Alternative eigenvalue problems

The eigenvalue problem as stated in Equation (2.12), is usually subject to numerical ill-conditioning. One of the possible sources of numerical errors is the inversion of the matrix D^*_LR in Equation (2.6) (Zhong and Williams, 1995). Moreover, the matrix of right eigenvectors, i.e.,

Φ_u =

[︃Φ_q Φ^⋆_q Φ_F Φ^⋆_F ]︃

, (2.28)

has largely disparate components as it involves displacement/rotation and force/moment compo-nents, which contributes to increase its condition number and, thereafter, according to the Bauer-Fike theorem (Golub and Loan, 1998; Mencik, 2010) recalled below, the eigenvalue sensitivity.

Theorem 2.1. (Bauer-Fike) If ˜𝜇_𝑗 is an eigenvalue of S + E ∈ C^2𝑛×2𝑛, where E is a perturbation matrix with small norm —i.e., ‖E‖_𝑝 ≪ ‖S‖_𝑝 —, and Φ⁻¹_u SΦ_u = 𝜇 = diag(𝜇₁, 𝜇₂, · · · , 𝜇_2𝑛),

then

min

𝜇∈𝜇(S)|𝜇 − ˜𝜇_𝑗| ≤ 𝜅(Φ_u)‖E‖_𝑝,

where ‖˙‖𝑝 denotes any of the 𝑝-norms⁶and 𝜅 denotes the condition number⁷of a given matrix.

Signorelli and von Flotow (1988) showed numerical errors in their study of periodic truss beams, and attributed them to computational inaccuracy (i.e., round-off errors). However, these errors might be the evidence of the numerical ill-conditioning of the eigenvalue problem. Later on, Zhong and Williams (1995) addressed the issue and proposed alternative formulations for the eigenvalue problem by making use of the symplectic property. The main idea behind the alter-native formulations is to write the eigenvalue problem as a function of the vectors of displace-ments/rotations only, in order to avoid the conditioning problems related to the consideration of force/moment components. In the following, alternative formulations for the eigenproblem stated in Equation (2.12) are presented. Afterwards, their final expressions are summarized in Table 2.2.

(N, L) eigenvalue problem

To begin with, the state vectors related to the left (right) cross-section of adjacent substruc-tures are written as a function of the vectors of displacements/rotations, as follows

u^(𝑘)_L = Lw^(𝑘), u^(𝑘)_R = Nw^(𝑘), (2.29) where

w^(𝑘)=[︁

q^(𝑘)𝑇_L q^(𝑘)𝑇_R ]︁𝑇

, (2.30)

L =

[︃ I 0

−D^*_LL −D^*_LR ]︃

and N =

[︃ 0 I D^*_RL D^*_RR

]︃

. (2.31)

Using the relation in Equation (2.11), which results from Bloch’s theorem, in Equations (2.29), an alternative eigenvalue problem is formulated

Nw_𝑗 = 𝜇_𝑗Lw_𝑗. (2.32)

6The 𝑝-norms are defined by ‖x‖𝑝= (|𝑥₁|^𝑝+ |𝑥₂|^𝑝+ · · · + |𝑥_𝑛|^𝑝)^1/𝑝.

7The condition number provides a measure of the sensitivity of the linear system Ax = b (Golub and Loan, 1998).

Here, the condition number is defined in terms of the 2-norm of A as 𝜅2(A) = ‖A‖2‖A‖⁻¹₂ .

Notice that the relation between the matrix S and matrices L and N is given by S = NL⁻¹. In Equation (2.32), {w_𝑗}_𝑗 refer to right eigenvectors which involve displacement/rotation compo-nents, only. They can be partitioned into 𝑛 × 1 vectors of displacement/rotation defined at the left cross-section of the substructure{︀𝜑_q𝑗}︀

𝑗 and 𝑛 × 1 vectors of displacement/rotation defined at the right cross-section of the substructure {︀𝜇𝑗𝜑_q𝑗}︀

𝑗. Thus, once these eigenvectors are computed by means of the eigenproblem in Equation (2.32), the wave shapes {𝜑_𝑗}_𝑗 can be easily retrieved as 𝜑_𝑗 = Lw_𝑗 (Mencik, 2014).

Zhong’s eigenvalue problem

A second companion eigenvalue problem has been proposed by Zhong and Williams (1995).

