- Email: [email protected]
M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity
Journal Pre-proof M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity
Jun He, Chaoqian Li, Yimin Wei
PII: DOI: Reference:
S0893-9659(19)30461-6 https://doi.org/10.1016/j.aml.2019.106137 AML 106137
To appear in:
Applied Mathematics Letters
Received date : 8 August 2019 Revised date : 7 November 2019 Accepted date : 7 November 2019 Please cite this article as: J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106137. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
*Manuscript Click here to view linked References
Journal Pre-proof
M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity Jun Hea , Chaoqian Lib,∗, Yimin Weic a School
of Mathematics, Zunyi Normal college, Zunyi, Guizhou, China 563006 of Mathematics and Statistics, Yunnan University, Yunnan, China 650091 c School of Mathematics Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, China 200433 b School
Abstract
na lP repr oo f
Some M-eigenvalue intervals of fourth-order structured partially symmetric tensors (PSTs) are established to provide some checkable sufficient conditions for the strong ellipticity (also the M-positive definiteness). These results are extended to general fourth-order PSTs case. Three numerical examples arising from anisotropic materials are given to show the effectivity of the proposed results. Keywords: Partially symmetric tensor, Strong ellipticity, M-positive definiteness, M-eigenvalue 2010 MSC: 15A18, 15A69, 65F15, 65F10
1. Introduction
ur
The strong ellipticity condition (abbr. SEC) plays important roles in nonlinear elasticity and materials [7, 9, 19, 20, 30]. It ensures the existence of solutions of basic boundary-value problems of elastostatics, and thus ensures an elastic material to satisfy some mechanical properties. As said in [7], unlike isotropic elastic materials [5, 10, 21, 22, 24, 29], there are very few necessary conditions, sufficient conditions or both for the strong ellipticity of general anisotropic elastic materials. Some existing SECs for anisotropic elastic materials mainly focus on transversely isotropic linearly elastic solids, see [2, 3, 4, 15, 16, 17]. For the general class of anisotropic elastic materials, Walton and Wilber [23] proposed some conditions for the strong ellipticity, which are only sufficient, but not necessary conditions. Recently, by introducing the M-eigenvalues of a fourth-order partially symmetric tensor Han et al. [7] provided a necessary and sufficient condition for the SEC for general anisotropic elastic materials as follows. ∗ Corresponding
Jo
author. Email addresses: [email protected] (Jun He), [email protected] (Chaoqian Li), [email protected] and [email protected] (Yimin Wei)
Preprint submitted to A.M.L.
November 7, 2019
Journal Pre-proof
Definition 1. [7] Let A = (aijkl ) be a fourth-order real partially symmetric tensor (PST), i.e., aijkl = ajikl = aijlk , i, j, k, l ∈ [n] := {1, 2, . . . , n}, denoted by A ∈ E4,n . A real number λ is called an M-eigenvalue of A if there are vectors x = (xi )ni=1 ∈ Rn \{0} and y = (yl )nl=1 ∈ Rn \{0} such that Axy2 = λx, Ax2 y = λy, x⊤ x = 1, y⊤ y = 1,
(1)
where Axy2 and Ax2 y are real vectors with the i-th and l-th components (Axy2 )i =
n ∑
aijkl xj yk yl and (Ax2 y)l =
j,k,l=1
n ∑
aijkl xi xj yk ,
i,j,k=1
respectively, and x and y are called the corresponding left and right M-eigenvectors. Theorem 1. [7] Let A ∈ E4,n . The SEC holds, i.e., aijkl xi xj yk yl > 0
(2)
i,j,k,l=1
na lP repr oo f
f (x, y) := Axxyy =
n ∑
for all nonzero vectors x, y ∈ Rn if and only if the smallest M-eigenvalues of A is positive.
