A Conservative Difference Scheme for Conservative Differential Equation with Periodic Boundary

1　ＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎａｎｄＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｙＰｒｏｐｅｒｔｉｅｓｏｆｔｈｅＳｏｌｕｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｃｏｎｓｅｒｖａｔｉｖｅｆｏｒｍａｎｄｓｉｎｇｕｌａｒｐｅｒｔｕｒｂｅｄｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｉｔｈｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ :Ｌｕ(ｘ) ≡ε(ｐ(ｘ)ｕ′(ｘ) )′ (ｑ(ｘ)ｕ(ｘ) )′-ｒ(ｘ)ｕ(ｘ) =ｆ(ｘ)　　( 0 <ｘ<1 ) ,( 1 )ｕ( 0 ) ≡ｕ( 1 ) ,ｌｕ≡ｕ′( 1 )…     

比例边界有限元二阶灵敏度设计及断裂力学分析

比例边界有限元是一种只需在边界上划分网格且无需基本解的半解析方法,能有效处理应力奇异性和无边界问题.论文提出了一种比例边界有限元的二阶灵敏度分析方法,可以准确而高效地求解响应关于参数的二阶梯度.首先通过建立仅需右特征向量的哈密顿矩阵特征灵敏度分析方程,发展了一种改进的比例边界有限元一阶灵敏度分析方法;其次,进一步通过构建二阶哈密顿矩阵特征灵敏度分析方程,并对比例边界有限元系统方程进行一系列二次直接微分,提出了一种半解析形式的比例边界有限元二阶灵敏度分析方法.该方法被应用于线弹性裂纹结构的形状灵敏度分析和不确定性传播分析.最后,给出了两个数值算例验证论文方法的有效性.     

Geometric shapes of the interface surface of bicomponent flows between two concentric rotating cylinders

In this paper, the shape problem of interface of bicomponent flows between two concentric rotating cylinders is investigated. With tensor analysis, the problem is reduced to an energy functional isoperimetric problem when neglecting the effects of the dissipative energy caused by viscosity. We derive the associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of the second order. Moreover, by considering the effects of the dissipative energy, we propose another total energy functional to characterize the geometric shape of the interface, and obtain the corresponding Euler-Lagrangian equation, which is also a nonlinear elliptic boundary value problem of the second order. Thus, the problem of the geometric shape is converted into a nonlinear boundary value problem of the second order in both cases.     

Equilibrium of layers weakened by loaded cuts

In this paper, a symmetric boundary value problem of the stress analysis for an equilibrium of layer with end-supports covered by diaphragms and weakened by several loaded stress raisers is investigated. The given boundary value problem is reduced to an infinite system of singular integral equations of the second kind. The expressions for stress components in an elastic layer weakened by stress raisers are presented. Based on the developed analytical procedure, extensive numerical investigations have been conducted. The results of these investigations are illustrated graphically exposing some novel qualitative and quantitative knowledge about stress concentration in the layer depending on some geometric parameters of stress raisers and Poisson’s ratio of a layer material.     

Third-order nonlinear singularly perturbed boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅ 1 958,Ｌｅｖｓｉｏｎａｎｄｓｅｖｅｒａｌｏｔｈｅｒａｕｔｈｏｒｓ[1]- [4 ]ｈａｖｅｍａｄｅａｓｕｂｓｔａｎｔｉａｌｓｔｕｄｙｆｏｒｅｘｉｓｔｅｎｃｅａｎｄｅｓｔｉｍａｔｅｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｔｈｉｒｄ＿ｏｒｄｅｒｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｗｉｔｈｔｗｏｐｏｉｎｔｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｘ( 0 ) =Ａ ,ｘ′( 0 ) =Ｂ ,ｘ′( 1 ) =Ｃ ,ｏｒｓｌｉｇｈｔｌｙｅｘｔｅｎｓｉｖｅｌｙｌｉｎｅａｒｂｏ…     

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .     

ON THE SECOND ORDER WAVE DIFFRACTION IN TWO LAYER FLUIDS

ＯＮ　ＴＨＥ　ＳＥＣＯＮＤ　ＯＲＤＥＲ　ＷＡＶＥ　ＤＩＦＦＲＡＣＴＩＯＮ　ＩＮ　ＴＷＯ　ＬＡＹＥＲ　ＦＬＵＩＤＳＷｕＪｉａｎｈｕａ（吴建华）；ＦａｎｇＹｉｎｇ（方颖）（ＲｅｃｅｉｖｅｄＭａｙ４，１９９５；ＲｅｓｉｖｅｄＪｕｎ．２１，１９９６；Ｃｏｍｍ...     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

Anti-plane dynamic problem of an electroelastic cylinder with thin rigid inclusions excited by a system of surface electrodes

Summary The anti-plane mixed boundary problem of electroelasticity for vibrations of an infinite piezoceramic cylinder with a thin rigid inclusion is considered. Using the developed integral representation of the solution, the boundary problem is reduced to a system of singular integro-differential equations of the second kind with resolvent kernels. Calculations yeild the amplitude-frequency characteristics of the piecewise homogeneous cylinder. The behaviour of electroelastic fields, both within the cylinder and on its boundary, is given.     

