Transactions on Modelling and Simulation vol 9, 1994 WIT Press, ISSN X

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1 Application of the boundary element method for analysis of scattering interactions between a sonar receiving array and neighbouring structures A. Monsallier", F. Lanteri* & C. Audoly" "DON Ingenierie/Sud/LSM, B.P. 30, Toulon Naval, France * ASM Ingenierie, ZE La Farlede, B.P. 240, ABSTRACT The paper is devoted to the theoretical analysis of interactions between a receiving array and a neighbouring structure as well as their consequences on the sonar function. The boundary element method is applied to solve the scattering problem for two bodies. Two configurations are considered: a cylindrical array placed in front of a thin rigid panel, both considered as two-dimensional bodies and the association of a spherical array and a thin soft disc. In both examples, the support of the sensors of the arrays are assumed perfectly rigid. The two-dimensional problem is solved using the Helmholtz integral representation; the threedimensional scattering problem using the axisymmetric formulation of the previous representation. Numerical results are used to compute directivity patterns as a function of the steering direction. Degradations produced by shielding and scattering phenomena such as the shift of the maximum level or the increase of the side lobe level are illustrated with diagrams. INTRODUCTION Sonar receiving arrays are often designed without considering carefully other structures that are placed near them onboard submarines or ships. However their presence can lead to severe degradations of the sonar function because of interactions between the acoustic signal and these bodies. The purpose of this paper is to put forward these phenomena and analyse them. In this paper, two configurations are studied: a two- and a three-dimensional geometry. We first consider a cylindrical array in the vicinity of a thin rigid panel. The threedimensional axisymmetric configuration consists in a spherical array placed near an acoustic barrier represented by a thin soft disc. The acoustic barrier is designed to reduce noise radiated from inside the submarine. A theoretical model using the Helmholtz integral representation for the two-dimensional problem and its

2 160 Boundary Element Methods for Fluid Dynamics II axisymmetric formulation for the three-dimensional problem is developed to obtain the response of the sensors of the array to any incident wave. In the next part numerical results for the response of the sensors are used to simulate array directivity patterns in presence of the neighbouring stucture and analyze their consequences on the sonar function. DESCRIPTION OF THE SYSTEM OF BODIES The surface of the cylindrical or spherical array is nearly uniformly filled with point-like sensors with equal sensitivity and spaced by half the wavelength of the incident wave. The geometry for the array and the neighbouring structure is shown in Figure 1. For the three dimensional (3D) geometry, it consists in an orthogonal projection in the plane (O,x,y) with the origin located at the center of the array. The common notations for both geometries are as follows: (<j),6) the azimuth and elevation angles of the sensors, (< >j,9j ) the angles of an incident acoustic plane wave, (( )g,6s ) the angles of the steering direction of the array, (6=81=65=0 for the 2D geometry) O the 2D or 3D infinite exterior domain, A and S respectively the surface of the array and the thin screen with an exterior unit normal n, 1^ the interior volume delimited by the surface A, M,Q any points which do not belong to SuA, m,q points on S or A, k the wavenumber of the incident pressure. incident wave shadow zone Figure 1: Geometry and notation of the problem Both the sphere and the disc can be obtained from 2D axisymmetric geometry (Fig.l) by rotating their generator about the x-axis. L* and L are respectively the generators of the sphere and the circular screen. MODELLING OF THE RESPONSE OF THE SENSORS General acoustic formulation We consider the scattering of a harmonic plane wave of angular frequency o. The time dependance is of the form exp(-ficot) which is suppressed from all the following formulae. Let p' be the incident field and p^ be the scattered field such that p the totalfieldat any point M in Q is given by: p(m) = p"*(m) + p*(m) The following integral representations for a rigid body and a soft or rigid screen are derived from the classical Helmholtz integral representation [1,2]. The scattered field from a thin screen is represented by means of layer potentials: a simple layer potential with a density a for the soft panel and a double layer potential with a density p, for the rigid panel. The densities a and (I can be interpreted respectively as the jump of the normal derivative of pressure and the jump of the pressure between both faces of the soft disc or the rigid panel.

