Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

doi:10.1007/s42967-023-00308-3

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1779-1800.doi: 10.1007/s42967-023-00308-3

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Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Clayton Little¹, Charbel Farhat^1,2,3

1 Department of Mechanical Engineering, Stanford University, Stanford, USA;
2 Department of Aeronautics and Astronautics, Stanford University, Stanford, USA;
3 Institute for Computational and Mathematical Engineering, Stanford University, Stanford, USA

收稿日期:2023-02-21 修回日期:2023-06-30 接受日期:2023-08-15 发布日期:2024-12-20
通讯作者: Clayton Little,crl11@stanford.edu;Charbel Farhat,cfarhat@stanford.edu E-mail:crl11@stanford.edu;cfarhat@stanford.edu

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Clayton Little¹, Charbel Farhat^1,2,3

1 Department of Mechanical Engineering, Stanford University, Stanford, USA;
2 Department of Aeronautics and Astronautics, Stanford University, Stanford, USA;
3 Institute for Computational and Mathematical Engineering, Stanford University, Stanford, USA

Received:2023-02-21 Revised:2023-06-30 Accepted:2023-08-15 Published:2024-12-20
Contact: Clayton Little,crl11@stanford.edu;Charbel Farhat,cfarhat@stanford.edu E-mail:crl11@stanford.edu;cfarhat@stanford.edu

摘要/Abstract

摘要： Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reducedorder models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wallclock speedup factors.

关键词: Adaptive mesh refinement (AMR), Computational fluid dynamics, Energyconserving sampling and weighting (ECSW), Model order reduction, Reduced-order model, Supermesh

Abstract: Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reducedorder models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wallclock speedup factors.

Key words: Adaptive mesh refinement (AMR), Computational fluid dynamics, Energyconserving sampling and weighting (ECSW), Model order reduction, Reduced-order model, Supermesh

中图分类号:

Clayton Little, Charbel Farhat. Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1779-1800.

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Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

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