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

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Abstract

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.

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Notes

  1. See Remark 4 in Sect. 5.2.

  2. Main source codes, as well as data sets, used in this section are available online at https://github.com/ykanno22/data_driven_kernel_manifold/.

  3. Therefore, the assertion “[i]n the linear elastic behavior the application of the just described technique results, as expected, in a flat manifold of dimension two” made in [23, section 2] is questionable.

  4. From a practical point of view, it seems to be unrealistic to suppose that millions data points of material experiments are available for a single material.

  5. The value of k for kNN was determined based on preliminary numerical experiments.

  6. In this example, any value of \(\varepsilon ^{(l+1)}\) satisfies the compatibility relation.

  7. This seems to be natural, because the reference solution assumes that all the Gauss evaluation points have the common Young’s modulus and Poisson’s ratio, while Young’s modulus and Poisson’s ratio of the data points used in the proposed method are different from each other.

  8. This large variance of the method using the local regression may possibly be mitigated if we use the local robust regression like Kanno [26].

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Acknowledgements

This work is supported by JST CREST Grant no. JPMJCR1911, Japan and the research grant from the Obayashi Foundation.

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  1. Mathematics and Informatics Center, The University of Tokyo, Hongo 7-3-1, Tokyo, 113-8656, Japan

    Yoshihiro Kanno

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Kanno, Y. A kernel method for learning constitutive relation in data-driven computational elasticity. Japan J. Indust. Appl. Math. 38, 39–77 (2021). https://doi.org/10.1007/s13160-020-00423-1

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