磁性液体-硅油两层流动界面的数学模拟

采用二维笛卡儿坐标系,数值求解了磁性液体和非磁性液体体系的连续性方程、双动量方程、运动方程以及磁势方程,模拟了磁性液体-硅油的两层流动。其中自由边界采用PLIC VOF方法跟踪．表面张力采用CSF方法模拟。通过对方程的数值求解,研究了外加磁场强度、进口速度分布、初始表面扰动对界面形状和界面张力的影响。     

区间上强奇异积分的一种近似计算方法

51．引言边界元方法是在经典的边界积分方程法的基础上吸取了有限元离散化技术而发展起来的一种偏微分方程数值解法，它已被广泛应用于弹性力学，断裂力学，流体力学，电磁场和热传导等领域的数值计算．由我国学者冯康等首创的自然边界元方法在各种边界元方法中占有特殊地位并具有许多优点［‘］．由于自然边界归化无一例外地导致强奇异积分方程，所以寻求计算强奇异积分的简单易行的数值方法便成为当前积分方程及边界元研究领域中的一个极为重要的课题．强奇异积分被定义为Hadamard有限部分积分，它是传统的Riemann积分和caucny主值积分的…     

二维多域弹性问题虚边界无网格伽辽金法分析

针对复杂的不同材料属性的多域组合问题（比如复合材料交界面上接触应力的计算）,虚边界无网格伽辽金法被进一步研究,提出了二维多域弹性问题虚边界无网格伽辽金法。简要介绍了多域组合思想、子域虚边界元法,详细推导了二维多域弹性问题分析的虚边界无网格伽辽金法,得到具体的离散格式,便于编程,推广研究。方程的加权系数为位移、面力、连续边界上的位移与面力关系式偏导,数值意义明确,公式具体。最后通过计算数值实例为复合材料交界面上接触应力的计算,给出了复合圆盘接触面上的法向、径向应力,分多种方案调整每个子域的虚边界半径值,所得结果与解析解、其他数值方法进行比较。结论是二维多域弹性问题虚边界无网格伽辽金法的方法计算可行、精确性与稳定性好。     

计算三维体积的四边形等参元方法

通过对三维有界区域的边界曲面作四边形网格剖分,用有限元方法处理高斯公式中的曲面积分,由等参变换及双线性插值导出任意四边形单元上曲面积分的数值求积公式．分析求积公式中三阶行列式意义,提出了简单五面体有向体积概念,推导出计算四边形网面所围立体的有向体积叠加方法．数值试验表明该方法对光滑边界三维体积计算有很好的数值逼近．     

一种有效的格子Boltzmann方法格点判断法

绕流问题中若物体边界不规则会给格子Boltzmann方法中离散边界的格点类型判断以及后续边界处理带来一定的困难.本文提出了简单的三角形不包含算法来有效判断不规则离散边界点的格点类型.针对离散边界,通过改进虚拟平衡态分布函数插值法中的虚拟速度,提高了格子Boltzmann方法在边界处的数值稳定性和精度.通过对经典的二维圆柱,方柱和椭圆柱绕流问题的数值模拟,验证了本文方法的有效性.     

TTI介质井间地震波场紧致交错网格高阶有限差分模拟及边界条件

研究井间地震波场的形成过程以及波场的传播机理、规律,对于指导实际井间地震勘探有着重要的意义．从具有倾斜对称轴的横向各向同性介质（TTI）的二维三分量一阶速度-应力弹性波方程出发,采用高阶紧致交错网格差分算子对方程进行差分离散,得到了TTI介质中井间地震波场正演的高阶有限差分格式．并推导了TTI介质完全匹配层吸收边界条件公式和相应的紧致交错网格高阶差分格式,在此基础上实现了二维三分量TTI介质中井间地震波场模拟．数值算例表明：紧致交错网格高阶有限差分方法模拟的记录精度高,数值频散小,该方法能够精确的模拟复杂各向异性介质中的地震波传播过程,可以得到高精度的正演记录．完全匹配层吸收边界能有效地解决人工边界问题,是一种高效的边界吸收算法．     

非均质问题的几种数值方法

计算力学在科学技术发展和工程分析中扮演着越来越重要的角色,其应用范围相当广泛,如航天、航空、机械、土木及材料等领域．本文仅简略介绍众多数值方法中的几种方法（有限元法、边界元法及边界域积分方程法等）及其在非均质问题中的应用以及所存在的问题．     

可压缩全速湍流流动的边界数值处理方法研究

计算流体力学问题的边界条件处理方法关系到数值仿真结果的精确度。为解决算法的精度,提出了三维可压缩湍流流动的边界条件数值处理方法,对所研究的边界类型包括进口边界、出口边界和固体壁面,流动的速度范围涉及亚音速、跨音速和超音速。流场数值仿真采用SIMPLE算法,湍流采用k-ε模型仿真。将边界总结为沟通型和孤立型边界两种类型,对每一控制方程分别阐述特定的数值处理方法。应用提出的边界处理方法对单圆弧凸包通道进行数值仿真获得了合理的结果,跨音速和超音速情形下准确地计算出了流场中存在的激波。     

凹角区域外问题基于椭圆弧人工边界的自然边界元与有限元耦合法

木文以具有凹角长条型内边界的调和方程外问题为例,研究一种以椭圆弧为人工边界的自然边界元与有限元耦合法,给出了耦合变分问题的适定性及近似解的误差估计．理论分析及数值结果表明,用该方法求解带凹角长条型内边界的外问题是十分有效的．     

基于结合扩展精度技术的基本解方法的非线性功能梯度材料热传导问题求解

采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量,将时域问题转化为频域问题求解;采用基本解方法计算得到高精度的频域数值解,再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例,分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点,能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.     

The falkneer-skan equation: Numerical solutions within group invariance theory

The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.     

First vorticity–velocity–pressure numerical scheme for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.     

A Roadmap to Well Posed and Stable Problems in Computational Physics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.     

Artificial boundary method for two-dimensional Burgers’ equation

The numerical solution of the two-dimensional Burgers equation in unbounded domains is considered. By introducing a circular artificial boundary, we consider the initial-boundary problem on the disc enclosed by the artificial boundary. Based on the Cole–Hopf transformation and Fourier series expansion, we obtain the exact boundary condition and a series of approximating boundary conditions on the artificial boundary. Then the original problem is reduced to an equivalent problem on the bounded domain. Furthermore, the stability of the reduced problem is obtained. Finally, the finite difference method is applied to the reduced problem, and some numerical examples are given to demonstrate the feasibility and effectiveness of the approach.     

A Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary

In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.     

A new numerical approach to a non-steady filtration problem

A free-boundary transient problem of seepage flow is studied from a numerical standpoint. From a suitable formulation of the problem in terms of variational inequality we introduce a new numerical approach of the implicit type and based on the finite element method. In this approach the problem is solved on a fixed region and the position of the free boundary is automatically found as part of the solution of the problem; so it is not necessary to solve a succession of problems with different positions of the free boundary. We prove stability and convergence for the approximate solution and we give several numerical results. Work supported by C. N. R. of Italy through the Laboratorio di Analisi Numerica of Pavia.     

Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral–differential equations and comparing with other methods, we find that the two methods are useful.     

A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method.     

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.