A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments

A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments | 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 A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments Zhenhao Zhou, Ruochun Zhang, Ganesh Arivoli, Dan Negrut This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9528335/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 We present the numerical methods and GPU-accelerated implementation underlying a Total Lagrangian finite element framework for finite-deformation flexible multibody dynamics, introduced in the companion paper of this series. The framework supports 10-node quadratic tetrahedral (T10) elements and ANCF beam and shell elements, with quadrature-based hyperelastic response (St. Venant–Kirchhoff and Mooney–Rivlin) and an optional Kelvin–Voigt viscous stress contribution. Time stepping employs a velocity-based implicit backward-Euler scheme, yielding a nonlinear residual in velocity that couples inertia, internal and external forces, and bilateral constraints. Constraints are enforced via an augmented Lagrangian method (ALM), alternating an inner velocity solve with a dual-ascent multiplier update. We introduce a two-stage GPU parallelization strategy for internal force and tangent stiffness evaluation, and provide two inner solvers: a first-order AdamW optimizer and a second-order Newton solver that assembles and factorizes a sparse global Hessian on the GPU using cuDSS. A fixed-sparsity matrix strategy eliminates repeated symbolic analysis and enables efficient numerical refactorization across Newton iterations. For collision detection, we present a GPU-native two-thread asynchronous algorithm operating on triangle soups, avoiding bounding-volume hierarchies entirely. Systematic scaling benchmarks across all three supported element types and six mesh resolutions show that the Newton solver, running on an NVIDIA GeForce RTX 5090, achieves approximately one order of magnitude lower real-time factor than CPU baselines run on an Intel i7-13700KF with MPI parallelization at the largest resolutions tested. The frictional contact model is validated against closed-form rigid-body predictions through quasi-static and dynamic impact unit tests; bilateral constraint enforcement is validated through revolute- and spherical-joint double-pendulum tests. Total Lagrangian finite element analysis multibody dynamics GPU acceleration augmented Lagrangian collision detection 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-9528335","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":630355695,"identity":"8bf2a242-33bc-4b7f-b1b8-c1a756a6730d","order_by":0,"name":"Zhenhao Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzklEQVRIiWNgGAWjYFACHiCusJHjZ2BsAHEhJGEtZ9KMJRtI0sLYdjhxwwEIl7AW+f6zBx/zsB1m3HwjufnDDwYbWZhenICx4VyyMQ9POrPZjcQ2yR6GNGOCWpgZe8ykeSSs2cxuJ7YxMzAgXIgTsDHzALUYMPMYz05s/szA8J+wFh42kJYEZwkD6cQGaQaGA4S1SPDwGBvOOZBmIHH/IdAvBsnGMwlpke8/Y/jg7T+b+v6e448//Kiwk+0jpAUNGJCmfBSMglEwCkYBDgAAuWw+pdr8k10AAAAASUVORK5CYII=","orcid":"","institution":"University of Wisconsin–Madison","correspondingAuthor":true,"prefix":"","firstName":"Zhenhao","middleName":"","lastName":"Zhou","suffix":""},{"id":630355699,"identity":"619f9087-39df-4d7b-9173-f50d525553f4","order_by":1,"name":"Ruochun Zhang","email":"","orcid":"","institution":"AEROCAE Digital Technology Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Ruochun","middleName":"","lastName":"Zhang","suffix":""},{"id":630355700,"identity":"76806c84-0a6e-4aa5-8eed-811a21518f4c","order_by":2,"name":"Ganesh Arivoli","email":"","orcid":"","institution":"University of Wisconsin–Madison","correspondingAuthor":false,"prefix":"","firstName":"Ganesh","middleName":"","lastName":"Arivoli","suffix":""},{"id":630355701,"identity":"5e86a24b-950c-4663-aab8-d3b790af649d","order_by":3,"name":"Dan Negrut","email":"","orcid":"","institution":"University of Wisconsin–Madison","correspondingAuthor":false,"prefix":"","firstName":"Dan","middleName":"","lastName":"Negrut","suffix":""}],"badges":[],"createdAt":"2026-04-25 21:38:41","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9528335/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9528335/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108181754,"identity":"ece9459a-0861-431c-974f-6552da586a43","added_by":"auto","created_at":"2026-04-30 08:58:53","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":9956483,"visible":true,"origin":"","legend":"","description":"","filename":"tlfeanum.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9528335/v1_covered_a7fa60e4-a56f-40f7-a695-e9f7ba56cdba.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments","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":"Total Lagrangian, finite element analysis, multibody dynamics, GPU acceleration, augmented Lagrangian, collision detection","lastPublishedDoi":"10.21203/rs.3.rs-9528335/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9528335/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We present the numerical methods and GPU-accelerated implementation underlying a Total Lagrangian finite element framework for finite-deformation flexible multibody dynamics, introduced in the companion paper of this series. The framework supports 10-node quadratic tetrahedral (T10) elements and ANCF beam and shell elements, with quadrature-based hyperelastic response (St. Venant–Kirchhoff and Mooney–Rivlin) and an optional Kelvin–Voigt viscous stress contribution. Time stepping employs a velocity-based implicit backward-Euler scheme, yielding a nonlinear residual in velocity that couples inertia, internal and external forces, and bilateral constraints. Constraints are enforced via an augmented Lagrangian method (ALM), alternating an inner velocity solve with a dual-ascent multiplier update. We introduce a two-stage GPU parallelization strategy for internal force and tangent stiffness evaluation, and provide two inner solvers: a first-order AdamW optimizer and a second-order Newton solver that assembles and factorizes a sparse global Hessian on the GPU using cuDSS. A fixed-sparsity matrix strategy eliminates repeated symbolic analysis and enables efficient numerical refactorization across Newton iterations. For collision detection, we present a GPU-native two-thread asynchronous algorithm operating on triangle soups, avoiding bounding-volume hierarchies entirely. Systematic scaling benchmarks across all three supported element types and six mesh resolutions show that the Newton solver, running on an NVIDIA GeForce RTX 5090, achieves approximately one order of magnitude lower real-time factor than CPU baselines run on an Intel i7-13700KF with MPI parallelization at the largest resolutions tested. 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