奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法.该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法.为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子.论文首次将该方法应用于求解平面弹性力学问题.数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度.     

基于时间依赖基本解的奇异边界法模拟二维狄利克雷边界标量波方程

奇异边界法是一个半解析边界配点强格式方法,具有无数值积分和无网格、编程容易以及数学简单等优点。本文首次将时间依赖基本解运用于奇异边界法,计算模拟二维标量波方程;结合确定源点强度因子的反插值技术,提出了二维狄利克雷边界标量波方程源点强度因子的一个经验公式;引进了解决波方程基本解G奇异性的一种无奇异积分处理方法。数值实验证明,基于时间依赖基本解的奇异边界法可精确高效地模拟二维狄利克雷边界标量波方程,在计算效率、精度、稳定性和适应性等方面有明显优势。     

一种新的求解对流扩散边值问题的无网格方法

基于配点法和楔形基函数,提出了一种新的求解对流扩散边值问题的无网格方法。通过一维和二维的问题验证了该数值方法的可行性;并根据数值算例和分析,可以看到该数值方法能达到满意的收敛效果。该数值方法的隐格式形式能够有效地消除对流占优问题的数值振荡现象,是一种真正的无网格方法。     

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)

In this paper the existence of solutions of the singularly perturbed boundary valueproblems on infinite interval for the second order nonlinear equation containing a smallparameterε>0 :is examined,whereα_i,βare constants,and i=0,1 .Moreover,asymptoticestimates of the solutions for the above problems are given.     

Nonsingular kernel boundary element method for thin-plate bending problems

In this paper,the nonsingular fundamental solutions are obtained from Fourierseries under some given conditions.These solutions can be taken as the kernels ofintegral equation.So a new boundary element method is presented,with which allkinds of thin-plate bending problems can’be solved,even with complicated loadings andsinuous boundaries.The calculation is much simpler and more accurate.     

Immersed boundary conditions method for unsteady flow problems described by the Laplace operator

An implicit, spectral algorithm for the analysis of unsteady flow problems governed by the Laplace operator in corrugated geometries is described. The algorithm treats the physical boundary conditions as constraints along lines internal to the solution domain. The method eliminates the need for coordinate generation and can be quickly adapted to changing geometries. Various tests confirm the spectral accuracy in space and the first‐ and second‐order accuracies in time. Copyright © 2007 John Wiley & Sons, Ltd.     

A Sinc‐collocation method with boundary treatment for the Stokes equations

In Stokes equations the velocity u and the pressure p are coupled together by the imcompressibility condition div u =0 which makes the equations difficult to solve numerically. In this paper, a method named Sinc‐collocation method with boundary treatment (SCMBT) is applied to the Stokes equations. The numerical results show that our method is of high accuracy, of good convergence with little computational effort. Copyright © 2006 John Wiley & Sons, Ltd.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

压电材料平面问题的虚边界元-等额配点解法

利用压电材料平面问题的基本解和弹性力学虚边界元方法的基本思想,提出了压电材料平面问题的虚边界元-等额配点解法。该解法继承了传统边界元方法的优点,而避免了传统边界元方法遇到的边界积分奇异性问题。最后给出了压电材料平面问题的一些具体算例,并与解析解作了比较。结果表明本文的方法有很高的精度,是该问题一个十分有效的数值求解方法。     

基于参数曲面三维势问题的边界面法

提出了边界元法(BEM)的一种新的实现方法——边界面法(BFM)。在传统的边界元法中,单元不仅用来进行边界积分和函数插值,而且用来近似几何体。当离散网格较稀疏时,会引起较大几何误差,因而影响计算精度。本文基于参数曲面,将几何实体的边界曲面离散为参数空间里的曲面单元,边界积分和场变量的插值都是在曲面参数空间里进行。积分点...     

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method (FMBEM) for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.     

