波动方程外区域问题基于自然边界归化的区域分解算法(英文)

本文以二维波动方程为例 ,研究基于自然边界归化的一种区域分解算法 .首先将控制方程对时间进行离散化 ,得到关于时间步长离散化格式 ,对每一时间步长求解一椭圆型外问题 ;然后引入两条人工边界 ,提出了 Schwarz交替算法 ,给出了算法的收敛性 ,并对圆外区域研究了压缩因子     

改进的无奇异局部边界积分方程方法

在局部边界积分方程方法中,当源节点位于分析域的整体边界上时,局部边界积分将出现奇异积分问题,这些奇异积分需要做特别的处理．为此,提出了对域内节点采用局部积分方程,而对边界节点直接采用移动最小二乘近似函数引入边界条件来解决奇异积分问题,这同时也解决了对积分边界进行插值引入近似误差的问题．作为应用和数值实验,对Laplace方程和Helmholtz方程问题进行了分析,取得了很好的数值结果．进而,在Helmholtz方程求解中,采用了含波解信息的修正基函数来代替单项式基函数进行近似．数值结果显示,这样处理是简单高效的,在高波数声传播问题的求解中非常具有前景．     

一类带裂缝散射问题的边界积分方程方法

声波的散射问题中,如果散射体由不可穿透障碍物和可穿透裂缝两部分组成,障碍物表面分别满足第一类和第三类边界条件,裂缝两边满足不同的第二类边界条件,通过位势理论,可以将此混合问题转化为边界积分方程,通过Fredholm算子理论可以得到这个边界积分方程解的存在性和唯一性,从而获得原问题解的存在和唯一性.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

二阶奇异积分边值问题的正解

By using the fixed point index in cone and the fixed theorem of cone expansion and compression, the existence of positive solutions to the singular second-order boundary value problem is considered.     

A posteriori estimate and asymptotic partial domain decomposition

The method of asymptotic partial decomposition of a domain aims at replacing a 3D or 2D problem by a hybrid problem 3D???1D; or 2D???1D, where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogeneous problems is asymptotically estimated in certain circumstances, but for numerical simulations it is important to be able to determine the location of the junction accurately. In this article, by reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. Some numerical results are presented for a toy problem.     

A New Boundary Element Method for the Biharmonic Equation with Dirichlet Boundary Conditions

We consider a scalar boundary integral formulation for the biharmonic equation based on the Almansi representation. This formulation was derived by the first author in an earlier paper. Our aim here is to prove the ellipticity of the integral operator and hence establish convergence of and error bounds for Galerkin boundary element methods. The theory applies both in two and three dimensions, but only for star-shaped domains. Numerical results in two dimensions confirm our analysis.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

发展方程初边值问题的高阶差分-边界积分方程法及误差分析

本文结合差分方法与边界积分方程方法,提出并研究了一类新的求解发展型方程初边值问题的高阶差分与边界积分方程耦合数值方法.对于有界区域问题与无界区域问题给出了数值计算格式及其误差的先验估计.     

积分上限函数相关问题的探讨

探讨积分上限函数的七个相关问题,给出对应例题和解析过程,旨在使学生对积分上限函数有更深的理解和掌握.     

An Artificial Boundary Condition for the Vortex Movements in Two Dimensions

1 Motion of vortices and cloud in cell method The motion of incompressible inviscid ?ow in two dimensions can be described by the equations ?u ?t (u ?) ρ1 ?P = f (1) ?u = 0 , (2) where u = (u, v), ρ, P , and f = (f1, f2) denote ?uid velocity, densit…     

关于薄板的无网格局部边界积分方程方法中的友解

无网格局部边界积分方程方法是最近发展起来的一种新的数值方法,这种方法综合了伽辽金有限元、边界元和无单元伽辽金法的优点,是一种具有广阔应用前景的、真正的无网格方法．把无网格局部边界积分方程方法应用于求解薄板问题,给出了薄板无网格局部边界积分方程方法所需要的友解及其全部公式．     

某些带变换的边值问题和奇异积分方程

本文主要考虑某些带变换的奇异积分方程。尽管用经典的正则化方法可以求解，我们还是给出一种新的分区跳跃方法，使它们化为较简单的可以系统地求解并可迅即讨论的Ｆｒｅｄｈｏｌｍ积分方程。     

Riemann-Stieltjes积分边值问题正解的存在唯一性

利用解的先验估计和极值原理,研究了一类具有Riemann-Stieltjes积分边值问题正解的存在唯一性．     

Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells

The study of vesicles, capsules and red blood cells (RBCs) under flow is a field of active research, belonging to the general problematic of fluid/structure interactions. Here, we are interested in modeling vesicles, capsules and RBCs using a boundary integral formulation, and focus on exact singularity subtractions of the kernel of the integral equations in 3D. In order to increase the precision of singular and near-singular integration, we propose here a refinement procedure in the vicinity of the pole of the Green-Oseen kernel. The refinement is performed homogeneously everywhere on the source surface in order to reuse the additional quadrature nodes when calculating boundary integrals in multiple target points. We also introduce a multi-level look-up algorithm in order to select the additional quadrature nodes in vicinity of the pole of the Green-Oseen kernel. The expected convergence rate of the proposed algorithm is of order$\mathcal{O}(1/N^2)$ while the computational complexity is of order$\mathcal{O}$($N^2$ln$N$), where $N$ is the number of degrees of freedom used for surface discretization. Several numerical tests are presented to demonstrate the convergence and the efficiency of the method.     

含曲线裂纹圆柱扭转问题的新边界元法

研究含曲线裂纹圆柱的Saint-Venant扭转,将问题化归为裂纹上边界积分方程的求解．利用裂纹尖端的奇异元和线性元插值模型,给出了扭转刚度和应力强度因子的边界元计算公式．对圆弧裂纹、曲折裂纹以及直线裂纹的典型问题进行了数值计算,并与用Gauss-Chebyshev求积法计算的直裂纹情形结果进行了比较,证明了方法的有效性和正确性．     

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement. The research project is supported by the National Basic Research Program under the Grant 2005CB321701 and the National Natural Science Foundation of China under the Grant 10771211.     

求解平片裂纹问题的有限部积分与边界元法

本文利用位移的Somigliana公式和有限部积分的概念,导出了求解三维弹性力学中的任意形状平片裂纹问题的超奇异积分方程组,进而联合使用有限部积分法与边界元法对所得方程建立了数值法.为验证本文的方法,计算了若干数值例子的裂纹面的位移间断及裂纹前沿的应力强度因子,它们与理论值相比符合很好.     

Effective Boundary Conditions for the Heat Equation with Interior Inclusion

Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the thin layer as a thickless surface, on which we impose "effective boundary conditions"(EBCs). These boundary conditions not only ease the computational burden, but also reveal the effect of the inclusion. In this paper, by considering the asymptotic behavior of the heat equation with interior inclusion subject to Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include a Poisson equation on the curve, and a non-local one. It turns out that the EBCs depend on the magnitude of the thermal conductivity in the thin layer,compared to the reciprocal of its thickness.