各向异性平面含斜裂纹的奇异积分方程方法

本文应用平面弹性复变方法，将无限各向异性平面中的任意斜裂纹问题归结为求解一组解析函数边值问题，通过构造适当的积分变换将边值问题转化为奇异积分方程，进而应用Lobotto-Chebyshev数值求积公式，求出该奇异积分方程的数值解，并得到了应力强度因子的近似表达式，最后，给出了一些实例的数值结果，对特例的数值结果与精确结果进行比较，吻合的很好。     

Asymptotic experssion of the solution of general boundary value problem for higher order elliptic equation with perturbation both in boundary and in operator

In this paper basing on (1) and (2), we study the singular perturbation of general boundary value problem for higher order elliptic equation with perturbation both in the boundary and in the operator, so as to establish the asymptotic expression involving two parameters. Thus, the iterative process of finding the asymptotic solution is derived and the estimation of the remainder term is given out, we extend and improve the previously published papers.     

On the mathematical problems of composite materials with a doubly periodic set of cracks

In this paper,the mathematical problem of the second fundamental problem of composite materials with a doubly periodic set of arbitrary shape cracks are investigated,and the interface are arbitrary smooth closed contours.At first,we establish mathematical models by using Muskhelisvili complex variable methods,change the primitive problems into searching complex stress functions which satisfy four boundary value problems and construct forms of the solution,then,under some general restrictions it is reduced to normal type singular integral equation,the unique solvability is proved mathematically     

A spectral approach for the stability analysis of turbulent open-channel flows over granular beds

A novel Orr–Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed is here derived, and the linear stability analysis is developed. The whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem is computed and analyzed. The fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. We propose a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular coefficients, some properties of Jacobi polynomials have been carefully blended with numerical integration of Gauss–Legendre type. The results show a positive agreement with the classical experimental data and allow one to relate the different types of instability to such parameters as the Froude number, wavenumber, and the roughness scale. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.     

Optimal design of vibrating rectangular plates

The thickness function of a simply supported rectangular plate is determined such that the fundamental natural frequency of transverse vibrations attains an optimum value. The volume of the solid plate, its size, and data pertaining to its material properties, are assumed to be given.The problem consists of determining the deflection function corresponding to the optimal plate thickness function from a non-linear, fourth order partial differential eigenvalue problem, derived by variational analysis.Second and higher order normal derivatives of the deflection function are singular along the boundary of the domain. This behaviour is investigated analytically, and taken into account in a finite difference formulation of the problem, which is solved numerically by successive iterations.     

On a Degenerate Free Boundary Problem and Continuous Subsonic–Sonic Flows in a Convergent Nozzle

This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic–sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux. Furthermore, the sonic curve of this continuous subsonic–sonic flow is a free boundary, where the flow is singular in the sense that while the speed is C ^1/2 Hölder continuous at the sonic state, the acceleration blows up at the sonic state.     

On the Principal Boundary Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are investigated.     

Mixed boundary value problems for a cylindrical shell

Two mixed boundary value problems in potential theory for a semi-infinite cylindrical shell are solved. The first is interpreted as a heat conduction problem for an insulated shell containing a circumferential obstruction. The second is the torsion of a cylindrical shell containing a circumferential crack. For both problems the normal derivative of the potential function is taken as zero on the inner and outer shell walls. The boundary conditions at the end of the shell are mixed with respect to the potential function and its normal derivative. The problems are formulated using integral transforms in a manner leading to a singular integral equation which can be solved by numerical means. Intensity factors along the circumference separating the mixed conditions are computed.     

Bernoulli Variational Problem and Beyond

The question of ‘cutting the tail’ of the solution of an elliptic equation arises naturally in several contexts and leads to a singular perturbation problem under the form of a strong cut-off. We consider both the PDE with a drift and the symmetric case where a variational problem can be stated. It is known that, in both cases, the same critical scale arises for the size of the singular perturbation. More interesting is that in both cases another critical parameter (of order one) arises that decides when the limiting behaviour is non-degenerate. We study both theoretically and numerically the values of this critical parameter and, in the symmetric case, ask if the variational solution leads to the same value as for the maximal solution of the PDE. Finally we propose a weak formulation of the limiting Bernoulli problem which incorporates both Dirichlet and Neumann boundary condition.     

Singular perturbation of initial value problem for a nonlinear second order systems

In this paper,the singular perturbation of initial value problem for nonlinearsecond order vector differential equationsε~rx″=f(t,x,x′,ε)x(0,ε)=a,x′(0,ε)=βis discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,aandβ∈R~n.Under suitable assumptions,by using the method of many-parameterexpansion and the technique of diagonalization,the existence of the solution of pertur-bation problem is proved and its uniformly valid asymptotic expansion of higher order isderived.     

Singular perturbation of first boundary value problem for higher order elliptic equations (II)

In this paper we discuss singular perturbations of first boundary value problem for higher order elliptic equations of two parameter and obtain the asymptotic expression for the formal solution containing two-parameter as well as the estimation of its remainder term. These results are the extensions of (2) and (3).