{"title":"A multigrid technique for Lagrange multiplier-based unfitted Finite Element contact issues.","authors":"Kothari Rolf, Krause Hardik","doi":"10.47679/ijasca.v2i2.22","DOIUrl":null,"url":null,"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","PeriodicalId":13824,"journal":{"name":"International Journal of Advanced Computer Science and Applications","volume":"43 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Advanced Computer Science and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.47679/ijasca.v2i2.22","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
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
期刊介绍:
IJACSA is a scholarly computer science journal representing the best in research. Its mission is to provide an outlet for quality research to be publicised and published to a global audience. The journal aims to publish papers selected through rigorous double-blind peer review to ensure originality, timeliness, relevance, and readability. In sync with the Journal''s vision "to be a respected publication that publishes peer reviewed research articles, as well as review and survey papers contributed by International community of Authors", we have drawn reviewers and editors from Institutions and Universities across the globe. A double blind peer review process is conducted to ensure that we retain high standards. At IJACSA, we stand strong because we know that global challenges make way for new innovations, new ways and new talent. International Journal of Advanced Computer Science and Applications publishes carefully refereed research, review and survey papers which offer a significant contribution to the computer science literature, and which are of interest to a wide audience. Coverage extends to all main-stream branches of computer science and related applications