European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS37: Innovative methods for contact problems (Part 1)
Chair: Stefan Frei
25 mins
On Nitsche’s Method for Elastic Contact Problems
Tom Gustafsson, Rolf Stenberg, Juha Videman
Abstract: In this talk, we present a priori and a posteriori error estimates for the frictionless contact problem between two elastic bodies. The analysis is built upon interpreting Nitsche's method as a stabilised finite element method for which the error estimates can be derived with minimal regularity assumptions and without a saturation assumption. The stabilising term corresponds to a master-slave mortaring technique on the contact boundary. The numerical experiments show the robustness of Nitsche's method and corroborates the efficiency of the a posteriori error estimators.
25 mins
Nitsche's method for contact and friction problems in linear elasticity
Franz Chouly, Patrick Hild, Vanessa Lleras, Yves Renard
Abstract: The aim of this presentation is to provide some mathematical results in applying Nitsche's method to some contact and friction problems in linear elasticity and studying both static and dynamic situations. Contact and friction conditions are usually formulated with a set of inequalities and non-linear equations on the boundary of each body, with corresponding unknowns that are displacements, velocities and surface stresses. Basically, contact conditions allow to enforce non-penetration on the whole candidate contact surface, and the actual contact surface is not known in advance. A friction law may be taken into account additionally, and various models exist that correspond to different surface properties, the most popular one being Coulomb's friction. For numerical computations with the FEM, various techniques have been devised to enforce contact and friction conditions at the discrete level, either penalty methods or mixed methods. The Nitsche's method originally proposed in [4] aims at treating the boundary or interface conditions in a weak sense, thanks to a consistent penalty term. It differs in this aspect from standard penalization techniques and from mixed methods since no Lagrange multiplier is needed and no discrete inf sup condition must be fullfilled. Most of the applications of Nitsche's method during the last two decades involved linear conditions on the boundary of a domain or at the interface between sub-domains. In [2] a new Nitsche-based FEM was proposed and analyzed for Signorini's problem, where a linear elastic body is in frictionless contact with a rigid foundation. Very few works deal with the adaptation of Nitsche's method to frictional contact : the Tresca's friction problem is only considered in [1] for the static case. The case of contact in elastodynamics is dealt with in [3]. In this presentation main results of the numerical analysis for the static and dynamic cases will be detailed: existence and uniqueness results under appropriate assumptions on physical (friction coefficient) and numerical parameters, numerical assessment of convergence .
25 mins
Hybrid High-Order discretizations combined with Nitsche's method for contact with Tresca friction in small strain elasticity
Michaël Abbas, Franz Chouly, Alexandre Ern, Nicolas Pignet
Abstract: We devise and analyze a Hybrid High-Order (HHO) method combined with Nitsche’s method for contact problems with Tresca friction in small strain elasticity. On the one hand, HHO methods have been introduced in [7] for linear diffusion and in [6] for linear elasticity; they support polyhedral meshes with nonmatching interfaces and have been bridged in [5] to Hybridizable Discontinuous Galerkin methods and nonconforming Virtual Element methods. On the other hand, Nitsche’s method [8] is a well-known primal boundary-penalty technique to enforce weakly and consistently Dirichlet boundary conditions. This technique has been recently extended to enforce weakly Signorini’s unilateral contact conditions [3, 4] and Tresca friction conditions [2], discretized with conforming finite elements. In a previous work [1], we first adapted these ideas to the setting of HHO discretization, for a scalar elliptic problem, and both for Dirichlet and for Signorini conditions. The present work extends the results of [1] to the Lamé system with unilateral contact and Tresca friction condition. Since HHO methods involve both cell unknowns and face unknowns, this leads to different formulations of Nitsche’s consistency and penalty terms, either using the trace of the cell unknowns (cell version) or using directly the face unknowns (face version). The face version uses equal order polynomials for cell and face unknowns, whereas the cell version uses cell unknowns of one order higher than the face unknowns. We derive error estimates for these methods, that are complemented with numerical experiments.
25 mins
Galerkin least-squares method for membrane contact
Erik Burman, Peter Hansbo, Mats Larson
Abstract: We give an overview of recent work on Galerkin least squares stabilised finite element methods for contact between a small deformation elastic membrane and a rigid obstacle. We consider both the classical obstacle problem and curved membranes in contact. We limit ourselves to friction-free contact, but the formulation is readily extendable to more complex situations.