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Sci."],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The simulation of viscoelastic time-evolution problems described by a large number of internal variables and with a large spectrum of relaxation times requires high computational resources for their resolution. Furthermore, the internal variables evolution is described by a set of linear differential equations which involves many time scales. In this context, the use of a space-time PGD approximation is proposed here to boost their resolution, where the temporal functions are constructed following a multi-scale strategy along with the partition of unity method, in order to catch each dynamic efficiently. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under cyclic loading.<\/jats:p>","DOI":"10.1186\/s40323-025-00294-6","type":"journal-article","created":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T13:45:09Z","timestamp":1746798309000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A reduced simulation applied to viscoelastic fatigue of polymers using a time multi-scale approach based on partition of Unity method"],"prefix":"10.1186","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4610-1440","authenticated-orcid":false,"given":"Sebastian","family":"Rodriguez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angelo","family":"Pasquale","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jad","family":"Mounayer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Diego","family":"Canales","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marianne","family":"Beringhier","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chady","family":"Ghnatios","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amine","family":"Ammar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francisco","family":"Chinesta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,9]]},"reference":[{"issue":"3","key":"294_CR1","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.jnnfm.2006.07.007","volume":"139","author":"A Ammar","year":"2006","unstructured":"Ammar A, Mokdad B, Chinesta F, Keunings R. 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