On the theoretical limitations in estimating thickness of a plate-like structure from a full-field single-tone response Lamb wave measurement

doi:10.1016/j.ultras.2020.106230

Ultrasonics

Volume 108, December 2020, 106230

https://doi.org/10.1016/j.ultras.2020.106230 Get rights and content

Highlights

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Full-field measurements in two dimensions simulated for calculations.
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Cramer-Rao bound is applied to thickness estimates from Lamb wave measurements.
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Simulation model used to drive theoretical sensitivity computations.
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Reflections accounted for in Cramer-Rao bound calculations.
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Relation of sensitivity to frequency, source distance, number of sources explored.

Abstract

A persistent question in wavenumber analysis in the estimation of thickness from a steady-state wave field is identifying the theoretical sensitivity of the system. A well-known trade-off between spatial frequency/wavenumber resolution and thickness resolution exists. The current work presents a calculation of the Cramer-Rao Lower bound (CRLB), specifically as applied to thickness estimates, for a 2-dimensional multi-mode waveform in Gaussian noise. Cases of near-field and far-field excitation are considered, and transducer position with respect to the scan area is also varied. Additionally, we consider the CRLB in a plate with multiple sources, simulated as sources placed on the boundary of the plate. We conclude by presenting the CRLB values in terms of frequency for various thicknesses, and by presenting optimal excitation frequencies for a nominal thickness, based on the CRLB.

Introduction

Laser Acoustic Wavenumber Spectroscopy (LAWS) is a process in which one excites a structure using a single-tone ultrasonic signal, often using an ultrasonic transducer. One then measures the resulting full-field steady-state response in the structure using, for instance, a scanning laser-Doppler vibrometer (LDV). To be more specific, this has been demonstrated (see [1]) using a single-tone ultrasonic excitation, with the response measured by a LDV, coupled with galvanometric mirrors to steer the LDV beam to measure the point-by-point response in the structure over a large scan area, at a standoff distance of up to several meters. The scan area is typically divided into pixels of dimension $1 mm \times 1 mm$ , over which the response is measured and averaged, and the output of the LDV is exported as a full-field multi-mode steady-state response to the chosen single-tone ultrasonic excitation. The measured wave field is composed of several Lamb wave modes, each corresponding to a distinct spatial frequency (wavenumber). The number of wave modes, as well as the wavenumber associated with each wave mode, is dependent on the excitation frequency, as well as the thickness and material properties of the plate, as well as material properties of surrounding media if the plate is immersed. This must be taken into account in order to fully describe the multi-mode steady-state response in a structure to ultrasonic excitation.

Recently, LAWS has been applied to the detection of delaminations in composite materials and localized corrosion in thin aluminum plate-like structures [1]. The original method proposed by Flynn et al. measures the wave form and estimates the wavenumber of the primary out-of-plane mode, $A_{0}$ . Corrosion or delaminations are then indicated by abrupt local changes in the wavenumber, which can then be used to estimate thickness using the dispersion curve associated with the $A_{0}$ Lamb wave mode. This method does not, however, account for the multi-mode nature of the Lamb wave response which is present in thicker aluminum or steel plate-like structures. Specifically, in thick steel plate-like structures, the $A_{0}$ and $S_{0}$ modes have nearly identical wavenumbers for large frequency-thickness products. Moreover, the wavenumber corresponding to the $A_{0}$ mode is always greater than that corresponding to the $S_{0}$ mode in steel structures. Because of these two facts, any estimate of thickness relying solely on the $A_{0}$ wavenumber will have bias introduced. In particular, the proximity of the $S_{0}$ wavenumber renders the two wave modes impossible to separate in practice in thick steel structures, which means any estimate of the $A_{0}$ wavenumber will actually be biased lower due to the presence of the $S_{0}$ wave mode. Thus, in order to properly quantify the theoretical sensitivity of thickness estimates based on full-field steady-state multi-mode wave form measurements, one must account for the presence of multiple modes and their behavior. Understanding the sensitivity of this system is crucial in evaluating and validating estimation algorithms developed to work in conjunction with the LAWS system.

Recent advances have allowed for non-contact measurement of dispersion relations, at least in the case of the $A_{0}$ mode [2]. If this method can be generalized to account for additional wave modes, which in practice is quite difficult, then the analysis presented in this work could be applied to a wide variety of materials without resorting to computationally intensive dispersion wave calculations. Additionally, the present work holds relevance in the context of new corrosion detection methodologies, such as that presented in [3], in which not only is the corrosion area detected, but the depth is also estimated using short-windowed Fourier Transform methods to locally estimate corrosion depth in a plate of known nominal thickness. Note that throughout the present work, material and geometric nonlinearities are not considered. As such, we do not include higher order harmonics in the present analysis. For a numerical examination of their contributions as modeled in COMSOL, the reader is referred to work by Charilla and Lissenden [4], and the references therein.

