Helmholtz声学边界积分方程中奇异积分的计算

提出了一种非等参单元的四边形坐标变换,它将积分的曲面单元映射为另一四边形单元,通过两次坐标变换引入的雅可比行列式可以消除Helmholtz声学边界积分方程中的弱奇异型O(1／r))积分．而且利用δr／δn以及坐标变换可以同时消除坐标变换无法消除的Cauchy型(O(1／r^2))奇异积分,并给出了消除奇异性的详细证明．该方法给Helmholtz声学边界积分方程中的弱奇异积分与Cauchy奇异积分的计算以及编程提供了极大便利。     

非线性奇摄动积分-微分发展方程Robin问题广义解

讨论了一类广义非线性奇异摄动积分-微分发展方程Robin问题.首先,利用广义Fredholm积分方程求解方法,得到了模型的外部解.其次,引入多重尺度变量,构造了Robin问题解的边界层校正项.然后利用伸长变量,得到了解的初始层校正项,并构造了奇异摄动问题的形式解的合成展开式.最后,用泛函分析不动点理论证明了广义解的渐近展开式的一致有效性.     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

带有Neumann边界波动方程初边值问题的达朗贝尔类解

D’Alembert方法通常应用于无限长弦自由振动初值问题的求解,基于这种思想研究带有Neumann边界波动方程初边值问题的达朗贝尔类精确解。对于有限长区间上的波动方程初边值问题,通常采用分离变量法求解。现用适当的延拓方法:一种是直接通过逐次延拓时间t,获得有限长区间带有Neumann边界的波动方程初边值问题按时间分段表示的解;另一种方法是通过对初始位移和速度的定义域进行延拓,获得有限长区间带有Neumann边界的波动方程初边值问题的D’Alembert类解。     

气动声学Lighthill方程的Kirchhoff积分解分析

Lighthill的声类比(acoustic analogy)是目前气动声定量预测中应用最为广泛的一种方法。使用非齐次波动方程的Kirchhoff积分公式对Ligthhill方程进行求解。Kirchhoff公式中的延迟时间表示不同位置点声源对场点声压叠加时的相位作用,推导时强调延迟时间函数的导数运算。基于Kirchhoff积分公式对于有物体存在于流场中的情况,详细推导了Curle解,并对Curle公式中的各声源项进行了分析。文章有助于气动声学初学者正确地认识声类比理论,加深对Curle公式的理解。     

半空间内二维圆柱声散射理论解析解

采用理论解析方法研究半空间内二维非紧致圆柱的声散射。齐次Helmholtz方程的解在柱坐标系内用自由空间格林函数的级数展开式表示。基于镜像源方法,利用刚性半空间边界反射圆柱散射声波来解决半空间边界和圆柱之间的多重散射。结合等效源原理,处理半空间边界质量型阻抗特性和刚度型阻抗特性对声传播的影响,推导单位强度简谐单极子点声源产生声场的理论表达式。总声场可以表示为四个分量的总和:入射声场、反射波以及圆柱和镜像圆柱的散射声场。采用边界积分方法对声散射进行计算,以验证理论公式的正确性。点声源模型的理论解析值与边界积分方法数值解在研究的波数和观察点角度范围内一致。     

Helmholtz方程基于变限积分法的数值求解

Helmholtz方程是一类描述电磁波的椭圆型偏微分方程，在力学、声学和电磁学等领域应用广泛。为了消除因高波数引起的污染效应，数值求解Helmholtz方程的传统方法是对网格进行加密，网格加密不仅增加了时间复杂度，且离散后的矩阵通常是病态的。因此，寻求对任意波数都有效的方法是必要的。在有限体积法的基础上，引入变限因子，将微分方程完全转换成积分方程，利用一元三点和二元九点Lagrange插值公式，构造含三对角矩阵的离散格式，分别对一维和二维Helmholtz方程进行变限积分法的数值求解。该方法适用于任意波数，求解过程物理意义明确，数值格式简单。对于一维Helmholtz方程研究了变限因子对误差的影响，利用Taylor展式及Lagrange插值余项公式进行误差估计，证明离散格式的截断误差达到二阶。数值实例表明该离散格式的变限因子和步长相等时，误差阶较低。对二维Helmholtz方程，探究不同波数对数值解的影响，证明离散格式的截断误差达到三阶。数值实例表明，对于不同的波数，数值格式都有较好的精度，高波数没有引起污染效应。     