A higher-order boundary element method for three-dimensional potential problems

A method of eliminating the singularities involved in boundary element methods for three-dimensional potential problems is presented and the non-singular expressions of integrals on an element on which the singular point is situated are given for linear and quadratic interpolation functions. Numerical examples are compared with analytical solutions to show that the higher-order interpolations have better precision.     

An effective boundary method for the analysis of elastoplastic problems

In this paper, a series of effective formulae of the boundary element method is presented. In these formulae, by using a new variable two kernels are only of the weaker singularity of Lnr (where r is the distance between a source point and a field point). Hence, the singularities in the conventional displacement formulation and stress formulation at internal points are reduced respectively so that the "boundary-layer" effect which strongly degenerates the accuracy of stress calculation by using original formulae is eliminated. Also the direct evaluation of coefficients C (boundary tensor), which are difficult to calculate, is avoided. This method is used in elastoplastic analysis. The results of the numerical investigation demonstrate the potential advantages of this method.     

On pressure boundary conditions for thermoconvective problems

Solving numerically hydrodynamical problems of incompressible fluids raises the question of handling first order derivatives (those of pressure) in a closed container and determining its boundary conditions. A way to avoid the first point is to derive a Poisson equation for pressure, although the problem of taking the right boundary conditions still remains. To remove this problem another formulation of the problem has been used consisting of projecting the master equations into the space of divergence‐free velocity fields, so pressure is eliminated from the equations. This technique raises the order of the differential equations and additional boundary conditions may be required. High‐order derivatives are sometimes troublesome, specially in cylindrical coordinates due to the singularity at the origin, so for these problems a low order formulation is very convenient. We research several pressure boundary conditions for the primitive variables formulation of thermoconvective problems. In particular we study the Marangoni instability of an infinite fluid layer and we show that the numerical results with a Chebyshev collocation method are highly correspondent to the exact ones. These ideas have been applied to linear stability analysis of the Bénard–Marangoni (BM) problem in cylindrical geometry and the results obtained have been very accurate. Copyright © 2002 John Wiley & Sons, Ltd.     

A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media

Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council.     

压电材料三维问题的虚边界元——最小二乘配点法

从压电材料三维问题的基本方程出发,利用已有的压电材料三维问题的基本解以及弹性力学虚边界元方法的基本思想和线性叠加原理,提出了压电材料三维问题的虚边界元——最小二乘配点解法。虚边界元解法继承了传统边界元方法的优点,并且有效避免了传统边界元方法中可能遇到的边界积分奇异性问题。最后,文章给出了压电材料三维问题的几个数值算例,并且与解析解做了比较,结果表明本文的方法具有较高的精度,是解决该问题一个十分有效的数值求解方法。     

A numerical implementation using mid-node collocation for the hypersingular boundary contour method

A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional problems does not require any numerical integration at all. In another development, a boundary contour implementation of a regularized hypersingular boundary integral equation (HBIE) using quadratic elements and end-node collocation was proposed and the technique is termed the hypersingular boundary contour method (HBCM). As reported in that work, the approach requires highly refined meshes in order to numerically enforce the stress continuity across boundary contour elements. This continuity requirement is very crucial since the regularized HBIE is only valid at collocation points where the stress tensor is continuous, while the computed stress at the endpoints of a boundary contour element, which is a non-conforming element, is generally not. This paper presents a new implementation of the HBCM for which the regularized HBIE is collocated at the mid-node of a boundary contour element. As the computed stress tensor is continuous at these mid-nodes, there is no need for unusually refined meshes. Some numerical tests herein show that, for the same mesh density, the HBCM using mid-node collocation has a comparable accuracy as the BCM.     

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

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求解位势问题的虚边界元法

本文提出了求解位势问题的虚边界元法，建立了位势问题的虚边界元的离散方程式，推导了离散化求系数的积分解析式。该方法与传统边界元法相比具有不存在奇异积分和边界附近精度较高等优点，可用来计算真空静电场、稳定温度场、流体绕流、介质中的渗流等各类位势问题。大量算例均获得了满意的结果。