It is well-known that a tensor A ∈ E4,n satisfying (2) is also called M-positive definite. Hence, the SEC and the M-positive definiteness can be checked by the smallest M-eigenvalue. One natural way is to compute all M-eigenvalues or the smallest M-eigenvalue for a tensor A ∈ E4,n . When n = 2, Qi et al.[19] presented a direct method to find all the M-eigenvalues. Although the SOS methods can be used to compute the least M-eigenvalue for the case that n = 3, this task is still not easy, for details, see Section 6 in [19]. We refer to Refs. [8, 14, 25, 26, 27, 28] for other methods of compute all the M-eigenvalues or the least M-eigenvalue. We in this paper present a new approach for checking the SEC, concretely, we first give some intervals which locate all M-eigenvalues, and then get lower bounds for the smallest M-eigenvalue. Furthermore, if a lower bound is larger than zero, then the smallest M-eigenvalue is positive, consequently, the SEC (the M-positive definiteness) holds. Hence, these lower bounds provide sufficient conditions for the SEC. 2. Main results
ur
The first two equations in (1) are not homogeneous on x and y. This makes some barriers to locate all M-eigenvalues by the traditional methods in [11, 12, 18]. To tackle this issue partly, we firstly try to locate all M-eigenvalues for a special class of PSTs.
Jo
Theorem 2. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi , i ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ1 (A) = {z ∈ R : z ∈ [αi − Ri (A), αi + Ri (A)]} , i∈[n]
2
Journal Pre-proof
where Ri (A) =
n ∑
j,k,l=1, k̸=l
|aijkl | + γi , γi =
n ∑
max{|aijll |}.
j=1, l∈[n] j̸=i
Proof. Assume that λ is an M-eigenvalue of A, x = (xi )ni=1 ∈ Rn \{0} and y = (yl )nl=1 ∈ Rn \{0} are the corresponding left and right M-eigenvectors. Let |xp | = max{|xj |}. Then |xp | ̸= 0. The p-th equation of λx = Axy2 is j∈[n]
λxp =
n ∑
apjkl xj yk yl , which can be rewritten as
j,k,l=1
(
) λ − (app11 y12 + . . . + appnn yn2 ) xp =
n ∑
apjkl xj yk yl +
j,k,l=1, k̸=l
n ∑
apjll xj yl2 . (3)
j,l=1, j̸=p
By taking absolute values on both sides of (3) and the fact that y⊤ y = 1, we have n ∑
n ∑
na lP repr oo f
|λ − αp ||xp | ≤
≤
j,k,l=1, k̸=l n ∑
j,k,l=1, k̸=l
|apjkl ||xj ||yk ||yl | +
|apjkl ||xp | +
Note that
n ∑
j,l=1, j̸=p
|apjll ||xp ||yl2 |
=
n ∑
j=1, j̸=p
≤
=
n ∑
j=1, j̸=p n ∑
j=1, j̸=p
(
n ∑ l=1
n ∑
j,l=1, j̸=p
j,l=1, j̸=p
|apjll ||xj ||yl2 |
|apjll ||xp ||yl2 |.
|apjll ||yl2 |
)
|xp |
) ( max{|apjll |} y12 + . . . + yn2 |xp | l∈[n]
max{|apjll |}|xp |. l∈[n]
Then |λ − αp ||xp | ≤ Rp (A)|xp | and |λ − αp | ≤ Rp (A). Thus λ ∈ Γ1 (A). The conclusion follows.
ur
From the proof of Theorem 2, it is easy to see that we only use the forms Axy2 = λx and y⊤ y = 1. Similarly, by the forms Ax2 y = λy and x⊤ x = 1, we can get the following M-eigenvalue interval, and omit its proof.
Jo
Theorem 3. Let A = (aijkl ) ∈ E4,n with a11ll = . . . = annll = βl , l ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ2 (A) = {z ∈ R : z ∈ [βl − Cl (A), βl + Cl (A)]} , l∈[n]
3
Journal Pre-proof
where Cl (A) =
n ∑
i,j,k=1, i̸=j
|aijkl | + δl , δl =
n ∑
max{|aiikl |}.
