新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势,     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势, 具有良好的工程应用前景.     

导热问题的有限单元边界积分方程方法

一、引言 边界元计算方法是七十年代迅速发展起来的一种数值计算方法,其主要优点是:将求解区域微分方程的问题转化成求解边界积分方程的问题,因而一般都把物理问题降了一维求解,使该方法计算效率和求解精度都较高.但它用于时关问题和非线性问题时,积分方程中还含有物理量的区域积分项,该方法的优点几乎全部消失.另外,在边界积分方程离散后,代数方程的系数矩阵为满阵.如果边界单元划分很多,其效率不如具     

三维Helmholtz方程外问题的自然积分方程及其数值解

用文[2,3]提出的自然边界归化方法来处理三维Helmholtz方程的外边值问题。在简要介绍如何用球谐展开的方法得到Helmholtz问题在外球域上的自然积分方程后,给出求解该自然积分方程的一种数值方法及相应的数值算例。     

势问题的径向基函数——局部边界积分方程方法

将基于径向基函数构造的具有插值特性的近似函数和局部边界积分方程方法相结合，建立了求解势问题的径向基函数——局部边界积分方程方法，推导了相应离散方程.与其他边界积分方程的无网格方法相比，本文方法具有数值实现过程简单、计算量小、精度高的优点，并可直接施加边界条件.最后通过算例说明了该方法的有效性. 关键词： 径向基函数 无网格方法 局部边界积分方程 势问题     

基于混合场积分方程的半空间上方金属目标电磁散射特性高效分析

研究一种可以高效求解半空间金属目标电磁散射积分方程方法,电场积分方程适用于任意结构电磁问题分析,但是生成的矩阵条件数大,迭代求解收敛性差;而磁场积分方程生成的矩阵条件数小,迭代收敛性好,但是仅能分析闭合结构问题,本文采用了混合场积分方程方法,同时具备电场积分方程的普适性与磁场积分方程的收敛性.由于混合场积分方程中涉及格林函数的梯度项,为了进一步加快计算效率,本文引入了一种针对半空间格林函数的高效四维空间插值方法,对组成半空间格林函数的索末菲积分进行列表和Lagrange插值,以实现高效的迭代求解,效率在传统混合场积分方程的基础上提高12.6倍.数值结果表明,该方法在保证精度的同时,可以显著降低求解问题的时间.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

在二维叶栅势流计算中边界元法的应用

一、前言 边界元法是近年来兴起的一种新的基于边界积分方程的数值计算方法.Brebbia将其归之为加权剩余法的一个分支,但该法比有限元和有限差分法更具有解析——数值计算特点.有别于区域计算法,边界元法通过引入一个满足场方程的奇异函数作为权函数,将问题的区域计算转化为边界计算.由于所获得的一组边界积分方程仅联系边界上各个     

热传导问题的边界元法

前言 边界元法是基于经典积分方程和有限元概念的一种数值方法,它主要应用于弹性力学和势理论问题。近些年来,用边界元法求解热传导问题已经引起注意。 本文从格林公式入手,对稳态和非稳态的热传导问题推出其边界元法的基本方程。对二个算例的计算结果表明,边界元法求解热传导问题节省计算工作量,精度较高。     

横向各向同性固体中圆柱状散射体对水平偏振剪切波的散射

给出横向各向同性介质中ＳＨ波的标量波动方程，并通过简单的坐标变换，将其化为标准的Ｈｅlｍｈｏｌｔｚ方程．建立了求解散射问题的积分方程，利用边界单元方法数值计算了横向各向同性固体中圆柱状空洞及刚性散射体对ＳＨ波的散射场分布。重点分析了各向异性对空洞散射体散射场指向性的影响．     

The Green's matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs

Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis.     

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.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

A NUMERICAL METHOD FOR SCATTERING FROM ACOUSTICALLY SOFT AND HARD THIN BODIES IN TWO DIMENSIONS

This paper presents a numerical method for predicting the acoustic scattering from two-dimensional (2-D) thin bodies. Both the Dirichlet and Neumann problems are considered. Applying the thin-body formulation leads to the boundary integral equations involving weakly singular and hypersingular kernels. Completely regularizing these kinds of singular kernels is thus the main concern of this paper. The basic subtraction-addition technique is adopted. The purpose of incorporating a parametric representation of the boundary surface with the integral equations is two-fold. The first is to facilitate the numerical implementation for arbitrarily shaped bodies. The second one is to facilitate the expansion of the unknown function into a series of Chebyshev polynomials. Some of the resultant integrals are evaluated by using the Gauss-Chebyshev integration rules after moving the series coefficients to the outside of the integral sign; others are evaluated exactly, including the modified hypersingular integral. The numerical implementation basically includes only two parts, one for evaluating the ordinary integrals and the other for solving a system of algebraic equations. Thus, the current method is highly efficient and accurate because these two solution procedures are easy and straightforward. Numerical calculations consist of the acoustic scattering by flat and curved plates. Comparisons with analytical solutions for flat plates are made.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

Approximate solutions to fractional subdiffusion equations

The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile. The solution was performed with Riemann-Liouville time-fractional derivative since the integral approach avoids the definition of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed through the M-Wright function. The solutions demonstrate some basic issues of the suggested integral approach, among them: a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical results allowing comparisons to the known solutions expressed to the M-Wright function and error estimations.     

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.     

任意边界的Stokes内流问题的数值计算

本文采用一种格林函数方法^[1],研究了包括任意形状的外边界和流场中任意形状的物体的Stokes内流问题。将方程化成边界积分,从而降低一个维度。数值地求解边界面上的应力分量,再进一步算出流动区域的速度场和压力场,各种不同形状边界的算例,都得到了满意的结果。在处理任意边界内的任意形状物体的绕流时,本方法较其它方法有着明显的优点。