平面定常Stokes问题的无奇异第一类边界积分方程

对无奇异边界积分方程归化法的研究,已有的结果都是针对直接变量的,其核心思想是利用刚体位移(包括刚体的转动和平移)或均匀场．然而,对第一类边界积分方程的无奇异边界归化法的研究,至今还未涉足．本文提交一种新方法,归化出平面定常Stokes问题的第一类无奇异边界积分方程,并建立完整的数值求解体系．一个简单的算例表明本文方法可获得理想的数值结果,特别是边界量的数值结果。     

三维横观各向同性介质界面裂纹的边界积分方程方法

基于两相三维横观各向同性介质的基本解和Somigliana恒等式,对三维横观各向同性介质中的任意形状的平片界面裂纹,以裂纹面上的不连续位移为待求参量建立了超奇异积分_微分方程,界面平行于横观各向同性面.根据发散积分的有限部积分理论,应用积分方程方法研究得到裂纹前沿的位移和应力场的表达式、奇性指数以及应力强度因子的不连续位移表达式.在非震荡情形下,超奇异积分_微分方程退化为超奇异积分方程,与均匀介质的超奇异积分方程形式完全相同.     

波动方程的边界单元法

本文从三维波动方程的Kirchhoff积分公式出发,首先经离散化给出无限均匀介质中的边界单元法公式.其后,在引入了波在不同介质面和自由面上的反射、折射系数以后,给出了三维波动方程在分区均匀介质中的边界单元法公式.     

积分算子方程的连续型配置方法

1．引言由于对积分算子方程来说，配置法比Galerkin法具计算量小的优点（少算一重积分），故配置法更受人们重视．但已有的文献几乎都是将配置空间取作非连续的分片多项式样条空间，以得到某种超收敛结果（如［1，2］）．这种方法存在下列不足：（a）光滑核Volterra积分方程与光滑核Fredholm积分方程具完全不同的收敛性质［1］，且需用不同的方法获得其加速收敛结果（比较［31与［4］），尽管Volterra积分方程在理论上被看作是Fredholm积分方程的特殊情形；（b）光滑核Volterra积分方程的配置解不具任何超收敛性，其迭代配置解也只在结点…     

一类共形不变摄动积分方程正解的存在性

本文讨论了一类共形不变摄动积分方程正解的存在性. 我们证明了:当参数对(p, q) 属于集合(-n, 0) × (0,∞) 且pq + p + 2n = 0 时, 对应摄动积分方程存在正解; 而当参数对(p, q) 属于集合(0,∞)×(-∞, 0) 也满足pq +p+2n = 0 时, 摄动积分方程不存在非负解. 这与原共形不变积分方程有着本质的不同, 此结果隐含着这类积分方程正解的存在性取决于解在无穷远处的性态.     

非线性弹性杆波动方程的摄动分析

针对计入横向惯性效应后的非线性弹性杆纵向波动方程进行了分析．在小振幅、长波长的一般情况下,根据远方场简单波理论,采用约化摄动法,得到了NLS方程,并讨论了存在NLS孤子的条件．     

非线性非完整系统Vacco动力学方程的积分方法*

本文给出积分非线性非完整系统Vacco动力学方程的积分方法.首先,将Vacco动力学方程表示为正则形式和场方程形式;然后,分别用梯度法,单分量法和场方法积分相应完整系统的动力学方程,并加上非完整约束对初条件的限制而得到非线性非完整系统Vacco动力学方程的解.     

横观各向同性电磁弹性介质中裂纹对SH波的散射

研究横观各向同性电磁弹性介质中裂纹和反平面剪切波之间的相互作用．根据电磁弹性介质的平衡运动微分方程、电位移和磁感应强度微分方程,得到SH波传播的控制场方程．引入线性变换,将控制场方程简化为Helmholtz方程和两个Laplace方程A·D2通过Fourier变换,并采用非电磁渗透型裂面边界条件,得到了柯西奇异积分方程组．利用Chebyshev多项式求解积分方程,得到应力场、电场和磁场以及动应力强度因子的表达,并给出了数值算例．     

复合多层介质在可变温度和集中荷载作用下的位移和应力

研究了多层介质中的热弹性位移和应力.多层介质具有不同厚度,各层又具有不同的弹性性质,最上层表面上作用热荷载和集中荷载.假设各层分别是均匀、各向同性弹性材料,各层相关的位移分量是轴对称的,对称轴为各层表面的垂线.因此,各层应力函数满足无体力的单一方程.利用积分变换法求解了该方程,对由任意多个层数构造的多层介质,给出了其相应层数基础热弹性位移和应力的解析表达式.并对3层介质和4层介质时的数值结果进行了比较.     

横观各向同性饱和地基与中厚圆板的非轴对称动力相互作用

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题,即首先利用Fourier展开和Hankel变换技术,求解了简谐激励下横观各向同性饱和土地基的非轴对称Biot波动方程,然后按混合边值问题建立地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程．文末给出了算例．数值结果表明,在一定频率范围内,地基表面的位移幅值随激振频率增加而增大,随距离的增大以振荡形式衰减变化．     

2.5维介质Born近似波速反演唯一性

考虑脉冲源引起的２．５维弱不均匀介质波速反演问题，利用线性化方法得到了波速的二维小扰动满足的积分方程，这是一个积分几何的问题，进而由Ｆｏｕｒｉｅｒ变换和脉冲函数的性质将此二维积分方程化为单变量的积分方程，最后用压缩映象理论证明了积分方程解的唯一性。本文给出了二给波速反演的一种新算法。同时，唯一性结果证明了已有的迭代算法的合理性。     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.     

