# New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

Authors: Zhuanzhou Zhang; Jun He; Yanmin Liu; Zerong Ren
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.


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## Body

## 1. Introduction

LetM=1,2,…,m and N=1,2,…,n; a real tensor A=aijkl∈ℝm×n×m×n is called an elasticity tensor, if(1)aijkl=akjil=ailkj,i,k∈M,j,l∈N.Consider the following optimization problem with an elasticity tensorA=aijkl [1, 2]:(2)maxfx,y=∑i,k=1m∑j,l=1naijklxiyjxkyl,s.t.xTx=1,yTy=1,x∈ℝm,y∈ℝn.Qi et al. introduced the following definition of M-eigenvalues of an elasticity tensor [3, 4].Definition 1.
(see [3, 4]). Let A=aijkl∈ℝm×n×m×n be an elasticity tensor, if there exist nonzero vectors, x∈ℝm and y=∈ℝn, and a real number λ∈ℝ, such that(3)Ayxy=λx,Axyx=λy,xTx=1,yTy=1,where(4)Ayxyi=∑k∈M∑j,l=1naijklyjxkyl,Axyxl=∑i,k∈M∑j=1naijklxiyjxk.
Then,λ is called an M-eigenvalue of A, and the nonzero vectors x and y are called the corresponding M-eigenvectors.Qi in [3, 5, 6] presented some basic studies for tensor computations and approximations. Li et al. [7–10], Bu et al. [11], Che et al. [12], and Zhao et al. [13, 14] worked on analyzing the M-eigenvalues for various elasticity tensors. The authors in [15] proposed a tensor-based FTV model for the three-dimensional image deblurring problem, and some properties for Z-eigenvalues of tensor are given in [16–18]. Let(5)fx,y=∑i,k=1m∑j,l=1naijklxiyjxkyl=∑i,k=1mxixkyTBiky=∑j,l=1nyjylxTCjlx,where Bik∈ℝn×n and Cjl∈ℝm×m are symmetric matrices with entries(6)Bikst=aiskt,Cjlst=asjtl.And, assume thatλminA is the minimal eigenvalue of a matrix A, λmaxA is the maximal eigenvalue of a matrix A, and ρA is the spectral radius of a matrix A. In 2021, Li et al. established the following bounds for M-eigenvalues of an elasticity tensor.Theorem 1.
(see [19]). LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor andλbe an M-eigenvalue ofA. Then,(7)maxδ1,δ2≤λ≤minθ1,θ2,where(8)δ1≕minl∈NλminCll−g1l,θ1≕minl∈NλmaxCll+g1l,δ2≕mini∈MλminBii−g2i,θ2≕mini∈MλmaxBii+g2i,and(9)g1l=∑j∈N,j≠lρCjl,g2i=∑k∈M,k≠iρBik.Theorem 2.
(see [19]). LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor andρMAbe the M-spectral radius ofA. Then,(10)ρMA≤γ≕minγ1,γ2,where(11)γ1=maxj,l∈N,j≠l12ρCll+ρ2Cll+4g1lg1j+ρCjj,γ2=maxi,k∈M,k≠i12ρBii+ρ2Bii+4g2ig2k+ρBkk.The following necessary and sufficient condition for strong ellipticity for general anisotropic elastic materials is presented by Han et al. [20].Theorem 3.
(see [20]). LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. The strong ellipticity condition holds, i.e.,(12)fx,y=∑i,k=1m∑j,l=1naijklxiyjxkyl>0,for all nonzero vectors x∈ℝm,y∈ℝn if and only if the smallest M-eigenvalue of A is positive.One application of the lower bound in Theorem1 is to identify the strong ellipticity condition of an elasticity tensor, and the upper bound in Theorem 2 is given to accelerate convergence of the WQZ-algorithm [19]. In this paper, by breaking N into disjoint subsets S and its complement, new S-type upper bounds for the M-spectral radius of an elasticity tensor are given in Section 2. In Section 3, S-type sufficient conditions are also given to identify the strong ellipticity condition of an elasticity tensor.


