二维Helmholtz边界超奇异积分方程解析研究

基于常规边界元法及超奇异边界积分方程复线性耦合的Burton-Miller方法应用于无限域声学问题的最大难点在于处理超奇异积分（二维问题）.目前,此类超奇异积分主要使用各种弱奇异/正则化方法求解,而这些弱奇异/正则化方法具有时间消耗大等弱点.基于围道积分定理,本文给出一种使用常值单元的二维Helmholtz边界超奇异积分的解析表达式.在有限部分积分意义下,所有的奇异和超奇异积分可以解析表达.数值算例表明该解析表达式是有效的.     

Analysis of mutiple-shepers radiation and scattering problems by using a null-field integral equation approach

A systematic approach using the null-field integral equation in conjunction with the degenerate kernel is employed to solve the multiple radiation and scattering problems. Our approach can avoid calculating the principal values of singular and hypersingular integrals. Although we use the idea of null-field integral equation, we can locate the point on the real boundary thanks to the degenerate kernel. The proposed approach is seen as one kind of semi-analytical methods, since the error is attributed from the truncation of spherical harmonics. Finally, the numerical examples including one and two spheres are given to verify the validity of proposed approach.     

基于Burton-Miller方程的轴对称结构声学边界元方法

提出了一种求取轴对称结构任意边界条件下声辐射特性的边界元方法。采用Burton和Miller改进型公式将高阶奇异项转化为弱奇异项之和,保证声辐射参数的唯一性,且计算简单精确。将结构表面声压与振速按照旋转轴角度进行Fourier级数展开,利用级数的正交性建立各项待定系数的求解公式;然后转化格林函数的法向偏导为切向偏导,方便直接计算各项积分,并将面积分公式表示为沿结构边界的线积分和沿旋转角度的积分;进一步采用二次等参单元离散结构边界线,建立声压与振速的关系矩阵,从而确定结构声辐射参数。以脉动球源和横向振动球源为例计算,与解析解和传统边界元法结果作对比,说明该方法的有效精确性。     

Multilevel fast multipole algorithm for acoustic wave scattering by truncated ground with trenches

The multilevel fast multipole algorithm (MLFMA) is extended to solve for acoustic wave scattering by very large objects with three-dimensional arbitrary shapes. Although the fast multipole method as the prototype of MLFMA was introduced to acoustics early, it has not been used to study acoustic problems with millions of unknowns. In this work, the MLFMA is applied to analyze the acoustic behavior for very large truncated ground with many trenches in order to investigate the approach for mitigating gun blast noise at proving grounds. The implementation of the MLFMA is based on the Nystrom method to create matrix equations for the acoustic boundary integral equation. As the Nystrom method has a simpler mechanism in the generation of far-interaction terms, which MLFMA acts on, the resulting scheme is more efficient than those based on the method of moments and the boundary element method (BEM). For near-interaction terms, the singular or near-singular integrals are evaluated using a robust technique, which differs from that in BEM. Due to the enhanced efficiency, the MLFMA can rapidly solve acoustic wave scattering problems with more than two million unknowns on workstations without involving parallel algorithms. Numerical examples are used to demonstrate the performance of the MLFMA with report of consumed CPU time and memory usage.     

Two-dimensional shear waves scattering by a large rigidity strip with interface crack

A two-dimensional problem of shear horizontal (SH) waves scattering by a finite width planar elastic (piezoelectric) inclusion partially debonded from its surrounding elastic matrix is investigated using the effective boundary conditions and singular integral equations technique. The case of large rigidity inclusions with blunted tips is considered, in which the upper face of the inclusion is perfectly bonded to the matrix. The debonding region is modeled as interface crack with non-contacting faces. Using the Green theorem the mixed boundary value problem is reduced to a system of the hypersingular integral equations. Numerical results of the scattering fields characteristics are presented. The effects of incidence direction, various material parameters of the strip on the scattering field are discussed and phenomenon of the non-specular reflection of SH waves is considered. The accuracy of the numerical results is confirmed by the use of analytical approximate problem solution of high-frequency SH waves scattering on a finite hard/soft inclusion.     

边界元奇异与近奇异数值积分方法及其应用于大规模声学问题

提出了综合处理Burton-Miller方法所导致的奇异积分与近奇异积分问题的数值求积方法,以此改进了基于常量元素的常规边界元和低频快速多极边界元方法。对于奇异积分问题,利用Hadamard有限积分方法进行解决;对于近奇异积分问题,则采用极坐标变换法和PART方法(Projection and Angular&;Radial Transformation)进行克服。与解析解和LMS Virtual.Lab商业软件的结果比较验证了方法的正确性,并对比分析了奇异积分与近奇异积分对计算精度的影响。采用低频快速多极子方法以加速常规边界元法的计算效率,计算分析了计算复杂度,并成功实现了34万自由度大规模问题的计算。结果表明,近奇异积分问题主要由超奇异核函数引起,对计算精度的影响不容忽略;快速多极边界元法的精度与常规边界元法一致,但计算复杂度要远低于后者。      

Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems

The numerical quadrature methods for dealing with the problems of singular and near-singular integrals caused by Burton-Miller method are proposed,by which the conventional and fast multipole BEMs(boundary element methods) for 3D acoustic problems based on constant elements are improved.To solve the problem of singular integrals,a Hadamard finite-part integral method is presented,which is a simplified combination of the methods proposed by Kirkup and Wolf.The problem of near-singular integrals is overcome by the simple method of polar transformation and the more complex method of PART(Projection and Angular Radial Transformation).The effectiveness of these methods for solving the singular and near-singular problems is validated through comparing with the results computed by the analytical method and/or the commercial software LMS Virtual.Lab.In addition,the influence of the near-singular integral problem on the computational precisions is analyzed by computing the errors relative to the exact solution.The computational complexities of the conventional and fast multipole BEM are analyzed and compared through numerical computations.A large-scale acoustic scattering problem,whose degree of freedoms is about 340,000,is implemented successfully.The results show that,the near singularity is primarily introduced by the hyper-singular kernel,and has great influences on the precision of the solution.The precision of fast multipole BEM is the same as conventional BEM,but the computational complexities are much lower.     

连续谱Faddeev积分方程的新解法

在动量空间中具有定域势的Faddeev方程是二维积分方程,在破裂过程和三体散射一类的连续谱情况下,方程的积分核是奇异的。本文根据奇异积分方程一般理论提出一种求解二维方程的数值方法。实践证明数值解是收敛的,全运动学微分截面的计算值与实验数据十分符合。     

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.     

热应力边界元法中几乎超奇异积分的计算

通过分部积分变换将热弹性力学应力边界积分方程中的超奇异积分转化为强奇异积分,然后与另一个强奇异积分求和,得到仅含几乎强奇异的热应力自然边界积分方程.再对其中的几乎强奇异积分施以正则化,消除了热弹性力学边界元法中的几乎奇异积分,可以准确计算出热弹性力学问题中近边界内点的热应力.算例证明了方法的有效性.     

Quarkonium bound-state problem in momentum space revisited

A semi-spectral Chebyshev method for solving numerically singular integral equations is presented and applied in the quarkonium bound-state problem in momentum space. The integrals containing both, logarithmic and Cauchy singular kernels, can be evaluated without subtractions by dedicated automatic quadratures. By introducing a Chebyshev mesh and using the Nystrom algorithm the singular integral equation is converted into an algebraic eigenvalue problem that can be solved by standard methods. The proposed scheme is very simple to use, is easy in programming and highly accurate.     

三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method

Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.     

New Way in Few-Body Scattering Calculations

The key features of the new general approach to solution of few-body scattering problems in hadronic, nuclear and atomic physics are presented and discussed in the paper. The approach is based on a general idea of the lattice-like discretization of few-body continuum using the stationary wave-packet basis in momentum space. The new technique includes an efficient averaging and smoothing of singular kernels of the scattering integral equations over the lattice cells. So, such an averaging procedure allows us to transform the complicated singular integral kernels into usual matrices with regular and smooth matrix elements. Such a transformation is shown to lead to an enormous simplification of the solving procedure for scattering equations.     

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.     

A Nyström method for weakly singular integral operators on surfaces

We describe a modified Nyström method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nyström and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in lieu of such representations. This results in a scheme which is mathematically equivalent to Galerkin discretization in that the resulting matrices are related to those obtained by Galerkin methods via conjugation with well-conditioned matrices, but which avoids the evaluation of double integrals. Moreover, we incorporate a new mechanism for approximating the singular integrals which arise from the discretization of weakly singular integral operators which is considerably more efficient than standard methods. We illustrate the performance of our method with numerical experiments.     

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.     

-Body Quantum Scattering Theory in Two Hilbert Spaces: -Body Integral Equations

Scattering for a nonrelativistic system of distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild H?lder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with . Received October 27, 1997; accepted for publication December 29, 1997     

弹性力学的重构核粒子边界无单元法

将重构核粒子法(RKPM)和边界积分方程方法结合，提出了一种新的边界积分方程无网格方法——重构核粒子边界无单元法(RKP-BEFM).对弹性力学问题，推导了其重构核粒子边界无单元法的公式，研究其数值积分方案，建立了重构核粒子边界无单元法离散化边界积分方程，并推导了重构核粒子边界无单元法的内点位移和应力积分公式.重构核粒子法形成的形函数具有重构核函数的光滑性，且能再现多项式在插值点的精确值，所以本方法具有更高的精度.最后给出了数值算例，验证了本方法的有效性和正确性. 关键词： 重构核粒子法 弹性力学 边界无单元法     

Numerical and Asymptotic Solution of the Problem of Oscillations of an Inhomogeneous Waveguide with an Annular Crack of Finite Width

Acoustical Physics - The problem of waves in an inhomogeneous cylindrical waveguide with an annular crack is considered. A system of integral equations with hypersingular kernels is obtained for...