MS37: Innovative methods for contact problems (Part 3)
Chair: Stefan Frei
Stabilized boundary elements for the wave equation with Signorini or friction contact
Abstract: (joint with G. Barrenechea, C. Ozdemir, J. Stocek)
We discuss a stabilized boundary element method for the time-dependent wave equation with contact, a scalar model problem for dynamic contact in elasticity. In a first step, the wave equation with Signorini boundary conditions is reduced to a variational inequality for the Poincare-Steklov operator on the contact boundary. the Poincare-Steklov operator is numerically realized in terms of retarded potentials. We review the existence of solutions to the dynamic contact problem and, assuming existence, discuss the a priori error analysis of an equivalent mixed formulation, as well as its minimal stabilization based on local projections. We further discuss extensions to boundary problems for the wave equation involving Tresca friction. Numerical results illustrate the efficiency of our methods in three dimensions.
An XFEM/DG approach for contact mechanics in fluid-structure interaction problems
Luca Formaggia, Federico Gatti, Stefano Zonca
Abstract: We present a numerical method for overlapping meshes that allows to simulate fluid-structure interaction problems in the case of immersed moving structures that undergo large displacements and come into contact.
The proposed method relies on a fixed fluid mesh which is arbitrarily overlapped by the structure ones, see [2, 1]. This numerical technique introduces two main issues: firstly, the generation of polyhedral fluid elements nearby the fluid-structure interface due to the intersection of the overlapping meshes; then, the treatment of the coupling at the unfitted interface between the fluid and the structure. The first issue is addressed via the eXtended Finite Element Method (XFEM) that permits to generalize the classical Finite Element approximation by locally enriching the discrete space in the fluid mesh elements crossed by the structure mesh. Moreover, XFEM has the advantage to be able to represent a discontinuity in the numerical solution within an element. In our context, this allows to handle structures which thickness is smaller than the fluid mesh element size. Regarding the second issue, we deal with a Discontinuous Galerkin/Nitsche approach to impose the kinematic and dynamic coupling conditions at the fluid-structure interface.
The same framework is extended to treat the contact between flexible structures. We consider a penalization approach in order to apply the non-penetration contact conditions at the non-matching structure-structure interface. Notice that the presence of the fluid in the proximity of the contact region requires a special treatment to control the fluid flow.
We present some 3D numerical results to show the effectiveness of the method by considering the particular case of immersed structures with a small thickness that come into contact, aiming at simulating the dynamics of cardiac valves.
 Vergara, C. and Zonca, S. Extended Finite Elements method for fluid-structure interaction with an immersed thick non-linear structure. Mathematical and Numerical Modeling of the Cardiovascular System and Applications. Ed. by D. Boffi, L. Pavarino, G. Rozza, S. Scacchi, and C. Vergara. SEMA SIMAI Springer Series (2018) pp. 209–243
 Zonca, S. and Vergara, C. and Formaggia, L. An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach. SIAM J. Sc. Comput. (2018) 40.1:B59–B84
A Nitsche-based formulation for fluid-structure interactions with contact
Erik Burman, Miguel Fernandez, Stefan Frei
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
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.