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Journal Articles Optimization Methods and Software Year : 2024

Second order cone programming for frictional contact mechanics using interior point algorithm

Abstract

We report experiments of an implementation of a primal-dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high precision and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modeling and simulating nonsmooth dynamical systems. Currently, Siconos uses mainly first order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too much slow to recover accurate solutions in a reasonable time. As these variational inequalities systems lead to the solution of an optimization problem with second order cone constraints, a natural idea is to apply second order optimization methods to speed up the convergence. We will present in detail a primal-dual interior point algorithm for minimizing a convex quadratic function with second order cone constraints. We will show, with some examples, that well known implementations of this algorithm such as SDPT3, do not provide solutions satisfactorily in terms of computation time and accuracy. The major difficulty in implementing this type of algorithm comes from the fact that at each iteration of the algorithm, a change of variable, called a scaling, must be performed to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. While this scaling strategy is very nice from a theoretical point of view, it leads to a huge deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems with rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.
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Dates and versions

hal-03913568 , version 1 (27-12-2022)
hal-03913568 , version 2 (30-04-2023)
hal-03913568 , version 3 (16-01-2024)

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Vincent Acary, Paul Armand, Hoang Minh Nguyen, Maksym Shpakovych. Second order cone programming for frictional contact mechanics using interior point algorithm. Optimization Methods and Software, 2024, pp.1-31. ⟨10.1080/10556788.2023.2296438⟩. ⟨hal-03913568v3⟩
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