BOUDARY ELEMET METHOD I REFRACTIVE MEDIA David R. Bergman Exact Solution Scientific Consulting LLC, Morristown J, USA email: davidrbergman@essc-llc.com Boundary Element Method (BEM), an application of the Method of Moments to acoustics, is a popular method for solving scattering problems. The method is used in electrodynamics to model Radar Cross Section (RCS) of targets and radiation patterns from phased arrays and circuit boards. The technique requires the use of a free space Greens Function integrated over the boundary of the scatterer, modeled as a tiled surface with simple response functions on each tile. Most presentations of this the technique assume, or even require, that the medium in which the scatterer is embedded be homogeneous, implying straight line propagation paths between pairs of points. In this paper we present a BEM which does not assume a homogeneous propagation medium. We focus on problems where a Greens function can be determined or approximated by some means other than a full FDTD or FEM solution of the Helmholtz equation.. Introduction This paper disuses the application of boundary element method (BEM) for determining the scattered field from a finite object immersed in a medium with non-trivial refractive properties. We are primarily interested in the exterior BEM applied to echo location in an underwater or aero environment where pressure and temperature gradients, as well as wind, can affect the propagation of sound from source to target and back. In BEM the wave equation, Helmholtz equation, is converted into an integral equation evaluated at the boundaries present in the domain of the problem []. Instead of having to solve for the fields at each point in space one solves for the fields on the boundary. From this information the field can be calculated at any point in space. To make practical use of BEM requires two levels of approximation of the true physical system; () replacing the true surface with a suitable set of triangular facets and, (2) expanding the surface fields in a set of basis function. The BEM is exact in the sense that, in principle, one gets an exact solution for the modified problem. The standard treatment of BEM assumes a static medium described by a constant sound speed, a seemingly trivial problem. However, BEM wasn t designed to describe propagation through a trivial medium, an exactly solved problem, but to describe in full detail the scattered field from a complex boundary. With focus on the scattering body there is a great investment in the description of the boundary to ensure accurate surface fields to be used in far field and near field calculations. In many real world situations we encounter scattering from hard boundaries that are immersed in a medium with variable properties [2], [3]. In theory one can treat the medium and scattering elements by extending BEM to the entire space or using a combination of BEM and FEM. The application of FEM couples neighbouring elements in the problem space leading to a very large but sparse coupling matrix. In contrast BEM deals only with the boundary surface. The trade-off comes in the fact that one must consider coupling between all facets in the surface leading to a smaller problem but one for which the coupling matrix is dense and all elements are needed for a complete solution. In underwater and aero acoustics problems the medium can be described to a good approximation as a layered medium. In such cases we look at a compromise suitable for problems encountered in radar and sonar performance prediction in which we want to perform a com-
plete BEM calculation but are unable to use the free space Greens function. We present a hybrid procedure that makes use of the mode expansion of the two point Greens function in the BEM integral equation. 2. Motivation The work presented is motivated by a desire and need to extend the concept of scattering cross section to situations involving non-trivial propagation. We encounter this problem in detection and discrimination of objects using phased arrays in underwater and aero-acoustics problems. In radar theory the Radar Cross Section (RCS) is used to estimate returns from a complex object [4]. The RCS is a specific example of the scattering cross section of a body which describes the angular deviation of monostatic returns as a function of look direction. The definition of scattering cross section can be found in most texts of acoustics or electromagnetism. Derivations make use of the free space Greens function and the far field limit by which the scattering amplitude of a finite body decouples from the far field Greens function. Through this process the medium is decoupled and one can think of the scattering amplitude as a description of the body or a property of the body. The procedure starts with a unit strength plane wave (source at infinity) incident on the scatterer. The BEM procedure is then used to solve for the fields on the surface of the scatterer required to maintain boundary conditions, which are then used as a source to determine the far field strength of the scattered field. By scanning the scatterer from all directions one builds up the angular distribution of the monostatic scattered field of the body (bi-static scattering is handled in the same manner by choosing