Material characterization of thin planar structures using full-field harmonic vibration response measured with stroboscopic holography

Abstract

We propose a novel material characterization method to estimate the Young’s modulus of thin 2-D structures using non-modal noisy single frequency harmonic vibration data measured with holography. The method uses finite-difference discretization to apply the plate equation to all measured pixels inside the boundary of the vibrating structure and then treats the problem as a Bayesian optimization process to find the value of the Young’s modulus by minimizing the Euclidian distance between the measured displacement field and repeatedly calculated displacement field using the plate equation. In order to assess the accuracy of the method, ground truth harmonic displacement magnitude fields of different plates were obtained using analytical solutions and the finite-element method and were used to estimate the Young’s moduli. We applied Gaussian and non-Gaussian noise with different intensities to assess the robustness and accuracy of the proposed material characterization method in the presence of noise. We demonstrated that for multiple benchmarks for signal to noise ratio of down to 0 dB, our proposed method had errors of less than 5%. We also quantified the effects of uncertainties in the geometrical and material parameters as well as boundary conditions on the estimated Young’s modulus. Furthermore, we studied the effects of the mesh size on the runtime and applied the method to experimental holography vibration measurement data of a copper plate.

Graphical Abstract

Introduction

Holography has been used in recent years to perform high-resolution vibrometry for different applications such as noise reduction [1], energy-flow measurements [2], measurement of ultrasonic surface wave fields [3], and piezoelectric surface vibration characterization [4]. It was also used to study the motions of the tympanic membrane with a high spatial resolution [5–8].

In the present work, we propose a method that will be the basis for further developments for estimating the Young’s modulus of the tympanic membrane as a diagnostic index for hearing pathologies. This method uses noisy 2-D full-field high-spatial-resolution holographic stroboscopic measurements of the harmonic motion at a single frequency. Although recently some researchers have worked on the transient response and modal parameters of the tympanic membrane [9,10], for material characterization in this work, we use harmonic measurements because they can provide better signal to noise ratio than transient measurements do and they are possible with less expensive hardware (e.g., no need for an expensive high-speed camera). Hence, the main novelty of the present work is to enable material characterization using only single frequency full-field noisy vibration measurements data. Because the proposed method does not require data with a high signal to noise ratio, no averaging is needed and the data can be collected quickly. This is especially important for applications such as medical applications where movement of the subject, breathing and heart beats can interfere with measurements.

Caillet et al. used the data from the nearfield acoustic holography to estimate the Young’s modulus of 2-D structures [11]. They used the plate equation and the finite-difference method to estimate the Young’s modulus using free-vibration holography data. Alfano et al. presented a method for estimating the Poisson’s ratio and Young’s modulus of thin plates using two of the first four natural frequencies [12].

Labelle et al. presented a method for estimating material properties from modal data extracted from video of the vibrating object [13]. To show the accuracy of their proposed method, they used modal data obtained using different techniques (visual vibrometry, laser Doppler vibrometry, and accelerometers) for estimating the mechanical properties of a steel cantilever and compared the results with each other. Hasheminejad et al. used scanning laser Doppler vibrometry measurements to develop a method for estimating the Young’s modulus of asphalt mixtures with modal data [14]. Material characterization using vibration data can be performed using finite-element model updating but much of the literature of finite-element model updating was developed for modal data [15] and therefore cannot be used with harmonic vibration data. More detailed reviews of the different vibration-based methods for material characterization can be found in [16] and [17]. However, to the best of our knowledge, no method exists in the literature that can find the material properties of 2-D structures using single frequency harmonic vibration measurements data especially for data affected by noise.

Various methods have been used to find the dynamic response of plates including some analytical methods for simple geometries [18] and numerical methods such as finite element and finite difference for more complicated geometries [19]. Because stroboscopic digital holography provides vibration data on a 2-D image of the structure, the vibration field is known on an equally-spaced structured grid that is easy to implement using the finite-difference method. The finite-difference method was successfully used by some researchers to study the vibrations of plates (e.g., the free vibration of stiffened plates [20], the vibrations of a beam-plate structure in the time domain [21], and axially-moving viscoelastic plates [22]).

To find the Young’s modulus, we formulated the parameter estimation as an optimization problem. Our long-term goal is to develop the material characterization method to be used for clinical hearing diagnosis and therefore to reduce the calculation time and make it relevant for real time clinical use, it is important to be able to find the global optimum using the fewest possible number of iterations. Bayesian optimization is a global optimization method which creates a surrogate model of the objective function that is cheaper to evaluate than the real objective function and is updated in each iteration. Bayesian optimization is known to be able to find the global optimum with a fewer number of iterations in comparison with many other optimization algorithms such as particle swarm, pattern search, and genetic algorithm [23–26]. A review of Bayesian optimization can be found in [25].

Bayesian finite-element model updating was also the subject of several studies for different applications such as parameter estimation and robust prediction of the desired outputs [27]. Yuen et al. used Bayesian model updating to find optimized values of the modal parameters as well as the uncertainties associated with them [28]. Moravej et al. used the Bayesian technique to calibrate a structural finite-element model based on modal vibration measurements for health monitoring purposes [29]. Unlike these studies that used experimental modal data, our proposed finite-difference-based method estimates material parameters using single frequency vibration measurement data and Bayesian optimization.

The paper is organized as follows:

Section 0 is dedicated to the methodology including the details of the material characterization procedure. In Section 0 we present the validation of the proposed method using analytical, numerical, and experimental benchmarks, respectively. We also studied the effects of different uncertainties as well as the effects of Gaussian and non-Gaussian noise on the accuracy of the proposed parameter-estimation method. In Section 4, we discuss the performance of the Bayesian optimization and the limitations of the proposed method. Section 0 provides a conclusion for the research findings.

