Fast Multipole Burton-Miller Boundary Element Method for Two and Three-Dimensional Acoustic Scattering

A multistage adaptive fast multipole method is used to accelerate the matrix-vector products arising from the Burton-Miller boundary integral equations, which are formed in a boundary element method. The present study considers the scattering of acoustic waves, generated by localized sources from bodies with rigid surfaces. Details on the implementation of a multistage adaptive fast multipole method are described for two and three-dimensional formulations. The code is veri ed through the solution of well documented test cases. The fast multipole method is tested for acoustic scattering problems of single and multiple bodies, and a discussion is provided on the performance of the method. Results for engineering problems with complex geometries, such as a multi-element wing, are presented in order to assess the implemented capability.


INTRODUCTION
The development of physics-based noise prediction tools for analyzing aerodynamic noise sources, such as the jet and airframe ones, is of paramount importance, since noise regulations have become more stringent, and more sophisticated methods are needed to achieve the required noise reductions numerical simulations for the study and mitigation of noise scattering codes, such as the fast scattering code (FSC) from frequencies and can be further accelerated by the method using a method that is easy to implement for any singular axisymmetric cylindrical and spherically symmetric sources by a cylinder and a sphere is computed and compared with the presented with details on the implementation of the adaptive acoustic scattering by complex geometries, e.g., a cylinder with a cavity, multiple cylinders and spheres, and a multiele-

THEORETICAL FORMULATION
The scattering of sound waves produced by a spatial distribution of concentrated sources is solved in the present represents the pressure disturbances induced by concentrated p represents the pressure, k is the wave number, Q si represents the i-th source strength, x x si is the i-th e -iwt is assumed for equation is the free space Green function, G(x , y ), which is the Helmholtz equation for the particular solution given by the

RESULTS
The present section presents results obtained by the -

Code veri cation
problem is compared with the analytical one presented by placed at a distance L the present computations, the distance L of the cylinder is set a k , which corresponds to a frequency of f Hz for a speed of sound c For this test problem, the boundary of the cylinder is method and by the analytical solution for a distance R a This solution is obtained for a cylinder discretized with elements per box is the maximum allowed in this computa- equation through the discretization of boundary integral equations by the boundary element high improvements in simulation time and memory storage will be used for the analysis of airframe noise sources and Among these, one can cite the advantage of requiring only the problems, the solution of the nonsymmetric dense matrices n solution of the linear systems and reduce the computational n log n However, the method was further developed and became approaches were developed based on the range of frequencies et al n ), the et al n log n ric dense linear systems are formed by the discretization of the products in the iterative solutions of the large-scale linear numerical solution is accelerated by a multilevel adaptive et al the matrix-vector products in the linear systems formed by are described in the literature for the solution of the Burtonet al et al the elements, and simplify the solutions of the interactions are computed with direct computations using Several approaches are found in the literature on how to singular integrals (Chien et al et and the integrals are computed Following this method, all the singular integrals can be computed by regular Gaussian quadrature formulae and withdifferent spatial lengths and using multipole expansions to evaluate the interactions among clusters, which are clusters as sets of elements that are circumscribed by circles in will reduce the performance of the method because more direct computed by the direct solution of the boundary integral equathe surface boundary is discretized into elements and the entire approach follows the same methodology, except that one should continues until the number of boundary elements inside all an airfoil with all the smaller boxes showing the adaptive and these are direct neighbors among themselves, which algorithm, one can write a quad-tree structure containing all ment, a table of nonempty boxes is maintained, so that once an empty box is encountered, its existence is forgotten and it multipole expansions and translations and the acceleration of the L 1 of a box b consists of box b itself and all boxes that do not contain children boxes and that share a node or edge with b b is a parent box, then L1 4 = L 2 of a box b is b parent neighbors, which L is empty if b is a parent box, and it consists of all children of b neighbors, at any level, which do not share a node or vertex with b L of box b is formed by all boxes c, such as that b L in L (b) are at higher levels than b, i e b, and all boxes in L (b) are at smaller levels than b, i e b L one can observe the lists associated with a box b b parent, which means that they are in L (b b from Cheng et al function, G(x,y one wants to expand in a suitable form in order to apply elements at y x directly, one can

