期刊
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
卷 389, 期 -, 页码 -出版社
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2021.114414
关键词
Hierarchical deep-learning neural networks; Proper generalized decomposition; Canonical tensor decomposition; Reduced order finite element method; Convergence study and error bound
资金
- National Natural Science Foundation of China [11890681, 11832001, 11521202, 11988102]
- National Science Foundation, USA [CMMI-1934367, CMMI-1762035]
This paper introduces a tensor decomposition (TD) based reduced-order model of hierarchical deep-learning neural networks (HiDeNN), which combines the advantages of both HiDeNN and TD methods. The proposed HiDeNN-TD method is shown to be more accurate than finite element method (FEM) and conventional methods, with significantly fewer degrees of freedom. The paper also presents a highly efficient solution strategy called HiDeNN-PGD, which achieves a balance between accuracy and efficiency.
This paper presents a tensor decomposition (TD) based reduced-order model of the hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-TD method keeps advantages of both HiDeNN and TD methods. The automatic mesh adaptivity makes the HiDeNN-TD more accurate than the finite element method (FEM) and conventional proper generalized decomposition (PGD) and TD, using a fraction of the FEM degrees of freedom. This work focuses on the theoretical foundation of the method. Hence, the accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, TD, HiDeNN and Deep Neural Networks. In addition, we have theoretically shown that the PGD/TD converges to FEM at increasing modes, and the PGD/TD solution error is a summation of the mesh discretization error and the mode reduction error. The proposed HiDeNN-TD shows a high accuracy with orders of magnitude fewer degrees of freedom than FEM, and hence a high potential to achieve fast computations with a high level of accuracy for large-size engineering and scientific problems. As a trade-off between accuracy and efficiency, we propose a highly efficient solution strategy called HiDeNN-PGD. Although the solution is less accurate than HiDeNN-TD, HiDeNN-PGD still provides a higher accuracy than PGD/TD and FEM with only a small amount of additional cost to PGD. (c) 2021 Elsevier B.V. All rights reserved.
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