k=1, i∈[n] k̸=l
Remark here that unlike the M-eigenvalue intervals provided in [1, 13], the intervals in Theorem 2 and Theorem 3 are not necessary to include zero. As we will see below, they can be used to identify the M-positive definite (the SEC). Furthermore, Theorem 2 together with Theorem 3 yield the following result. Corollary 1. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi = a11ii = . . . = annii , i ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ(A) = {z ∈ R : z ∈ [αi − min {Ri (A), Ci (A)} , αi + min {Ri (A), Ci (A)}]} . i∈[n]
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By using the M-eigenvalue intervals above, we next give some sufficient conditions for the M-positive definite (the SEC) of structured PSTs. Before that we need a lemma as follows. Lemma 4. Let A = (aijkl ) ∈ E4,n be M-positive definite. Then aiill > 0, i, l ∈ [n]. Proof. Let ei be the n dimension real vector, whose i-th entry is 1 and others 0, el be the n dimension real vector, whose l-th entry is 1 and others 0. Since the partially symmetric tensor A is M-positive definite, then Aei ei el el = aiill > 0, for any i, l ∈ [n]. Lemma 4 provides a necessary condition for the M-positive definiteness, also the SEC. Theorem 5. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi > 0, i ∈ [n]. If for all i ∈ [n], αi > Ri (A), then A is M-positive definite. Proof. Assume that λ ≤ 0 is an M-eigenvalue of A. From Theorem 2, we have λ ∈ Γ1 (A). Hence, there is i0 ∈ [n] such that |λ − αi0 | ≤ Ri0 (A). From αi0 > 0 for all i0 ∈ [n], we get |λ − αi0 | ≥ αi0 > Ri0 (A), This is a contradiction. Hence, λ > 0. Then A is M-positive definite. Similarly to Theorem 5, we can get the following results, and omits the proof. Theorem 6. Let A = (aijkl ) ∈ E4,n with a11ll = . . . = annll = βl > 0, l ∈ [n]. If for all l ∈ [n], βl > Cl (A), then A is M-positive definite.
ur
Corollary 2. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi = a11ii = . . . = annii , i ∈ [n]. If for all i ∈ [n], αi > min {Ri (A), Ci (A)} , then A is M-positive definite.
Jo
Theorem 5, Theorem 6 and Corollary 2 provide, respectively, an easily checkable sufficient condition of the SEC for three classes of structured PSTs. Next, these conditions can be extended to general PSTs. 4
Journal Pre-proof
Theorem 7. Let A = (aijkl ) ∈ E4,n with aiill > 0 for any i, l ∈ [n], and min {aii11 , . . . , aiinn } = αi , min {a11ll , . . . , annll } = βl . If for all i, l ∈ [n],
i∈[n]
l∈[n]
αi > Ri (A), or βl > Cl (A), then A is M-positive definite. Proof. Let B1 be a PST whose (ijkl)-th entry is defined as follows: { bii11 = . . . = biinn = αi , i ∈ [n], (B1 )ijkl = aijkl , otherwise, If for each i ∈ [n], αi > Ri (B1 ) holds, then by Theorem 5, B1 is M-positive definite. Hence B1 xxyy∑> 0 for any nonzero vectors x ∈ Rn , y ∈ Rn . Note that Axxyy = B1 xxyy + (aiill − biill )x2i yl2 > 0. Thus A is M-positive definite. i,l∈[n]
Let B2 be
(B2 )ijkl =
{
b11ll = . . . = bnnll = βl , l ∈ [n], aijkl , otherwise.
na lP repr oo f
By Theorem 6 we can similarly prove that when βl > Cl (A) for l ∈ [n], A is also M-positive definite. The proof is complete. 3. Numerical example
Example 1. Consider the class of rhombic elastic materials characterized by the conditions a1123 = a1131 = a1112 = a2223 = a2231 = a2212 = 0, a3323 = a3331 = a3312 = a2331 = a2312 = a3112 = 0,
(4)
see Page 4 in [3]. Take such an elasticity tensor with the only non-zero components: a1111 = 4.2, a2222 = 5, a3333 = 4.5, a1122 = 4.3, a2233 = 4.5, a3311 = 5, a2323 = 1, a1313 = 1 and a1212 = 1. It can be verified that α1 = 4.2, α2 = 4.3, α3 = 4.5, and Ri (A) = 4 for i = 1, 2, 3. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity holds. Example 2. Consider the tetragonal system (see Page 6 in [3]) with the related elasticity tensor satisfying (4) and non-zero components: a1111 = a2222 = 5, a3333 = 4.5, a1122 = 4.3, a2233 = a3311 = 5, a2323 = a1313 = 0.5 and a1212 = 1. It can be verified that α1 = α2 = 4.3, α3 = 4.5, R1 (A) = R2 (A) = 3, and R3 (A) = 2. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity holds.