美式期权价格的积分表示及其数值方法

本文利用Laplace变换方法得到带连续红利的美式看涨期权价格的积分表示,以及最优执行边界满足的一个非线性的第二类Volterra积分方程.然后用数值积分公式给出了积分方程的数值解,从而得到了带连续红利的美式看涨期权价格及其执行边界的数值解.     

解Reissner板弯曲问题的一个新的边界元法

本文从虚功原理出发,以胡海昌导出的E、Reissner板弯曲理论归结为求解两个位移函数作为中间变量,推导出三个广义位移和三个广义力表示的边界积分方程。本文提出的方法适用于任意边界,任意载荷的薄板,中厚板弯曲。文未给出了固支。简支和自由三类边界的算例,均得到满意的结果。     

粘弹性分析的复变边界积分方程法

本文应用弹性－粘弹性对应原理提出了基于复位势基本解的二维粘弹性分析的复变边界积分方程方法，给出了所有基本关系式，编制了相应的计算程序并给出计算实例，与已有工作相比，本方法具有公式统一，程序简洁通用，边界单元数少和效率高等特点。     

车内声场的数学模型建立

本文首先利用Ｈｅｌｍｈｏｌｔｚ方程和Ｇｒｅｅｎ定理推导出适合多种边界条件的车内声场边界积分方程，然后利用边界元数值分析技术离散方程，得到已知某一封闭空间边界的振动特性求解其内部声压的边界元数学模型。作为验证，本文还对两个实例进行了试验，结果表明边界元计算值与理论值和试验实测值吻合良好。     

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Accelerating boundary integral equation method using a special‐purpose computer

We present a new solution to accelerate the boundary integral equation method (BIEM). The calculation time of the BIEM is dominated by the evaluation of the layer potential in the boundary integral equation. We performed this task using MDGRAPE‐2, a special‐purpose computer designed for molecular dynamics simulations. MDGRAPE‐2 calculates pairwise interactions among particles (e.g. atoms and ions) using hardwired‐pipeline processors. We combined this hardware with an iterative solver. During the iteration process, MDGRAPE‐2 evaluates the layer potential. The rest of the calculation is performed on a conventional PC connected to MDGRAPE‐2. We applied this solution to the Laplace and Helmholtz equations in three dimensions. Numerical tests showed that BIEM is accelerated by a factor of 10–100. Our rather naive solution has a calculation cost of O(N² × N_iter), where N is the number of unknowns and N_iter is the number of iterations. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary integral methods in low frequency acoustics

Abstract

This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

Dual boundary element analysis of oblique incident wave passing a thin submerged breakwater

In this paper, the dual integral formulation for the modified Helmholtz equation in solving the propagation of oblique incident wave passing a thin barrier (a degenerate boundary) is derived. All the improper integrals for the kernel functions in the dual integral equations are reformulated into regular integrals by integrating by parts and are calculated by means of the Gaussian quadrature rule. The jump properties for the single layer potential, double layer potential and their directional derivatives are examined and the potential distributions are shown. To demonstrate the validity of the present formulation, the transmission and reflection coefficients of oblique incident wave passing a thin rigid barrier are determined by the developed dual boundary element method program. Also, the results are obtained for the cases of wave scattering by a rigid barrier with a finite or zero thickness in a constant water depth and compared with those of experiment and analytical solution using eigenfunction expansion method. Good agreement is observed.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.