## 2. S-Type Upper Bounds

In this section, we give S-type upper bounds for the largest M-eigenvalues of an elasticity tensor, and the relationship between the S-type upper bounds and existed upper bounds is also established. The setsSm, S¯m, Sn, and S¯n are defined by M=Sm∪S¯m and Sm∩S¯m=∅, N=Sn∪S¯n, and Sn∩S¯n=∅.Theorem 4.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor andρMAbe the M-spectral radius ofA. Then,(13)ρMA≤τ≕minτ1,τ2,where(14)τ1=maxi∈Sm,k∈S¯m12g2Smi+g2S¯mk+g2Smi−g2S¯mk2+4g2S¯mig2Smk,τ2=maxj∈Sn,l∈S¯n12g1Snj+g1S¯nl+g1Snj−g1S¯nl2+4g1S¯njg1Snl,g1Snl=∑j∈SnρCjl,g1S¯nl=∑j∈S¯nρCjl,g2Smi=∑k∈SmρBik,g2S¯mi=∑k∈S¯mρBik.Proof.
Letλ be an M-eigenvalue of A with the M-eigenvectors x, y,(15)xp=maxk∈Smxk,xs=maxk∈S¯mxk.
Obviously, at least one ofxp and xs is nonzero.
Case I. If xp‖xs≠0, from the p-th equation of λx=Ayxy, we have(16)λxp=∑k=1m∑j,l=1napjklyjxkyl.
Then, we can get(17)λxp=∑k∈Sm∑j,l∈Napjklyjxkyl+∑k∈S¯m∑j,l∈Napjklyjxkyl=∑k∈Smxk∑j,l∈Napjklyjyl+∑k∈S¯mxk∑j,l∈Napjklyjyl=∑k∈SmxkyTBpky+∑k∈S¯mxkyTBpky.
Taking modulus in the above equation, we have(18)λ‖xp≤∑k∈Smxk‖yTBpky+∑k∈S¯mxk‖yTBpky≤g2Smpxp+g2S¯mpxs.
Then,(19)λ−g2Smpxp≤g2S¯mpxs.
Ifλ−g2Smp>0, similarly we can get(20)λ−g2S¯msxs≤g2Smsxp.
Multiplying (20) with (21), we have(21)λ−g2Smpλ−g2S¯ms≤g2S¯mpg2Sms.
Therefore,(22)λ≤12g2Smp+g2S¯ms+g2Smp−g2S¯ms2+4g2S¯mpg2Sms.
Ifλ−g2Smp<0, then(23)λ<g2Smp,which means that (23) also holds.
Case II. xp‖xs=0. If xs=0, by inequality (5), then λ−g2Smp≤0; it yields that (7) also holds. If xp=0, by inequality (6), then λ−g2S¯ms≤0; it yields that (7) also holds.
Letyq=maxj∈Snyj and yt=maxj∈S¯nyj, from the q-th equation of λy=Axyx, we have(24)λyq=∑i,k∈M∑j∈Snaijkqxiyjxk+∑i,k∈M∑j∈S¯naijkqxiyjxk,and similarly, we can get(25)λ≤12g1Snq+g1S¯nt+g1Snq−g1S¯nt2+4g1S¯nqg1Snt.We compare the S-type upper bounds in Theorem4 with the results in [19], which shows that our new S-type upper bounds are always tighter than the results in [19].Theorem 5.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. Then,(26)ρMA≤τ≤γ.Proof.
IfρMA≤τ, then(27)ρMA≤12g2Smi+g2S¯mk+g2Smi−g2S¯mk2+4g2S¯mig2Smkor(28)ρMA≤12g1Snj+g1S¯nl+g1Snj−g1S¯nl2+4g1S¯njg1Snl.
We only proof the following case, and the other case can be proved similarly. If(29)ρMA≤12g2Smi+g2S¯ms+g2Smi−g2S¯ms2+4g2S¯mig2Sms,from the proof of Theorem 4,(30)λ−g2Smiλ−g2S¯ms≤g2S¯mig2Sms.
LetSm=i, S¯m=M\i, then(31)λ−ρBiiλ−g2s≤g2iρBss.
