A modified Burton-Miller algorithm for treating the uniqueness of representation problem for exterior acoustic radiation and scattering problems

A modification to the Burton-Miller algorithm is formulated for exterior acoustic radiation and scattering problems which resolves the uniqueness of representation problem associated with the Helmholtz integral equation method (HIEM) at the interior eigenvalues. In particular, this modification reduces the required number of integral equation evaluations and allows for the continued use of the popular higher-order Lagrangian shape functions. Several example problems are considered to demonstrate the difficulties and subtleties of the uniqueness of representation problems associated with the Helmholtz integral equation method and to demonstrate the effectiveness of the modified Burton-Miller algorithm in overcoming these problems.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

Computation of scattering from a class of bodies of unrestricted size

Summary A method is presented for computing the scattering from any size of totally reflecting body by the inversion of one finite matrix, provided that the shape of the body can be derived by inwardly deforming a finite part of a body from which the scattering is known explicitly. Only the size of the deformed surface is limited by available computational facilities. The method, which is applicable to acoustic and electromagnetic scattering problems, is illustrated by applying it to a deformed infinite wedge for both electric polarization (Dirichlet problem: sound-soft boundary in acoustics), and magnetic polarization (Neumann problem: sound-hard boundary in acoustics). Numerical results are presented to demonstrate the convergence of the computations.     

The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation

The precorrected-FFT acceleration technique is successfully applied in the boundary element method for the simulation of 3-D acoustic scattering problems. The composite Helmholtz integral equation presented by Burton and Miller is employed to overcome the nonuniqueness problem occurring in the simulation of exterior acoustic problems by the boundary element method. Since the triangular constant element is employed, the hypersingular boundary integral equation is reduced into a weakly singular boundary integral equation with the application of a modified Burton and Miller's formulation. The computational cost, the consumed memory and the convergence of the current method are demonstrated and analyzed through the simulation of a plane acoustic wave scattering from a rigid sphere and from an axisymmetrical rigid structure.     

Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method

This work presents a novel finite-element solution to the problem of scattering from a finite and an infinite array of cylindrical objects with arbitrary shapes and materials over perfectly conducting ground planes. The formulation is based on using the surface integral equation with Green's function of the first or second kind as a boundary constraint. The solution region is divided into interior regions containing the cylindrical objects and the region exterior to all the objects. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. Using two-boundary formulation, the surface integral equation is then applied at the truncation boundary as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate in comparison with previously published methods. The near and far fields are generated for a finite and an infinite array of objects. While the surface integral equation in combination with the finite-element method was applied before to the problem of scattering from objects in free space, the application of the method to the important problem of scattering from objects above infinite flat ground planes is presented here for the first time, to our knowledge.     

Scattering of a point generated field by a multilayered spheroid

Summary The point source excitation acoustic scattering problem by a multilayer isotropic and homogeneous spheroidal body is presented. The multilayer spheroidal body is reached by an acoustic wave emanated by an external point source. The core spheroidal region is inpenetrable and rigid. The exterior interface and the interfaces separating the interior layers are penetrable. The scattered field is determined given the geometrical and physical characteristics of the spheroidal body, the location of the point source and the form of the incident field. The approach is not limited in a certain region of frequencies.     

Modified fundamental solutions for the scattering of elastic waves by a cavity: Numerical results

The boundary integral equation method is very often used to solve exterior problems of scattering of waves (elastic waves, acoustic waves, water waves and electromagnetic waves). It is known, however, that this method fails to provide a unique solution at the so-called irregular frequencies. This difficulty is inherent to the method used rather than the nature of the problem. In the context of elastodynamics. we proposed, in a recent work¹, two methods for eliminating these irregular frequencies. Both are based on modifying the fundamental solution. Here we present numerical results pertaining to the solutions of the modified and unmodified integral equations.     

