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doi: 10.3934/jimo.2020122

C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices

Xifu Liu 1,, , Shuheng Yin 2, and  Hanyu Li 2,

1. 

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Xifu Liu

Received  December 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant No. 11661036), the Top-notch talent Foundation of Chongqing Normal University (Grant No. 0071), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant Nos. KJQN201800506, KJZD-M201800501), the Natural Science Foundation of Chongqing (Grant No. cstc2018jcyjAX0794), and the Program of Chongqing Innovation Research Group Project in University (Grant No. CXQT19018). The third author is supported by the National Natural Science Foundation of China (Grant No. 11671060) and the Natural Science Foundation Project of Chongqing (Grant No. cstc2019jcyj-msxmX0267)

C-eigenvalues of piezoelectric-type tensors play an crucial role in piezoelectric effect and converse piezoelectric effect. In this paper, by the partial symmetry property of piezoelectric-type tensors, we present three intervals to locate all C-eigenvalues of a given piezoelectric-type tensor. Numerical examples show that our results are better than the existing ones.

Keywords: Piezoelectric-type tensor, C-eigenvalue, interval.
Mathematics Subject Classification: Primary: 12E10, 15A18, 15A69.
Citation: Xifu Liu, Shuheng Yin, Hanyu Li. C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020122
References:
[1]

H. T. Che, H. B. Chen and Y. J. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.  Google Scholar

[2]

Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1. Google Scholar

[3]

Y. N. Chen, L. Q. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20 pp. doi: 10.1088/1751-8121/aa98a8.  Google Scholar

[4]

J. Curie and P. Curie, Développement, par compression de l'éctricité polaire dans les cristaux hémièdres à faces inclinées, Bulletin de Minéralogie, 3, 4 (1880), 90-93.  doi: 10.3406/bulmi.1880.1564.  Google Scholar

[5]

M. De Jong, W. Chen, H. Geerlings, M. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053. doi: 10.1038/sdata.2015.53.  Google Scholar

[6]

S. Haussl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007. doi: 10.1002/9783527621156.  Google Scholar

[7]

A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008. Google Scholar

[8]

C. Q. Li, Y. J. Liu and Y. T. Li, C-eigenvalues intervals for piezoelectric-type tensors, Applied Mathematics and Computation, 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.  Google Scholar

[9]

D. Lovett, Tensor Properties of Crystals, 2$^{nd}$ edition, Institute of Physics Publishing, Bristol, 1989. Google Scholar

[10]

J. F. Nye, Physical properties of crystals: Their representation by tensors and matrices, Physics Today, 10 (1957), 26 pp. doi: 10.1063/1.3060200.  Google Scholar

[11]

L. Qi, Transposes, L-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327. Google Scholar

[12]

L. Q. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[13]

W. J. Wang, H. B. Chen and Y. J. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035, 6 pp. doi: 10.1016/j.aml.2019.106035.  Google Scholar

[14]

W.-N. Zou, C.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755. doi: 10.1098/rspa.2012.0755.  Google Scholar

show all references

References:
[1]

H. T. Che, H. B. Chen and Y. J. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.  Google Scholar

[2]

Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1. Google Scholar

[3]

Y. N. Chen, L. Q. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20 pp. doi: 10.1088/1751-8121/aa98a8.  Google Scholar

[4]

J. Curie and P. Curie, Développement, par compression de l'éctricité polaire dans les cristaux hémièdres à faces inclinées, Bulletin de Minéralogie, 3, 4 (1880), 90-93.  doi: 10.3406/bulmi.1880.1564.  Google Scholar

[5]

M. De Jong, W. Chen, H. Geerlings, M. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053. doi: 10.1038/sdata.2015.53.  Google Scholar

[6]

S. Haussl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007. doi: 10.1002/9783527621156.  Google Scholar

[7]

A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008. Google Scholar

[8]

C. Q. Li, Y. J. Liu and Y. T. Li, C-eigenvalues intervals for piezoelectric-type tensors, Applied Mathematics and Computation, 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.  Google Scholar

[9]