3 Boundary Element Methods for Fluid Dynamics II 161 Integral representation for the rigid object/soft thin screen representation [p(q).9n(q)g(m,q)]ds(q) - J[a(q).G(M,q)]dS(q) J[ _ Jp(M) for Me Q (1) " \0 for Me O* (2) G is the Green function: o(m,q) = - ^ ^(k.mq) for the 2D problem [G(M,q) = exp(-ikmq)/mq for the 3D problem Integral representation for the rigid object/rigid thin screen configuration J[p(q)A(q)G(M,q)]dS(q) + f [tfq) A(q)G(M,q)]dS(q) _ Jp(M) for Me Q (3) " [0 for Me O& (4) The pressure p(q) for q on A and the density function a(q) or ja(q) for points q on S are unknown functions. For such points, the following Helmholtz integral equations can be written. Integral equation for the rigid object/soft thin screen configuration J[p(q).9n(q)G(m,q)]dS(q) - J[o(q).G(m,q)]dS(q) Integral equation for the rigid object/rigid thin screen {2 p(m) for me A (5) 0 for me S (6) Pf. J [ p(q) an(m)3n(q)g(m,q)]ds(q) + ^ J [ ^(q) 9n(m)an(q)G(m,q)]dS(q) = - 3n(m)P*^(m) ^r me Au S (7) Remark: the symbol Pf. is used to be understood as the limit value of the integral. For the Helmholtz integral representation, an irregular frequency for one of both bodies is not an irregular frequency when we consider the problem in its entirety. Let kg and \ /a be respectively a wavenumber and an associated eigen function related to an irregular frequency for the problem of the array alone. We can demonstrate that \ /a is not a null function on S and also doesn't verify the associated homogeneous equations for Equations 5 and 6 (or 7). Thus k% is not an eigen wavenumber for the array/screen geometry. However numerical experiments show that an instability occurs when the distance between the scatterers increases. This instability can be overcome using Schenck's method [3]. The unknown functions p and a (or (i) must also verify at once the integral equations 5 and 6 (or 7) and the integral representation 2 (or 4) for any points in the interior domain Q.

4 162 Boundary Element Methods for Fluid Dynamics II Application to the two-dimensional cylindrical array/panel configuration For the two-dimensional geometry, the total pressure p and the density (I on the panel or cylindrical array surfaces are directly computed using the equation 7. These functions are then inserted in the integral representation (Equation 3) in order to evaluate the total pressure p in the exterior domain Q. To numerically solve Equations 7 and 3 a collocation method is used. This method consists in dividing the surface S (or A) into N arcs Sj, the centers of which are named Qj. Sj is also replaced by a segment Sj located in the plane tangent to Sj at the point Qj. S is approximated by the union of the segments Sj. The unknown functions over each segment are approximated by a constant value. The approximated function is also piecewise constant. Application to the three-dimensional spherical array/soft disc geometry Axisymmetric formulation By taking advantage of the axisymmetric properties of the system of bodies the surface integrals can be reduced to line integrals along the generator. The acoustic pressure in the exterior region can be expressed in terms of the Fourier pressure components p^(m) with m on LA and the components (Jn(m) with m on L$ of the density function G on the disc p(m)=p (M)+ n=-oo J where: g (M,q)= JG(M,q)exp(in8q)d8q and g^(m,q)= j9n 0 0 The integrals g and g/ can be expressed as the sum of a contribution with a non singular integrand and a contribution that involves elliptic integrals of the first and second kind [4,5]. Further, only the positive components n need to be considered. In practice, the summation is truncated to a number NMAX that ensures the convergence of the Fourier series of the incident pressure. The unknown Fourier components p% and (? are the solutions of a system of line integral equations derived from (8) and the overdetermination equation. p^(m)=a(m)pn(m)- J g;(m,q)pn(q)yqdl(q) + Jg/m,q)(?n(q)yqdl(q)) (9) LA with a(m) =yif me L^ and oc(m) = 0 if me Lg or m is inside the sphere. Numerical Implementation The integral equations involve integrals of the form: I=jf(q).K(m,q)dl(q) L where f is an unknown function. The line L and the unknown function f are approximated using quadratic shape functions [4,5]. By applying the same procedure to all terms of the integral equations we obtain a set of linear systems AnXn=Bn where X^ is the vector of unknown Fourier components pn(m;) and Gn(mj), An is a complex matrix involving integrals of the previous form; B^ is a complex matrix whose coefficients are the Fourier components of the incident field. With the Fourier components Pn(mj) and (^(m;) the acoustic pressure can be computed at any points of the exterior region thanks to equation (8). LS (8)