Jo
ur
Example 3. Consider the cubic system (see Page 7 in [3]) with the related elasticity tensor satisfying (4) and non-zero components: a1111 = a2222 = a3333 = 4.5, a1122 = a2233 = a3311 = 5, and a2323 = a1313 = a1212 = 1. It can be verified that αi = 4.5 > Ri (A) = 4 for i = 1, 2, 3. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity condition holds.
5
Journal Pre-proof
Remark here that (I) besides the rhombic, tetragonal, cubic systems, there are other systems, such as triclinic system, monoclinic system, hexagonal system and so on, for details, see [3, 4, 5, 6]. One could use Theorem 7 to verify the strongly ellipticity condition; (II) The new sufficient conditions for the strongly ellipticity all depend only on the entries of the involved tensor, and hence are computable and checkable; (III) The authors in [1, 13] provided some Meigenvalue intervals, it cannot be used to verify the strongly ellipticity. However, our M-eigenvalue intervals can perform. Acknowledgements
na lP repr oo f
The authors are grateful to the referees for their useful and constructive suggestions. This work is supported in part by National Natural Science Foundations of China (11771099,11861077), new academic talents and innovative exploration fostering project(Qian Ke He Pingtai Rencai[2017]5727-21), Program for Excellent Young Talents of Yunnan University, Yunnan Provincial Ten Thousands Plan Young Top Talents, Shanghai Key Laboratory for Contemporary Applied Mathematics (KBH1411209), International Cooperation Project of Shanghai Municipal Science and Technology Commission under grant 16510711200, and Innovation Program of Shanghai Municipal Education Commission.
References
[1] Che H.T., Chen H.B., and Wang Y.J. On the M-eigenvalue estimation of fourthorder partially symmetric tensors, Journal of Industrial and Management Optimization, 2020, 16(1):309-324. [2] Chirit¸a ˇ S. On the strong ellipticity condition for transversely isotropic linearly elastic solids. An. St. Univ. Iasi, Matematica f, 2006, 2(52):113-118. [3] Chirit¸a ˇ S., Danescu A., and Ciarletta M. On the strong ellipticity of the anisotropic linearly elastic materials. Journal of Elasticity, 2007, 87:1-27. [4] Chirit¸a ˇ S., and Danescu A. Strong ellipticity for tetragonal system in linearly elastic solids. International Journal of Solids and Structures, 2008, 45(17):48504859. [5] Dacorogna B. Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension. Discrete Contin. Dyn. Syst. Ser. B, 2001, 1:257-263. [6] Gurtin M.E. The linear theory of elasticity. In: Truesdell, C. Handbuch der Physik, vol 2. Springer, Berlin Heidelberg New York, 1972.
ur
[7] Han D., Dai H.H., and Qi L.Q. Conditions for strong ellipticity of anisotropic elastic materials. Journal of Elasticity, 2009, 97(1):1-13.
Jo
[8] Hu S.L. and Huang Z.H. Alternating direction method for bi-quadratic programming. Journal of Global Optimization, 2011, 51(3):429-446.
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[9] Huang Z.H. and Qi L.Q. Positive definiteness of paired symmetric tensors and elasticity tensors. Journal of Computational and Applied Mathematics, 2018, 338: 22-43. [10] Knowles J.K. and Sternberg E. On the ellipticity of the equations of non-linear elastostatics for a special material. Journal of Elasticity, 1975, 5:341-361. [11] Li C.Q., Li Y.T., and Kong X. New eigenvalue inclusion sets for tensors. Numerical Linear Algebra with Applications, 2014, 21:39-50. [12] Li C.Q. and Li Y.T. An eigenvalue localization set for tensors with applications to determine the positive (semi-) deffiniteness of tensors. Linear and Multilinear Algebra, 2016, 64:587-601. [13] Li S.H., Li C.Q., and Li Y.T. M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 2019, 356:391-401.
