PhD thesis,
Finite Element Methods for Contact Problems
Department of Mathematics and Systems Analysis, Aalto University, (2018)
Abstract
This thesis addresses finite element methods for saddle point problems related to the mechanics of contact. In particular, we consider the discretization of second- and fourth-order obstacle problems and a second-order domain decomposition problem. The main focus is on stabilized finite element methods and their reinterpretation as Nitsche-type methods. We study the stability of the proposed methods and prove optimal a priori and a posteriori error estimates. The performance of the proposed methods is demonstrated through numerical examples.
For second-order obstacle problems, we study mixed and stabilized methods. The mixed method is shown to be stable if a bubble-enriched finite element space is used for the primal variable and the stabilized method for any pair of discrete spaces. The Lagrange multiplier of the stabilized formulation is eliminated locally to yield a Nitsche-type method which is more straightforward to implement.
The results on the finite element approximation of the second-order obstacle problem are generalized and applied to the adaptive solution of the Reynolds cavitation problem, modeled as a second-order elliptic variational inequality with variable coefficients. As a numerical example we consider the hydrodynamic lubrication of journal bearings. The location of the cavitation zone is unknown prior to solving the problem and an adaptive method is necessary for properly resolving the free boundary.
The fourth-order Kirchhoff plate obstacle problem is discretized using a similar stabilized method for which optimal a priori and a posteriori estimates are derived. The implementation is done by reinterpreting the stabilized method as a novel Nitsche-type method. In addition, for the unconstrained Kirchhoff plate bending problem, we present a more general a posteriori analysis than is currently available in the existing literature.
The second-order domain decomposition problem is discretized using a stabilized method with properly weighted stabilization terms so that the method is robust with respect to large jumps in the material and mesh parameters over the interface between the domains. We present an a priori error analysis of the method which avoids unnecessary assumptions on the regularity of the exact solution. Moreover, we present an a posteriori error analysis that does not rely on the saturation assumption and, for the implementation, recast the method as a Nitsche-type mortar method.