5 NUMERICAL RESULTS Boundary Element Methods for Fluid Dynamics II 163 We study the perturbations due to the presence of the neighbouring screen on the response of the sensors and on the array directivity. Let a, b and c represent respectively the array radius, the distance from the center of the array to the screen and the disc radius or panel length. The numerical results have been computed using the parameters found in Table 1. 2D geometry 3D geometry ka 8 10 kb kd Table 1 Perturbations on the sensors response due to the screen These perturbations are studied on the 3D geometry. HI O NORMALIZED PRESSURE HI OC o (/) LU CO SENSOR AZIMUTHAL ANGLE Figure 2: response of the sensors of the spherical array to an incident wave of direction (<h=120,9i=0 ). LU O 100-, ~n 200 NORMALIZED PRESSURE > o- LU LLJ cr O -50- LU -100 ' I ' I SENSOR AZIMUTHAL ANGLE Figure 3: response of the sensors of the spherical array to an incident wave of direction (c )j=120,0i=0 ) in presence of the acoustic barrier. "

6 164 Boundary Element Methods for Fluid Dynamics II Figure 2 shows the response of the sensors of the array (natural values of the modulus of normalized pressure) when the acoustic barrier is removed and the incident plane wave comes from the direction (< >pl20,0po ). Therefore, sensors near the point (( )=-60,6=0 ) are in the shadow region of the sphere whereas sensors located near the position (< )= 120,0=0 ) receive about two times the level of the incident pressure. When it is placed near the spherical array the acoustic barrier creates a shadow zone (Fig.l). The sensors in (0=120,6=0 ) do not receive the maximum pressure level any more and a loss of signal is also observed on sensors which are close to the acoustic barrier (Fig.3). Perturbations on the directivity function due to the screen We consider a array composed of N sensors. The response of each sensor is proportional to the total pressure it receives. The beamforrning in a steering direction (4>g,8g ) consists in adding the sensors response in phase in order to obtain the maximum of the resultant signal when the incident direction ((j)i,0i) corresponds to the steering direction (<t>s>0s ) This signal called the directivity function has the following expression: n=l Shp : sensor sensitivity p«: pressure received by the sensor located at Mp kgi wave vector in the steering direction)) In practice not all the sensors are used to compute the array response for a given steering direction. Only those which are located in an angular aperture symmetrical about the steering direction and equal to 120 are considered. Figures 4 and 5 represent the level (in decibels) of the directivity pattern normalized to its corresponding maximum value. Figure 4 shows the 2D directivity patterns as a function of the incident direction for various steering directions. Figure 5 represents the same results in the azimuthal plane (63=81=0 ) for the 3D geometry. When the cylindrical (Figure 4) and spherical (Figure 5) array are steered in the direction ( )g=0 the same patterns symmetrical about the steering direction are approximately obtained whether the panel or the disc is present or removed (curves 1 and 2). Fluctuations of the pressure level caused by reflections on the neighbouring structure and scattering interactions between the bodies do not induce any strong degradation of the array characteristics. The difference between the main lobe level and the side-lobe level is significant: 12 db for the 2D problem and 18 db for the 3D problem. For a steering direction (^=90, we can note an increase of the side-lobes (curves 3, Figures 4 and 5). The diagrams are no longer symmetrical. Indeed a degradation of the side-lobes overlooking the screen can be noted for both geometries. However the screen effects are not yet significant. Much more severe degradations occur when the arrays are steered in the direction <j)g=120 (curve 4, Figures 4-5). For such steering directions the maximum of the directivity patterns corresponds no longer with a direction of the incident field equal to the steering direction. Another consequence of the presence of the neighbouring structure is the increase of the side-lobes level. Figure 5 shows a difference between the main lobe level and the side-lobes level only equal to 5dB.

