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A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

Article dans une revue avec comité de lecture
Author
PASQUALE, Angelo
211916 Laboratoire Angevin de Mécanique, Procédés et InnovAtion [LAMPA]
ccAMMAR, Amine
211916 Laboratoire Angevin de Mécanique, Procédés et InnovAtion [LAMPA]
FALCÓ, Antonio
307554 Universidad Cardenal Herrera-CEU [CEU-UCH]
PEROTTO, Simona
125443 Politecnico di Milano [Milan] [POLIMI]
ccCUETO, Elias
161327 Aragón Institute of Engineering Research [Zaragoza] [I3A]
DUVAL, Jean-Louis
564849 ESI Group [ESI Group]
ccCHINESTA SORIA, Francisco
86289 Laboratoire Procédés et Ingénierie en Mécanique et Matériaux [PIMM]

URI
http://hdl.handle.net/10985/23265
DOI
10.1186/s40323-021-00211-7
Date
2021-11-26
Journal
Advanced Modeling and Simulation in Engineering Sciences

Abstract

Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

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