One method of estimating the theoretical thickness resolution of the system is the Cramer-Rao Lower Bound (CRLB) (see [5], [6]). The CRLB is the minimum value of the variance that may be attained by an unbiased estimator. However, the existence of an unbiased estimator which achieves the CRLB is not guaranteed [7]. Several studies have been performed analyzing estimators of sinusoids in white noise, and their associated CRLB’s (see, for instance, [8], [9], [10], [11]). However, these studies are not sufficient for a thorough study of the AWS system and its associated thickness resolution. In order to properly analyze the CRLB for thickness estimation based on the AWS system, one must account for the characteristics of Lamb wave modes, including number of modes, wavenumbers of individual modes, and the relations between thickness, excitation frequency, and wavenumber.

The remainder of the current work is organized as follows. In Section 2, we will review the theory of Lamb waves and the fundamentals of the Fisher information matrix and its relation to the CRLB. In Section 3, we will specialize the Fisher information matrix and the CRLB to the situation of full-field Lamb wave measurements. In Section 4, we will present calculations of the CRLB and discuss the significance of the results, as well as experimental design considerations indicated by these calculations. Section 5 will present some concluding remarks and indicate directions for future work.

Section snippets

Review of Lamb Wave Theory

Lamb waves are a specific class of elastic waves in solid media, which propagate through plate-like structures. Upon applying stress-free boundary conditions at $z = \pm d / 2$ to the wave equation, one obtains the Rayleigh-Lamb equations [12]: $\frac{\tan qd / 2}{\tan pd / 2} = - \frac{{(q^{2} - k^{2})}^{2}}{4 k^{2} pq}$ for anti-symmetric Lamb wave modes and $\frac{\tan qd / 2}{\tan pd / 2} = - \frac{4 k^{2} pq}{{(q^{2} - k^{2})}^{2}}$ for symmetric modes, where $p = \sqrt{{(\frac{ω}{c_{L}})}^{2} - k^{2}} and q = \sqrt{{(\frac{ω}{c_{T}})}^{2} - k^{2}},$

k is the wavenumber (with units rad/m), $ω / 2 π$ is the transducer excitation frequency, d is the plate thickness, $c_{L}$ is the

CRLB as applicable to LAWS

It is important to emphasize that the current work does not depend on any particular measurement technique, as the CRLB is independent of experimental setup or data acquisition technique. Instead, the CRLB is calculated in order to determine the smallest possible standard error of an unbiased estimator given a particular model or set of experimental observations (in this case, simulated full-field multi-mode Lamb wave response in a steel plate). However, we begin by very briefly describing an

Methodology

Mathematical models indicated in (7), (18) were used to simulate full-field Lamb wave responses assuming single-tone ultrasonic excitation. All simulations were implemented in Matlab, under the assumptions described above. Specifically, phase shift was considered either to be deterministic (value 0) or randomized and averaged, and amplitude of each wave mode was assumed to be a deterministic parameter with value 1. Additionally, in the case where reflections were considered in the simulation,

Conclusion and future work

CRLB calculations were carried out for estimation of thickness of a mild steel plate using simulations of the full-field multi-mode response to ultrasonic excitation. The calculations do not indicate the ideal design of an estimator, nor do they indicate a methodology behind the estimation. The results do, however, provide guidance on experimental design, specifically indicating that a frequency-thickness product of approximately 10 MHz-mm should be chosen for small scan areas in order to

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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¹: This work was sponsored in part by Los Alamos National Laboratory’s Laboratory Directed Research and Development funds, and by a commercial partner who wishes to remain anonymous. Neither funding source interfered with or otherwise influenced this research.

View full text

On the theoretical limitations in estimating thickness of a plate-like structure from a full-field single-tone response Lamb wave measurement

Highlights

Abstract

Introduction

Section snippets

Review of Lamb Wave Theory

CRLB as applicable to LAWS

Methodology

Conclusion and future work

Declaration of Competing Interest

NDT and E Int.

Ultrasonics

Ultrasonics

Ultrasonics

Ultrasonics

Ultrasonics

Mathematical Methods of Statistics

Information and accuracy attainable in the estimation of statistical parameters

Bull. Calcutta Math. Soc.