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and, then, compute multipole approximations representing the multipole expansions around the centroids of all childless boxes, represented by z which holds for distances | x z | larger than | y z |, one can where, S is a set of discrete elements inside the box with centroid z multipole expansion and all the multipoles are summed to Subsequently, all the multipole expansions from the childless boxes at all levels are shifted to the centroids of multipoles from centroids of boxes of level l parents centroids at level l is commonly called multipole-the boxes in L (b) are converted to local expansions about b centroid and added up forming a local expansion around b well-separated boxes at the same level of b the same ideas, the multipole expansions for the boxes in L (b) are converted to local expansions about b centroid representations around centroids of well-separated boxes L and L are then shifted to b -Finally, all the calculations can be performed in order computed for the centroids of the boxes in the highest level sources to each of the elements contained inside box b For each childless box b, we compute the direct interactions among all elements inside b and those on L (b) and L (b the interactions from elements on L (b) are calculated using L (b), we compute multipole expansions for all elements inside the boxes in L (b), and, then, compute the effects of these multipoles to each of the elements inside b All theorems and analytical tools, which prove the validity of the multipole expansions and translations, and several formal et al computed the multipole expansions and translations using plane wave expansions and in the latter, Gumerov the translation operators and rotation-coaxial translation decomposition of the translation operators to reduce the for the n th order r and are the polar coordinates of some vector X , which can be ox or oy , ox and oy point from some box center, o x y should truncate the series using p terms, where p k oy > and o o is located closely to the boundary element S and, then, |ox | > max|oy -

and
the center of a parent box and o is the center o is assumed to be closer to x compared to o boundary and hyper-singular integral equations as functions and3D FormulationFor the same reason as in the previous subsection, we pres-In m is the complex conjugate of In m terms r, , z represent the spherical coordinates of some vector X, which can be ox or oy ox and oy point from some box center, o location, the center of a parent box and o is the ol v is assumed to be closer to x compared to o expressions, the summations in l, term Wn,n',m,m',l are computed (b) Sphere rigid cylinder and sphere for R kawave number and dimension of the scatterer in this problem is n ka, this number has to be increasedThe convergence residue of the CGS iterative solver is directivity matches perfectly with the analytical solution This is a consequence of the several recursive divisions and the number of boundary elements is increased, the compu-tional to O(n O(nThe second problem considers the acoustic scattering around a rigid sphere due to a spherically symmetric point source is placed at a distance L from the center of the L For this test problem, the boundary of the sphere is method and by the analytical solution for a distance R of boundary elements per box is set equal to one hundred, paper, except for the cavity problem, since we are considersolution for pressure directivity matches perfectly with the ing order of accuracy and memory usage of the direct and the -L , is E| , p is the acoustic pressure computed by the numerical method and p can be controlled by using more terms in the truncated series a plot for the memory requirements for the direct and the fast order O(n ) in terms of memory requirements, while the fast O(n), where n is the number ofCylinder with cavityThe third test case presents results for the scattering of pressure waves around a rigid cylinder with a cavity, due to a monopole source located non-symmetrically with surface of the cylinder is plotted for a wave number k m a L h d and its length is l due to a monopole source located non-symmetrically accuracy and memory requirements for the acoustic scat-one can see the solution for the same problem, but for a cylinder x y z For the cavity case, the scattering has a preferential direction, tions obtained for the cylinder with no cavity show the expected effects of the cavity on the scattering of pressure waves can be observer at a radial distance R z the effects of the cylinder with cavity compared to the one with time spent by each method for achieving the desired convercavity due to a monopole source located non-symmetri-the required convergence while the two, three and four-level box should be always chosen to increase the performance of the pole computations and translations) nor too big (many directMultiple bodiesThe fourth studied problem assesses the implemented capability in dealing with the scattering of pressure waves around multiplex yThese have centers on x y a the spheres is plotted for a wave number k (b) R z each method for achieving the desired convergence, obtained in distributed inside the computational box if compared with the Cavity -Multi-element wing This test case consists of acoustic scattering around a two monopole sources are placed in the gaps formed between the multi-element wing surface for a wave number k radial distance R z the desired convergence, obtained with ten steps by all the trated in the center of the computational box in the y have a smaller number of operations compared to cases where the scattering body is more uniformly distributed around number of empty boxes distant from the center of the compu-CONCLUSIONS neous Helmholtz equation is solved using a boundary integral is implemented to deal with the non-uniqueness problems of only regular Gaussian quadrature is presented for the solution easier, and the equations are discretized along the boundaries the matrix-vector products arising from the method become provides excellent results matching analytical solutions computational time are shown and the multistage algorithm has proved to be more efficient than the single-stage an increasing number of boundary elements in the discretithere is always an optimum number of levels of refinement depending on the number of boundary elements and the elements uniformly distributed over the computational box, the improvements in computational cost achieved distribution of the boundary elements is concentrated over a specific region of the computational box, e.g., a wing located along the center of the box, the results show larger problems, a factor of two orders of magnitude reduction