Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910748630843392 |
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| author | Li, Mingwu Thurnher, Thomas Xu, Zhenwei Jain, Shobhit |
| author_facet | Li, Mingwu Thurnher, Thomas Xu, Zhenwei Jain, Shobhit |
| contents | The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds Li, Mingwu Thurnher, Thomas Xu, Zhenwei Jain, Shobhit Numerical Analysis Computational Engineering, Finance, and Science Dynamical Systems The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom. |
| title | Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds |
| topic | Numerical Analysis Computational Engineering, Finance, and Science Dynamical Systems |
| url | https://arxiv.org/abs/2409.10126 |