Numerical analysis of the Biot equations coupled to frictional contact mechanics

Jul 16, 2026

The document studies a poro-visco-elastic medium in frictional contact with a rigid obstacle, focusing on numerical approximation of a coupled multiphysics model. The formulation combines Biot equations with contact conditions expressed via normal compliance and Coulomb friction, yielding a variational problem: a linear PDE coupled to a nonlinear variational inequality. A fully discrete scheme using conformal finite elements in space and implicit Euler time stepping is proposed. Discrete existence and uniqueness, stability, and a priori error estimates are proved, and a numerical experiment validates theoretical results.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages26

Reading Time1 h 6 min

Introduction

Mathematical model and contact conditions

Variational inequality structure and prior difficulties

Numerical analysis background and related poroelastic work

arXiv :2607 .02030v1 [math .NA] 2 Jul 2026 Numerical analysis of the Biot equations coupled to frictional contact mechanics Marius Nevland∗†, Kundan Kumar∗, Inga Berre∗, Jakub Wiktor Both∗, Eirik Keilegavlen∗ Abstract 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. 1 Introduction In this paper, we consider a mathematical model of a linearly viscoelastic, porous body that is saturated with a fluid (also called a poroelastic body), and which is subject to contact conditions including friction. This can be viewed as an extension of classical models of linear elasticity subject to contact conditions, introducing a coupling of such models to fluid flow, resulting in a multiphysics problem. Problems of linear elasticity subject to contact conditions lead to so-called variational inequalities [13, 19] . The modeling of contact with friction typically leads to two types of constraints. The first type of constraint restricts the movement of the elastic body in the normal direction, and it is often formulated as a KarushKuhn-Tucker type inequality [19], in which impenetrability between the body and obstacle is enforced (also called a Signorini condition) . An alternative model is the normal compliance model, in which there is a constitutive relationship between the normal traction and the penetration [22] . The second type of constraint regards the tangential motion of the body, modeling the tendency of the body to either stick or slip when there is contact, with the Coulomb friction law being a common model. The model of a linearly elastic body subject to the Signorini condition and the Coulomb friction law is notoriously difficult to analyze [10], with the central caveat being that this combination results in only aquasi-variational inequality [21] . However, if the Signorini condition is replaced by a normal compliance law, we obtain a classical variational inequality, which is more tractable for analysis. For this combination of the normal compliance and Coulomb friction laws, existence and uniqueness can be proven, given that aviscoelastic term is included in the stress tensor [22]; this term is necessary to control the time derivative of displacement appearing in the Coulomb friction law. Numerical analysis of this model, including error estimates and convergence rates, have also successfully been conducted in works by Sofonea and Han [17, 18] . Using the implicit Euler method, they obtained first order convergence in time, while the spatial convergence ∗ Center for Modeling of Coupled Subsurface Dynamics, Department of Mathematics, University of Bergen, All´egaten 41, Bergen, 5020, Norway. †Corresponding [author: marius.nevland@uib.no](author: marius.nevland@uib.no) order is reduced by a square root. The latter is common in contact problems, due to the contact conditions reducing the regularity of the solution. While there is a large body of literature on numerical analysis of an elastic body subject to contact conditions, including several books written on the subject [10, 18, 19], the case of a poroelastic body subject to contact conditions remains relatively unexplored. Coupled flow and elasticity in porous media (excluding contact mechanics) is frequently m

FAQs

What physical and mathematical model is studied in the document?

It considers a poro-visco-elastic medium (a porous body saturated with fluid) subject to frictional contact with a rigid obstacle, coupling Biot equations with contact constraints.

How are the frictional contact conditions incorporated?

Normal interaction is modeled using a normal compliance law, while tangential behavior follows Coulomb friction, resulting in a linear PDE coupled to a nonlinear variational inequality.

What numerical scheme and theoretical results are provided?

A fully discrete method is introduced with conformal finite elements in space and implicit Euler in time. The work proves existence and uniqueness of the discrete solution and derives stability and a priori error estimates, supported by a numerical experiment.

Jul 16, 2026

Upload by Ava Thompson