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
Nitsche’s Master-Slave 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
3D-2D Stokes-Darcy coupling for the modelling of seepage with an application to fluid-structure interaction with contact
Erik Burman, Miguel Fernandez, Stefan Frei, Fannie Gerosa
Abstract: In this talk we discuss a Nitsche-based formulation for fluid-structure interaction (FSI) problems with contact. The approach is based on the work of Chouly and Hild [SIAM Journal on Numerical Analysis. 2013;51(2):1295--1307] for contact problems in solid mechanics. We present two numerical approaches, both of them formulating the FSI interface and the contact conditions simultaneously in equation form on a joint interface-contact surface $\Gamma(t)$. The first approach uses a relaxation of the contact conditions to allow for a small mesh-dependent gap between solid and wall. The second alternative introduces an artificial fluid below the contact surface. The resulting systems of equations can be included in a consistent fashion within a monolithic variational formulation, which prevents the so-called ``chattering'' phenomenon. To deal with the topology changes in the fluid domain at the time of impact, we use a fully Eulerian approach for the FSI problem. We compare the effect of slip and no-slip interface conditions and study the performance of the method by means of numerical examples.
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
Augmented Lagrangian Method for Thin Plates with Signorini Boundaries
Erik Burman, Peter Hansbo, Mats Larson
Abstract: We consider $C^1$-continuous approximations of the Kirchhoff plate problem in combination with a mesh dependent augmented Lagrangian method on a simply supported Signorini boundary.