From inequalities (20) or (21), there is an i∈M with λ−ρBii≤g2i; for this index i, we have(32)λ−ρBiiλ≤λ−ρBiig2s+g2iρBss≤g2ig2s+ρBss,and therefore, ρMA≤γ.In 2009, the following WQZ-algorithm was presented to compute the largest M-eigenvalue of an elasticity tensor [4].(33)eijkl=1,ifi=kandj=l,eijkl=0,otherwise.(34)x¯t+1=A¯ytxtyt,xt+1=x¯t+1x¯t+1,y¯t+1=A¯xt+1ytxt+1,yt+1=y¯t+1y¯t+1,t=t+1.(35)λmaxA=fx∗,y∗−υ,(36)fx∗,y∗=∑i,k=1m∑j,l=1na¯ijklxi∗yj∗xk∗yl∗,υ=∑1≤s≤t≤mnAst.Algorithm 1: WQZ-algorithm.
Step 0: given a tensorA=aijkl, vectors x0∈ℝm and y0∈ℝn. Set t=0 and A¯=υℐ+A, where ℐ=eijkl∈ℝm×n×m×n with the entries as follows:Step 1: computeOutputx∗, y∗.Step 2: find the largest M-eigenvalueλmaxA of the tensor A:whereThe following example in [4] is taken to show that the tighter upper bound can accelerate convergence of the WQZ-algorithm.Example 1.
Consider the tensorA=aijkl of Example 4.1 in [4, 21], where(37)A:,:,1,1=−0.97270.3169−0.3437−0.6332−0.78660.4257−0.3350−0.9896−0.4323,A:,:,2,1=−0.6332−0.78660.42570.73870.6873−0.3248−0.7986−0.5988−0.9485,A:,:,3,1=−0.3350−0.9896−0.4323−0.7986−0.5988−0.94850.58530.59210.6301,A:,:,1,2=0.31690.6158−0.0184−0.78660.01600.0085−0.9896−0.66630.2559,A:,:,2,2=−0.78660.01600.00850.68730.5160−0.0216−0.59880.04110.9857,A:,:,3,2=−0.9896−0.66630.2559−0.59880.04110.98570.5921−0.2907−0.3881,A:,:,1,3=−0.3437−0.01840.56490.42570.0085−0.1439−0.43230.25590.6162,A:,:,2,3=0.42570.0085−0.1439−0.3248−0.0216−0.0037−0.94850.9857−0.7734,A:,:,3,3=−0.43230.25590.6162−0.94850.9857−0.77340.6301−0.3881−0.8526.
In [4], υ is taken as follows:(38)∑1≤s≤t≤mnAst=23.3503.
LetSm=Sn=1; by Corollary 2 in [22], we have(39)ρA≤11.7253.
By Theorem2, we have(40)ρA≤4.2523.
LetSn=1,3; by Theorem 4, we have(41)ρA≤4.1528.Example 2.
Consider the elasticity tensorA=aijkl of CaMg(CO3)2-dolomite [21], whose nonzero entries are(42)a2222=a1111=196.6,a3311=a2233=83.2,a3333=110,a2323=a3232=a1313=a3131=54.7,a1212=a2121=64.4,a2223=a2232=−a1213=−a2131=−31.7,a1122=132.2,a2132=a1223=−35.84,a3112=a1321=44.8,a2321=a1232=−a1311=−a1131=−25.3.
In [4], υ is taken as follows:(43)∑1≤s≤t≤mnAst=23.3503.
LetSm=Sn=1; by Corollary 2 in [22], we have(44)ρA≤491.7400.
By Theorem2, we have(45)ρA≤462.2316.
LetSm=2,3; by Theorem 4, we have(46)ρA≤211.4729.
In Figure1, we can find that, when taking υ=211.4729, the sequence generated in the WQZ-algorithm converges to the largest M-eigenvalue more rapidly than taking υ=1998.6000 and υ=462.2316.Figure 1
Numerical results for the WQZ-algorithm with differentτ.