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[14] Ling C., Nie J.W., Qi L.Q., and Ye Y.Y. Bi-quadratic optimization over unit spheres and semidefinite programming relaxations. SIAM Journal on Optimization, 2009, 20(3):1286-1310. [15] Merodio J. and Ogden R.W. Instabilities and loss of ellipticity in fiber-reinforced compressible nonlinearly elastic solids under plane deformation. Int. J. Solids Struct. 2003, 40:4707-4727. [16] Padovani C. Strong ellipticity of transversely isotropic elasticity tensors. Meccanica 2002, 37:515-525 [17] Payton R.G. Elastic Wave Propagation in Transversely Isotropic Media. Nijhoff, Dordrecht 1983. [18] Qi L.Q. Eigenvalues of a real supersymmetric tensor. Journal of Symbolic Computation. 2005, 40(6):1302-1324. [19] Qi L.Q., Dai H.H., and Han D.R. Conditions for strong ellipticity and Meigenvalues. Frontiers of Mathematics in China, 2009, 4(2):349-364. [20] Qi L.Q. and Luo Z.Y. Tensor analysis spectral theory and special tensors. SIAM, USA, 2017. [21] Rosakis P. Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch. Ration. Mech. Anal. 1990, 109:1-37. [22] Simpson H.C. and Spector S.J. On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Ration. Mech. Anal. 1983, 84:55-68.
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[23] Walton J.R. and Wilber J.P. Sufficient conditions for strong ellipticity for a class of anisotropic materials. Int. J. Non-Linear Mech. 2003, 38:44-455.
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[24] Wang Y. and Aron M. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. Journal of Elasticity, 1996, 44:89-96.
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[25] Wang Y.J., Qi L.Q., and Zhang X.Z. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numerical Linear Algebra with Applications, 2009, 16(7):589-601. [26] Wang X.Z., Che M.L., and Wei Y.M. Best rank-one approximation of fourthOrder partially symmetric tensors by neural network. Numer. Math. Theor. Meth. Appl. 2018, 11(4):673-700. [27] Xiang H., Qi L.Q., and Wei Y.M. M-eigenvalues of the Riemann curvature tensor. Communications in Mathematical Sciences. 2018, 16(8):2301-2315. [28] Xiang H., Qi L.Q., and Wei Y.M. A note on the M-eigenvalues of the elasticity tensor and strong ellipticity. arXiv preprint arXiv:1708.04876, 2017. [29] Zee L. and Sternberg E. Ordinary and strong ellipticity in the equilibrium theory of impressible hyperelastic solids. Arch. Ration. Mech. Anal. 1983, 83:53-90.
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ur
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[30] Zubov L.M. and Rudev A.N. On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. ZAMM-Journal of Applied Mathematics and Mechanics, 2016, 96(9):1096-1102.
8
Jun He, Chaoqian Li, Yimin Wei
PII: DOI: Reference:
S0893-9659(19)30461-6 https://doi.org/10.1016/j.aml.2019.106137 AML 106137
To appear in:
Applied Mathematics Letters
Received date : 8 August 2019 Revised date : 7 November 2019 Accepted date : 7 November 2019 Please cite this article as: J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106137. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
*Manuscript Click here to view linked References
Journal Pre-proof
M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity Jun Hea , Chaoqian Lib,∗, Yimin Weic a School
of Mathematics, Zunyi Normal college, Zunyi, Guizhou, China 563006 of Mathematics and Statistics, Yunnan University, Yunnan, China 650091 c School of Mathematics Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, China 200433 b School
Abstract
na lP repr oo f
Some M-eigenvalue intervals of fourth-order structured partially symmetric tensors (PSTs) are established to provide some checkable sufficient conditions for the strong ellipticity (also the M-positive definiteness). These results are extended to general fourth-order PSTs case. Three numerical examples arising from anisotropic materials are given to show the effectivity of the proposed results. Keywords: Partially symmetric tensor, Strong ellipticity, M-positive definiteness, M-eigenvalue 2010 MSC: 15A18, 15A69, 65F15, 65F10
1. Introduction
ur
The strong ellipticity condition (abbr. SEC) plays important roles in nonlinear elasticity and materials [7, 9, 19, 20, 30]. It ensures the existence of solutions of basic boundary-value problems of elastostatics, and thus ensures an elastic material to satisfy some mechanical properties. As said in [7], unlike isotropic elastic materials [5, 10, 21, 22, 24, 29], there are very few necessary conditions, sufficient conditions or both for the strong ellipticity of general anisotropic elastic materials. Some existing SECs for anisotropic elastic materials mainly focus on transversely isotropic linearly elastic solids, see [2, 3, 4, 15, 16, 17]. For the general class of anisotropic elastic materials, Walton and Wilber [23] proposed some conditions for the strong ellipticity, which are only sufficient, but not necessary conditions. Recently, by introducing the M-eigenvalues of a fourth-order partially symmetric tensor Han et al. [7] provided a necessary and sufficient condition for the SEC for general anisotropic elastic materials as follows. ∗ Corresponding
Jo
author. Email addresses: [email protected] (Jun He), [email protected] (Chaoqian Li), [email protected] and [email protected] (Yimin Wei)
Preprint submitted to A.M.L.
