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Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor

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Abstract

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Qi et al. (Front Math China 4(2):349–364, 2009) pointed out that the SE-condition of the displacement equations of equilibrium can be equivalently transformed into the SE-condition of a fourth-order real partially symmetric tensor \({\mathcal {A}}\), and that the SE-condition of \({\mathcal {A}}\) holds if and only if the smallest M-eigenvalue of \({\mathcal {A}}\) is positive. In order to judge the strong ellipticity of \({\mathcal {A}}\), we propose a shifted inverse power method for computing the smallest M-eigenvalue of \({\mathcal {A}}\) and give its convergence analysis. And then, we borrow and fine-tune an existing initialization strategy to make the sequence generated by the shifted inverse power method rapidly converge to a good approximation of the smallest M-eigenvalue of \({\mathcal {A}}\). Finally, we by numerical examples illustrate the effectiveness of the proposed method in computing the smallest M-eigenvalue of \({\mathcal {A}}\) and judging the SE-condition of the displacement equations of equilibrium.

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Acknowledgements

The authors are very grateful to the anonymous reviewers for their insightful comments and constructive suggestions, which considerably improve our manuscript. The authors also thank Associate Professor Qilong Liu (Guizhou Normal University) for his discussions and suggestions during the revision of the manuscript.

Funding

This work of Jianxing Zhao is supported by Guizhou Provincial Science and Technology Projects (Grant No. QKHJC-ZK[2022]YB215). This work of Caili Sang is supported by Guizhou Provincial Science and Technology Projects (Grant No. QKHJC-ZK[2021]YB013) and Natural Science Research Project of Department of Education of Guizhou Province (Grant Nos. QJJ[2023]012; QJJ[2023]061; QJJ[2023]062).

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  1. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, People’s Republic of China

    Jianxing Zhao, Pin Liu & Caili Sang

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  1. Jianxing Zhao
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  2. Pin Liu
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Correspondence to Caili Sang.

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Communicated by Liqun Qi.

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Zhao, J., Liu, P. & Sang, C. Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor. J Optim Theory Appl 200, 1131–1159 (2024). https://doi.org/10.1007/s10957-023-02369-z

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