Doctoral thesis

Multigrid methods for stress-based contact problems in linear elasticity

  • 2021

PhD: Università della Svizzera italiana, 2021

English Problems of contact mechanics arise in many engineering applications. The contact conditions make the problem constrained and are the main challenge to be tackled. Different weak forms can be used for the modeling of contact problems. The primal weak form solves for the only displacement u ∈ H1. After the finite element (FE) discretization, the problem can be solved by means of the monotone multigrid (MMG) method, a solver for constrained problems with optimal complexity. However, the primal formulation is affected by locking and can compute the stress σ, a physical quantity of primary interest, only by means of differentiation of the displacement. In contrast to the primal formulation, the stress-based formulations are not affected by locking, as they are based on the stresses as main unknowns. Thus they are quite attractive for nearly incompressible and incompressible materials. In this thesis, we study the first-order system least squares (FOSLS) and the dual formulations for the Signorini problem, i.e., a unilateral contact problem. In the first approach, an energy functional subject to only box-constraints has to be minimized. In the second case, also global equality constraints must be enforced and an LBB (Ladyzhenskaya-Babuška-Brezzi) condition must be satisfied. This thesis extends the MMG method for the primal formulation to the stress-based formulations applied to the Signorini problem for nearly incompressible and incompressible materials. However, in the stress-based formulations, the stress σ belongs to the space Hdiv and therefore special care is needed for the finite element discretization and the corresponding solution methods. Linear multigrid methods which work for Hdiv spaces have been already investigated. To the author’s knowledge, this thesis is the first attempt to generalize the MMG from the primal formulation to the stress-based ones. To this purpose, we generalize the Arnold-Falk-Winther smoother patch smoother for Hdiv-regular problems, so that the contact constraints are solved locally. We show several numerical experiments that illustrate the performance of our new multigrid method for both the FOSLS and the dual cases.
  • English
Computer science and technology
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