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

Xifu Liu; Shuheng Yin; Hanyu Li

doi:10.3934/jimo.2020122

Article Contents

2021, Volume 17, Issue 6: 3349-3356. Doi: 10.3934/jimo.2020122

This issue Previous Article Modeling and computation of mean field game with compound carbon abatement mechanisms Next Article New inertial method for generalized split variational inclusion problems

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

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
^* Corresponding author: Xifu Liu

Received: November 30, 2019

Revised: February 29, 2020

Early access: June 2020

Published: November 2021

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)

Abstract Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 12E10, 15A18, 15A69.

Citation:

Full Text(HTML)

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.
[2]	Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1.
[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.
[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.
[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.
[6]	S. Haussl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007. doi: 10.1002/9783527621156.
[7]	A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008.
[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.
[9]	D. Lovett, Tensor Properties of Crystals, 2$^{nd}$ edition, Institute of Physics Publishing, Bristol, 1989.
[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.
[11]	L. Qi, Transposes, L-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327.
[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.
[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.
[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.