A. Pinto da Costa

Cone-constrained eigenvalue problems: theory and algorithms

Equilibria in mechanics or in transportation models are not always expressed through a system of equations, but sometimes they are characterized by means of complementarity conditions involving a convex cone. This work deals with the analysis of cone-constrained eigenvalue problems. We discuss some theoretical issues like, for instance, the estimation of the maximal number of eigenvalues in a cone-constrained problem. Special attention is paid to the Paretian case. As a short addition to the theoretical part, we introduce and study two algorithms for solving numerically such type of eigenvalue problems.

Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction

Abstract Necessary and sufficient conditions are established for the occurrence of dynamic instabilities in finite dimensional linearly elastic systems in unilateral frictional contact with a rigid flat surface. These conditions apply, in particular, to the systems that result from the finite element discretization of linearly elastic bodies. From a numerical point of view, these conditions lead to studying eigenproblems relative to a non-symmetric (tangent) stiffness matrix that incorporates the effect of the current state of the contract candidate particles. Illustrative small-sized examples are presented, together with an application to the case of an experimentally tested block of polyurethane, where friction induced instability phenomena were observed.

Numerical resolution of cone-constrained eigenvalue problems

Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity system K ∋ x ⊥(Ax-λ Bx) ∈ K+. This problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.

A Complementarity Eigenproblem in the Stability Analysis of Finite Dimensional Elastic Systems with Frictional Contact

In this paper a mixed complementarity eigenproblem (MCEIP) is formulated and a method is proposed for its numerical solution. This mathematical problem is motivated by the study of divergence instabilities of static equilibrium states of finite dimensional mechanical systems with unilateral frictional contact. The complementarity eigenproblem is solved by transforming it into a non-monotone mixed complementarity problem (MCP), which is then solved by using the algorithm PATH. The proposed method is used to study some small sized examples and some large finite element problems.

Stability of finite-dimensional nonlinear elastic systems with unilateral contact and friction

This paper addresses some questions in the general area of the instability of finite-dimensional elastic systems in unilateral frictional contact with rigid obstacles. We study the occurrence of dynamic solutions in the neighborhood of a given equilibrium state which might tend to diverge from that state. Some of the results obtained by Martins et al. (1998) are generalized here to encompass the effects of the system nonlinear elastic behavior and of the obstacle curvature.

The evolution and rate problems and the computation of all possible evolutions in quasi-static frictional contact

The present work studies several issues related to the quasi-static behavior of finite dimensional systems in frictional contact with rigid curved obstacles. Conditions for existence of quasi-static evolutions are discussed, the occurrence of angular bifurcations on a quasi-static trajectory is studied as well as the possibility of continuation of an equilibrium trajectory along the direction of a solution to the rate problem. Several complementarity formulations are presented for the quasi-static rate problem and the computation of all the solutions of that problem is performed.

Bifurcations and Instabilities in Some Finite Dimensional Frictional Contact Problems

The present work is part of a research effort devoted to the study of bifurcation and instability phenomena in frictional contact problems. Several situations have been considered in these studies: (i) the occurrence of bifurcations in quasi-static paths; this is a stiffness and friction induced phenomenon of non-uniqueness of quasi-static solutions; (ii) the initiation of dynamic solutions at equilibrium positions, with no initial perturbations, but with initial acceleration and reaction discontinuities; this is a mass and friction induced phenomenon of non-uniqueness of dynamic solutions; (iii) the existence of smooth non-oscillatory growing dynamic solutions with perturbed initial conditions arbitrarily close to equilibrium configurations, i.e. the divergence instability of equilibrium states; (iv) the existence of non-oscillatory or oscillatory growing dynamic solutions with perturbed initial conditions arbitrarily close to portions of quasi-static paths; (v) the occurrence of non-oscillatory (divergence) or oscillatory (flutter) instabilities of steady sliding equilibrium states.

Surface Instabilities in Linear Orthotropic Half-Spaces With a Frictional Interface

This paper studies the friction induced vibrations that may develop in the neighborhood of steady sliding states of elastic orthotropic half-spaces compressed against a rigid plane moving tangentially with a prescribed speed. These vibrations may lead to flutter instability associated to a surfacelike oscillation. The system of dynamic differential equations and boundary conditions that governs the small plane oscillations of the half-space about the steady sliding state is established. The general form of the surface solutions to the plane strain case is given. The way how the coefficient of friction varies with changes in some of the systems parameters is investigated. It is shown that for certain combinations of material data, low coefficients of friction are found for surface flutter instability (lower than in the isotropic case).

Complementarity eigenvalue problems for nonlinear matrix pencils

This work deals with a class of nonlinear complementarity eigenvalue problems that, from a mathematical point of view, can be written as an equilibrium model [A(λ)B(λ)C(λ)D(λ)][uw]=[v0],u≥0,v≥0,uTv=0,where the vectors u and v are subject to complementarity constraints. The block structured matrix appearing in this partially constrained equilibrium model depends continuously on a real scalar λ ∈ Λ. Such a scalar plays the role of a non-dimensional load parameter, but it may have also other physical meanings. The symbol Λ stands for a given bounded interval, possibly non-closed. The numerical problem at hand is to find all the values of λ (and, in particular, the smallest one) for which the above equilibrium model admits a nontrivial solution. By using the so-called Facial Reduction Technique, we solve efficiently such a numerical problem in various randomly generated test examples and in two mechanical examples of unilateral buckling of columns.

Finite Dimensional Frictional Contact Quasi-Static Rate and Evolution Problems Revisited

In this paper we revisit the quasi-static rate and the quasi-static evolution problems for finite dimensional nonlinear elastic systems, in frictional contact with curved obstacles. Our objectives are: (i) to present complementarity formulations for the quasi-static rate problem; (ii) to discuss conditions for existence and uniqueness of solution for that problem; (iii) to present a differential inclusion formulation and related mathematical results for the quasi-static evolution problem; (iv) to present numerical results for a specific finite element example of (non-) uniqueness of solution for the quasi-static rate problem.