Numerical analysis of the Biot equations coupled to frictional contact mechanics
This work provides a rigorous numerical analysis for a coupled problem in poromechanics, which is important for applications like geomechanics and biomechanics, but the results are theoretical and incremental.
The authors developed and analyzed a fully discrete numerical scheme for a poro-visco-elastic medium with frictional contact, proving existence, uniqueness, stability, and error estimates, with numerical experiments confirming the theoretical rates.
We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.