A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.     

Hertz接触问题的精确的积分方程

本文给出Hertz接触问题的精确积分方程,也就是将压力作用下接触面中各点的水平位移也考虑在内而得到的积分方程.     

Solution of a Contact Elasticity Problem with a Rigid Inclusion

Computational Mathematics and Mathematical Physics - An equilibrium problem for an elastic body containing a rigid inclusion is solved. There is a delamination crack on a portion of the interface...     

A Contact Problem for a Half-Space and a Rigid Base with an Axially Symmetric Recess

An axially symmetric problem of frictionless contact interaction of an elastic half-space and a rigid base that has a small surface recess is considered. A corresponding problem of elasticity theory is formulated. The problem is solved by the method of double integral equations. The contact pressure, the surface shape of the half-space after compression, and the size of the gap are found in closed form.     

具有非单调摩擦热弹性接触问题的有限元法

给出了一个变形体和刚性基础之间用双边摩擦表达其接触性质的、静态热弹性问题的方程式及其近似解法.以非单调、多值性表示该摩擦定律.忽略了问题的耦合效应,则问题的传热部分与弹性部分各自独立处理.位移矢量公式化为非凸的次静态问题,用局部Lipschitz连续函数来表示变形体的总势能.用有限单元法近似求解全部问题.     

Discontinuous Galerkin Methods for Solving a Frictional Contact Problem with Normal Compliance

Abstract

Several discontinuous Galerkin (DG) methods are introduced for solving a frictional contact problem with normal compliance, which is modeled as a quasi-variational inequality. Consistency, boundedness, and stability are established for the DG methods. Two numerical examples are presented to illustrate the performance of the DG methods.     

Existence and Uniqueness to a Dynamic Bilateral Frictional Contact Problem in Viscoelasticity

In this paper we examine an evolution problem which describes the dynamic bilateral contact of a viscoelastic body and a foundation. The contact is modeled by a friction multivalued subdifferential boundary condition which incorporates the Coulomb law of friction, the SJK model and the orthotropic friction law. The main result concerns the existence and uniqueness of weak solutions to the hyperbolic variational inequality when the friction coefficient is sufficiently small. The proof is based on a surjectivity result for multivalued operators and a fixed point argument. Research supported in part by the State Committee for Scientific Research of the Republic of Poland (KBN) under Grants no. 2 P03A 003 25 and 4 T07A 027 26.     

求解摩擦接触问题的一个非内点光滑化算法

给出了一个求解三维弹性有摩擦接触问题的新算法,即基于NCP函数的非内点光滑化算法．首先通过参变量变分原理和参数二次规划法,将三维弹性有摩擦接触问题的分析归结为线性互补问题的求解;然后利用NCP函数,将互补问题的求解转换为非光滑方程组的求解;再用凝聚函数对其进行光滑化,最后用NEWTON法解所得到的光滑非线性方程组．方法具有易于理解及实现方便等特点．通过线性互补问题的数值算例及接触问题实例证实了该算法的可靠性与有效性．     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

借助经典平面弹性复变函数方法,研究了单个刚性凸基底压头作用下,带任意形状裂纹十次对称二维准晶半平面弹性的无摩擦接触问题.利用十次对称二维准晶位移、应力的复变函数表达式,带任意形状裂纹的准晶半平面弹性无摩擦接触问题被转换为可解的解析函数复合边值问题,进而简化成一类可解的Riemann边值问题.通过求解Riemann边值问题,得到了应力函数的封闭解,并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式.从压头下方接触应力的表达式可以看出,接触应力在压头边缘和裂纹端点处具有奇异性.当忽略相位子场影响时,该文所得结论与弹性材料对应结果一致.数值算例分别给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的十次对称二维准晶下半平面的结果.该文所得结论为准晶材料的应用提供了理论参考.     

圆形界面刚性线夹杂的反平面问题

研究了在反平面集中力和无穷远纵向剪切作用下,不同弹性材料圆形界面上有多条刚性线夹杂的问题．运用Riemann-Schwarz解析延拓技术与复势函数奇性主部分析方法,首次获得了该问题的一般解答,求出了几种典型情况的封闭解,并给出了刚性线夹杂尖端的应力场分布A·D2结果表明,在反平面加载的情况下圆形界面刚性线夹杂尖端应力具有平方根奇异性,无奇异性应力振荡;应力场与刚性线夹杂的形状,加载方式和材料性质有关．退化结果与已有的解答完全吻合．     

“接触式轮廓仪的自动标注”的问题解析

就2020年"高教社杯"全国大学生数学建模竞赛D题"接触式轮廓仪的自动标注"给出了一种求解方法,并针对学生在参赛论文中出现的问题作了简要的说明与点评.为保证求解的连贯性,论文的前一部分是问题的求解,后一部分是参赛论文的点评.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact-wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.     

冲击接触问题的一种双共振投影梯度算法

根据冲击接触计算模型所需满足的基本控制方程和非线性互补条件,应用非线性互补问题与约束优化的等价关系将非线性互补接触问题转变成一个非线性规划问题,系统地推导建立了冲击接触问题的一种双共轭投影梯度计算方法．增广Lagrange乘子法克服了罚函数要求减小迭代步长以达到计算稳定的限制,即使对于冲击接触问题亦可以采用较大迭代步长,在形成的与原互补问题等价的无约束规划模式下,应用双共轭投影梯度算法提高非线性搜索速度和计算效率．算法模型计算结果表明,所建立的双共轭投影梯度计算理论及方法是正确有效的．     

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.     

Quasistatic Thermoviscoelastic Frictional Contact Problem with Damped Response

We analyze a problem which describes the frictional contact between a thermoviscoelastic body and a rigid foundation. The process is assumed to be quasistatic and the contact is modeled by a general normal damped response condition with friction law and heat exchange. Then we present a variational formulation of the problem, which is set in an abstract form as a system of evolution equations for the displacements and temperature. We establish the existence and uniqueness of the weak solution, using general results on evolution equations with monotone operators and fixed point arguments. Finally, we study the continuous dependence of the solution with respect to the initial data and contact conditions.     

一类具有耦合转移条件Sturm-Liouville问题的逆问题

考虑了具有耦合转移条件Sturm-Liouville(简称S-L)问题的逆问题,在一定条件下,通过利用S-L方程右边的函数f_j(x)确定方程的解,并由数据{u_j(x_0)}_j~∞=i或{p(x_0)(du_j(x_0))-(dx)}_j~∞=1唯一确定S-L算子中的系数p(x)和q(x).其中u_j(x)满足S-L方程,分离边界条件和耦合转移条件,而{f_j(x)}_(j-i)~∞构成L~2(I)的一个基.