A kernel method for learning constitutive relation in data-driven computational elasticity

For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to...

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Bibliographic Details
Published in:Japan journal of industrial and applied mathematics 2021-02, Vol.38 (1), p.39-77
Main Author: Kanno, Yoshihiro
Format: Article
Language:English
Subjects:
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Constitutive equations
Constitutive relationships
Data points
Datasets
Eigenvectors
Equilibrium analysis
Kernels
Learning
Least squares method
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix methods
Original Paper
Robustness (mathematics)
Static equilibrium
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Summary:For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents a kernel method for extracting this manifold. We formulate a regularized least-squares problem for learning a manifold, and show that its optimal solution corresponds to an eigenvector of a real symmetric matrix. Therefore, the method requires only simple computational task, and is easy to implement. We also give a description how to use the obtained solution in static equilibrium analysis of an elastic structure. Numerical experiments on two-dimensional continua are performed to demonstrate effectiveness and robustness of the proposed method.
ISSN:0916-7005
1868-937X
DOI:10.1007/s13160-020-00423-1