VEM-Nitsche Fully Discrete Polytopal Scheme for Frictionless Contact-Mechanics

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 81-102, February 2025.
Abstract. This work targets the discretization of contact-mechanics accounting for small strains, linear elastic constitutive laws, and fractures or faults represented as a network of co-dimension one planar interfaces. This type of model coupled with Darcy flow plays an important role typically for the simulation of fault reactivation by fluid injection in geological storage or the hydraulic fracture stimulation in enhanced geothermal systems. To simplify the presentation, a frictionless contact behavior at matrix fracture interfaces is considered, although the scheme developed in this work readily extends to more complex contact models such as the Mohr–Coulomb friction. To account for the geometrical complexity of subsurface, our discretization is based on the first order virtual element method (VEM), which generalizes the [math] finite element method to polytopal meshes. Following previous works in the finite element framework, the contact conditions are enforced in a weak sense using Nitsche’s formulation based on additional consistent penalization terms. We perform, in a fully discrete framework, the well-posedness and convergence analysis showing an optimal first order error estimate with minimal regularity assumptions. Numerical experiments confirm our theoretical findings and exhibit the good behavior of the nonlinear semismooth Newton solver.