On properties and numerical computation of critical points of eigencurves of bivariate matrix pencils

On the quadratic two-parameter eigenvalue problem and its linearization

Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in GEW ZGV computation (doi: 10.5281/zenodo.7537442).

On linearizations of the quadratic two-parameter eigenvalue problem

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

Multiple zero-group velocity resonances in elastic layered structures

On the singular two-parameter eigenvalue problem II

This paper studies the propagation of symmetric and antisymmetric Lamb waves along a 1 mm thick iodic acid plate (HIO3) in the 1–50 MHz frequency range. The Lamb mode propagation along three crystallographic planes was theoretically investigated, for two mutually orthogonal propagation directions. Several frequencies were found that correspond to Lamb modes with zero group velocity (ZGV) and non-null phase velocity values. The first symmetric Lamb mode, S1, was found to possess only one ZGV point, regardless of the propagation direction; higher order symmetric and antisymmetric modes with up to four ZGV points were found, depending on the propagation plane. The dependence of the ZGV frequencies on each elastic constant (c11, c13, c33, c55) of the HIO3 plate material was also investigated by changing the constant values by 5% and 10%. It was found that c33 and c55 affect the number of the ZGV points, while c11 and c13 affect the frequency of the ZGV points. The existence and frequencies of the ZGV points are strongly dependent on the proximity of standing longitudinal and transverse waves at nearby cutoff frequencies.

It is known that guided modes in isotropic hollow cylinders exhibit backward wave propagation with negative group velocity. And interferences between backward and forward waves generate zero-group-velocity (ZGV) resonances with a finite wavenumber but vanishing group velocity. These ZGV resonances can be applied for non-destructive evaluation (NDE) of hollow pipes. In this paper, influences of a fluid-loading on ZGV resonances in pipes are studied for the possible application of integrity inspections of oil transportation pipelines. From numerically simulated frequency-wavenumber spectra of axisymmetric guided modes, in addition to the backward mode with a single ZGV point, certain branches change the sign of their slopes for twice (i.e., two ZGV points in one branch). Such multiple ZGV modes might be caused by the strong repulsion between the backward mode with a single ZGV point that is propagating in the hollow pipe and a number of longitudinal modes in the fluid cylinder. It is found that, from wave structure analyses, ZGV points correspond to relatively large displacement amplitudes at the pipe’s inner and outer interfaces. It indicates that guided modes with multiple ZGV points can be sensitive to the surface features of fluid-filled pipes, which is useful for NDE application.

The propagation of the Lamb-like modes along a silicon-on-insulator (SOI)/AlN thin supported structure was simulated in order to exploit the intrinsic zero group velocity (ZGV) features to design electroacoustic resonators that do not require metal strip gratings or suspended edges to confine the acoustic energy. The ZGV resonant conditions in the SOI/AlN composite plate, i.e. the frequencies where the mode group velocity vanishes while the phase velocity remains finite, were investigated in the frequency range from few hundreds of MHz up to 1900 MHz. Some ZGV points were found that show up mostly in low-order modes. The thermal behaviour of these points was studied in the −30 to 220 °C temperature range and the temperature coefficients of the ZGV resonant frequencies (TCF) were estimated. The behaviour of the ZGV resonators operating as gas sensors was studied under the hypothesis that the surface of the device is covered with a thin polyisobutylene (PIB) film able to selectively adsorb dichloromethane (CH2Cl2), trichloromethane (CHCl3), carbontetrachloride (CCl4), tetrachloroethylene (C2Cl4), and trichloroethylene (C2HCl3), at atmospheric pressure and room temperature. The sensor sensitivity to gas concentration in air was simulated for the first four ZGV points of the inhomogeneous plate. The feasibility of high-frequency, low TCF electroacoustic micro-resonator based on SOI and piezoelectric thin film technology was demonstrated by the present simulation study.

The propagation of the Lamb-like modes along a diamond/AlN thin supported structure was simulated in order to exploit the intrinsic zero group velocity (ZGV) features to design high frequency electroacoustic resonators. As the ZGV points are associated with an intrinsic energy localization under the metal electrodes, acoustic micro-resonators can be designed that employ only one interdigital transducer (IDT) and no reflectors, thus reducing both the device size and technological complexity. The ZGV resonant conditions in the diamond/AlN composite plate, i.e., the frequencies where the mode group velocity vanishes while the phase velocity remains finite, were investigated in the frequency range from few hundreds of MHz up to 3500 MHz. Thin film bulk acoustic resonators (TFBARs) based on c-AlN and on 45° c-axis tilted AlN film on diamond suspended membrane were simulated that operate in longitudinal and shear mode: the former is a thickness-extensional mode, while the latter is a thickness-in plane-shear mode that is suitable for liquid sensing applications. A smart structure based on diamond/AlN composite suspended membrane was modelled that provides several integrated functions including sensing in gaseous and liquid environment, and stable frequency source.

Algebraic polynomial system solving and applications

Local resonances formed by zero-group velocity (ZGV) and cutoff frequency points usually demonstrate sharp resonance peaks in frequency spectra, which can be utilized for nondestructive evaluation (NDE) and Structural Health Monitoring (SHM). The existence and application of those local resonances have been extensively reported in plate and pipe structures. However, local resonances in rails are rarely studied. The team recently reported that impulse dynamic tests can promote the local resonances in rails up to 40 kHz, and the results were verified using both semi-analytical finite element (SAFE) analysis and frequency-domain fully discretized finite element analysis. In this work, we present the discovery of ZGV modes and cutoff frequency resonances in free rails up to 80 kHz using piezoelectric elements. A miniature low-cost PZT patch works as a consistent excitation source compared with the impulse dynamic testing method. First, we implement the SAFE analysis to compute dispersion curves of a standard AREMA 115RE rail and to identify potential ZGV and cutoff frequency points up to 80 kHz. Then, to understand the existence and detectability of identified ZGV and cutoff points in a free rail, we install one PZT patch on the side of the rail head. A chirp signal covering 20 to 120 kHz is selected as the excitation to cover the desired frequency range. Finally, we perform a spatial sampling of wave propagation using three receivers along the wave propagation direction to calculate the dispersion relations experimentally via two-dimensional Fourier Transforms (2D-FFT). This study verifies the existence of ZGV modes in free rail up to 80 kHz and demonstrates the feasibility of using piezoelectric elements to generate local resonances.

Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections.

Appearing ZGV point in the first flexural branch of Lamb waves in multilayered plates