Numerical Analysis for a piezoelectric contact problem with long-memory and wear

Numerical Analysis for a piezoelectric contact problem with long-memory and wear | 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 Numerical Analysis for a piezoelectric contact problem with long-memory and wear Abderrahmane Oultou, Othmane Baiz, Hicham Benaissa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3858204/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 In this paper, we consider a new mathematical model that describes a frictional quasi-static contact between a piezoelectric body with a long memory effect and a foundation. The contact is described with the normal compliance condition with unilateral constraint and a version of Archard’s wear law. We introduce the model and prove its weak formulation, which is composed of a history-dependent variational equality and an integral equation. Using abstract history-dependent variational inequality and fixed point theorem, we show the existence and uniqueness solution of the model. In addition, we introduce a fully discrete scheme for the numerical solution of the problem, based on the finite element method to approximate the spatial derivative, and on the Euler scheme to discretize the time derivative. Finally, we derive the optimal-order error estimate under appropriate regularity hypotheses. MSC Classification: 74D10 , 74F15 , 74G30 , 47J20 , 49J40 , 35K86 , 74M10 , 74S05 Piezoelectric materials parabolic variational inequality quasi-static process frictional contact problem wear and long memory error estimate 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-3858204","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":267549390,"identity":"c9d5e1dd-b2d5-48ee-baba-0850972775b7","order_by":0,"name":"Abderrahmane Oultou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxUlEQVRIiWNgGAWjYDACCQYDIGkBxMwHSNIiIcHAwJZAshYeA+J08M9u3sDwsU2ijl8i55vkz5w6Bt12Ag6UuHOsgHFmm4SE5IzcbdK82w4zmJ0h5MAbOQbMvEAtBjeAWhi3HWAwO0BAizxCS84zyZ/b6hjMzj/Ar8UASQubBO82ZgazGwRsMQT65eCMcxKSM3ueGVsD/cJjdoOALXK3mzc++FBmw8/PnvzwJtBhcmbnCdgCAgeQOTyE1Y+CUTAKRsEoIAgAVfBAZmA3W4AAAAAASUVORK5CYII=","orcid":"","institution":"Université Sultan Moulay Slimane","correspondingAuthor":true,"prefix":"","firstName":"Abderrahmane","middleName":"","lastName":"Oultou","suffix":""},{"id":267549391,"identity":"ff59ed2b-0b9d-44c6-a6fb-65a3ea93be7f","order_by":1,"name":"Othmane Baiz","email":"","orcid":"","institution":"Université Ibn Zohr","correspondingAuthor":false,"prefix":"","firstName":"Othmane","middleName":"","lastName":"Baiz","suffix":""},{"id":267549392,"identity":"dd58187c-f1c6-4984-bcb9-40f60e597c10","order_by":2,"name":"Hicham Benaissa","email":"","orcid":"","institution":"Université Sultan Moulay Slimane","correspondingAuthor":false,"prefix":"","firstName":"Hicham","middleName":"","lastName":"Benaissa","suffix":""}],"badges":[],"createdAt":"2024-01-12 21:14:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3858204/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3858204/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50256002,"identity":"6162e856-3e6d-4be6-bc54-d2ecf92a1ba2","added_by":"auto","created_at":"2024-01-27 18:25:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":393888,"visible":true,"origin":"","legend":"","description":"","filename":"NumericalAnalysisforapiezoelectriccontactproblemwithlongmemoryandwear.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3858204/v1_covered_52e9c7d8-08bc-4ca5-bf0f-8a270da4807a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Numerical Analysis for a piezoelectric contact problem with long-memory and wear","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":"Piezoelectric materials, parabolic variational inequality, quasi-static process, frictional contact problem, wear and long memory, error estimate","lastPublishedDoi":"10.21203/rs.3.rs-3858204/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3858204/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we consider a new mathematical model that describes a frictional quasi-static contact between a piezoelectric body with a long memory effect and a foundation. The contact is described with the normal compliance condition with unilateral constraint and a version of Archard’s wear law. We introduce the model and prove its weak formulation, which is composed of a history-dependent variational equality and an integral equation. Using abstract history-dependent variational inequality and fixed point theorem, we show the existence and uniqueness solution of the model. In addition, we introduce a fully discrete scheme for the numerical solution of the problem, based on the finite element method to approximate the spatial derivative, and on the Euler scheme to discretize the time derivative. 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