It takes advantage of the fact that the eigenvalues come in pairs as 𝜇𝑗 and 1/𝜇𝑗. Here, Equa-tion (2.32) is left multiplied, independently, by L^𝑇J and by N^𝑇J,which gives

L^𝑇JNw_𝑗 = 𝜇_𝑗L^𝑇JLw_𝑗, (2.33a)

N^𝑇JNw_𝑗 = 𝜇_𝑗N^𝑇JLw_𝑗. (2.33b)

Then, using the fact that

L^𝑇JL = N^𝑇JN =

[︃ 0 −D^*_LR D^*_RL 0

]︃

(2.34) in the second equation, one can write

N^𝑇JLw_𝑗 = 1/𝜇_𝑗L^𝑇JLw_𝑗. (2.35)

Finally, adding Equations (2.33a) and (2.35), yields

Z₁𝜙_𝑗 = 𝜆_𝑗Z₂𝜙_𝑗, (2.36)

where

Z1 = (L^𝑇JN + N^𝑇JL) =

[︃ (D^*_RL− D^*_LR) (D^*_LL+ D^*_RR)

−(D^*_LL+ D^*_RR) (D^*_RL− D^*_LR) ]︃

(2.37) and

Z₂ = L^𝑇JL =

[︃ 0 −D^*_LR D^*_RL 0

]︃

. (2.38)

This alternative eigenvalue problem has double eigenvalues in the form 𝜆_𝑗 = (𝜇_𝑗+ 1/𝜇_𝑗). Thus, as-suming that 𝜇_𝑗 ̸= 0, the original eigenvalues are the solutions of the quadratic polynomial equation 𝜇²_𝑗 − 𝜆𝑗𝜇𝑗+ 1 = 0, i.e.,

Moreover, each double eigenvalue 𝜆𝑗has two corresponding linearly independent eigenvectors 𝜙⁽¹⁾_𝑗 and 𝜙⁽²⁾_𝑗 of size 𝑛 × 1 each. The original right eigenvectors of Equation (2.12) can be viewed as a linear combination of the eigenvectors related to double eigenvalues of Equation (2.36) (Zhong and Williams, 1995). Thus, for an eigenvalue 𝜇_𝑗, the corresponding eigenvector can be expressed as

w_𝑗 = 𝜙_𝑗a_𝑗 = 𝑎⁽¹⁾_𝑗 𝜙⁽¹⁾_𝑗 + 𝑎⁽²⁾_𝑗 𝜙⁽²⁾_𝑗 , (2.40) scalar coefficients 𝑎⁽¹⁾_𝑗 and 𝑎⁽²⁾_𝑗 . Substituting this equation into Equation (2.32), yields

(N − 𝜇_𝑗L)𝜙_𝑗a_𝑗 = 0, (2.41)

A natural way of obtaining the relationship between coefficients 𝑎⁽¹⁾_𝑗 and 𝑎⁽²⁾_𝑗 consists in left multi-plying Equation (2.41) by 𝜙^(1)𝐻_𝑗 (or, alternatively, by 𝜙^(2)𝐻_𝑗 ), which allows one to write

𝑎⁽¹⁾_𝑗

𝑎⁽²⁾_𝑗 = −𝜙^(1)𝐻_𝑗 (N − 𝜇_𝑗L)𝜙⁽²⁾_𝑗

𝜙^(1)𝐻_𝑗 (N − 𝜇_𝑗L)𝜙⁽¹⁾_𝑗 . (2.42) Although algebraically correct, this expression may suffer from numerical difficulties when 𝜙⁽¹⁾_𝑗 is almost parallel to w_𝑗. An alternative way of determining w_𝑗 involves the use of singular value decomposition (SVD), as proposed by Waki et al. (2009b). For the sake of clarity and precision, the procedure is presented hereafter.

Let A𝑗 = (N − 𝜇𝑗L)𝜙_𝑗. Using SVD, this matrix may be written as A𝑗 = UA𝑗ΣA𝑗V^𝐻_A𝑗, where UA𝑗 is a square 2𝑛 × 2𝑛 matrix, ΣA𝑗 is a 2𝑛 × 2 diagonal matrix of singular values and VA𝑗 is a square 2 × 2 matrix, expressed as

[︃(︀𝑉A𝑗

Then, Equation (2.41) can be re-written as

where ¯𝑉_𝑝𝑞indicates the complex conjugate of the matrix entry 𝑉𝑝𝑞. As w𝑗is a linear combination of the eigenvectors related to double eigenvalues 𝜆𝑗, A𝑗 is rank-deficient, which makes 𝜎_A⁽¹⁾_𝑗 ≫ 𝜎_A⁽²⁾_𝑗 ≈ 0, then the equality in Equation (2.43) is guaranteed by making(︁