Uniform asymptotic solutions for the two-dimensional potential field problem with joining relations on the surface of a slender body

The 2-dimensional potential field in the whole plane with joining relations on the surface of a slender cylindrical body is studied. The inner solution is obtained by means of a regular perturbation problem and the outer solution is represented as a super position of potentials of point sources and point currents distributed on a segment inside the body. The problem is then reduced to a system of integral equations whose solution is obtained by using the method of Handelsman and Keller. As particular cases the Dirichlet and Neumann boundary-value problems for the domain exterior to a slender body are considered. The paper gives the asymptotic expansion of the logarithmic potential on a segment and some quadrature formulae useful for computing the solution as well.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

声场分解的均匀圆阵实值MUSIC算法

经典方位角估计算法中未考虑阵列安装支架对阵列接收信号的影响,实际中阵列安装支架必然会对阵列接收信号产生影响。以环绕在刚性球上的均匀圆阵为阵列模型。在对声场特性的分析中将声学原理和阵列信号处理技术相结合,探讨了存在刚性球形障碍物时的声源方位角估计问题。首先从声学理论出发,分析了刚性球体散射声场及声场分解,讨论了刚性球体对圆阵响应的影响;进而结合阵列信号处理技术,在对声场分解所得到的特征波束空间,利用实值MUSIC算法实现了声源方位角估计。计算机仿真表明,该算法能较好地估计出空间多个声源的方位角,且计算量小,估计精度高,具有解相关声源的能力。     

Material parameter optimization for interior and exterior fluid-structure acoustic problems

We present an efficient adjoint-based framework for computing sensitivities of quantities of interest with respect to material parameters for coupled fluid-structural acoustic systems with explicit interface coupling. The fluid is modeled using the Helmholtz equation and the structure is modeled using the Navier-Cauchy equations. Sensitivities are used to drive a gradient based optimization algorithm to solve important problems in structural acoustics, viz noise minimization and vibration isolation. For each problem, we consider two different priors: one where the optimal solution has a smooth variation and another with a bimaterial distribution. These priors are imposed with the help of suitable regularization terms. The effectiveness of this approach is demonstrated on both interior and exterior structural acoustic problems.     

A scattering problem by means of the spectral representation of Green's function for a layered acoustic half-space

 The problem in this paper is for scattering waves caused by an object and a plane wave in a layered acoustic half space. The boundary integral equation method as well as the spectral representation of Green's function for a layered acoustic half space are introduced to the present analyses. The spectral form of Green's function developed here is expressed in terms of the eigenfunctions for the point and the continuous spectra, that is the extension form of Green's function expressed by Ewing, Jardetsky and Press (1957). The advantage of the spectral representation of Green's function is that it enables us to decompose the scattering waves into eigenfunctions for the layered medium. Several numerical calculations are carried out to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the spectral form of Green's function provides accurate values and is applicable to the boundary element analysis for a layered medium. The spectral structures of the scattering waves are also found to be able to explain their properties. Received 2 November 1999     

A family of second‐order boundary dampers for finite element analysis of two and three‐dimensional problems in transient exterior acoustics

Numerical modelling of exterior acoustics problems involving infinite medium requires truncation of the medium at a finite distance from the obstacle or the structure and use of non‐reflecting boundary condition at this truncation surface to simulate the asymptotic behaviour of radiated waves at far field. In the context of the finite element method, Bayliss–Gunzburger–Turkel (BGT) boundary conditions are well suited since they are local in both space and time. These conditions involve ‘damper’ operators of various orders, which work on acoustic pressure p and they have been used in time harmonic problems widely and in transient problems in a limited way. Alternative forms of second‐order BGT operators, which work on ṗ (time derivative of p) had been suggested in an earlier paper for 3D problems but they were neither implemented nor validated. This paper presents detailed formulations of these second‐order dampers both for 2D and 3D problems, implements them in a finite element code and validates them using appropriate example problems. The developed code is capable of handling exterior acoustics problems involving both Dirichlet and Neumann boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Dual boundary integral equations for exterior problems

A dual integral formulation for the interior problem of the Laplace equation with a smooth boundary is extended to the exterior problem. Two regularized versions are proposed and compared with the interior problem. It is found that an additional free term is present in the second regularized version of the exterior problem. An analytical solution for a benchmark example in ISBE is derived by two methods, conformal mapping and the Poisson integral formula using symbolic software. The potential gradient on the boundary is calculated by using the hypersingular integral equation except on the two singular points where the potential is discontinuous instead of failure in ISBE benchmarks. Based on the matrix relations between the interior and exterior problems, the BEPO2D program for the interior problem can be easily reintegrated. This benchmark example was used to check the validity of the dual integral formulation, and the numerical results match the exact solution well.     

