Matrix-Free Finite Element Methods with Arbitrary Quadrature Point Locations

Matrix-Free Finite Element Methods with Arbitrary Quadrature Point Locations | 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 Matrix-Free Finite Element Methods with Arbitrary Quadrature Point Locations Zachary R. Atkins, Jed Brown, Rezgar Shakeri, Jeremy L. Thompson This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7130992/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract The Material Point Method (MPM) is a numerical method commonly used for solid mechanics simulations involving large deformations, history-dependent materials, or complex interactions such as contact and fracture. MPM can be treated as a finite element method where the integration points are allowed to move independently of the mesh. Matrix-free implementations of finite element operators provide higher performance on modern supercomputing hardware due to the higher arithmetic intensity, as measured by FLOPs per byte of memory transferred from memory, and the lower memory bandwidth requirements per degree of freedom when compared to assembled sparse matrices. In this paper, we derive a mathematical formulation for evaluation of finite element bases at arbitrary quadrature points per element and demonstrate the performance of matrix-free implementation of these MPM operators on GPU hardware. The measured performance compares favorably with matrix-free finite element operators on elements without a tensor product structure. FEM MPM matrix-free GPU Full Text Additional Declarations Competing interest reported. J.B. is an editor for PDESoft. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 09 Dec, 2025 Reviews received at journal 23 Nov, 2025 Reviews received at journal 11 Nov, 2025 Reviewers agreed at journal 07 Oct, 2025 Reviewers agreed at journal 08 Sep, 2025 Reviewers invited by journal 08 Sep, 2025 Editor assigned by journal 20 Jul, 2025 Submission checks completed at journal 18 Jul, 2025 First submitted to journal 15 Jul, 2025 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-7130992","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":511778956,"identity":"6775cb4e-58e3-4e58-9cb7-f546a82e1903","order_by":0,"name":"Zachary R. 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