Optimised Tensor Contractions and Vectorised Execution for Efficient Finite Element Solvers in Computational Solid Mechanics: A Benchmark Study

Optimised Tensor Contractions and Vectorised Execution for Efficient Finite Element Solvers in Computational Solid Mechanics: A Benchmark Study | 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 Optimised Tensor Contractions and Vectorised Execution for Efficient Finite Element Solvers in Computational Solid Mechanics: A Benchmark Study P G Kubendran Amos This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9382789/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 Tensor operations recur throughout computational solid mechanics, yet the optimisation strategies that have transformed similar computations in deep learning have not been examined systematically at the level of representative mechanics kernels. This work addresses that gap through a controlled benchmark study of three critical operations: anisotropic fourthorder elasticity tensor rotation, algorithmic tangent evaluation for J2 plasticity, and high-order matrix-free operator application on hexahedral elements. Through CPU execution and the use of open-source Python tools (NumPy, opt_einsum, and JAX), the study shows that runtime can be reduced by more than two orders of magnitude without altering the underlying mechanics, with speedups of 53:1 for anisotropic tensor rotation, 209:5× for nonlinear constitutive tangent evaluation, 19:6× for fused dense high-order operator application, and 8:4× for sum-factorisation. The results further show that different bottlenecks require different remedies. Particularly, the contractionpath optimisation is effective only when combined with batched compiled execution, branchless vectorisation is especially powerful for constitutive updates with pointwise logic, and tensor-product reformulation becomes important for high-order operators. Beyond the individual benchmarks, the work provides a transferable kernel-level blueprint for building faster finite element workflows, showing how substantial gains can be achieved through computational restructuring alone, before invoking specialised hardware. tensor contraction finite element method contraction path optimisation computational mechanics sum-factorisation J2 plasticity Full Text Additional Declarations The authors declare no competing interests. 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-9382789","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":621171196,"identity":"4b885072-41a9-4dc2-ac8d-dda91d8a5b7b","order_by":0,"name":"P G Kubendran Amos","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVRIiWNgGAWjYFACxgYgYcPABuawEa8ljYGNjZloLWBwGKiaWC3m0ofbHvyoOJ/HJ99/gOFD2WHCWiz7EtsNe87cLgY5jHHGOSK0GJxhbJNmbLud2AbUwszbRrSWf+cgWv4Sr6XhAEQLIzFaLHsY2yR7jiUDtSQbHOw5l05YizkP+zOJHzV2ifObDz588KPMmgiHIXMOEFaPrmUUjIJRMApGAVYAACkkNBFKVWbGAAAAAElFTkSuQmCC","orcid":"","institution":"National Institute of Technology Tiruchirappalli","correspondingAuthor":true,"prefix":"","firstName":"P","middleName":"G Kubendran","lastName":"Amos","suffix":""}],"badges":[],"createdAt":"2026-04-10 19:54:42","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9382789/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9382789/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106994359,"identity":"e33caaea-1496-4d8a-9277-26771a549246","added_by":"auto","created_at":"2026-04-15 15:07:58","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":379298,"visible":true,"origin":"","legend":"","description":"","filename":"cpc1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9382789/v1_covered_7b8b6f36-d7ad-4ab4-b473-cf4c6c104e84.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eOptimised Tensor Contractions and Vectorised Execution for Efficient Finite Element Solvers in Computational Solid Mechanics: A Benchmark Study\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"National Institute of Technology Tiruchirappalli","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":"tensor contraction, finite element method, contraction path optimisation, computational mechanics, sum-factorisation, J2 plasticity","lastPublishedDoi":"10.21203/rs.3.rs-9382789/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9382789/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTensor operations recur throughout computational solid mechanics, yet the optimisation strategies that have transformed similar computations in deep learning have not been examined systematically at the level of representative mechanics kernels. This work addresses that gap through a controlled benchmark study of three critical operations: anisotropic fourthorder elasticity tensor rotation, algorithmic tangent evaluation for J2 plasticity, and high-order matrix-free operator application on hexahedral elements. Through CPU execution and the use of open-source Python tools (NumPy, opt_einsum, and JAX), the study shows that runtime can be reduced by more than two orders of magnitude without altering the underlying mechanics, with speedups of 53:1 for anisotropic tensor rotation, 209:5× for nonlinear constitutive tangent evaluation, 19:6× for fused dense high-order operator application, and 8:4× for sum-factorisation. The results further show that different bottlenecks require different remedies. Particularly, the contractionpath optimisation is effective only when combined with batched compiled execution, branchless vectorisation is especially powerful for constitutive updates with pointwise logic, and tensor-product reformulation becomes important for high-order operators. Beyond the individual benchmarks, the work provides a transferable kernel-level blueprint for building faster finite element workflows, showing how substantial gains can be achieved through computational restructuring alone, before invoking specialised hardware.\u003c/p\u003e","manuscriptTitle":"Optimised Tensor Contractions and Vectorised Execution for Efficient Finite Element Solvers in Computational Solid Mechanics: A Benchmark Study","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-14 12:49:34","doi":"10.21203/rs.3.rs-9382789/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":"a661425f-f59b-49bd-a9c5-f08342bd540f","owner":[],"postedDate":"April 14th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-14T12:49:34+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-14 12:49:34","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9382789","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9382789","identity":"rs-9382789","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}