Thermoelastic rolling contact problems for multilayer materials

The paper deals with the existence of solutions to the thermoelastic rolling contact problems for nonhomogeneous materials. One of the contacting surfaces is assumed to be covered with a graded material coating. The thermal and mechanical features of the coating material depend on its depth. The thermoelastic contact problem is governed by the system of mildly coupled evolutionary boundary value problems with discontinuous coefficients. Quasistatic approach is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The Faedo–Galerkin approach combined with the penalization and smoothing approach are used to show the existence of solutions to this contact problem. The operator splitting method is used to solve the problem numerically. Numerical results indicating the reduction of mechanically and/or thermally induced stresses are provided.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Alternative formulations of mixed integer programs

We study the formation of mixed-integer programming (MIP) constraints through the development of constructions which syntactically parallel the set operations of union, intersection, Cartesian product, and linear affine transformation. In this manner, we are able to both modularize the work of constructing representations (as the representations for base sets of a composite construction need not be disjunctively derived) and to make connections to certain of the logic-based approaches to artificial intelligence which utilize the intersection (and) and union (or) operations. We provide results which allow one to calculate the linear relaxation of a composite construction, in terms of set operations on the relaxation of the base sets. We are also able to compare the size of the relaxations for different formulations of the same MIP set, when these different formulations arise from one another through distributive laws. Utilizing these results, we generalize the Davis-Putnam algorithm of propositional logic to an MIP form, and answer a question regarding the relative efficiency of two versions of this algorithm. In this context, the subroutines of the logic algorithm correspond to list processing subroutines for MIP to be used prior to running linear programs. They are similar in nature to preprocessing routines, wherein entire MIP constraint sets are manipulated as formal symbols of logic.This work has been partially supported by National Science Foundation Grant MCS-8304075.     

Generalized variational formulations of contact problems of the dynamic theory of elasticity and thermoelasticity

We state generalized variational formulations of dynamic contact problems in the presence of zones of nonideal contact. For a problem of the theory of elasticity we write a generalized functional of Hamiltonian type. The variational formulations of contact problems of coupled dynamic thermoelasticity are stated using a generalized functional of Gurtin type.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 71–76.     

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established. The research of this author was based on work supported by the National, Science Foundation under grants DDM-9104078 and CCR-9213739. The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, by the Texas Advanced Research Program grant 999903-078, and by the Texas Advanced Technology Program under grant 999903-095.     

Boundary integral equation method in rolling contact problems involving roughness,wear and frictional heating

The method of boundary integral equations is developed to the solution of the rolling contact problems for rough bodies. A model of boundary roughness is proposed. The method permits to involve wear and frictional heating due to the slip in rolling contact.     

Alternative problems and symmetry

The results of Hale [1, 2] for systems which have the property E are extended to general nonlinear equations and to general (finite) symmetry groups.     

Dual formulations of optimal control problems

First-order necessary conditions are obtained for a dual formulation, in terms of a class of direction fields, of an optimal control problem of Hestenes. The control functions do not appear. A maximum principle for a boundary arc of a class of admissible arcs is stated in a similar dual formulation.This research is part of a dissertation submitted in partial satisfaction of the requirements for the Ph.D. degree in mathematics at the University of California at Los Angeles. The author would like to express his appreciation to Dr. M. R. Hestenes for his guidance.     

Enhanced lumped-differential formulations of diffusion problems

This paper is directly related to the task of modelling diffusion problems, prior to the choice of solution strategies. The approach presented is in fact a reformulation tool, aimed at reducing, as much as possible and within prescribed accuracy requirements, the number of dimensions in a certain diffusion formulation. It is shown how appropriate integration strategies can be employed to deduce mathematical formulations of comparable simplicity and improved accuracy in comparison with well-established classical lumping procedures. The approach is demonstrated through representative heat conduction problems, and the enhancement characteristics are examined against the classical lumped system analysis (CLSA) and the exact solutions of the fully differential systems.     

Extended formulations for stochastic lot-sizing problems

In this paper, extended formulations for stochastic uncapacitated lot-sizing problems with and without backlogging are developed in higher dimensional spaces that provide integral solutions. Moreover, physical meanings of the decision variables in the extended formulations are explored and special cases with more efficient formulations are studied.     

Model formulations for hub covering problems

Improved model formulations for hub covering problems are proposed. We discuss multiple and single allocation problems, including non-increasing quantity-dependent transport time functions for transport links for the latter case. Computational results are presented, which show that the new solution approaches for problems with quantity-independent transport times clearly outperform previous work on this topic.     

Integer programming formulations of vehicle routing problems

Various mathematical formulations are available for situations represented by vehicle routing problems. The assignment-based integer programming formulations of these problems are more common and easy to understand. Such formulations are discussed in this paper and a much simpler formulation for the vechicle routing problem is presented for the case, when all the vehicles have the same load capacity and maximum allowable cost per route.     

Thermoelastic rolling contact problem with temperature dependent friction

The paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The actual contact area in viscoelastic contact problems

The effect of the viscoelastic properties of polymers on the formation of the actual contact area is analyzed with reference to a rod model of the rough surface. The proposed time dependence of the actual contact area has been verified experimentally by an optical method for surfaces with different finishes ( 4, 8) and for different materials (polycaprolactam, polypropylene).Institute of Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel. Translated from Mekhanika Polimerov, No. 1, pp. 172–175, January–February, 1970.     

Modelling of the frictional heating generated in rolling contact

The transformation of the frictional energy into the heating is a problem of continued technical interest. This problem in the case of the pure sliding contact is widely discussed in many papers. In the case of rolling contact, the frictional heating is due to the slip between the rollers. Since the value of slip is unknown, the problem is more complicated, it becomes non-linear. The paper presents a new mathematical model of the generation of the heat during the rolling problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Alternative approaches for one-dimensional slitting problems

Dynamic programming has been used to find the optimal solution for one-dimensional slitting problems in which the value of a piece is nonlinearly dependent on both the width of the piece and the number of defects on it. In this paper, linear programming is proposed to solve this problem. Additionally, it is shown that this problem can be converted to be a shortest path problem and can be solved easily by an acyclic algorithm. Either a linear programming approach or a shortest path approach are both simpler and faster than the use of dynamic programming.