双周期裂纹场的不同材料弹性平面焊接问题

本文讨论双周期胞腔中含若干个任意形状裂纹的不同材料的弹性平面焊接问题,根据路见可和方法,对这类弹性平面问题建立起了数学模型,将求解弹性平衡问题化归为寻求复应力函数的问题,并把路见可给出的复Airy函数用于推广方法,更进一步地将寻求复应力函数的问题归结为求解正则型的奇异积分方程,最后证明了其解存在且唯一。     

具双周期孔洞的不同材料弹性平面焊接的第二基本问题

本文讨论双周期胞腔中含任意形状孔洞的不同材料弹性平面焊接的第二基本问题.引进函数ρ_1(z)=∑'{ρ/(z-ρ)~2-2z(ρ/ρ~3)-ρ/ρ~2},构造了推广的变换,将这类弹性平面问题归结为求解正则型的奇异积分方程,最后证明了其解的存在和唯一.所用方法简单、直观,而且由于是构造性的,因而有利于具体的数值求解.     

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.     

双周期平面弹性基本问题

周期平面弹性基本问题,路见可教授早已解决。至于双周期平面弹性问题,W.T.Koiter在基本周期胞腔仅含一个洞的情况下,对第一基本问题曾作过讨论,但是求解复应力函数时理论上有明显的漏洞,问题并没有解决。本文首先建立复应力函数的一般表达式,然后用类似于的方法,把寻求复应力函数的问题归结为求解一唯一可解的第二类Fredholm积分方程,从而完全解决了基本周期胞腔含任意个任意形状的洞的一般情况下的第一、第二基本问题。     

带周期裂缝的弹性半平面问题

本文把具有任意形状和个数的周期裂缝的弹性半平面基本问题化为了某种特殊类型的奇异积分方程,证明了其解的存在和唯一。并对带周期共线直裂缝的弹性半平面问题,给出了封闭形式的解。     

一种获得各向异性平面梁在任意荷载作用下弹性解的新方法

通过求解函数方程,给出了一种获得各向异性线性平面梁弹性解的新方法,该方法可以考虑任意形式的荷载以及各种端部支撑条件．将该方法与传统的逆解法或者半逆解法比较,其最大的好处在于不需要猜测应力函数的形式而直接获得问题的精确解．算例验证了该方法的正确性,同时也提供了一种求解平面梁承受任意荷载的新思路.     

复合材料焊接线出现裂缝的平面弹性基本问题

本文用复变方法讨论了复合材料任意形状焊接线上出现若干条裂缝时的平面弹性第一和第二基本问题，把寻求复应力函数的问题分别归结为求解某种正则型奇异积分方程和正则型奇异积分方程组，并证明了其解存在且唯一。     

几类带双周期裂缝的无限平面弹性问题

本文讨论了基本周期胞腔内含一条任意形状光滑裂缝时的三类双周期平面弹性问题。本文采用复变方法求解,把寻求复应力函数的问题归结求解某种正则型奇异积分方程,证明了这种方程的解存在且唯一。     

带任意裂纹的一维六方准晶弹性半平面第一基本问题

研究了周期平面内含任意裂纹的一维六方准晶的弹性半平面第一基本问题.首先借助保角变换将半平面第一基本问题转化为单位圆内带任意裂纹的第一基本问题;再利用复变函数方法将求有界域内的弹性平衡问题转化为奇异积分方程的求解,并证明方程是唯一可解的.该问题的求解为研究工程断裂问题提供了理论方法.     

弹性长条的周期基本问题

Howland、唐立民、Isida 等对一个周期带内含有一个圆孔的弹性平面周期基本问题作了研究.把具有一个任意形状周期孔的平面弹性问题化为 Fredholm 积分方程.森口繁一在文献[4]§10中给出了在应力是周期的条件下函数的一般表达式,并分离出周期部分和非周期部分.路见可教授在文献[1]中把弹性平面或半平面中     

A class of nondifferentiable control problems

Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.     

A method for reduction of differential problems to integral equations

A differential problem in an arbitrary domain is replaced with a problem with the same differential operator posed in a canonical domain for which the solution is either easily written out or is relatively easy to construct in terms of a function which is already known on the boundary of the canonical domain. The function is determined using the boundary conditions of the original problem, which in general leads to an integral equation. Some typical problems of mathematical physics are considered as an example.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 8–14, 1987.     

Reference point approximation method for the solution of bicriterial nonlinear optimization problems

This paper presents a reference point approximation algorithm which can be used for the interactive solution of bicriterial nonlinear optimization problems with inequality and equality constraints. The advantage of this method is that the decision maker may choose arbitrary reference points in the criteria space. Moreover, a special tunneling technique is given for the computation of global solutions of certain subproblems. Finally, the proposed method is applied to a mathematical example and a problem in mechanical engineering.     

On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.     

The many-to-many location-routing problem

In this paper the many to many location routing problem is introduced, and its relationship to various problems in distribution management is emphasised. Useful mathematical formulations which can be easily extended to cater for other related problems are produced. Techniques for tackling this complex distribution problem are also outlined.     

Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

Investigation of boundary value problems for plane-parallel fluid flows in an anisotropic porous medium

We pose and consider the first and second boundary value problems and the transmission boundary value problem for plane-parallel steady flows in an anisotropic porous medium characterized by the permeability tensor, which is not necessarily symmetric. If the anisotropic medium is homogeneous, then the solutions of the problems in the case of canonical boundaries (a straight line or an ellipse) can be found in closed form, and in the case of arbitrary smooth boundaries, the study of these problems can be reduced with the use of Cauchy type integrals to the solution of inhomogeneous integral equations of the second kind. These problems are mathematical models of topical practical problems that arise, for example, in fluid (water or oil) recovery from natural soil strata of complicated geological structure.     

Dynamic adaptation for parabolic equations

A dynamic adaptation method is presented that is based on the idea of using an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. Using some model problems as examples, the generation of grids that adapt to the solution is considered for parabolic equations. Among these problems are the nonlinear heat transfer problem concerning the formation of stationary and moving temperature fronts and the convection-diffusion problems described by the nonlinear Burgers and Buckley-Leverette equations. A detailed analysis of differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dynamic adaptation method universal, effective, and algorithmically simple. The universality is achieved due to the use of an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent problems of mathematical physics with various mathematical features. Among these features are large gradients, propagation of weak and strong discontinuities in nonlinear transport and heat transfer problems, and moving contact and free boundaries in fluid dynamics. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of the parabolic equations allows one to automatically control the nodes’ motion so that their trajectories do not intersect. This mechanism can be applied to all parabolic equations in contrast to the hyperbolic equations, which do not include repulsive components. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation, which is independent of the form and type of the differential equations.     

Weighted proximity search

Journal of Heuristics - Proximity search is an iterative method to solve complex mathematical programming problems. At each iteration, the objective function of the problem at hand is replaced by...     

The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187?C1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex relaxations and needed parameters are constructed with ideas of the ??BB method of global optimization and interval methods. It is shown that after finitely many steps an ${\epsilon}$ -stationary point of the original semi-infinite problem is reached. A numerical example illustrates the performance of the proposed method.