Computation of Leaky Waves in Layered Structures Coupled to Unbounded Media Using Multiparameter Eigenvalue Problems

Core Concepts

The authors present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media by formulating the problem as a multiparameter eigenvalue problem.

Abstract

The authors present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media. The problem is formulated as a nonlinear eigenvalue problem, which is then cast into the form of a multiparameter eigenvalue problem. This allows for the efficient and robust numerical computation of the wave modes and corresponding horizontal wavenumbers.
The key aspects are:

The layered structure is discretized using a semi-analytical finite element method, leading to a quadratic eigenvalue problem for the mode shapes and wavenumbers in the free layer.

Incorporating the coupling conditions at the interfaces with the unbounded media introduces additional nonlinear terms in the eigenvalue problem.

The nonlinear eigenvalue problem is rewritten as a multiparameter eigenvalue problem, which can be solved efficiently using established numerical algorithms.

The approach is demonstrated for various configurations, including plates coupled to fluid or solid half-spaces on one or both sides. The results are validated against alternative methods and show excellent agreement.

The computational efficiency of the proposed approach is highlighted, with the ability to compute dispersion curves for complex setups in reasonable time.

Stats

The plate thickness is 1 mm.
The maximum frequency of interest is 4 MHz for the brass plate in water, and 7 MHz for the brass plate coupled to Teflon.

Quotes

"We present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media."
"Rigorously incorporating the coupling conditions to account for the adjacent half-spaces gives rise to additional terms that are nonlinear in the wavenumber."
"We show that the resulting nonlinear eigenvalue problem can be cast in the form of a multiparameter eigenvalue problem whose solutions represent the wave numbers in the plate and in the half-spaces."

Key Insights Distilled From

Computation of leaky waves in layered structures coupled to unbounded media by exploiting multiparameter eigenvalue problems

by Hauke Graven... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15277.pdf

Deeper Inquiries

How can the proposed approach be extended to handle more complex layered structures, such as those with anisotropic or viscoelastic materials

The proposed approach can be extended to handle more complex layered structures by incorporating anisotropic or viscoelastic materials. For anisotropic materials, the stiffness tensor in the governing equations needs to be modified to account for the material's directional dependence. This would involve introducing additional parameters to describe the material properties in different directions. The finite element discretization would need to be adjusted to accommodate the anisotropic behavior, potentially requiring different element types or basis functions to capture the material's directional characteristics accurately.
In the case of viscoelastic materials, the time-dependent behavior of the material must be considered in the formulation. This would involve modifying the governing equations to include the viscoelastic properties, such as relaxation times or damping coefficients. The multiparameter eigenvalue problem would need to be adapted to handle the additional parameters related to the viscoelastic behavior. The numerical solution would also need to account for the time-dependent nature of the material response, potentially requiring more sophisticated time integration schemes.
Overall, extending the approach to handle more complex layered structures with anisotropic or viscoelastic materials would involve modifying the governing equations, adjusting the finite element discretization, and updating the multiparameter eigenvalue problem to include the additional material parameters.

What are the limitations of the multiparameter eigenvalue problem formulation, and how can it be further improved in terms of computational efficiency

The multiparameter eigenvalue problem formulation has certain limitations that can impact its computational efficiency. One limitation is the increase in computational complexity as the number of parameters grows. The size of the operator determinants in the multiparameter eigenvalue problem increases rapidly with the number of parameters, leading to longer computation times and higher memory requirements. This can make solving the multiparameter eigenvalue problem challenging for large and complex systems.
To improve the computational efficiency of the multiparameter eigenvalue problem formulation, several strategies can be employed. One approach is to optimize the numerical algorithms used to solve the generalized eigenvalue problem, taking advantage of parallel computing and efficient linear algebra techniques. Additionally, reducing the number of parameters by exploiting symmetries or simplifying assumptions can help streamline the solution process. Implementing efficient data structures and algorithms tailored to the specific problem structure can also enhance computational efficiency.
Furthermore, utilizing model order reduction techniques or adaptive strategies to focus computational resources on the most critical modes can help improve efficiency. By carefully selecting the parameters and refining the numerical methods, the computational efficiency of the multiparameter eigenvalue problem formulation can be enhanced.

Can the insights gained from this work on leaky wave modes be applied to the analysis of other types of wave propagation problems, such as those encountered in seismic engineering or structural health monitoring

The insights gained from the study on leaky wave modes can be applied to the analysis of other wave propagation problems encountered in seismic engineering or structural health monitoring. The understanding of how waves propagate and interact with different materials and interfaces can be leveraged to develop models for seismic wave propagation in layered soil structures or structural health monitoring of composite materials.
In seismic engineering, the principles of leaky wave modes can be used to study the behavior of seismic waves in complex geological formations, such as layered soils or rock formations. By considering the interaction of seismic waves with different material interfaces and boundaries, researchers can better predict wave propagation patterns and potential seismic hazards.
In structural health monitoring, the knowledge of leaky wave modes can aid in the development of wave-based damage detection techniques for composite structures. By analyzing how guided waves interact with defects or damage in the material, researchers can identify changes in wave patterns that indicate structural anomalies. This can enable early detection of damage and improve the overall health monitoring of civil infrastructure and mechanical systems.