Reduced Basis Method for Variational Inequalities in Contact Mechanics

We present an efficient model order reduction method [1] for parametrized elliptic variational inequalities of the first kind: find u ∈ K such that: a(u, v − u;μ) ≥ f(v − u;μ), ∀v ∈ K(μ) where K(μ) := {v ∈ H1(Ω)|Bv ≤ g(μ)}. Motivated by numerous engineering applications that involve contact between elastic body and rigid obstacle, e.g. the obstacle problem [2], we develop a primaldual reduced basis approach to construct offline-online efficient yet certified reduced order models. Such models find application in the real-time or many query context of PDE-constrained optimization, control, or parameter estimation. They can also be easily extended to parabolic systems [3]. Firstly, we develop a primal-dual certified reduced basis method for bijective constraint operator B that provides sharp and inexpensive a posteriori error bounds. We compare both the proposed error bounds and the computational costs with the proposal in [4], demonstrating the quality and effectivity of the approximation and the error bounds. Then, we extend our results to more generalized problems, namely an injective constraint operator B, e.g. for a generalized obstacle problem, or a surjective constraint operator B, e.g. in Signorini’s Problem. We present both a priori and a posteriori analysis for the generalized formulation [2] and the saddle point formulation [5]. Lastly, we discuss the construction and sampling procedure [1] for the given method.