November 7, 2019
Journal Pre-proof
Definition 1. [7] Let A = (aijkl ) be a fourth-order real partially symmetric tensor (PST), i.e., aijkl = ajikl = aijlk , i, j, k, l ∈ [n] := {1, 2, . . . , n}, denoted by A ∈ E4,n . A real number λ is called an M-eigenvalue of A if there are vectors x = (xi )ni=1 ∈ Rn \{0} and y = (yl )nl=1 ∈ Rn \{0} such that Axy2 = λx, Ax2 y = λy, x⊤ x = 1, y⊤ y = 1,
(1)
where Axy2 and Ax2 y are real vectors with the i-th and l-th components (Axy2 )i =
n ∑
aijkl xj yk yl and (Ax2 y)l =
j,k,l=1
n ∑
aijkl xi xj yk ,
i,j,k=1
respectively, and x and y are called the corresponding left and right M-eigenvectors. Theorem 1. [7] Let A ∈ E4,n . The SEC holds, i.e., aijkl xi xj yk yl > 0
(2)
i,j,k,l=1
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f (x, y) := Axxyy =
n ∑
for all nonzero vectors x, y ∈ Rn if and only if the smallest M-eigenvalues of A is positive.
It is well-known that a tensor A ∈ E4,n satisfying (2) is also called M-positive definite. Hence, the SEC and the M-positive definiteness can be checked by the smallest M-eigenvalue. One natural way is to compute all M-eigenvalues or the smallest M-eigenvalue for a tensor A ∈ E4,n . When n = 2, Qi et al.[19] presented a direct method to find all the M-eigenvalues. Although the SOS methods can be used to compute the least M-eigenvalue for the case that n = 3, this task is still not easy, for details, see Section 6 in [19]. We refer to Refs. [8, 14, 25, 26, 27, 28] for other methods of compute all the M-eigenvalues or the least M-eigenvalue. We in this paper present a new approach for checking the SEC, concretely, we first give some intervals which locate all M-eigenvalues, and then get lower bounds for the smallest M-eigenvalue. Furthermore, if a lower bound is larger than zero, then the smallest M-eigenvalue is positive, consequently, the SEC (the M-positive definiteness) holds. Hence, these lower bounds provide sufficient conditions for the SEC. 2. Main results
ur
The first two equations in (1) are not homogeneous on x and y. This makes some barriers to locate all M-eigenvalues by the traditional methods in [11, 12, 18]. To tackle this issue partly, we firstly try to locate all M-eigenvalues for a special class of PSTs.
Jo
Theorem 2. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi , i ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ1 (A) = {z ∈ R : z ∈ [αi − Ri (A), αi + Ri (A)]} , i∈[n]
2
Journal Pre-proof
where Ri (A) =
n ∑
j,k,l=1, k̸=l
|aijkl | + γi , γi =
n ∑
max{|aijll |}.
