Mechanical Analysis of Functionally Graded Materials with Complex Singularities by Singularity-Informed Deep Finite Element Method

Mechanical Analysis of Functionally Graded Materials with Complex Singularities by Singularity-Informed Deep Finite Element Method | 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 Mechanical Analysis of Functionally Graded Materials with Complex Singularities by Singularity-Informed Deep Finite Element Method Huiping Wang, Junjie Fan, Alatancang Chen, Lianhe Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9461515/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract Functionally graded materials (FGMs) containing interacting cracks and holes exhibit strong local singularities and steep displacement gradients, which pose significant challenges to conventional numerical analysis. This study proposes a Singularity-Informed Deep Finite Element Method (SIDFEM) for the elastic analysis of FGM plates with a central elliptical hole and asymmetric collinear cracks. Built on the Deep Finite Element Method (DFEM), the proposed method reconstructs the approximation space by embedding crack-tip distance information, near-tip square-root asymptotics, crack-surface topology, and hole-boundary enrichment, so that the neural approximation is better aligned with the local singular structure of the mechanical field. The displacement solution is obtained through discrete potential-energy minimization with FEM-style quadrature and a two-stage Adam/L-BFGS optimization strategy. Compared with Abaqus reference solutions, the benchmark case yields a relative $L_2$ error of $0.026$ for the displacement vector and a maximum absolute error of $0.015$. In addition, compared with standard DFEM, SIDFEM reduces the relative $L_2$ errors of $u_x$, $u_y$, and the displacement vector by more than $92\%$. Further studies on ablation, geometric degeneration, defect rotation, load reversal, and generalized $J$-integral evaluation demonstrate the robustness and mechanical applicability of the proposed framework for fracture analysis of nonhomogeneous materials. MSC Classification: 74S05 , 74R10 , 74G70 Singularity-Informed Deep Finite Element Method functionally graded materials complex singularities discrete potential-energy minimization cracks and elliptical holes Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 23 Apr, 2026 Reviewers agreed at journal 22 Apr, 2026 Reviewers invited by journal 21 Apr, 2026 Editor assigned by journal 21 Apr, 2026 Submission checks completed at journal 20 Apr, 2026 First submitted to journal 19 Apr, 2026 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-9461515","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":628897110,"identity":"ec030677-65f4-445f-9c52-f3d22c823cf6","order_by":0,"name":"Huiping Wang","email":"","orcid":"","institution":"Inner Mongolia Normal University","correspondingAuthor":false,"prefix":"","firstName":"Huiping","middleName":"","lastName":"Wang","suffix":""},{"id":628897116,"identity":"d92f4fcf-52c9-40c4-9270-30280b263335","order_by":1,"name":"Junjie Fan","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Junjie","middleName":"","lastName":"Fan","suffix":""},{"id":628897119,"identity":"91ba0c4f-fe02-4c1c-8837-83c66db40570","order_by":2,"name":"Alatancang Chen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYDACdijNzwymmInQwszA2ACiJZtJ1mJwgFgt/M3Mzx/83GEnb3ycO02CocI6sYH97AG8WiQOsxk29p5JNtx2mHebBMOZ9MQGnrwE/NYcZjBs4G07wAjWwth2OLFBgscArw75w+wfG/+2HbDf3AzS8o8ILQaHeQybgbYkbmAGaWkgQovhYZ7C2bJtyckzDvNutkg4lm7cxpODX4vc8fYNH9+22dn295/deONDjbVsP/sZ/FpQQQIQs5GgfhSMglEwCkYBDgAAbmFCXeHL078AAAAASUVORK5CYII=","orcid":"","institution":"Inner Mongolia Normal University","correspondingAuthor":true,"prefix":"","firstName":"Alatancang","middleName":"","lastName":"Chen","suffix":""},{"id":628897122,"identity":"86d672b5-7c7a-497d-b266-ef1fbd87136c","order_by":3,"name":"Lianhe Li","email":"","orcid":"","institution":"Inner Mongolia Normal University","correspondingAuthor":false,"prefix":"","firstName":"Lianhe","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2026-04-19 11:08:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9461515/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9461515/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108491210,"identity":"d2f422a3-926e-4d3d-a2d8-0d75439a1910","added_by":"auto","created_at":"2026-05-05 09:52:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":19934519,"visible":true,"origin":"","legend":"","description":"","filename":"sidfemEngineeringwithComputers.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9461515/v1_covered_c4404750-34c2-424c-af9a-e217e280be92.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Mechanical Analysis of Functionally Graded Materials with Complex Singularities by Singularity-Informed Deep Finite Element Method","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"engineering-with-computers","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ewco","sideBox":"Learn more about [Engineering with Computers](http://link.springer.com/journal/366)","snPcode":"366","submissionUrl":"https://submission.nature.com/new-submission/366/3","title":"Engineering with Computers","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Singularity-Informed Deep Finite Element Method, functionally graded materials, complex singularities, discrete potential-energy minimization, cracks and elliptical holes","lastPublishedDoi":"10.21203/rs.3.rs-9461515/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9461515/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFunctionally graded materials (FGMs) containing interacting cracks and holes exhibit strong local singularities and steep displacement gradients, which pose significant challenges to conventional numerical analysis. This study proposes a Singularity-Informed Deep Finite Element Method (SIDFEM) for the elastic analysis of FGM plates with a central elliptical hole and asymmetric collinear cracks. Built on the Deep Finite Element Method (DFEM), the proposed method reconstructs the approximation space by embedding crack-tip distance information, near-tip square-root asymptotics, crack-surface topology, and hole-boundary enrichment, so that the neural approximation is better aligned with the local singular structure of the mechanical field. The displacement solution is obtained through discrete potential-energy minimization with FEM-style quadrature and a two-stage Adam/L-BFGS optimization strategy. Compared with Abaqus reference solutions, the benchmark case yields a relative $L_2$ error of $0.026$ for the displacement vector and a maximum absolute error of $0.015$. In addition, compared with standard DFEM, SIDFEM reduces the relative $L_2$ errors of $u_x$, $u_y$, and the displacement vector by more than $92\\%$. Further studies on ablation, geometric degeneration, defect rotation, load reversal, and generalized $J$-integral evaluation demonstrate the robustness and mechanical applicability of the proposed framework for fracture analysis of nonhomogeneous materials.\u003c/p\u003e\n\u003cp\u003eMSC Classification: 74S05 , 74R10 , 74G70\u003c/p\u003e","manuscriptTitle":"Mechanical Analysis of Functionally Graded Materials with Complex Singularities by Singularity-Informed Deep Finite Element Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-30 13:26:03","doi":"10.21203/rs.3.rs-9461515/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"45084660710222686259553001633908626470","date":"2026-04-23T06:29:07+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"233469075197477330820340173515108389342","date":"2026-04-22T14:48:25+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-21T17:14:48+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-21T05:04:35+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-20T06:46:31+00:00","index":"","fulltext":""},{"type":"submitted","content":"Engineering with Computers","date":"2026-04-19T11:04:10+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"engineering-with-computers","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ewco","sideBox":"Learn more about [Engineering with Computers](http://link.springer.com/journal/366)","snPcode":"366","submissionUrl":"https://submission.nature.com/new-submission/366/3","title":"Engineering with Computers","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"47f4f4c6-e393-487f-a1a6-8350860b3d15","owner":[],"postedDate":"April 30th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-30T13:26:05+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-30 13:26:03","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9461515","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9461515","identity":"rs-9461515","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}