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Article Dans Une Revue Advanced Modeling and Simulation in Engineering Sciences Année : 2021

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

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

Angelo Pasquale
  • Fonction : Auteur
Amine Ammar
  • Fonction : Auteur
Antonio Falcó
  • Fonction : Auteur
Simona Perotto
Elías Cueto
  • Fonction : Auteur
Jean-Louis Duval
  • Fonction : Auteur
Francisco Chinesta
  • Fonction : Auteur

Résumé

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.
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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Dates et versions

hal-03501301 , version 1 (01-02-2023)

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Angelo Pasquale, Amine Ammar, Antonio Falcó, Simona Perotto, Elías Cueto, et al.. A separated representation involving multiple time scales within the Proper Generalized Decomposition framework. Advanced Modeling and Simulation in Engineering Sciences, 2021, 8 (1), pp.267-293. ⟨10.1186/s40323-021-00211-7⟩. ⟨hal-03501301⟩

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