j=1, l∈[n] j̸=i
Proof. Assume that λ is an M-eigenvalue of A, x = (xi )ni=1 ∈ Rn \{0} and y = (yl )nl=1 ∈ Rn \{0} are the corresponding left and right M-eigenvectors. Let |xp | = max{|xj |}. Then |xp | ̸= 0. The p-th equation of λx = Axy2 is j∈[n]
λxp =
n ∑
apjkl xj yk yl , which can be rewritten as
j,k,l=1
(
) λ − (app11 y12 + . . . + appnn yn2 ) xp =
n ∑
apjkl xj yk yl +
j,k,l=1, k̸=l
n ∑
apjll xj yl2 . (3)
j,l=1, j̸=p
By taking absolute values on both sides of (3) and the fact that y⊤ y = 1, we have n ∑
n ∑
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|λ − αp ||xp | ≤
≤
j,k,l=1, k̸=l n ∑
j,k,l=1, k̸=l
|apjkl ||xj ||yk ||yl | +
|apjkl ||xp | +
Note that
n ∑
j,l=1, j̸=p
|apjll ||xp ||yl2 |
=
n ∑
j=1, j̸=p
≤
=
n ∑
j=1, j̸=p n ∑
j=1, j̸=p
(
n ∑ l=1
n ∑
j,l=1, j̸=p
j,l=1, j̸=p
|apjll ||xj ||yl2 |
|apjll ||xp ||yl2 |.
|apjll ||yl2 |
)
|xp |
) ( max{|apjll |} y12 + . . . + yn2 |xp | l∈[n]
max{|apjll |}|xp |. l∈[n]
Then |λ − αp ||xp | ≤ Rp (A)|xp | and |λ − αp | ≤ Rp (A). Thus λ ∈ Γ1 (A). The conclusion follows.
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From the proof of Theorem 2, it is easy to see that we only use the forms Axy2 = λx and y⊤ y = 1. Similarly, by the forms Ax2 y = λy and x⊤ x = 1, we can get the following M-eigenvalue interval, and omit its proof.
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Theorem 3. Let A = (aijkl ) ∈ E4,n with a11ll = . . . = annll = βl , l ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ2 (A) = {z ∈ R : z ∈ [βl − Cl (A), βl + Cl (A)]} , l∈[n]
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where Cl (A) =
n ∑
i,j,k=1, i̸=j
|aijkl | + δl , δl =
n ∑
max{|aiikl |}.
k=1, i∈[n] k̸=l
Remark here that unlike the M-eigenvalue intervals provided in [1, 13], the intervals in Theorem 2 and Theorem 3 are not necessary to include zero. As we will see below, they can be used to identify the M-positive definite (the SEC). Furthermore, Theorem 2 together with Theorem 3 yield the following result. Corollary 1. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi = a11ii = . . . = annii , i ∈ [n]. If λ is an M-eigenvalue of A, then ∪ λ ∈ Γ(A) = {z ∈ R : z ∈ [αi − min {Ri (A), Ci (A)} , αi + min {Ri (A), Ci (A)}]} . i∈[n]
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By using the M-eigenvalue intervals above, we next give some sufficient conditions for the M-positive definite (the SEC) of structured PSTs. Before that we need a lemma as follows. Lemma 4. Let A = (aijkl ) ∈ E4,n be M-positive definite. Then aiill > 0, i, l ∈ [n]. Proof. Let ei be the n dimension real vector, whose i-th entry is 1 and others 0, el be the n dimension real vector, whose l-th entry is 1 and others 0. Since the partially symmetric tensor A is M-positive definite, then Aei ei el el = aiill > 0, for any i, l ∈ [n]. Lemma 4 provides a necessary condition for the M-positive definiteness, also the SEC. Theorem 5. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi > 0, i ∈ [n]. If for all i ∈ [n], αi > Ri (A), then A is M-positive definite. Proof. Assume that λ ≤ 0 is an M-eigenvalue of A. From Theorem 2, we have λ ∈ Γ1 (A). Hence, there is i0 ∈ [n] such that |λ − αi0 | ≤ Ri0 (A). From αi0 > 0 for all i0 ∈ [n], we get |λ − αi0 | ≥ αi0 > Ri0 (A), This is a contradiction. Hence, λ > 0. Then A is M-positive definite. Similarly to Theorem 5, we can get the following results, and omits the proof. Theorem 6. Let A = (aijkl ) ∈ E4,n with a11ll = . . . = annll = βl > 0, l ∈ [n]. If for all l ∈ [n], βl > Cl (A), then A is M-positive definite.
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Corollary 2. Let A = (aijkl ) ∈ E4,n with aii11 = . . . = aiinn = αi = a11ii = . . . = annii , i ∈ [n]. If for all i ∈ [n], αi > min {Ri (A), Ci (A)} , then A is M-positive definite.