D. Lovett, Tensor Properties of Crystals, 2$^{nd}$ edition, Institute of Physics Publishing, Bristol, 1989. Google Scholar

[10]

J. F. Nye, Physical properties of crystals: Their representation by tensors and matrices, Physics Today, 10 (1957), 26 pp. doi: 10.1063/1.3060200.  Google Scholar

[11]

L. Qi, Transposes, L-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327. Google Scholar

[12]

L. Q. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[13]

W. J. Wang, H. B. Chen and Y. J. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035, 6 pp. doi: 10.1016/j.aml.2019.106035.  Google Scholar

[14]

W.-N. Zou, C.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755. doi: 10.1098/rspa.2012.0755.  Google Scholar

Table 1.  Numerical comparison between our methods with the methods from [1,8,13]
$ \mathcal{A}_{\mathrm{VFeSb}} $ $ \mathcal{A}_{\mathrm{SiO_2}} $ $ \mathcal{A}_{\mathrm{Cr_2 AgBiO_8}} $ $ \mathcal{A}_{\mathrm{RbTaO_3}} $ $ \mathcal{A}_{\mathrm{NaBiS_2}} $ $ \mathcal{A}_{\mathrm{LiBiS_2 O_5}} $ $ \mathcal{A}_{\mathrm{KBi_2 F_7}} $ $ \mathcal{A}_{\mathrm{BaNiO_3}} $
$ \lambda^{*} $ 4.2514 0.1375 2.6258 12.4234 11.6674 7.7376 13.5021 27.4628
$ \rho_{min} $ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$ \rho_{\Gamma} $ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$ \rho_{\mathcal{L}} $ 7.3636 0.2744 4.8058 23.5377 16.5640 11.0127 18.8793 27.5109
$ \rho_{\mathcal{M}} $ 7.3636 0.2834 4.7861 23.5377 16.8464 11.0038 19.8830 27.5013
$ \rho_{\gamma} $ 7.3636 0.2737 3.3543 21.9667 16.0233 9.4595 16.7483 27.5012
$ \tilde{\rho}_{min} $ 7.3636 0.2393 4.6717 22.7163 14.5723 12.1694 18.7025 27.5396
$ \rho_{\sigma} $ 6.3771 0.1943 3.7242 16.0259 11.9319 7.7540 13.5113 27.4629
$ \rho_{A} $ 5.2069 0.1938 3.7025 14.9344 11.9319 7.7523 13.5054 27.4629
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$ \mathcal{A}_{\mathrm{VFeSb}} $ $ \mathcal{A}_{\mathrm{SiO_2}} $ $ \mathcal{A}_{\mathrm{Cr_2 AgBiO_8}} $ $ \mathcal{A}_{\mathrm{RbTaO_3}} $ $ \mathcal{A}_{\mathrm{NaBiS_2}} $ $ \mathcal{A}_{\mathrm{LiBiS_2 O_5}} $ $ \mathcal{A}_{\mathrm{KBi_2 F_7}} $ $ \mathcal{A}_{\mathrm{BaNiO_3}} $
$ \lambda^{*} $ 4.2514 0.1375 2.6258 12.4234 11.6674 7.7376 13.5021 27.4628
$ \rho_{min} $ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$ \rho_{\Gamma} $ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$ \rho_{\mathcal{L}} $ 7.3636 0.2744 4.8058 23.5377 16.5640 11.0127 18.8793 27.5109
$ \rho_{\mathcal{M}} $ 7.3636 0.2834 4.7861 23.5377 16.8464 11.0038 19.8830 27.5013
$ \rho_{\gamma} $ 7.3636 0.2737 3.3543 21.9667 16.0233 9.4595 16.7483 27.5012
$ \tilde{\rho}_{min} $ 7.3636 0.2393 4.6717 22.7163 14.5723 12.1694 18.7025 27.5396
$ \rho_{\sigma} $ 6.3771 0.1943 3.7242 16.0259 11.9319 7.7540 13.5113 27.4629
$ \rho_{A} $ 5.2069 0.1938 3.7025 14.9344 11.9319 7.7523 13.5054 27.4629
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