American Institute of Mathematical Sciences

Advanced
November  2021, 17(6): 3349-3356. 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  November 2021 Early access  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, 2021, 17 (6) : 3349-3356. 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
Table Options

Download as excel

$ \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
[1]

Jianxing Zhao, Jincheng Luo. Properties and calculation for C-eigenvalues of a piezoelectric-type tensor. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021162
[2]

Yannan Chen, Antal Jákli, Liqun Qi. The C-eigenvalue of third order tensors and its application in crystals. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021183
[3]

Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042
[4]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[5]

Fan Wu. Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor. Evolution Equations & Control Theory, 2021, 10 (3) : 511-518. doi: 10.3934/eect.2020078
[6]

Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, 2022, 18 (1) : 157-172. doi: 10.3934/jimo.2020147
[7]

Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054
[8]

Hao Li, Hai Bi, Yidu Yang. The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1775-1789. doi: 10.3934/dcdsb.2020002
[9]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[10]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002
[11]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[12]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859
[13]

Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2551-2562. doi: 10.3934/jimo.2019069
[14]

Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure & Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63
[15]

Qi Li, Hong Xue, Changxin Lu. Event-based fault detection for interval type-2 fuzzy systems with measurement outliers. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1301-1328. doi: 10.3934/dcdss.2020412
[16]

Ramasamy Kavikumar, Boomipalagan Kaviarasan, Yong-Gwon Lee, Oh-Min Kwon, Rathinasamy Sakthivel, Seong-Gon Choi. Robust dynamic sliding mode control design for interval type-2 fuzzy systems. Discrete & Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022014
[17]

Marius Tucsnak. Control of plate vibrations by means of piezoelectric actuators. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 281-293. doi: 10.3934/dcds.1996.2.281
[18]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the exponential stabilization of a thermo piezoelectric/piezomagnetic system. Evolution Equations & Control Theory, 2012, 1 (2) : 315-336. doi: 10.3934/eect.2012.1.315
[19]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
[20]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021015

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (226)
  • HTML views (623)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy