基于三维弹性问题的多极边界元法核函数分解

多极边界元法已经成功地应用于大规模工程计算中.得到并且证明了基于三维弹性问题的多极边界元法核函数分解的定理(定理1),完善了多击边界元法的数学理论.     

三维快速多极边界元法分析地埋管群传热问题北大核心CSCD

基于三节点三角形线性单元,为克服单元跨叶子积分难题,将三维位势问题快速多极边界元法与几乎奇异积分的半解析算法相结合,实现了三维边界元法中几乎奇异积分的准确计算,该方法适用于U型地埋管薄体结构的换热分析.在制冷、制热两种工况下研究了U型地埋管壁厚对换热量的影响,并进一步分析了管群间的热相互作用.计算结果显示,当管壁导热系数一定时,管壁越厚,对管内流体和土壤之间的换热影响越大.当钻孔间距一定时,管群中埋管数量越多,热干扰现象越强烈,提高管群换热量的主要措施是降低管群间热干扰.因准确计算了几乎奇异积分,三维快速多极边界元法可以有效计算薄体和厚体耦合的三维热传导问题.该文方法和分析结果可为地埋管换热器系统的工程应用提供参考.     

薄体结构温度场的高阶边界元分析

分析了二维问题边界元法3节点二次单元的几何特征,区分和定义了源点相对高阶单元的Ⅰ型和Ⅱ型接近度.针对二维位势问题高阶边界元中奇异积分核,构造出具有相同Ⅱ型几乎奇异性的近似核函数,在几乎奇异积分单元上分离出积分核中主导的奇异函数部分.原积分核扣除其近似核函数后消除几乎奇异性,成为正则积分核函数,并采用常规Gauss数值方法计算该正则积分;对奇异核函数的积分推导出解析公式,从而建立了一种新的边界元法高阶单元几乎奇异积分半解析算法.应用该算法计算了二维薄体结构温度场算例,计算结果表明高阶单元半解析算法能充分发挥边界元法优势,显著提高计算精度.     

多裂纹问题计算分析的本征COD边界积分方程方法

针对多裂纹问题，若采用常规的数值求解技术，计算效率较低.为实现多裂纹问题的大规模数值模拟，建立了本征裂纹张开位移（crack opening displacement, COD）边界积分方程及其迭代算法，并引入Eshelby矩阵的定义，将多裂纹分为近场裂纹和远场裂纹来处理裂纹间的相互影响.以采用常单元作为离散单元的快速多极边界元法为参照，对提出的计算模型和迭代算法进行了数值验证.结果表明，本征COD边界积分方程方法在处理多裂纹问题时取得较大的改进，其计算效率显著高于传统的边界元法和快速多极边界元法.     

速多极边界元法解的存在唯一性

本文对求解3维弹性摩擦接触问题的快速多极边界元法(FM- BEM)在数学理论上作了深入探讨.首先,利用向量和子空间理论找出快速优化广义极小残余算法(GMRES(m) )求解边界元方程组所满足的代数条件,使对工程用FM- BEM解的研究转化为对代数问题的讨论,然后,分三步证明了FM- BEM解的存在唯一性,为FM- BEM求解弹性摩擦接触工程问题提供强有力的数学支撑.     

凸二次函数优化问题在快速多极边界元法中的应用

利用快速多极边界元法(FMM-BEM)求解大规模工程问题最终结为稀疏线性方程组的求解,因此,采用更好的方法求解线性方程组可以提高边界元法的计算效率.本文利用最优化数值技术处理,将稀疏线性方程组的求解等价为求解一个凸二次函数极小化的问题,并利用最优化理论及相关数学理论证明了其解的存在唯一性,为该理论的形成和发展奠定了理论基础.     

求解平面双调和方程边界元法的误差分析

边界元法是求解数学物理方程的一种新的数值计算方法。它与有限元法及有限差分法比较,有很多优点。边界无法特别适合干求解无限域的问题,对干这类问题,有限元法与有限差分法将遇到许多困难。边界元法使处理问题的维数降低一维,即三维问题变成二维问题来处理,二维问题变成一维问题来处理,从而解算一个问题所需要的方程少,求解工作大为简化。边界元法的误差限制在边界上,数值精度一般高干有限元法。 边界元法已逐渐应用于力学和工程科学,但理论上的误差分析较少。在文[1]中,祝家麟给出了求解平面双调和方程的边界元法及其数值实验结果,但没有做误差分析。本     

快速多极边界元法解的存在唯一性

本对求解3维弹性摩擦接触问题的快速多极边界元法(FM—BEM)在数学理论上作了深入探讨．首先，利用向量和子空间理论找出快速优化广义极小残余算法(GMRES(m))求解边界元方程组所满足的代数条件．使对工程用FM—BEM解的研究转化为对代数问题的讨论，然后．分三步证明了FM-BEM解的存在唯一性，为FM-BEM求解弹性摩擦接触工程问题提供强有力的数学支撑．     

三维位势问题快速多极算法整体误差的收敛定理

快速多极算法是加速计算由许多物理问题得出的大型稠密线性方程组的一种有效算法.本文研究了求解三维位势问题快速多极算法整体误差的收敛性问题.首先推导了整体误差的表达式,然后给出了误差上界.其次将结果应用于自适应八叉树结构,得到具体的误差收敛阶.最后通过具体的数值算例验证了本文的结果.本文的方法和结论也可以推广到计算弹性静力学问题和斯托克斯流问题的快速多极算法的误差分析中.     

POISSON方程新的边界积分方程

POISSON方程边界值问题边界元法所应用的边界积分方程,其类型,关于未知位势导数是第一类积分方程,关于未知位势是第二类积分方程。本本文从格林公式出发,通过建立位势的单、双场守恒积分公式,推导出POISSON方程新的边界积分方程,其类型与经典方程相反,关于未知位势是第一类积分方程,关于未知位势导数是第二类积分方程。     

A regular fast multipole method for geometric numerical integrations of Hamiltonian systems

The Fast Multipole Method (FMM) has been widely developed and studied for the evaluation of Coulomb energy and Coulomb forces. A major problem occurs when the FMM is applied to approximate the Coulomb energy and Coulomb energy gradient within geometric numerical integrations of Hamiltonian systems considered for solving astronomy or molecular-dynamics problems: The FMM approximation involves an approximated potential which is not regular, implying a loss of the preservation of the Hamiltonian of the system. In this paper, we present a regularization of the Fast Multipole Method in order to recover the invariance of energy. Numerical tests are given on a toy problem to confirm the gain of such a regularization of the fast method.     

Approximating three‐dimensional potential problems using the complex variable boundary element method (CVBEM)

In the current research, the primary focus is to extend the CVBEM to solving potential problems in three dimensions (3D). This is achieved by applying the CVBEM to three coupled projections of the 3D problem domain, in 2D planes, and then superimposing the resulting corresponding 2D CVBEM solutions. The new 3D CVBEM technique is also applied towards improving 3D problem approximations, which are based on the usual 3D boundary element method (BEM) techniques, by approximating the 3D BEM residual error. Finally, a technique to extend a 3D problem geometry into higher geometric dimensions is introduced, and a corresponding numeric error reduction technique is advanced for use in superimposing multiple dimension approximations to improve 3D approximations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 535–560, 2000.     

A convergence theorem for the fast multipole method for 2 dimensional scattering problems

The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.

     

Adaptive Cross Approximation in an elastodynamic Boundary Element formulation

Elastodynamic phenomena can be effectively analyzed by using the Boundary Element Method (BEM), especially in unbounded media. However, for the simulation of such problems, beside others, two difficulties restrict the BEM to rather small or medium–sized problems. Firstly, one has to deal with dense matrices and secondly the treatment of the kernel functions is very costly. Several approaches have been developed to overcome these drawbacks. Approaches, such as Fast Multipole and Panel Clustering etc. gain their efficiency basically from an analytic kernel approximation. The main difficulty of these methods is that the so called degenerate kernel has to be known explicitly. Hence, the present work focuses on a purely algebraic approach, the adaptive cross approximation (ACA). By means of a geometrical clustering and a reliable admissibility condition, first, a so called hierarchical matrix structure is set up. Then each admissible block can be represented by a low–rank approximation. The advantage of the ACA is based on the fact that only a few of the original matrix entries have to be generated. As will be shown numerically, the presented approach is suitable for an efficient simulation of elastodynamic problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含界面电极压电介质的Green函数

研究了由两个不同压电材料和一半无限长电极组成的复合材料系统的广义二维问题·　基于Stroh公式,提供了当一个线力、线电荷和一个线电偶极子施加在电极端附近时,精确的Green函数解·　进一步地,获得了相应的场强度系数·　这些结果可作为边界元的基本解,以分析更加复杂的压电复合材料断裂问题·     

有势场逆问题的边界元法

本文给出了位势方程逆问题的一种最小二乘边界元解法。控制方程为Ｌａｐｌａｃｅ方程，但一部分边界上未给出任何边值，而只在某些内点上给出了势函值。这一问题在数学上属不适定问题，但在一定条件下存在唯一解。本文同时给出了一种估计解的可靠性的方法。数值试验表明，这类逆问题采用边界元法是非常有效的。     

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fast nonlinear stratified flow over an obstacle

Fast 2-D stratified flow over a hard obstacle is considered. The problem is reduced to a linear boundary value problem by a nonlinear substitution. The linear problem is studied by potential theory. The solution of the nonlinear problem is justified by some estimates.