(︀𝑉¯_A𝑗)︀

It is also possible to solve a companion quadratic eigenvalue problem. This is possible through consideration of Bloch’s theorem, i.e., u^(𝑘)_𝑗 = 𝜇_𝑗u^(𝑘−1)_𝑗 , which allows one to write q_R𝑗 = 𝜇_𝑗q_L𝑗 and fR𝑗 = −𝜇_𝑗f_L𝑗. Using these relations in Equation (2.3), yields

F_L𝑗 = D^*_LLq_L𝑗+ 𝜇_𝑗D^*_LRq_L𝑗 (2.45a)

−𝜇_𝑗F_L𝑗 = D^*_RLq_L𝑗+ 𝜇_𝑗D^*_RRq_L𝑗 (2.45b) Substituting Equation (2.45a) into Equation (2.45b), a quadratic eigenvalue problem of the form

C_q𝜑_q𝑗 = 0_(𝑛×1) (2.46)

is formulated, where C_q=(︀𝜇²_𝑗D^*_LR+ 𝜇_𝑗(D^*_LL+ D^*_RR) + D^*_RL)︀.

Eigenvalue problem with symmetric matrices

From Equation (2.46), an eigenvalue problem with symmetric matrices can be formulated (Arruda et al., 2007), which enhances the performance of the eigenvalue problem computation.

The key idea here is to associate to Equation (2.46) additional equations which would make the matrices A and B of a generalized eigenvalue problem Az = 𝜆Bz symmetric. To begin with,

Equation (2.46) is re-written in terms of w_𝑗, which yields

To this equation the following equation is associated [︁

which allows one to express the following eigenvalue problem

Nw¯ 𝑗 = 𝜇𝑗Lw¯ 𝑗, (2.49)

Table 2.2: Overview of the eigenproblems used to compute the wave modes of a periodic structure.

Eigenvalue Problem Associated Eigenvector Reference Equation

Additional comments regarding the eigenvalue problem formulation for periodic structures

The mathematical formulation presented in this chapter is stated in the frequency domain.

The displacements/rotations and force/moments vectors (q and F, respectively) are harmonic so-lutions of the dynamic problem under concern — i.e., the free wave propagation through a 1D periodic system —, with the harmonic term 𝑒^i𝜔𝑡 being omitted for the sake of conciseness. Thus, the associated eigenvalue problems are functions of the angular frequency 𝜔 and the wavenumber 𝛽. There are two ways of solving these eigenvalue problems: the direct and the inverse methods.

The inverse method is considered to solve the numerical problems that will be presented in this thesis. It consists in solving the eigenvalue problem with respect to the wavenumber 𝛽𝑗, for speci-fied angular frequencies within the range of analysis. One of the advantage of this approach is that complex wavenumbers are possible.

On the other hand, the direct method consists in solving an eigenproblem, for prescribed val-ues of the wavenumber 𝛽, with respect to the angular frequency 𝜔. This approach is typically used to evaluate the dispersion relation of undamped periodic systems. Here, phase constant surfaces (𝜔 = 𝑓 (𝛽) in the 1D case, 𝜔 = 𝑓 (𝛽_𝑥, 𝛽_𝑦) in the 2D case, 𝜔 = 𝑓 (𝛽_𝑥, 𝛽_𝑦,, 𝛽_𝑧) in the 3D case) are obtained, which are convenient to the identification of attenuation zones (or band gaps), typical in periodic structures. The eigenproblem into consideration here is formulated without condensing internal degrees of freedom as it must be linear with respect to 𝜔². This yields a problem of large size, compared to those solved via inverse method (where the condensation of internal DOFs is possible). From Bloch’s theorem (stated in Section 2.2.3), qR𝑗 = 𝜇_𝑗q_L𝑗 and FR𝑗 = −𝜇_𝑗F_L𝑗. Using these relations in Equation (2.3) and by assuming that damping is not present, it yields

(︁ ˜K(𝜇) − 𝜔²M(𝜇)˜ )︁

𝜑_q = 0, (2.50)

where

K(𝜇) =˜

[︃(𝜇²K_LR+ 𝜇 (K_LL+ K_RR) + K_RL) K_RI+ 𝜇K_LI

K_IL+ 𝜇K_IR K_II

]︃

and

M(𝜇) =˜

[︃(𝜇²M_LR+ 𝜇 (M_LL+ M_RR) + M_RL) M_RI+ 𝜇M_LI

M_IL+ 𝜇M_IR M_II

]︃

, both of them are symmetric matrices.