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

Idrissa Niakh (1, 2, 3) , Guillaume Drouet (3) , Virginie Ehrlacher (1, 4) , Alexandre Ern (1, 2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 EDF Labs
4 MATHERIALS - MATHematics for MatERIALS

We consider model reduction for linear variational inequalities with parameterdependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameterdependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-spheres with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

Mots clés: Contact problem; Obstacle problem; Inf-sup condition; Model reduction; Variational inequalities; Reduced basis method

Domaines: Analyse numérique [math.NA]

Idrissa Niakh, Guillaume Drouet, Virginie Ehrlacher, Alexandre Ern. Stable model reduction for linear variational inequalities with parameter-dependent constraints. ESAIM: Mathematical Modelling and Numerical Analysis, 2023, 57 (1), pp.167--189. ⟨10.1051/m2an/2022077⟩. ⟨hal-03611982v3⟩