different incoming and outgoing propagation directions). Several things prevent this process from being easily extended in non-trivial media; () in general it is not possible to insonify the scatterer with a plane wave as the medium will distort the wave, (2) defining the cross section involves taking the limit of the scattered field viewed at infinity, far field approximation, and it is not generally true that the field will decay as expected, (3) even isotropic environments will introduce anisotropy in the definition of the cross section, just to name a few. The last issue mentioned implies that in such cases it may be difficult to define the cross section as an intrinsic property of the body. 3. Computational approaches 3. Weakly refracting environment In special cases the local sound speed profile (SSP) can be modelled by a linear depth (or height) dependent function. The effects of a linear SSP are well studied. In particular it is known that the ray paths are circles, geometric spread is similar to a homogeneous environment, and that for a point source caustics will not form as a result of this profile. The effect on beamformers has been studied where it has been shown that the primary effect is a deterministic displacement or shift of the predicted location of an object [5], [6]. In cases where the propagation distances are small compared to the distance over which changes in the SSP are severe a linear model may suffice for prediction models. This suggests that one can approximate both the incident field and the return from a target by propagating the known amplitude function along the circular paths connecting the body to the receiver. In some sense one can view the linear SSP as simply bending the results obtained for homogeneous media, at least in the case of weak refraction. The only real issue the reader should keep in mind that the usual definition of look direction will not coincide with the propagation direction. This is a concern in general, especially when multi paths are present, but not for the linear SSP since there is a unique circular arc connecting each pair of points. It should also be noted that the exact form of the Greens function is known for a linear SSP making it possible to employ the exact form in the BEM integrals directly [7]. 2 ICSV23, Athens (Greece), 0-4 July 206
3.2 Mode expansion of the Greens function For more general layered media, e.g. depth dependent SSP encountered in underwater acoustics, the full wave equation reduces to a one dimensional problem describing the pressure field in terms of normal modes. The specific modes are determined by the nature of the SSP and for a hand full of cases the modes can be found exactly in terms of special functions. In general one needs to solve the one dimensional wave equation numerically for the Eigen functions and Eigen values of the problem. We apply the normal mode expansion for the two point Greens function to the problem. Rather than evaluate a plane wave at the boundary we use the Greens function to produce the field at the surface due to a monochromatic point source somewhere in the medium. This same Greens function is used to evaluate the coupling between boundary elements in the surface of the scatterer. G(r, r ; k) = C(k n, k)ψ n (r ; k n )ψ n (r ; k n ) n= () Equation () can be used in general but for layered media in the x and y dependence can be written explicitly and the summation is over vertical modes. The coefficient is a function of the Eigen value and is singular when the source wave number matches and Eigen value. The boundary integral form of the Helmholtz equation is, αp(r ) = {n G(r, r ; k)p(r ) G(r, r ; k)n p(r )}ds + p 0 (r ). (2) The Greens function G(r, r ; k) depends on both source and receiver points and is a solution to the Helmholtz equation for a point source. The field point r resides on the surface and r is the source point, all primed quantities are evaluated on the surface and integrated over the source. The outward normal of S is n, p the pressure and p 0 an incident field due to another exterior source. The gradient of the pressure field and the particle velocity are related on the surface by n p = iωρn υ. Equation (2) is discretized by introducing a set of flat patches on the surface. The fields of the problems are expanded in a set of basis function defined on the patches where the basis functions can be defined on one or more connected patches. To fully discretize the problem a testing function set is introduced to sample the surface elements. Though in general any set of functions can be used as a testing function set a common choice is to use the same functions used in the expansion of the fields. This is known as the Galerkin method. Writing the pressure field as an expansion over a basis set p(r ) = n= f n (r )p n, the functions are defined only on the patches that contribute to that particular basis and {p n } are a set of undetermined constants. One often encounters a definition where the basis functions span one patch each and are constant on the patch (referred to as a pulse basis). Since the derivative of the pressure field is also required one can expand these fields in the basis set. The pressure and the normal derivative are typically treated as independent fields each requiring measurement. Applying the expansion to Eq. (2) leads to Eq. (3). 2 p(r ) = (L n(r )p n L 2n (r )( n p) n ) + p 0 (r ) (3) n= We have introduced the operators L,2n (r ). ICSV23, Athens (Greece), 0-4 July 206 3