7 Boundary Element Methods for Fluid Dynamics II 165 For such steering directions the cylindrical and spherical arrays are not operational. 90 r^\ Figure 4: cylindrical array directivity patterns (in decibels) as a function of the incident direction and for various steering directions. 270 Figure 5: spherical array directivity patterns (in decibels) in the azimuth plane as a function of the incident direction and for various steering directions. In order to illustrate more synthetically the influence of the presence of the neighbouring structure we study the evolution of the directivity index. This index is the ratio (in decibels) of the value of the directivity function for an incident direction equal to the steering direction to the mean value of this function on the three-dimensional space or in a limited sense on the two-dimensional space. A significant decrease in index values corresponds with an increase of the maximum side-lobes level and the emergence of an angle difference between the steering direction and the direction that corresponds to the maximum level of the directivity function. Figure 6 represents the level (in decibels) of the directivity index for the two- and the three-dimensional problem. We can note a significant decrease for a steering direction equal to 120 for the 2D problem and 110 for the 3D problem. In a geometrical point of view we can easily obtain the limit steering direction value so that none of the sensors of the aperture is shaded by the neighbouring structure when the incident direction corresponds with the limit steering direction. The limit steering direction value is equal to 126 for the 2D geometry and 100 for the 3D geometry. The agreement between these limit values and the values which correspond to a decrease in index is very good. The geometrical criterion is a good estimation, in a first approximation.

8 166 Boundary Element Methods for Fluid Dynamics II I i Steering direction (degrees) Figure 6: cylindrical and spherical patterns (in decibels) as a function of the steering direction CONCLUSION Results show that scattering phenomena should not be ignored when studying the characteristics of a sonar receiving array placed near other bodies. In the paper the Helmholtz integral representation for a 2D geometry and its axisymmetric formulation for a 3D geometry have been used to compute the response of the sensors and obtain the characteristics of the array. The main effects of the presence of a neighbouring stucture are the increase of the side-lobe level and the angle difference between the steering direction and the direction that corresponds to the maximum level of the directivity function. Although a simple geometrical criterion can be used to obtain the limit steering direction, scattering phenomena have been considered to compute accurate directivity patterns. This has been accomplished using the boundary integral method. REFERENCES I.COPLEY, L.G. Integral Equation Method for Radiation from Vibrating Bodies. J.Acoust.Soc.Am.41(4), pages , FILIPPI, P. and DUMERY, G. Etude Theorique et Numerique de la Diffraction par un Ecran Mince. Acustica (21), pages , SCHENCK, H.A. Improved Integral Formulation for Acoustic Radiation Problems. J.Acoust.Soc.Am.44(l), pages 41-58, SOENARKO, B. A Boundary Element Formulation for Radiation of Acoustic Waves from Axisymmetric Bodies with Arbitrary Boundary Conditions. J.Acoust.Soc.Am.93(2), pages , KIM, H.S. and KIM, J.S. and KANG HJ. Acoustic Wave Scattering from Axisymmetric Bodies. Journal of Sound and Vibration 163(3), pages , URICK, R.J. Principles of Underwater Sound. Me Gaw-Hill, New-York, 1975.