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

Dynamic stress intensity factor of a cracked dielectric medium in a uniform electric field

Summary We consider the scattering of normally incident longitudinal waves by a finite crack in an infinite isotropic dielectric body under a uniform electric field. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor versus frequency is computed, and the influence of the electric field on the normalized values is displayed graphically.     

Fictitious frequency revisited

The nonexistence and nonuniqueness problems associated with integral equation methods for exterior acoustics are revisited. The Fredholm alternative theorem in conjunction with the singular value decomposition updating technique is used to simultaneously determine the fictitious frequencies and corresponding modes in exterior acoustics. After selecting the combined Helmholtz interior integral equation formulation (CHIEF) points, the influence row vectors are obtained. A criterion in selecting the minimum number of CHIEF points and their positions is proposed by testing the orthogonal condition between the influence row vector and the right unitary vector. It is proved in the discrete system for arbitrary-shape problems that the source of numerical instability of irregular frequency originates from the zero divided by zero using the generalized coordinates of unitary vectors. The mathematical structures of the four influence matrices in the dual boundary element method (BEM) are examined by using the left and right unitary matrices. Extracting the true eigenvalue and filtering out the fictitious frequency can be unified by using the updating term and updating document, respectively. Radiation problems of a cylinder and a square rod are demonstrated to see the validity of the present formulation.     

声学多层结构介质的研究进展

首先回顾了声学多层结构介质的研究历史,在此基础上简要介绍了近几年在声学多层结构介质特异声传输调控和原型声学功能器件方面的一些工作进展。从声学的角度,系统介绍了多层结构介质的有效介质理论,系统声学参数的各向异性近似,在声波隐身斗篷中的应用。通过严格散射理论推导和有限元数值模拟研究了声斗篷系统的近场声压分布和远场散射强度,发现该设计可在低频较宽的频带范围内显著降低被遮蔽区域的声散射截面;在此频率区间内,声信号散射截面随着频率的提高而增加,其截止频率由薄层厚度决定。最后对该领域的研究作了若干展望。     

Plane wave basis in Galerkin BEM for bidimensional wave scattering

This paper considers the problem of scattering of a time-harmonic acoustic incident wave by a bidimensional hard obstacle. The numerical solution to this problem is found using a Galerkin wave boundary integral formulation whereby the functional space is built as the product of conventional low order piecewise polynomials with a set of plane waves propagating in various directions. In this work we improve the original method by presenting new strategies when dealing with irregular meshes and corners. Numerical results clearly demonstrate that these improvements allow the handling of scatterers with complicated geometries while maintaining a low discretization level of 2.5–3 degrees of freedom per full wavelength. This makes the method a reliable tool for tackling high-frequency scattering problems.     

Ultrasonic inverse scattering of multidimensional objects buried in multilayered elastic background structures

The authors focus on the multidimensional inverse scattering of objects buried in an inhomogeneous elastic background structure. The medium is probed by an ultrasonic force and the scattered field is observed along a receiver array. The goal is to retrieve both the geometry (imaging problem) and the constitutive parameters (inverse problem) of the object through an appropriate multiparameter direct linear inversion. The problem is cast in terms of a vector integral equation elastic scattering framework. The multidimensional inverse scattering problem, being nonlinear and ill-posed, is linearized within the Born approximation for inhomogeneous background, and a minimum-norm least-square solution to the discretized version of the vector integral formulation is sought. The solution is based on a singular value decomposition of the forward operator matrix. The method is illustrated on a 2-D problem where constrained least-square inversion of the object is performed from synthetic data. A Tikhonov regularization scheme is examined and compared to the minimum-norm least-square estimate.