A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted Finite Element method

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-L-2 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. (C)& nbsp;2022 The Author(s). Published by Elsevier B.V.& nbsp;

Automated summary

The article presents an unfitted Finite Element method for internal interfaces within a domain and a multigrid algorithm for solving contact problems on these interfaces. By utilizing structured background meshes and the method of Lagrange multipliers for discretizing non-penetration conditions, and constructing a hierarchy of nested FE spaces using pseudo-L-2 projection-based transfer operators, the method ensures global convergence while enforcing linear constraints locally.