## 3. S-Type M-Eigenvalue Inclusion Sets and Strong Ellipticity Conditions

In this section, based on the S-type M-eigenvalue inclusion sets of an elasticity tensor, S-type sufficient conditions for strong ellipticity conditions are given. LetAx2jl=∑j,l=1naijklxixk and Ay2ik=∑i,k=1maijklyjyl, we need the following lemma.Lemma 1.
(see [23]). LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. Then, the strong ellipticity condition holds if and only if the matrixAx2∈ℝn×n(orAy2∈ℝm×m) is positive definite for each nonzerox∈ℝm(ory∈ℝn).Theorem 6.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor andλbe an M-eigenvalue ofAwith the M-eigenvectorsx,y. Then,(47)λ∈Δ1A∩Δ2A,where(48)Δ1A=∪j∈Sn,l∈S¯nz∈ℝ:z−xTCjjx−h1Snjz−xTCllx−h1S¯nl≤g1S¯mjg1Sml∪∪j∈Snz∈ℝ:|z−xTCjjx|≤h1Snj,Δ2A=∪i∈Sm,k∈S¯mz∈ℝ:z−yTBiiy−h2Smiz−yTBkky−h2S¯mk≤g2S¯mig2Smk∪∪i∈Smz∈ℝ:z−yTBiiy≤h2Smi,h1Snl=∑j∈Sn,j≠lρCjl,h1S¯nl=∑j∈S¯n,j≠lρCjl,h2Smi=∑k∈Sm,k≠iρBik,h2S¯mi=∑k∈S¯m,k≠iρBik.Proof.
Letλ be an M-eigenvalue of A with the M-eigenvectors x and y,(49)xp=maxk∈Smxk,xs=maxk∈S¯mxk.
Obviously, at least one ofxp and xs is nonzero.
Case I. If xp‖xs≠0, from the p-th equation of λx=Ayxy, we have(50)λxp=∑k=1m∑j,l=1napjklyjxkyl.
Then, we can get(51)λxp−yTBppyxp=∑k∈Sm,k≠pxkyTBpky+∑k∈S¯mxkyTBpky.
Taking modulus in the above equation, we have(52)λ−yTBppy‖xp≤∑k∈Sm,k≠pxk‖yTBpky+∑k∈S¯mxk‖yTBpky≤h2Smpxp+g2S¯mpxs.
Then,(53)λ−yTBppy−h2Smpxp≤g2S¯mpxs.
Ifλ−yTBppy−h2Smp>0, similarly we can get(54)λ−yTBssy−h2S¯msxs≤g2Smsxp.
Multiplying (53) with (54), we have(55)λ−yTBppy−h2Smpλ−yTBssy−h2S¯ms≤g2S¯mpg2Sms,so that λ∈Δ2A. If λ−yTBppy−h2Smp≤0; then,(56)λ−yTBppy≤h2Smp,which means that λ∈Δ2A.
Case II. xp‖xs=0. Without loss of generality, let xs=0, by inequality (8), then λ−yTBppy−h2Smp≤0; it yields that λ∈Δ2A.
Letyq=maxj∈Snyj and yt=maxj∈S¯nyj, from the q-th equation of λy=Axyx, similarly we can get λ∈Δ1A.Theorem 7.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. If there existsSmorSnsuch that(57)λminBii>h2Smifor alli∈Sm,λminBii−h2SmiλminBkk−h2S¯mk>g2S¯mig2Smkfor alli∈Sm,k∈S¯m,or(58)λminCjj>h1Snjfor allj∈Sn,λminCjj−h1SnjλminCll−h1S¯nl>g2S¯njg2Snlfor allj∈Sn,l∈S¯n,then the strong ellipticity condition holds.Proof.
Letλ be an M-eigenvalue of A and λ≤0. From Theorem 6, we obtain λ∈Δℬ. If λ∈Δ2ℬ, there are i∈Sm and k∈S¯m such that(59)λ−yTBiiy−h2Smiλ−yTBkky−h2S¯mk≤g2S¯mig2Smkor(60)λ−yTBiiy≤h2Smi.
Then,(61)λ−yTBiiy−h2Smiλ−yTBkky−h2S¯mk≥yTBiiy−h2SmiyTBkky−h2S¯mk≥λminBii−h2SmiλminBkk−h2S¯mk>g2S¯mig2Smkand(62)λ−yTBiiy≥yTBiiy≥λminBii>h2Smi,which contradicts λ∈Δℬ. Therefore, λ>0. Then, by Theorem 3, the strong ellipticity condition holds for the elasticity tensor A.
