Numerical errors related to the eigenvalue problem formulation

(a) (b)

Figure 2.7: Schematics of the solid waveguide used to analyze numerical errors related to the WFE method: (a) full waveguide, (b) FE model of the substructure.

Consider a linear elastic and damped waveguide as shown in Figure 2.7(a). A substructure of length ∆ = 0.004/36 m is modeled by means of 3D solid hexahedral elements with eight nodes and three DOFs per node — i.e., translations along the 𝑥, 𝑦 and 𝑧 axes —, the SOLID45 elements from ANSYS^®(see Figure 2.7(b)). The structure is made of steel (Young’s modulus 𝐸 = 2.1×10¹¹ Pa, density 𝜌 = 7800 kg.m^-3, Poisson’s ratio 𝜈 = 0.3, and internal loss factor 𝜂 = 0.01) and it has a rectangular cross-section (height ℎ𝑦 = 0.003 m and width ℎ_𝑧 = 0.004 m). The numerical errors involved in the computation of the numerical wave modes of this waveguide by means of the WFE method are analyzed. The frequency range of interest here is 𝛽_𝑓 = [200 Hz —2 MHz], evaluated at every 2000 Hz. It is important to point out here that substructure model was built using a fine enough FE mesh for the maximum frequency within the range under concern.

Here, the different eigenproblems presented in Section 2.2.5 are compared in terms of the accuracy of the computed numerical wave modes. To begin with, low-order numerical wave modes are compared to the analytical ones computed by means of the elementary theory of rods and Timoshenko’s beam theory. The comparison is restricted to the frequency range for the validity of these theories, which according to Doyle (1997), would be below

𝑓_max≈ 3 4

𝑐_𝑠

ℎ, (2.56)

where 𝑐𝑠 is the speed of shear S-waves given by √︀𝐺/𝜌, with 𝐺 being the shear modulus, and ℎ, the cross-section length. This frequency is chosen to be close to the first cut-on frequencies of the Lamb symmetric and antisymmetric modes. For the structural model into consideration here, 𝑓_max≈ 0.4 MHz. In Figure 2.8, the dispersion curves — |ℜ (𝛽_𝑗∆) | in the positive 𝑦-axis, ℑ (𝛽_𝑗∆) in the negative 𝑦-axis — for some low-order wave modes computed by means of the various nu-merical formulations and analytical relations are plotted as well as the relative errors associated to the computation of the propagation constants 𝜇𝑗 and the wave mode shapes 𝜑_𝑗. These errors are expressed as

𝜖_𝜇𝑗(𝜔_𝑖) = |𝜇_𝑗(𝜔_𝑖) − 𝜇_{𝑗 ref}(𝜔_𝑖)|

|𝜇_{𝑗 ref}(𝜔_𝑖)| , 𝜖_𝜑𝑗(𝜔_𝑖) = ‖𝜑_𝑗(𝜔𝑖) − 𝜑_{𝑗 ref}(𝜔𝑖)‖2

‖𝜑_{𝑗 ref}(𝜔_𝑖)‖₂ , (2.57) respectively, where the subscript ref is used to denote reference, which, in this case, is the analyt-ical value. From this analysis, one may notice that all eigenproblem formulations provide almost the same accuracy, with exception of the (N,L) eigenproblem (Equation (2.32)), which seems to be more sensitive to errors at very low frequencies. Moreover, a good agreement is observed until the first cut-on frequency of the shear mode w.r.t. the 𝑦-axis.

0 1 2 3 4 5 6

Figure 2.8: Comparison of numerical wave modes, related to the longitudinal wave (a,c,e) and the shear wave w.r.t. the 𝑦-axis (b,d,f), computed by means of WFE method with the corresponding analytical values. (a,b) Dispersion curves, (c,d) 𝜖_𝜇𝑗, (e,f) 𝜖_𝜑𝑗 (Equation (2.57)). The following ap-proaches are compared: (—) analytical solution, (- - -) Equation (2.36), (∘) Equation (2.12), (∙) Equation (2.32), (x) Equation (2.49).

