American Institute of Mathematical Sciences

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November  2021, 17(6): 3685-3694. doi: 10.3934/jimo.2020139

New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors

Haitao Che 1, , Haibin Chen 2,, and  Guanglu Zhou 3,

1. 

School of Mathematics and Information Science, Weifang University, Weifang 261061, Shandong, China
2. 

School of Management Science, Qufu Normal University, Rizhao 276800, Shandong, China
3. 

Department of Mathematics and Statistics, Curtin University, Perth 6102, Western Australia, Australia

* Corresponding author: Haibin Chen

Received  January 2020 Revised  June 2020 Published  November 2021 Early access  September 2020

Fund Project: This paper is supported by NSF grant(11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2019MA022), Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J14LI52), and China Postdoctoral Science Foundation (Grant No. 2017M622163)

M-eigenvalues of fourth-order partially symmetric tensors play an important role in nonlinear elasticity and materials. In this paper, we present some M-eigenvalue intervals to locate all M-eigenvalues of fourth-order partially symmetric tensors. It is proved that the new interval is tighter than the one proposed by He, Li and Wei [16]. Furthermore, we obtain some new checkable sufficient conditions for the strong ellipticity of fourth-order partially symmetric tensors. Three numerical examples arising from anisotropic materials are presented to verify the efficiency of the proposed results.

Keywords: M-eigenvalue, fourth-order partially symmetric tensors, strong ellipticity, M-positive definiteness.
Mathematics Subject Classification: Primary: 15A06, 74B20; Secondary: 47J25.
Citation: Haitao Che, Haibin Chen, Guanglu Zhou. New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3685-3694. doi: 10.3934/jimo.2020139
References:
[1]

H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[2]

H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp. doi: 10.1186/s13660-019-1986-x.

[3]

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

[4]

H. Chen, Y. Wang and G. Zhou, High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.  doi: 10.1007/s11464-020-0833-1.

[5]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.

[6]

H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp. doi: 10.1002/nla.2125.

[7]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.

[8]

H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020). doi: 10.1080/10556788.2020.1768389.

[9]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[10]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[11]

H. Chen, L. Qi, L. Caccetta and G. Zhou, Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.  doi: 10.1016/j.laa.2019.08.027.

[12]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[13]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[14]

M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020.

[15]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[16]

J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137. doi: 10.1016/j.aml.2019.106137.

[17]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[18]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.  doi: 10.1007/BF00279991.

[19]

S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.

[20]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.

[21]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[23]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.

[24]

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

[25]

C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678. doi: 10.1016/j.cam.2019.112678.

[26]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.

[27]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[28]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[29]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[30]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[31]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

[32]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134. doi: 10.1002/nla.2134.

show all references

References:
[1]

H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[2]

H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp. doi: 10.1186/s13660-019-1986-x.

[3]

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

[4]

H. Chen, Y. Wang and G. Zhou, High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.  doi: 10.1007/s11464-020-0833-1.

[5]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.

[6]

H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp. doi: 10.1002/nla.2125.

[7]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.

[8]

H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020). doi: 10.1080/10556788.2020.1768389.

[9]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.

[10]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.

[11]

H. Chen, L. Qi, L. Caccetta and G. Zhou, Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.  doi: 10.1016/j.laa.2019.08.027.

[12]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[13]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[14]

M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020.

[15]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[16]

J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137. doi: 10.1016/j.aml.2019.106137.

[17]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[18]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.  doi: 10.1007/BF00279991.

[19]

S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.

[20]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.

[21]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[23]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.

[24]

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

[25]

C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678. doi: 10.1016/j.cam.2019.112678.

[26]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.

[27]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[28]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.

[29]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.

[30]

Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[31]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.

[32]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134. doi: 10.1002/nla.2134.

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