seen (Fig. 6) despite the fact that the experi- ment could not achieve true free-field conditions at the lowest frequencies. A further feature of these iiuinerical methods is the ability to evaluate dynamic stresses within the structure. There is a potential area of tech- nological risk when considering segmented rings driven at high power and numerical methods can help assess this by predicting stresses for a given frequency and array configuration.

E 7

$1001 , , I 4 I ' j c I 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0

frequency, kHz

Fig.6 rings a Damped PAFEC results (radial) b Damped PAFEC results (axial) c Experimental results measured in near field rig (radial)

Projector sensitivity for uctive line urruy uf four pee-flooding

4 Passive arrays

Passive sonars are designed to have the greatest possi- ble array gain. They are usually composed of very many identical receivers arranged in a regular fashion. Each of these individual hydrophones is usually of sim- ple form being much smaller than a typical wavelength. Finite-element methods have proved to be of value in predicting the dynamic response and sensitivity of indi- vidual pressure hydrophones including piezoelectric ceramic sensors and novel optical fibre devices [68]. The behaviour of passive acoustic velocity and pressure gradient hydrophones with inherent directivity can be accurately described using boundary element tech- niques [69]. Passive arrays operate over a wide band. An array may be very many wavelengths in extent even at the lowest frequency of interest (kL can exceed 20). This can present significant computational difficulties in considering explicitly a whole array. Most treat- ments using analytical methods adopt simplifying assumptions such as taking the array to be effectively very long in one or more of its dimensions. These can

IEE Proc.-Radar, Sonar Navig.. Vol. 143, No. 3, June 1996 200

be used to advantage in reducing the complexity of more elaborate numerical schemes.

Somewhat paradoxically, the pure finite-element method can be applied to consider strucl-ures of infinite span with regular spatial periodicity [70]. Here invok- ing Bloch’s theorem, the problem is reduced to consid- ering only a single unit cell containing a single hydrophone module. Boundary-element methods based on similar ideas exist [71]. Owing to the effectively infi- nite domain of the structure’s surface 110 problematic critical frequencies are present. Consideration of a fully coupled approach allows for the prediction of array sensitivity to incoming acoustic radiation. The compu- tational effort can be reduced when adopting such a scheme for estimating array response by simply evalu- ating the harmonic displacements or stresses at the hydrophone sites, ignoring the physical presence of the hydrophones. For practical cases and at low frequen- cies this may be a useful approximation but arrays with significant amounts of stiff ceramic may require a more elaborate treatment. Here the active piezoelectric mate- rial must be included explicitly.

0 10 20 30 40

i

50 60 70 80 90 incident angle.deg

unit cell

I

Fig. 7 H = hydrophone

Reduction of large passive array

To demonstrate this technique, con,sider a simple passive planar array consisting of rectangular plates of ceramic embedded in a polyurethane matrix. This is considered to be consisting of an infinite number of regularly spaced units (Fig. 7). The transmission loss and reflectivity of the array are derived for a given inci- dent direction and frequency. These factors indicate whether a fully coupled calculation is appropriate. Pie- zoelectric coupling yields a voltage response at the hydrophones (Fig. 8). This is compared to that derived for a single omnidirectional hydrophone. The presence of fluid below the array represents an ideal anechoic baffle fitted behind the array. The characteristic (sin x)l x response due to the finite size of the plates is well reproduced. This reduction of the receiver array prob- lem to that of a single unit is attractive and its use is recommended if possible. Large-scale planar, linear and cylindrical arrays should benefit from this approach. The finite size of the array can be accornmodated by a convolution with a suitable spatial Fourier transform.

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Not all passive array designs can be treated so conven- iently and an explicit approach may be obligatory. A fully conformable hull mounted array of generally curved shape will need careful consideration if the modelling is to be computationally feasible!

\ I I &-40

2 -50

Hull mounted passive arrays require acoustic baffles to increase self noise rejection. The dynamic and hydrostatic behaviour of the baffle can be involved with many competing mechanisms depending on its design and construction. Finite-element methods pre- dict the gross behaviour of even complicated baffles well [72]. The potential for accurate estimation of self noise rejection using this approach is clear.

5 Further aspects

Sonar arrays can be fitted with acoustically transparent structures or domes. Their function is to isolate the array from the flow past the vessel. In passive sonars, flow noise can be a severe limit to the array’s ability to detect acoustic sources and the high wavenumber com- ponents of the flow noise are attenuated by the stand- off distance provided by the dome. For high frequencies the refraction due to the solid dome can disrupt array directivity response. A solid dome cannot be made absolutely transparent. A coupled Gnite-ele- ment and boundary-element calculation can be used to determine the sound field at the hydrophone sites [73] . It is important to note that a thin-shell finite-element description of the dome structure is not generally suffi- cient. To include all possible structureborne waves a full elastic description is necessary. When considering interaction effects in a domed array wherein elastic effects couple the projectors a more elaborate numeri- cal scheme has been proposed [74]. Good results are achieved with a combined 2D and 3D finite-element and boundary-element approach.

Potentially optimisation procedures can be used to aid array design. A number of commercial codes con- tain some capability to optimise designs. The optimisa- tion process is described in terms of design variables, which are the input parameters subject to change (e.g. shell thickness, elastic modulus, etc.) and response vari- ables which are chosen to assess the design. Resonance frequency and overall weight are possible responses. The whole procedure attempts to maximise the multidi- mensional evaluation function having formulated an initial design. Obviously the closer the initial design to

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an agreed optimum the better. These facilities enable the transducer engineer to automate his design proce- dures.

More specifically, proposals for optimising acoustic performance of both active and passive arrays have been investigated [75]. Macey [76] has proposed a pro- cedure using a combined boundary-element and finite- element method to design a volumetric active array with the maximum possible source level over a selected frequency band. Here the variables are the relative dis- tances between projectors. Designing receivers with a desirable directional discrimination has been demon- strated using a similar numerical approach [77]. These methods are computationally intense and require care- ful consideration by the sonar engineer to see whether the design issues can be resolved by simpler means and good judgment.

The introduction of new active materials promises greater performance for future sonar systems. The cur- rently commercially available ceramics such as PZT4 are ferroelectric with small active domains rather than truly piezoelectric as are quartz or barium titanate crys- tals. Nevertheless the electric strain tensors are equiva- lent for ferroelectric and piezoelectric materials. There are composite ferroelectrics available with grains (0,3 and 3,3 composites) or rods (1,3 composites) embedded in a matrix with desirable acoustic performance partic- ularly for passive sonars [78]. Finite-element descrip- tions of these materials are under development as are material models of PVDF and similar piezoelectric pol- ymers. The desire for higher power densities for acous- tic projectors has heightened interest in magnetostrictive [79] and electrostrictive materials. The inherently nonlinear response of these materials poses no problem for the finite-element technique provided the amplitude of the motion does not exceed the applied bias field [80]. A feature of rare-earth magneto- strictive devices are their poor material permeability. Hence in a closely packed active array stray magnetic fields can constitute severe interaction between neigh- bouring projectors. Finite-element codes for predicting magnetic and electric fields are available and will allow for detailed scrutiny of possible magnetic interaction effects of a particular array design.

6 Conclusions

A brief review of numerical methods used for sonar array perforinance has been given. Specifically, the popular finite-element and boundary-element methods have been considered. While no attempt has been made at a detail mathematical presentation, certain salient points concerning the various attributes of the numeri- cal techniques have been examined. Active arrays gen- erally have few projectors and not a large frequency bandwidth. Pure finite element or combined finite-ele- ment and boundary-element approaches have proved to be very useful in predicting their acoustic perform- ance. For passive arrays care must be taken since the computational effort to address the array’s response over all the frequency range of interest may become excessively large. Some ways for reducing the problem based on simplifying assumptions have been proposed.

7 Acknowledgements

The authors are indebted to Dr. P.C. Macey of PAFEC Ltd. for writing the PAFEC acoustic software

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and providing invaluable support. They would also like to thank Drs. D. Boucher and J. Garcin for their help- ful discussions during the preparation of this paper.

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