We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented.

We consider the class of two or three-dimensional discrete contact problems in which a set of contact nodes can make frictional contact with a corresponding set of rigid obstacles. Such a system might result from a finite element discretization of an elastic contact problem after the application of standard static reduction operations. The Coulomb friction law requires that the tractions at any point on the contact boundary must lie within or on the surface of a 'friction cone', but the exact position of any 'stuck' node (i.e., a node where the tractions are strictly within the cone) depends on the initial conditions and/or the previous history of loading. If the long-term loading is periodic in time, we anticipate that the system will eventually approach a steady periodic cycle.Here we prove that if the elastic system is 'uncoupled', meaning that changes in slip displacements alone have no effect on the instantaneous normal contact reactions, the time-varying terms in this steady cycle are independent of initial conditions. In particular, we establish the existence of a unique 'permanent stick zone' T comprising the set of all nodes that do not slip after some finite number of cycles. We also prove that the tractions and slip velocities at all nodes not contained in T approach unique periodic functions of time, whereas the (time-invariant) slip displacements in T may depend on initial conditions.Typical examples of uncoupled systems include those where the contact surface is a plane of symmetry, or where the contacting bodies can be approximated locally as half spaces and Dundurs' mismatch parameter beta = 0. An important consequence of these results is that systems of this kind will exhibit damping characteristics that are independent of initial conditions. Also, the energy dissipated at each slipping node in the steady state is independent of initial conditions, so wear patterns and the incidence of fretting fatigue failure should also be so independent. (C) 2014 Elsevier Ltd. All rights reserved.

This paper explores the effect of initial conditions on the behavior of coupled frictional elastic systems subject to periodic loading.Previously, it has been conjectured that the long term response will be independent of initial conditions if all nodes slip at least onceduring each loading cycle. Here, this conjecture is disproved in the context of a simple two-node system. Counter examples are presentedof “unstable” steady-state orbits that repel orbits starting from initial conditions that are sufficiently close to the steady state.The conditions guaranteeing stability of such steady states are shown to be more restrictive than those required for the rate problem tobe uniquely solvable for arbitrary derivative of the external loading.In cases of instability, the transient orbit is eventually limited either by slip occurring at both nodes simultaneously, or by one nodeseparating. In both cases a stable limit cycle is obtained. Depending on the slopes of the constraint lines, the limit cycle can involvetwo periods of the loading cycle, in which case it appears to be unique, or it may repeat every loading cycle, in which case distinctlimit cycles are reached depending on the sign of the initial deviation from the steady state. In the case of instability an example isgiven of a loading for which a quasi-static evolution problem with multiple solutions exists, whereas all rate problems are uniquely solvable.

Subdivision surfaces permit a designer to specify the approximate form of a surface defining an object and to refine and smooth the form to obtain a more useful or attractive version of the surface. A considerable amount of mathematical theory is required to understand the characteristics of the resulting surfaces, and this book provides a careful and rigorous presentation of the mathematics underlying subdivision surfaces as used in computer graphics and animation, explaining the concepts necessary to easily read the subdivision literature. It also organizes subdivision methods in a unique and unambiguous hierarchy in order to provide insight and understanding. The material is not restricted to questions related to regularity of subdivision surfaces at so-called extraordinary points but instead gives a broad discussion of the various methods. It is excellent preparation for reading more advanced texts that delve more deeply into special questions of regularity. The authors provide exercises and projects at the end of each chapter. Course material, including solutions to the exercises, is available on an associated Web page. Audience: This book is written for mathematically inclined Ph.D. students in computer science and researchers with advanced graduate-level expertise. Contents: List of Figures; List of Tables; Preface; Notation, Conventions, Abbreviations; Chapter 1: Introduction; Chapter 2: B-Spline Surfaces; Chapter 3: Box-Spline Surfaces; Chapter 4: Generalized-Spline Surfaces; Chapter 5: Convergence and Smoothness; Chapter 6: Evaluation and Estimation of Surfaces; Chapter 7: Shape Control; Appendix; Notes; Bibliography; Index

The problem of maintaining consistent representations of solids in computer-aided design and of giving rigorous proofs of error bounds for operations such as regularized Boolean intersection has been widely studied for at least two decades. One of the major difficulties is that the representations used in practice not only are in error but are fundamentally inconsistent. Such inconsistency is one of the main bottlenecks in downstream applications. This paper provides a framework for error analysis in the context of solid modeling, in the case where the data is represented using the standard representational method, and where the data may be uncertain. Included are discussions of ill-condition, error measurement, stability of algorithms, inconsistency of defining data, and the question of when we should invoke methods outside the scope of numerical analysis. A solution to the inconsistency problem is proposed and supported by theorems: it is based on the use of Whitney extension to define sets, called Quasi-NURBS sets, which are viewed as realizations of the inconsistent data provided to the numerical method. A detailed example illustrating the problem of regularized Boolean intersection is also given.

The paper treats thermoelastic contact problems, where a variable contact heat flow resistance as well as frictional heating are considered. Existence and uniqueness of steady state solutions, for both a one-dimensional and a three-dimensional system, are investigated. Existence is guaranteed if the contact heat flow resistance goes to zero as the pressure goes to infinity or if the frictional heating is sufficiently small. Uniqueness holds in the vicinity of zero frictional heating and thermally insulated contact.

Volino and Thalmann have published a conjecture proposing sufficient conditions for non-selfintersection of surfaces. Such conditions may be used in solid modeling, computer graphics, and other application areas, as a basis for collision-detection algorithms. In this paper we clarify certain of the hypotheses of the proposed theorem, and give a proof. A brief summary of possible pitfalls related to using the conditions, when the hypotheses of the formal theorem given here are not satisfied, is also given. We also give examples, and show that the theorem can be extended to domains that are not simply connected.

We present a sufficient condition for the existence of solutions to noncoercive incremental friction problems for discrete systems in contact with obstacles. By discrete system we mean that the displacement of the object can be discribed by a finite number of displacement variables, and by noncoercive we mean that the stiffness matrix of the object is semidefinite. We have a noncoercive friction problem when the object is not fixed to a support. This means that the friction forces need to balance the applied forces if the object is to remain stationary. This is manifested in our condition for existence of solutions. This condition is a compatibility condition on the applied force field, and if it is violated there exists a nontrivial solution to a corresponding dynamical problem. (C) 2606 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim.

For static or incremental contact problems with Coulomb friction there are satisfactory and well known existence results for the coercive case, i.e., when the elastic body is anchored so that rigid body motions are not possible, see [3, 1, 6, 7, 2]. The articles by Jaruusek and Cocu, [7, 2] indeed contain results for the noncoercive case, i.e., when rigid body motions are possible. However, the compatibility conditions which are used to ensure the existence of a solution, are the same that guarantee that the corresponding contact problem without friction has a solution. The condition is essentially that the applied force field should push the elastic body towards the obstacle. One of few previous articles containing friction-dependent compatibility conditions is [1].

In the present paper results on existence and uniqueness of solutions to discrete frictional quasi-static unilateral contact problems are given under a condition that the coefficients of friction are smaller than a certain upper bound. This upper bound is defined in terms of an influence matrix for the contact nodes. The results of existence and uniqueness may be ordered into two classes depending on whether regularity conditions for the applied forces are imposed or not. For general loading which has a time derivative almost everywhere it is shown that a solution exists which satisfies governing equations for almost all times. Uniqueness of the solution has been shown only when the problem is restricted to two degrees of freedom. For a loading which is right piecewise analytic, additional results can be obtained. For instance, if each contact node has only two degrees of freedom a unique solution which satisfies governing equeations for all times exists. For the constructed solutions a priori estimates of the displacement field and its time derivate in terms of the applied forces are also given.