European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   MS37: Innovative methods for contact problems (Part 2)
Chair: Stefan Frei
25 mins
A reduced basis method for parameterized variational inequalities applied to contact mechanics
Amina Benaceur, Virginie Ehrlacher, Alexandre Ern
Abstract: We investigate Reduced-Basis methods for parameterized variational inequalities with nonlinear constraints. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using non-matching meshes. We propose a reduced-basis scheme in a saddle-point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced-basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. The key idea is to construct the latter using a cone-projected greedy algorithm that conserves the non-negativity of the dual basis vectors. The numerical examples confirm the efficiency of the reduction strategy. For further insight, we refer the reader to \cite{BEE:19}. \begin{thebibliography}{5} \bibitem{BEE:19} A.~Benaceur, V.~Ehrlacher and A.~Ern. \newblock A reduced basis method for parameterized variational inequalities applied to contact mechanics \newblock HAL preprint \texttt{}, 2019. \end{thebibliography}
25 mins
Adaptive discretization methods for the numerical simulation of static and time-dependent contact problems
Mirjam Walloth, Rolf Krause, Andreas Veeser
Abstract: In this talk we consider the adaptive numerical simulation of static and quasi-static as well as dynamic contact problems. First, we present residual-type a posteriori estimators for the adaptive discretization of static and quasi-static contact problems. Second, we present a time discretization which resolves the local impact times for each node in the numerical simulation of dynamic contact problems. An adaptation of the finite element mesh by means of a posteriori error estimators is indispensable for an efficient and accurate numerical simulation of contact problems. Therefore we develop residual-type a posteriori estimators for the linear finite element solution of the Signorini problem [2]. The estimator is designed for controlling the H^1-error of the displacements plus the H^{-1}-error of a suitable approximation of the contact force. The estimator generalizes the standard residual-type estimator for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non-linearity. The estimator contributions addressing the nonlinearity are related to the contact stresses, the complementarity condition, and the approximation of the gap function. Reliability and efficiency of the estimator is proven. In the case of quasi-static Signorini problems, modeling the contact between a viscoelastic and a rigid body, we derive a residual-type a posteriori estimator [1] which constitutes upper and lower bounds with respect to an error notion which measures the error in the displacements, the velocities and a suitable approximation of the contact forces. The estimator splits in temporal and spatial contributions which can be used for the adaptation of the time step as well as the mesh size. Remarkably, the estimators perceive that in the interior of the actual (time-dependent) contact zone, adaptive mesh refinement cannot improve the solution. Thus overestimation is avoided while the free boundary zone is appropriately well refined. For the discretization in time of dynamic Signorini problems we use a space-time connecting discretization method which enables an implicit prediction of the individual impact times of each contact boundary node [3]. It avoids oscillations in the contact stresses, is provably dissipative and allows for a physically motivated update of the velocities. In each time step we have to solve a discrete variational inequality for which we use an a posteriori error estimator like for static Signorini problems. Since for subsequent time-steps we might end up with different spatial meshes, a special treatment of the numerically computed values of the foregoing time step is required. The resulting method can resolve non-smooth effects at the contact boundary in space and time. Referenzen: [1] M. Walloth, Residual-type a posteriori estimator for a quasi-static Signorini contact problem. Preprint 2721, Fachbereich Mathematik, TU Darmstadt (2018), accepted for publication in IMA Journal of Numerical Analysis [2] R. Krause, A. Veeser, M. Walloth, An efficient and reliable residual-type a posteriori error estimator for the Signorini problem. Numerische Mathematik 130, pp. 151-197 (2015) [3] R. Krause, M. Walloth, A family of space-time connecting discretization schemes with local impact detection for elastodynamic contact problems, Computer Methods in Applied Mechanics and Engineering, 200, pp. 3425--3438 (2011).
25 mins
Local Flux Reconstruction for a Frictionless Unilateral Contact Problem
Daniela Capatina, Robert Luce
Abstract: We first present a uniform framework for the computation of conservative local fluxes for an elliptic problem for classical (conforming, nonconforming and discontinuous) finite element methods of arbitrary order on triangular meshes. The computation of these H(div)-fluxes is done by local post-processing of the finite element solution, avoiding the solution of any mixed problem. Optimal error estimates for the reconstructed fluxes are proved and illustrated numerically. Then we extend the approach to a frictionless unilateral contact problem, modeled by Signorini's equation. We consider a Nitsche formulation of the contact condition and a piecewise linear continuous finite element approximation. A residual-based a posteriori error estimator has been previously proposed and analyzed under a saturation assumption. We show here that the reconstructed local conservative flux yields a robust a posteriori error estimator, without any additional assumption.
25 mins
Construction of stable Implicit-explicit (IMEX) schemes for the impact of elastic solids
Elie Bretin, Yves Renard
Abstract: The aim of the presentation is to propose some new explicit-implicit (IMEX) time integration schemes for the dynamics with impact of deformable bodies thanks to a Nitsche based approximation of the contact condition. Some unconditionnaly stable IMEX schemes which necessitate only the solve of a linear system at each time step will be exhibited. We perform a comparison with explicit and implicit schemes in term of energy conservation, convergence and occurrence of spurious oscillations.