Journal
ADVANCED MODELING AND SIMULATION IN ENGINEERING SCIENCES
Volume 8, Issue 1, Pages -Publisher
SPRINGERNATURE
DOI: 10.1186/s40323-021-00211-7
Keywords
Model Order Reduction; Time multiscale; Proper Generalized Decomposition; Thermal problems; Elastodynamics
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Funding
- University of Zaragoza [2019-0060]
- Spanish Ministry of Science [CICYT-DPI2017-85139-C2-1-R]
- Regional Government of Aragon
- European Social Fund
- European Union's Horizon 2020 research and innovation programme [872442]
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The paper presents an alternative approach to solving partial differential equations by using a time-separated representation that allows for fine time discretization without impacting computational efficiency. By transforming the time coordinate into multidimensional time through separated coordinates, the method circumvents prohibitive meshes and captures different scales effectively. The formulation is explored and verified numerically on thermal and elastodynamic problems, showcasing its effectiveness in handling multiple time scales.
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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