L n (r ) n G(r, r ; k)f n (r )ds n (4) L 2n (r ) G(r, r ; k)f n (r )ds n (5) When the Galerkin method is employed the discretization is completed by integrating Eq. (3) over each basis elements. 2 f m f n p n = ( f m L n p n f m L 2n ( n p) n ) + f m p 0 (6) n= The bra-ket notation is introduced in Eq. (6), f m p f m (r )p(r )ds m. Equation (6) is now a matrix form of Eq. (2) suitable for scattering problems. The rest of the work centres around evaluating the quantities f m L,2n. Integrals of G(r, r ; k) involve a singularity requiring special treatment such as singularity extraction. Integrals of G(r, r ; k) have a more severe singularity, R 3, but may also be handled with singularity extraction techniques. Integrals are now expressed in terms of the mode expansion. We are essentially interested in the integration of G(r, r ; k) and its normal derivative over one or two patches (one is always required, two if Galerkin is employed). { f m (r )G(r, r ; k)f n (r )ds n } ds m = C(k j ; k) f m (r )ψ j (r ; k j )ds m f n (r ) ψ j (r ; k j )ds n = C(k j ; k) f m ψ j f n ψ j (7) { f m (r )n G(r, r ; k)f n (r )ds n } ds m = C(k j ; k) f m (r )ψ j (r ; k j )ds m f n (r ) n ψ j (r ; k j )ds n = C(k j;k ) f m ψ j f n n ψ j (8) Equation (6) may now be expressed in terms of the modes of the background environment. 2 f m f n p n = C(k j ; k) { ( f m ψ j f n ψ j p n f m ψ j f n n ψ j ( n p) n )} + f m p 0 (9) n= By the unique form of the mode representation of the Greens function the integral over two patches decouples into two single patch integrals multiplied together. It is assumed that the basis functions are real. In general the Greens function is not separable and a quadruple integral is required. The form of Eq. (9) implies a reduction in work since one does not have to attempt 4-dim quadrature, and explicit singularity extraction will not be necessary. However it comes at the cost of needing to 4 ICSV23, Athens (Greece), 0-4 July 206
do these calculations for as many modes as required to ensure convergence of the mode expansion and the need to regulate the singularity present in C(k j ; k). For a purely depth dependent SSP problems exhibit considerable symmetry. While one does not know ahead of time where patches will be placed one can sample the inner products f n ψ j for a variety of patch sizes, orientations, and depths. From this information one can develop lookup tables similar to those used in normal mode propagation codes to save time. One can then build matrices for a variety of sensor and target positions in the same environment without needing to explicitly calculate numerical integrals for each new case. 4. Concluding remarks We have presented a procedure for implementing BEM for sources and targets immersed in a variable refractive medium. The approach makes use of the mode expansion representation of the Greens function appropriate for the background medium and uses this to evaluate the matrices involved in an acoustic BEM. The use of BEM in non-trivial situations is not new. Authors have looked at imbedding targets in ideal waveguides, using the method of images to build a Greens function for the integral equations [8]. In fact Wu did implement a mode expansion for evaluating long range waveguide effects on scattering cross sections. Previous work has focused on the effect of global boundaries on propagation in an otherwise trivial medium while our interest is in the development of the same for refractive waveguides. A second integral field equation is derived by taking the normal derivative of the first integral equation. The same process can be applied to these equations to develop their form in terms of normal modes. REFERECES Kirkup, Stephen. (2007). The Boundary Element Method in Acoustics, A development in Fortran, [Online.] available: http://www.boundary-element-method.com. 2 Robinson, Allen R., Lee, Ding, Ed., Oceanography and Acoustics Prediction and Propagation Models, AIP Series in Modern Acoustics and Signal Processing, AIP Press, ew York, (994). 3 Tolstoy, I., Clay, C. S., Ocean Acoustic Theory and Experiment in Underwater Sound, McGraw & Hill, USA, (966). 4 Skolnik, M., Radar Handbook Second Edition, McGraw Hill, (990). 5 Bergman, D. R., Beamformer Performance in Variable Environments, Proceedings SPIE.DSS, Ground/Air Multisensor Interoperability, Integration, and etworking for Persistent ISR VI, 94640X, Baltimore MD, United States, 2-23 April, (205). 6 Bergman, D. R., Modeling Beamformers in Refractive Media, Proceedings of the 6 th Berlin Beamforming Conference (BeBeC 206), Berlin, Germany, February 29 March, (206). 7 Bergman, D. R., Exact Green s function for the wave equation in a linear graded inhomogeneous layered medium using geodesic polar coordinates, Proceedings of the 8 th International Congress on Sound and Vibration, Rio de Janeiro, Brazil, 0 4 July, (20). 8 Wu, T. W., On computational aspects of the boundary element method for acoustic radiation and scattering in a perfect waveguide, Journal of the Acoustical Society of America, 96 (6), pp 3733-3743, (994). ICSV23, Athens (Greece), 0-4 July 206 5