Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches

Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches Mahmoud Idan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7703219/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Fracture and crack propagation are critical challenges in computational mechanics, as they directly affect the safety and durability of structural components in aerospace, automotive, and civil engineering. The extended finite element method (XFEM) has become a powerful tool for modeling discontinuities without remeshing. However, enrichment functions in XFEM are often introduced in an ad hoc manner, leading to issues of computational cost, numerical instability, and difficulty in extending the framework to optimization and nonlinear dynamic problems. This study develops a systematic enrichment strategy for XFEM by integrating module theory into the enrichment framework. The approach decomposes enrichment functions into modular components, enabling consistent management of crack-tip singularities and discontinuities while ensuring stability and scalability. A weak form of equilibrium is formulated with enrichment functions satisfying the partition of unity (POU). A MATLAB implementation is developed for two-dimensional linear elastic fracture problems, with validation performed against analytical linear elastic fracture mechanics (LEFM) benchmarks. The method is also extended to topology optimization using the solid isotropic material with penalization (SIMP) scheme, where material distribution evolves under fracture-driven constraints. The modular enrichment strategy accurately reproduces stress intensity factors and crack paths under various loading conditions, with results closely matching LEFM solutions. Figures demonstrate the evolution of displacement fields, crack propagation trajectories, and stress distributions, confirming the robustness of the enrichment functions. Compared with conventional XFEM, the proposed method reduces computational overhead and improves numerical stability in crack growth simulations. Integration with SIMP topology optimization further enables automatic crack-aware material redistribution, illustrating the framework’s potential for dynamic structural design applications. The integration of module theory with XFEM provides a mathematically consistent and computationally efficient approach for fracture analysis. The framework addresses key limitations of traditional enrichment methods, including stability and scalability, while enabling natural extension to topology optimization. This work establishes a foundation for future studies on nonlinear fracture, three-dimensional crack propagation, and lightweight design optimization under dynamic loading conditions. Extended Finite Element Method Crack Propagation Module Theory Enrichment Functions Stress Intensity Factor Fracture Mechanics Computational Mechanics Structural Health Monitoring Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7703219","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":524988137,"identity":"06f290cb-ef47-4d51-af07-672908c29de1","order_by":0,"name":"Mahmoud Idan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVRIiWNgGAWjYDCCAyDChqHe/ngDkGFgQayWNIYEhjMgloEEKVpuJIBYRGjhu32AdcOPBLs8xpnPr274USDBwN/enYBXi+S5BLabPQnJxczSOWU3e4AOkzhzdgNeLQZnGNhu8P5gZmyTzkm7wQPUYiCRS1jLzT8J9Yw9kmfSbv4hVsttnoTDiTMk2I/dJsoWyTOMbbdlEo4bG/DksN2WMZDgIegXvjPMx26+SaiWM2A//uzmmz82cvztvfi1MDAwNkAZPAZgkoByFMD+gBTVo2AUjIJRMIIAAOsXSWjW8YmvAAAAAElFTkSuQmCC","orcid":"","institution":"University of Al Maarif","correspondingAuthor":true,"prefix":"","firstName":"Mahmoud","middleName":"","lastName":"Idan","suffix":""}],"badges":[],"createdAt":"2025-09-24 11:38:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7703219/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7703219/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93130181,"identity":"4b5e2d7f-6305-4f57-8bfe-0bfc9d43e607","added_by":"auto","created_at":"2025-10-09 11:23:51","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":630534,"visible":true,"origin":"","legend":"","description":"","filename":"blinededmanuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7703219/v1_covered_669a05b6-74d6-4fd2-bc8c-4319788854e8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Extended Finite Element Method, Crack Propagation, Module Theory, Enrichment Functions, Stress Intensity Factor, Fracture Mechanics, Computational Mechanics, Structural Health Monitoring","lastPublishedDoi":"10.21203/rs.3.rs-7703219/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7703219/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Fracture and crack propagation are critical challenges in computational mechanics, as they directly affect the safety and durability of structural components in aerospace, automotive, and civil engineering. The extended finite element method (XFEM) has become a powerful tool for modeling discontinuities without remeshing. However, enrichment functions in XFEM are often introduced in an ad hoc manner, leading to issues of computational cost, numerical instability, and difficulty in extending the framework to optimization and nonlinear dynamic problems. This study develops a systematic enrichment strategy for XFEM by integrating module theory into the enrichment framework. The approach decomposes enrichment functions into modular components, enabling consistent management of crack-tip singularities and discontinuities while ensuring stability and scalability. A weak form of equilibrium is formulated with enrichment functions satisfying the partition of unity (POU). A MATLAB implementation is developed for two-dimensional linear elastic fracture problems, with validation performed against analytical linear elastic fracture mechanics (LEFM) benchmarks. The method is also extended to topology optimization using the solid isotropic material with penalization (SIMP) scheme, where material distribution evolves under fracture-driven constraints. The modular enrichment strategy accurately reproduces stress intensity factors and crack paths under various loading conditions, with results closely matching LEFM solutions. Figures demonstrate the evolution of displacement fields, crack propagation trajectories, and stress distributions, confirming the robustness of the enrichment functions. Compared with conventional XFEM, the proposed method reduces computational overhead and improves numerical stability in crack growth simulations. Integration with SIMP topology optimization further enables automatic crack-aware material redistribution, illustrating the framework’s potential for dynamic structural design applications. The integration of module theory with XFEM provides a mathematically consistent and computationally efficient approach for fracture analysis. The framework addresses key limitations of traditional enrichment methods, including stability and scalability, while enabling natural extension to topology optimization. This work establishes a foundation for future studies on nonlinear fracture, three-dimensional crack propagation, and lightweight design optimization under dynamic loading conditions.","manuscriptTitle":"Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-09 11:15:44","doi":"10.21203/rs.3.rs-7703219/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"03faa908-56aa-42e5-8b42-a0dead7f5903","owner":[],"postedDate":"October 9th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-10-09T11:15:44+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-09 11:15:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7703219","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7703219","identity":"rs-7703219","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}