[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84644-en":3,"doc-seo-84644-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84644,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities","A fully variational, model-independent Cut Finite Element Method (CutFEM) formulation for finite-strain elasticity is presented, built from a single augmented energy functional. The formulation weakly enforces boundary conditions via Nitsche terms and stabilizes small-cut effects using ghost-penalty. Residuals and the symmetrised tangent are obtained by successive variations, while automatic differentiation generates the first Piola–Kirchhoff stress and elasticity tensors directly from scalar energy density. An analysis of the linearised Newton subproblem proves cut-independent coercivity, continuity, and an O(h−2) condition number bound, yielding quasi-optimal convergence for regular solutions. Numerical tests show optimal h-convergence and quantified accuracy limits at mixed Dirichlet–Neumann junctions, where corner singularities cap rates for both fitted and unfitted methods.","arXiv :2607 .02334v1 [math .NA] 2 Jul 2026  \nA Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities  \nMichał Tomasz Wichrowskia,, Ella Godiva Noomena  \na Ruprecht-Karls-Universität Heidelberg, Faculty of Mathematics and Computer Science, Germany  \nAbstract  \nWe present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is the stationarity condition of a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. The residual and the (symmetrised) tangent follow from this functional by successive variations. Automatic differentiation (AD) generates the first Piola–Kirchhoff stress tensor and the elasticity tensor directly from the scalar energy density, avoiding manual re-derivation when exchanging hyperelastic models. To our knowledge, this is the first unfitted finite-strain scheme combining an energy-only, model-independent construction with AD and an accuracy analysis at unfitted boundaries.  \nAnalysis of the linearised problem solved at each Newton step establishes cut-independent coercivity, continuity, and an O (h−2) condition number bound, yielding a quasi-optimal convergence theorem for regular solutions through the Brezzi–Rappaz–Raviart framework. Numerically, the method attains optimal h-convergence for linear, quadratic, and cubic elements on a smooth test case. Furthermore, we quantify the method’s accuracy limit at mixed Dirichlet–Neumann junctions using the Kolosov–Muskhelishvili characteristic equation. The exact solution’s corner singularity caps the convergence rate identically for fitted and unfitted methods. We demonstrate that local mesh refinement removes this bound, with the unfitted discretisation inheriting the recovered optimal rates and cut-independent constants.  \nKeywords: CutFEM, Nonlinear Elasticity, Automatic Differentiation, Finite Element Method  \n1. Introduction  \nFinite-strain elasticity provides the framework for a wide spectrum of materials and applications in which the assumption of infinitesimally small displacements is no longer tenable [1, 2] . Once large deformations enter, the linear relation between stress and strain of Hooke’s law is lost and the governing equations become nonlinear, both geometrically and, in general, materially [3] . Since analytical solutions for such problems are rarely available, one relies on numerical methods, primarily the Finite Element Method (FEM), which discretises the weak (or, for hyperelastic materials, the variational) formulation of the problem [4, 5] . Two practical difficulties, however, persistently accompany finite element simulations at finite strain: the treatment of complex geometries, and the consistent derivation of the linearised operators required by Newton-type solvers. This paper addresses both within a single variational framework.  \nIn classical, body-fitted FEM the mesh matches the geometry of the domain, so that the overall complexity of a simulation is concentrated in the meshing procedure [6], which becomes expensive for intricate or evolving geometries [7] . Non-matching (unfitted, immersed) methods avoid this bottleneck by embedding the physical domain into a fixed background mesh and describing its boundary implicitly, typically by a level-set function [8] . The family of such methods includes the Partition of Unity FEM [9], the Extended/Generalised FEM [10, 11], the Immersed FEM [12], the Shifted Boundary Method [13], and the Cut Finite Element Method (CutFEM), introduced in [14] and reviewed comprehensively in [8, 15] . In this work we adopt CutFEMin its classical form: Dirichlet boundary conditions are imposed weakly by Nitsche’s method [16], closely  \n∗ Corresponding [author.](author. mwichro@mimuw.edu.pl)[ mwichro@mimuw.edu.pl](author. mwichro@mimuw.edu.pl)  \nrelated ","cbCaiuzd9dyFdVhV","https://ap.wps.com/l/cbCaiuzd9dyFdVhV","pdf",2115102,1,35,"English","en",105,"# Introduction\n## Finite-strain elasticity and FEM challenges\n## Unfitted (immersed) methods and CutFEM background\n## Variational formulation and stabilisation\n## Contributions and scope","[{\"question\":\"What is the core idea of the proposed CutFEM scheme for finite-strain elasticity?\",\"answer\":\"The scheme is fully variational and model-independent, derived from a single augmented energy functional. Boundary conditions are imposed weakly through Nitsche terms and small-cut instabilities are controlled using ghost-penalty stabilisation.\"},{\"question\":\"How are stresses and the elasticity tensor obtained in the method?\",\"answer\":\"Automatic differentiation generates the first Piola–Kirchhoff stress tensor and the elasticity tensor directly from the scalar strain-energy density. This avoids manual re-derivation when changing hyperelastic material models.\"},{\"question\":\"How do corner singularities affect convergence accuracy?\",\"answer\":\"Accuracy is limited by the exact solution’s corner singularity at mixed Dirichlet–Neumann junctions, which caps the convergence rate identically for fitted and unfitted methods. Local mesh refinement removes this bound by recovering optimal rates and cut-independent constants.\"}]",1784197433,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-unified-cutfem-formulation-for-finite-strain-elasticity-energy-minimisation-and-corner-singularities","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-unified-cutfem-formulation-for-finite-strain-elasticity-energy-minimisation-and-corner-singularities/84644/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the core idea of the proposed CutFEM scheme for finite-strain elasticity?","Question",{"text":75,"@type":76},"The scheme is fully variational and model-independent, derived from a single augmented energy functional. Boundary conditions are imposed weakly through Nitsche terms and small-cut instabilities are controlled using ghost-penalty stabilisation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are stresses and the elasticity tensor obtained in the method?",{"text":80,"@type":76},"Automatic differentiation generates the first Piola–Kirchhoff stress tensor and the elasticity tensor directly from the scalar strain-energy density. This avoids manual re-derivation when changing hyperelastic material models.",{"name":82,"@type":73,"acceptedAnswer":83},"How do corner singularities affect convergence accuracy?",{"text":84,"@type":76},"Accuracy is limited by the exact solution’s corner singularity at mixed Dirichlet–Neumann junctions, which caps the convergence rate identically for fitted and unfitted methods. Local mesh refinement removes this bound by recovering optimal rates and cut-independent constants.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]