Topology Optimization with Learned Displacement Fields: An Analysis of Model-Driven Errors

Topology Optimization with Learned Displacement Fields: An Analysis of Model-Driven Errors | 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 Topology Optimization with Learned Displacement Fields: An Analysis of Model-Driven Errors Ricardo Bastos, Mafalda Gonçalves, João Dias-de-Oliveira This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8116105/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 Topology optimization enables the creation of highly efficient structures by optimally distributing material within a design domain. However, traditional methods rely on repeated Finite Element Method (FEM) analyses, leading to high computational costs, particularly for large-scale or complex problems. To mitigate this, recent research has focused on integrating machine learning methodologies to replace costly steps with faster predictions using metamodels. This study presents a metamodel for predicting displacement fields in a 2D cantilever beam topology optimization problem. A U-Net model architecture, originally developed for image segmentation, presented an excellent trade-off between accuracy and training time, measured using the Mean Squared Error (MSE). The dataset was generated using a parameterized Solid Isotropic Material with Penalization (SIMP) method, requiring 35 hours of computation and approximately 20 hours of offline metamodel training. For a domain of size 180 by 60, the metamodel achieved a 6.5 times speedup over traditional methods. The predicted displacement fields closely resembled the ground truth values, presenting a relatively low error. However, pixel-wise differences led to notable deviations in final optimized topologies. Sensitivity analysis showed that, due to localized high displacement gradients, the metamodel often overestimated the importance of the wrong locations, namely, the load application location. Despite these challenges, the metamodel showed strong potential to accelerate topology optimization. It may also significantly reduce computation time in standalone FEM simulations across various engineering applications, since the metamodel achieved reasonable accuracy in its predictions. Topology Optimization Metamodeling Finite Element Method Machine Learning 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-8116105","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":552263404,"identity":"5721e9a9-5b07-42fb-ac2c-585e85e6470a","order_by":0,"name":"Ricardo Bastos","email":"","orcid":"","institution":"University of Aveiro","correspondingAuthor":false,"prefix":"","firstName":"Ricardo","middleName":"","lastName":"Bastos","suffix":""},{"id":552263405,"identity":"b556f624-43da-465a-aca4-214417fc3ab4","order_by":1,"name":"Mafalda Gonçalves","email":"","orcid":"","institution":"Institute of Science and Innovation in Mechanical and Industrial Engineering (INEGI)","correspondingAuthor":false,"prefix":"","firstName":"Mafalda","middleName":"","lastName":"Gonçalves","suffix":""},{"id":552263406,"identity":"c3e95f79-4f5b-467a-b986-659d14fca285","order_by":2,"name":"João Dias-de-Oliveira","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnElEQVRIiWNgGAWjYBACxgbmBgaGCtK0MAK1nAExmUnQxMDYRooW5vaDjY955x2WZ5DuP0CkHT2JzYYztx02bJA5TLSzEtskPm47zNggkUyslv6H7T8S5xy2J0HLjMQ2ho8NhxNJ0fKwWXLGsfTkNpnDBsRpMexPPviZp8batl+68QGRWhqgDDYJIt3FIA9nEa1lFIyCUTAKRhwAAN46K94ymKFEAAAAAElFTkSuQmCC","orcid":"","institution":"University of Aveiro","correspondingAuthor":true,"prefix":"","firstName":"João","middleName":"","lastName":"Dias-de-Oliveira","suffix":""}],"badges":[],"createdAt":"2025-11-14 14:38:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8116105/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8116105/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":97104983,"identity":"8672f299-974a-4094-b431-67870b88af2e","added_by":"auto","created_at":"2025-12-01 04:36:42","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5664,"visible":true,"origin":"","legend":"","description":"","filename":"d3ae72d3aa4c481cb844ebfa1489784c.json","url":"https://assets-eu.researchsquare.com/files/rs-8116105/v1/69bc6303ec77d4ca2c3264ae.json"},{"id":106831310,"identity":"ae18fe57-cd47-4a93-a9a5-41182b4e63be","added_by":"auto","created_at":"2026-04-13 23:54:26","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":795077,"visible":true,"origin":"","legend":"","description":"","filename":"IJMMDanonymised.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8116105/v1_covered_a9ccb675-d0b6-4544-9c28-008ff55b6989.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Topology Optimization with Learned Displacement Fields: An Analysis of Model-Driven Errors\n\n\n","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"Topology Optimization, Metamodeling, Finite Element Method, Machine Learning","lastPublishedDoi":"10.21203/rs.3.rs-8116105/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8116105/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Topology optimization enables the creation of highly efficient structures by optimally distributing material within a design domain. 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