The various formulations for the WFE-based eigenproblem are also compared with respect to the symplectic and symmetry conditions between left- and right-going wave modes. As discussed in Section 2.2.4, due to the symplectic property of the eigenvalue problems the eigenvalues might be in pairs 𝜇𝑗 and 1/𝜇𝑗, which means that if a wave travels to the right direction with a propagating constant 𝜇𝑗, the corresponding left-going wave would travel with a propagating constant 𝜇^⋆_𝑗 = 1/𝜇_𝑗. With exception to the generalized eigenproblem with antisymmetric matrices which provides pairs of repetitive eigenvalues (Equation (2.36)), we expect that all other formulations provide pairs (𝜇_𝑗, 1/𝜇_𝑗) as eigenvalues. The error associated to the guarantee of obtaining as eigenvalues 𝜇_𝑗 and 1/𝜇_𝑗, or double eigenvalues in the case of Equation (2.36), is presented in Figure 2.9(a). The expressions for the error in the eigenvalue computation are given by

𝜖_𝜇(𝜔_𝑖) =

, for the eigenproblem stated in Equation (2.36), (2.58a)

, for all other eigenproblems. (2.58b)

Notice that the errors are really small for almost all formulations, with exception to the (N,L) eigenproblem (Equation (2.32)). For this eigenproblem, the error regarding the eigenvalues is in-vestigated with respect to the wave mode rank in Figure 2.10. The magnitude of the corresponding right-going propagating constants is also plotted as a function of the wave mode rank and frequency.

From the results shown in Figure 2.10, one may conclude that the eigenvalues which are prone to more errors are neither purely evanescent or propagating. The propagation of those errors can be avoided by enforcing the property of the eigenvalues related to left- and right-going waves.

Regarding the wave mode shapes, the symmetry condition stated in Equation (2.22) is evalu-ated by means of the following error expression

𝜖_Φ(𝜔_𝑖) =

This error analysis is performed for all eigenproblem formulations with exception of the Zhong’s eigenproblem (Equation (2.36)), as in this case the symmetry condition is always enforced. The results are presented in Figure 2.9(b). As in the previous analysis, the eigenproblem of

Equa-tion (2.32) provides the worst results, thus requiring the enforcement of the symmetry condiEqua-tion

Figure 2.9: Verification of the symmetry relation, by means of : (a) 𝜖𝜇 (Equation (2.58)), (b) 𝜖Φ

(Equation (2.59)), for the following approaches: (- - -) Zhong’s (Z₁,Z₂) eigenproblem (Equa-tion (2.36)), (∘) S eigenproblem (Equation (2.12)), ( ∙) (N,L) eigenproblem (Equation (2.32)), (x)( ¯N, ¯L) eigenproblem (Equation (2.49)).

Figure 2.10: (a)Magnitude of right-going propagation constants (|𝜇_𝑗|𝑗,=1,··· ,𝑛), (b) error 𝜖_𝜇 com-puted via the (N,L) eigenproblem stated in Equation (2.32) as a function of the wave mode rank and frequency.

The WFE eigenproblems have also been compared in terms of the sensitivity of their cor-responding eigenvalues to numerical errors. The Bauer-Fike theorem (Section 2.2.5) demonstrates

that the sensitivity of the eigenvalues can be measured by evaluating the condition number of the eigenvector matrix, which is plotted in Figure 2.11(a). As expected, for small perturbations, the eigenvalues computed by means of Equation (2.12) seem to be the highly prone to errors. The alternative formulations are related to small condition numbers of almost the same order of magni-tude.

Moreover, WFE-based eigenproblems are compared in terms of computational time in Figure 2.11(b). The results show that the eigenproblem stated in Equation (2.12) is the fastest one, it saves 72.6 % of the time used for solving Zhong’s eigenproblem (Equation (2.36)). However, as just discussed, it may be prone to numerical errors. The same justification holds for avoiding the L/N eigenproblem in Equation (2.32) although it saves 12.6 % of computational time. In order to meet a good compromise in terms of performance and accuracy, the eigenproblems in Equations (2.36) and (2.49) are preferred.

0 0.5 1 1.5 2

x 10⁶ 10²

10⁴ 10⁶ 10⁸ 10¹⁰

Frequency (Hz)

κ(Φ)

1 2 3 4

0 500 1000 1500 2000

Eigenvalue Problem

Elapsed time (s)

(a) (b)

Figure 2.11: (a) Condition number of the matrix of eigenvectors and (b) elapsed times are compared for the following eigenproblems: (- - -) or 1 Equation (2.36) , (∘) or 2 Equation (2.12), (∙) or 3 Equation (2.32), (x) or 4 Equation (2.49).