Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications

D. Goeleven, G. E. Stavroulakis, G. Salmon and P. D. Panagiotopoulos

Journal of Optimization Theory and Applications, 1997, vol. 95, issue 2, No 2, 263-293

Abstract: Abstract The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form $${\text{(P) }}u \in C:\left\langle {Mu - q,v - u} \right\rangle \geqslant 0,{\text{ }}\forall v \in C$$ , where C is a closed convex set of $$\mathbb{R}^N $$ (kinematically admissible set), $$q \in \mathbb{R}^N $$ (loading strain vector), and $$M \in \mathbb{R}^{N \times N} $$ (stiffness matrix). If rigid body displacements and rotations cannot be excluded from these applications, then the resulting matrix M is singular and serious mathematical difficulties occur. The aim of this paper is to discuss the existence and the numerical computation of the solutions of problem (P) for the class of cocoercive matrices. Our theoretical results are applied to two concrete engineering problems: the unilateral cantilever problem and the elastic stamp problem.

Keywords: Variational inequalities; positive-semidefinite matrices; cocoercive matrices; recession cones; rigid body motions; unilateral contact problems (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022679020242

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