稳态问题混合边界积分方程的高精度求积法与分裂外推

提出了求积法解稳态问题的混合边界积分方程,它拥有高精度,低复杂度.通过并行地解粗网格上的离散方程,根据误差的多参数渐近展开,应用分裂外推算法得到高精度的近似解,同时获得后验误差估计.     

定常Stokes问题的介积分方程的高精度求积方法与外推

借助Ｓｉｄｅ＝Ｉｓｒａｅｌｉ求积公式,给出解Ｓｔｏｋｅｓ问题的边界积分方程的机械求积方法。此方法精度较高、计算量少,近似解的误差有奇数幂的渐近展开,这表明使用Ｒｉｃｈａｒｄｏｎ外推能够改善近似解的精度阶。     

第二类弱奇异积分方程的高精度Nystrom方法与外推

借助奇异函数的新型求积公式，建立了解第二类弱奇异积分方程的高精度Ｎｙｓｔｒｏｍ算法及渐近展开式。数值试验表明本文的算法较常用的投影法计算量大大减少，崦精度却很高，其外推法打丰了Ｆ．Ｃｈａｔｅｌｉｎ认为非光滑核积分方程的 解外推缺少理论依据的断定。     

宽带最优空域矩阵滤波器设计

本文首先针对窄带,分别设计出通带误差或阻带响应满足设定条件,以及通阻带归一化误差之和最小的最优空域矩阵滤波器.利用Lagrange乘子理论,给出了最优解,以及确定最优Lagrange乘子的方程,显著提高了空域矩阵滤波器的设计效率.通过广义奇异值分解,进一步提高了设计效率.其次,从理论上分析了特定频带空域矩阵滤波器对其他...     

基于平面问题的位移压力混合配点法

引入压力变量,将弹性力学控制方程表达为位移和压力的耦合偏微分方程组,采用重心插值近似未知量,利用重心插值微分矩阵得到平面问题控制方程的矩阵形式离散表达式.采用重心插值离散位移和应力边界条件,采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法求解过约束方程组,得到平面问题位移数值解.数值算例验证了所提方法的有效性和计算精度.     

有限差分法中的贴体坐标变换

分析了有限差分法曲线贴体坐标系下守恒型控制方程的推导过程,认为在离散条件下所采用的数学恒等式不成立,推断目前CFD广泛采用的齐次方程是原始Descartes直角坐标系下方程的近似,提出增加源项的非齐次方程作为离散等价方程.采用数值实验研究了源项,结论是大部分情况下源项不等于0,且对数值解的影响大于差分格式的截断误差,在分析了引起源项非0的原因和推导过程后,提出源项离散的相容性准则.利用坐标变换系数和守恒型方程对流通量的特性,建立了源项隐式计算的耦合算法,通过数值实验证明耦合算法有效消除了坐标变换引起的误差.      

YG算法设计分数傅里叶变换衍射光学光束整形器件

针对分数傅里叶变换衍射光学光束整形器件,比较了按离散点与按菲涅耳积分计算得到的变换矩阵间的差距,从中分析得出按离散点计算的变换矩阵引入的离散化误差与分数傅里叶变换系统参数的关系.模拟计算结果表明,该离散化误差不能忽略.为了在输出面上得到真实的光强分布,应采用菲涅耳积分计算其变换矩阵.鉴于该变换矩阵非幺正,本文利用YG算法进行了光束整形器件的设计.     

基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程

波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.     

用线化和校正法近似求解一种非谐振动

利用拉格朗日方程建立了单质点弹簧振子非线性振动方程,作出了回复力、势能随坐标的变化曲线以及相图.应用第一类完全椭圆积分求出了非线性振动周期的精确解.应用线化和校正法对单质点弹簧振子的周期和近似解进行了求解.利用Maple计算机绘图,分别作出了它们的周期近似解与周期精确解随振幅的变化曲线以及近似解与数值解的变化曲线.利用线化和校正法所求得的近似解与数值解比较,具有简单实用、精度高、相对误差低等优点,在求解非线性振动中具有一定的实用价值.     

基于磁荷面分布的舰船磁场预测方法

利用等效源观点, 把钢铁建造的舰船船体视为等效磁荷面分布, 建立了该分布与磁场传感器测量值之间应满足的第一类Fredholm积分方程, 然后利用广义逆矩阵对积分方程离散化后得到的线性方程组进行求解, 并以矩阵条件数来衡量方程组的病态程度, 最后利用求解得到的磁荷分布来预测舰船在空间中任意点上产生的磁场. 数值模拟表明该方法可以准确地识别磁性从而得到精确的磁场预测值. 最后利用一个船模实验验证了本方法. 关键词： 舰船 磁场 磁荷 积分方程     

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.     

High-order quadratures for the solution of scattering problems in two dimensions

We construct an iterative algorithm for the solution of forward scattering problems in two dimensions. The scheme is based on the combination of high-order quadrature formulae, fast application of integral operators in Lippmann–Schwinger equations, and the stabilized bi-conjugate gradient method (BI-CGSTAB). While the FFT-based fast application of integral operators and the BI-CGSTAB for the solution of linear systems are fairly standard, a large part of this paper is devoted to constructing a class of high-order quadrature formulae applicable to a wide range of singular functions in two and three dimensions; these are used to obtain rapidly convergent discretizations of Lippmann–Schwinger equations. The performance of the algorithm is illustrated with several numerical examples.     

Differential Quadrature Analysis of Moving Load Problems

The differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Time-Domain Scattering Modelling of Two-Dimensional Cylinder Located on a Rough Surface

Numerical modelling on the transient electromagnetic scattering by a two-dimensional （21）） cylinder located on a time-evolving rough surface is presented by using time-domain integral equations. The proposed special choice of a tapered Gauss pulse incident wave removes the truncation error from the rough surface. Additionally, a two-level averaging technique is utilized to overcome the instability from the time marching procedure of solving integral equations. Excellent correspondences between the surface current distributions, as well as the far-zone fields, computed by the proposed method and that obtained by the traditional method of moments associated with the inverse discrete Fourier transformation scheme demonstrate the accuracy of the modelling.     

Symmetry in extended phase space for singular Lagrangian with subsidiary constraints

A generalization of the Noether theorem to a singular nonholonomic system in the canonical formalism is given and its inverse theorem is presented. Based on the canonical action integral, a generalization of the Poincaré-Cartan integral invariant of a singular nonholonomic system is obtained. It is shown that this invariant is equivalent to the canonical equations of a singular constrained system. Some confusions in the literature are corrected. An example is given.     

Stress fields caused by a dislocation in an anisotropic 3-layer system

We investigated the stress fields caused by a dislocation in an anisotropic 3-layer system. Based on the image method, the original 3-layer system is firstly decomposed into three infinite homogenous systems. The image dislocation densities used as unknowns are then strategically distributed in order to satisfy the boundary conditions. The resulting governing equations are singular Cauchy integral ones. Removing the singular terms yields non-linear Fredhom integral equations of the second kind. The obtained...     

Efficient discretization of Laplace boundary integral equations on polygonal domains

We describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments. While the procedure has applications to the solution of boundary value problems on a wide class of complicated domains, we concentrate in this paper on a particularly simple case: the rapid solution of boundary value problems for Laplace’s equation on two-dimensional polygonal domains. We view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions. The performance of the method is illustrated with several numerical examples.