Ifλ∈Δ1ℬ, the second conclusion can be obtained similarly.The following sufficient conditions for strong ellipticity are given by Li et al. [19].Theorem 8.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. If(63)λminBii>g2i,for alli∈M,or(64)λminCll>g1l,for alll∈N,then the strong ellipticity condition holds.Based on the above theorems, we introduce the definitions strictly diagonally dominated (M-SDD) and S-type strictly diagonally dominated (M-SSDD) elasticity tensors, which are based on the eigenvalues of matrices ofBik and Cjl.Definition 2.
LetA=aijkl∈ℝm×n×m×n be an elasticity tensor. If(65)λminBii>g2i,for alli∈M,or(66)λminCll>g1l,for alll∈N,then the elasticity tensor A is called strictly diagonally dominated(M-SDD).Definition 3.
LetA=aijkl∈ℝm×n×m×n be an elasticity tensor. If there exists Sm or Sn such that(67)λminBii>h2Smifor alli∈Sm,λminBii−h2SmiλminBkk−h2S¯mk>g2S¯mig2Smkfor alli∈Sm,k∈S¯m,or(68)λminCjj>h1Snjfor allj∈Sn,λminCjj−h1SnjλminCll−h1S¯nl>g2S¯njg2Snlfor allj∈Sn,l∈S¯n,then the elasticity tensor A is called S-type strictly diagonally dominated(M-SSDD).Next, we give the relationships between the M-SDD elasticity tensor and the M-SSDD elasticity tensor.Theorem 9.
LetA=aijkl∈ℝm×n×m×nbe an elasticity tensor. IfAis an M-SDD elasticity tensor, thenAis an M-SSDD elasticity tensor.Proof.
IfA is an M-SDD elasticity tensor, we only prove the following case; the other case can be proved similarly. For all i∈M,(69)λminBii>g2i.
Then, for alli∈Sm and k∈S¯m,(70)λminBii>g2i>h2Smi,λminBii−h2Smi>g2S¯mi,λminBkk−h2S¯mk>g2Smk,which imply that(71)λminBii>h2Smi,λminBii−h2SmiλminBkk−h2S¯mk>g2S¯mig2Smk,and then A is an M-SSDD elasticity tensor.Now, the following example is explored to show the efficiency of the results in Theorems8 and 9.Example 3.
LetA=aijkl∈ℝ2×2×2×2 be an elasticity tensor, where(72)a1111=a1212=2.5,a2121=a2222=4,a1112=a2122=a1122=−a1121=−a1222=1,and other aijkl=0.
Obviously, we have(73)B11=2.5112.5,B22=4114,B12=B21=−111−1,C11=C22=2.5−1−14,C12=C21=1111.
LetSm=1, by direct computation, we have(74)λminB11=1.5>0=h2Sm1,λminB22=3>0=h2S¯m2,and(75)λminB11−h2Sm1λminB22−h2S¯m2=4.5>4=g2S¯m1g2Sm2.
Then,A satisfies the sufficient conditions of Theorem 7, and the conditions of Theorem 8 do not hold by λminB11=1.5<2=g21 and λminC11=1.6707<2=g11. Therefore, the strong ellipticity condition holds for the elasticity tensor A by Theorem 7. In fact, the smallest M-eigenvalue of A is 3.5.
LetSm=Sn=1; by Theorem 11 in [22], we have(76)α1−r¯11Aα2−r¯22A=7.5<r¯12Ar¯21A=20,where α1,r¯11A,α2,r¯22A,r¯12A,r¯21A are defined in [22], which shows that the conditions of Theorem 11 in [22] do not hold.



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