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Theorem 5, Theorem 6 and Corollary 2 provide, respectively, an easily checkable sufficient condition of the SEC for three classes of structured PSTs. Next, these conditions can be extended to general PSTs. 4
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Theorem 7. Let A = (aijkl ) ∈ E4,n with aiill > 0 for any i, l ∈ [n], and min {aii11 , . . . , aiinn } = αi , min {a11ll , . . . , annll } = βl . If for all i, l ∈ [n],
i∈[n]
l∈[n]
αi > Ri (A), or βl > Cl (A), then A is M-positive definite. Proof. Let B1 be a PST whose (ijkl)-th entry is defined as follows: { bii11 = . . . = biinn = αi , i ∈ [n], (B1 )ijkl = aijkl , otherwise, If for each i ∈ [n], αi > Ri (B1 ) holds, then by Theorem 5, B1 is M-positive definite. Hence B1 xxyy∑> 0 for any nonzero vectors x ∈ Rn , y ∈ Rn . Note that Axxyy = B1 xxyy + (aiill − biill )x2i yl2 > 0. Thus A is M-positive definite. i,l∈[n]
Let B2 be
(B2 )ijkl =
{
b11ll = . . . = bnnll = βl , l ∈ [n], aijkl , otherwise.
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By Theorem 6 we can similarly prove that when βl > Cl (A) for l ∈ [n], A is also M-positive definite. The proof is complete. 3. Numerical example
Example 1. Consider the class of rhombic elastic materials characterized by the conditions a1123 = a1131 = a1112 = a2223 = a2231 = a2212 = 0, a3323 = a3331 = a3312 = a2331 = a2312 = a3112 = 0,
(4)
see Page 4 in [3]. Take such an elasticity tensor with the only non-zero components: a1111 = 4.2, a2222 = 5, a3333 = 4.5, a1122 = 4.3, a2233 = 4.5, a3311 = 5, a2323 = 1, a1313 = 1 and a1212 = 1. It can be verified that α1 = 4.2, α2 = 4.3, α3 = 4.5, and Ri (A) = 4 for i = 1, 2, 3. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity holds. Example 2. Consider the tetragonal system (see Page 6 in [3]) with the related elasticity tensor satisfying (4) and non-zero components: a1111 = a2222 = 5, a3333 = 4.5, a1122 = 4.3, a2233 = a3311 = 5, a2323 = a1313 = 0.5 and a1212 = 1. It can be verified that α1 = α2 = 4.3, α3 = 4.5, R1 (A) = R2 (A) = 3, and R3 (A) = 2. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity holds.
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Example 3. Consider the cubic system (see Page 7 in [3]) with the related elasticity tensor satisfying (4) and non-zero components: a1111 = a2222 = a3333 = 4.5, a1122 = a2233 = a3311 = 5, and a2323 = a1313 = a1212 = 1. It can be verified that αi = 4.5 > Ri (A) = 4 for i = 1, 2, 3. Hence, from Theorem 7, A is M-positive definite, and the strongly ellipticity condition holds.
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Remark here that (I) besides the rhombic, tetragonal, cubic systems, there are other systems, such as triclinic system, monoclinic system, hexagonal system and so on, for details, see [3, 4, 5, 6]. One could use Theorem 7 to verify the strongly ellipticity condition; (II) The new sufficient conditions for the strongly ellipticity all depend only on the entries of the involved tensor, and hence are computable and checkable; (III) The authors in [1, 13] provided some Meigenvalue intervals, it cannot be used to verify the strongly ellipticity. However, our M-eigenvalue intervals can perform. Acknowledgements
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The authors are grateful to the referees for their useful and constructive suggestions. This work is supported in part by National Natural Science Foundations of China (11771099,11861077), new academic talents and innovative exploration fostering project(Qian Ke He Pingtai Rencai[2017]5727-21), Program for Excellent Young Talents of Yunnan University, Yunnan Provincial Ten Thousands Plan Young Top Talents, Shanghai Key Laboratory for Contemporary Applied Mathematics (KBH1411209), International Cooperation Project of Shanghai Municipal Science and Technology Commission under grant 16510711200, and Innovation Program of Shanghai Municipal Education Commission.
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