A reduced basis method for parametrized variational inequalities applied to contact mechanics

Abstract

Summary We investigate new developments of the reduced‐basis method for parametrized optimization problems with nonlinear constraints. We propose a reduced‐basis scheme in a saddle‐point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced‐basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone‐projected greedy algorithm that conserves the non‐negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using nonmatching meshes. The numerical examples confirm the efficiency of the reduction strategy.

On the limitations of low‐rank approximations in contact mechanics problems

“…Low-rank approaches can potentially address this issue by reducing the number of constraints. 11,14 The idea behind ROMs is to split the cost of computation into two stages. The offline stage, where most of the computational complexity is resolved, consists of extracting the underlying structure of the system.…”

“…8 Application of ROMs to variational problems with inequality constraints (also referred to as variational inequalities) has been more recent. [9][10][11][12][13][14][15] Although the scope of this article is limited to contact mechanics, several other applications of variational inequalities are found in porous media flow problems, 16 cavitation problems in lubrication systems, 17 anti-plane frictional problems, 18 and even in financial trading problems. 10 Inequality constraints appear in mechanical problems with obstacles or multi-body mechanical problems where there is a possibility of contact between bodies and obstacles, or with other bodies.…”

Stable model reduction for linear variational inequalities with parameter-dependent constraints

“…Let us also mention [16] where a hypereduction method for contact problems is proposed. Instead of the NMF, one can also consider the Angle Greedy [22,11] and the Cone Projected Greedy (CPG) [9] algorithms for the compression of the dual basis. However, whatever the compression technique, if the primal and dual reduced bases are generated in a decorrelated way, the stability of the reduced problem is not guaranteed a priori.…”

A Multiscale Method for the Heterogeneous Signorini Problem

“…In [9], the authors generalized the reduction technique in [10], which constructs the reduced basis by combining the greedy algorithm and a posterior error indicator, to solve parametrized variational inequalities. Applications to contact mechanics, optimal control, and some improvements based on this method have been proposed in [11,12,13,14] and the references therein. Another family of model reductions applied to variational inequalities is based on the proper orthogonal decomposition (POD) methodology (see [15,16] and the references therein).…”