A multigrid technique for Lagrange multiplier-based unfitted Finite Element contact issues.

Abstract

Internal  interfaces  in a  domain  could exist as a  material defect or they can appear  due to propagations  of  cracks. Discretization  of such geometries  and solution  of the contact problem  on the internal  interfaces  can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally  convergent, and efficient  multigrid method for solving contact problems on the internal interfaces.  In the unfitted FE methods, structured background meshes are used and only the underlying finite element spaces are modified to incorporate the discontinuities. The non-penetration conditions  on the embedded interfaces  of the domains are discretized  using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization  problem with linear inequality constraints.  Our multigrid method can solve such problems by employing  a tailored multilevel  hierarchy  of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-L2  projection- based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear  constraints  using an orthogonal transformation. The decoupled constraints  are handled by a modified variant of the projected  Gauss–Seidel  method, which we employ  as a smoother in the multigrid method.  These components of the multigrid method allow us to enforce  linear constraints locally and ensure the global convergence.  We will demonstrate the robustness, efficiency,  and level independent convergence property of the proposed method for Signorini’s  problem and two-body contact problems