Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
153 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A Generalized Multigrid Method for Solving Contact Problems in Lagrange Multiplier based Unfitted Finite Element Method (2106.16107v1)

Published 30 Jun 2021 in math.NA and cs.NA

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 space has to be 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 of the basis. 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 of our method. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini's problem and two-body contact problems.

Citations (13)

Summary

We haven't generated a summary for this paper yet.