Linearization Ill-Posedness for 2.5-D Ware Equation Inversion Model

Abstract For the weakly inhomogeneous acoustic medium in Ω={(x,y,z):z>0},we consider the inverse problemof determining the density function ρ(x,y).The inversion input for our inverse problem is the wave field givenon a line.We get an integral equation for the 2-D density perturbation from the linearization.By virtue of theintegral transform,we prove the uniqueness and the instability of the solution to the integral equation.Thedegree of ill-posedness for this problem is also given.     

Boundary element method and wave equation

In this paper the boundary integral expression for a one-dimensional wave equation with homogeneous boundary conditions is developed. This is done using the time dependent fundamental solution of the corresponding hyperbolic partial differential equation. The integral expression developed is a generalized function with the same form as the well-known D'Alembert formula. The derivatives of the solution and some useful invariants on the characteristics of the partial differential equation are also calculated.The boundary element method is applied to find the numerical solution. The results show excellent agreement with analytical solutions.A multi-step procedure for large time steps which can be used in the boundary element method is also described.In addition, the way in which boundary conditions are introduced during the time dependent process is explained in detail. In the Appendix the main properties of Dirac's delta function and the Heaviside unit step function are described.     

A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

This paper presents a meshless method, which replaces the inhomogeneous biharmonic equation by two Poisson equations in terms of an intermediate function. The solution of the Poisson equation with the intermediate function as the right-hand term may be written as a sum of a particular solution and a homogeneous solution of a Laplace equation. The intermediate function is approximated by a series of radial basis functions. Then the particular solution is obtained via employing Kansa’s method, while the homogeneous solution is approximated by using the boundary radial point interpolation method by means of boundary integral equations. Besides, the proposed meshless method, in conjunction with the analog equation method, is further developed for solving generalized biharmonic-type problems. Some numerical tests illustrate the efficiency of the method proposed.     

The efficient solution of electromagnetic scattering for inhomogeneous media

We consider an electromagnetic scattering problem for inhomogeneous media. In particular, we focus on the numerical computation of the electromagnetic scattered wave generated by the interaction of an electromagnetic plane wave and an inhomogeneity in the corresponding propagation medium. This problem is studied in the VV polarization case, where some special symmetry requirements for the incident wave and for the inhomogeneity are assumed. This problem is reformulated as a Fredholm integral equation of second kind, which is discretized by a linear system having a special form. This allows to compute efficiently an approximate solution of the scattering problem by using iterative techniques for linear systems. Some numerical examples are reported.     

Collocation discretization for an integral equation in ocean acoustics with depth‐dependent speed of sound

We analyze two collocation schemes for the Helmholtz equation with depth‐dependent sonic wave velocity, modeling time‐harmonic acoustic wave propagation in a three‐dimensional inhomogeneous ocean of finite height. Both discretization schemes are derived from a periodized version of the Lippmann‐Schwinger integral equation that equivalently describes the sound wave. The eigenfunctions of the corresponding periodized integral operator consist of trigonometric polynomials in the horizontal variables and eigenfunctions to some Sturm‐Liouville operator linked to the background profile of the sonic wave velocity in the vertical variable. Applying an interpolation projection onto a space spanned by finitely many of these eigenfunctions to either the unknown periodized wave field or the integral operator yields two different collocation schemes. A convergence estimate of Sloan [J. Approx. Theory, 39:97–117, 1983] on non‐polynomial interpolation allows to show converge of both schemes, together with algebraic convergence rates depending on the smoothness of the inhomogeneity and the source. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundary integral method for scattering problems with cracks buried in a piecewise homogeneous medium

We consider the scattering of time‐harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. The integral representation for a solution is obtained in the form of potentials by using Green's formula. The density in potentials satisfies the uniquely solvable Fredholm integral equation. Then we obtain the existence and uniqueness of the solution. Copyright © 2011 John Wiley & Sons, Ltd.     

圆柱壳的轴对称平面应变弹性动力学解

给出一种圆柱壳的轴对称平面应变弹性动力学问题的解析方法。首先通过引入一特定函数将非齐次边界条件化为齐次边界条件,然后利用分离变量法将位移减去特定函数的量展开为关于贝塞尔函数和时间函数乘积的级数,并由贝塞尔函数的正交性,导出时间函数的方程,容易求得此方程的解。将两者叠加可得弹性动力学问题的位移解。运用此方法,可以避免积分变换,并适宜于各种载荷。文中给出了各向同性和柱面各向同性圆柱壳内表面和实心圆柱外表面受冲击荷载作用以及内表面固定的柱面各向同性圆柱壳外表面受冲击荷载作用的数值结果。     

Coupling of numerical Green''s matrix and boundary integral equations for the elastodynamic analysis of inhomogeneous bodies on an elastic half-space

A coupled system of integral equations (of the domain and boundary types) is formulated for the elastodynamic response analysis of a locally inhomogeneous body on a homogeneous elastic half-space. The method uses the fundamental solution for homogeneous elastostatics in the inhomogeneous domain owing to the lack of a fundamental solution in inhomogeneous elastodynamics.

The integral representation of displacements in the inhomogeneous domain is formulated with the help of this elastostatic fundamental solution by considering the term induced by the inhomogeneity of materials and the acceleration term as the body force term. Then the Green's matrix is obtained numerically from this integral representation and combined with the ordinary boundary integral equations, which are valid in the exterior homogeneous half-space.

Some numerical examples show the efficiency and the versatility of this coupled method.