Materials and Methods

Figure 1 shows the flowchart of the proposed material characterization method. This method uses the holography measurements of the harmonic vibration at different stroboscopic phases to obtain the displacement magnitude and phase. It then solves the governing equation of the motion of the structure for all of the nodes inside the boundary at each frequency and eventually finds the optimum Young’s modulus which yields a finite-difference solution of the plate equation with a displacement magnitude field as close as possible to the one measured using holography. Details of the proposed method are as follows:

Figure 1 -.

Flowchart of the proposed material characterization method. The magnitude and phase of the displacements of the nodes inside the vibrating domain are calculated from holography data and the full-field displacement is then used to find the boundaries of the vibrating domain. Our proposed material characterization method is based on the Bayesian optimization process. It begins by considering four initial random guesses for the Young’s modulus, applying the plate equation to all nodes inside the detected boundary for these random guesses and evaluating the objective function (the Euclidean distance between the measured and calculated fields). Then the Gaussian process regression model and the acquisition function are formed and updated in each step in order to find the next value of the Young’s modulus for which the objective function needs to be calculated. This procedure continues until the stopping criteria (which is set to be a specific number of model evaluations in this work) is reached. Also, in each iteration, if overexploiting is detected, the kernel function is modified and a new Young’s modulus point is calculated.

Governing equation

In the present study, we used the classical plate theory to describe the transversal linear deformation of thin elastic objects [30]. In this work, we use the words displacement and deformation interchangeably and both as the short forms of transversal deformation. The plate theory has the following assumptions: the thickness of the plate is relatively small in comparison to its other dimensions; the thickness of the plate does not change during deformation and straight lines normal to mid-surface remain straight and normal after deformation [18,30]. The classical plate transversal deformation is expressed as:

$$D\left(\frac{{\partial }^{4}w(x,y,t)}{\partial x^4}+2\frac{{\partial }^{4}w(x,y,t)}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w(x,y,t)}{\partial y^4}\right)+\rho h\frac{{\partial }^{2}w(x,y,t)}{\partial t^2}=p(x,y,t)$$ 1

where t is time and x and y are lateral coordinates of the plate and w represents the transversal deformation. The plate is subject to the uniform harmonic transverse excitation pressure p of frequency Ω

$$p(x,y,\text{Ω},t)=P(x,y,\text{Ω})cos(\text{Ω}t)$$ 2

where the pressure amplitude is considered to be constant for all values of x and y and for all excitation frequencies. D, the flexural rigidity of the plate, is defined as:

$$D=\frac{E{h}^3}{12\left(1-v^2\right)}$$ 3

where E, h, and υ are the Young’s modulus, thickness, and Poisson’s ratio of the plate, respectively. Due to linearity, the frequency of the transversal deformation response is the same as the frequency of excitation:

$$w(x,y,\Omega ,t)=W(x,y,\Omega )cos(\Omega t+\phi )$$ 4

W and φ represent the amplitude and phase of the harmonic response, respectively. By substituting Equations 2 and 4 in Equation 1, the time-dependent terms cancel out from both sides:

$$D\left(\frac{{\partial }^{4}W(x,y,\Omega )}{\partial x^4}+2\frac{{\partial }^{4}W(x,y,\Omega )}{\partial x^2\partial y^2}+\frac{{\partial }^{4}W(x,y,\Omega )}{\partial y^4}\right)-\rho h{\text{Ω}}^{2}W(x,y,\Omega )=P(x,y,\Omega )$$ 5

Because in this paper we focus on systems with negligible damping, the discussions are concentrated on the deformation magnitude rather than the phase which is always 0 or 180 degrees for undamped structures.

Finite-difference formulations

In digital holography, the vibration data are obtained by post-processing the images gathered using a camera and therefore the data are available on structured equally spaced grids formed by treating the pixels of the digital images as nodes. The classical plate equation (Equation 5) can be easily discretized and solved on a structured grid using the finite-difference method.

In our holographic vibration measurements, we collect data of both the moving object and some parts of the stationary boundary around it and the boundary of the moving object can be delineated in the measurement data. We used a second-order central difference scheme to discretize Equation 5 [31] and we had to consider auxiliary nodes out of the boundary (Figure 2-a) to be able to use this scheme. For each of the x and y directions, the first node out of the boundary (in the stationary surrounding structure) was considered to be fixed (green nodes in Figure 2-a) and the next node (red nodes in Figure 2-a) further away from the moving object (in the stationary surrounding structure) was considered as an auxiliary node. In Figure 2-a, d is the uniform grid spacing in both x and y directions. The plate equation was solved for the nodes inside the boundary (yellow nodes in Figure 2-a).

Figure 2 -.

(a) Schematic diagram of the numerical grid used in the finite-difference method. The classical plate equation is applied to the yellow nodes that are located in the vibrating region indicated in blue, the green nodes are considered to be the boundaries of the moving plate and the red nodes are auxiliary nodes. (b) Visual explanation of the progression of the Bayesian optimization process represented by the value of the objective function for iterations 5, 20, 60, and 80 for estimation of the Young’s modulus of a simply-supported square plate (discussed in section 0) using the measured displacement field. The black dots represent the points at which the objective function was evaluated, the gray shaded areas show the confidence interval which becomes smaller by each iteration. Before the main optimization procedure starts, four initial random guesses for the Young’s modulus are made, the plate equation is solved for the structure four times and the objective function is calculated for these four guesses. The main optimization process starts from iteration #5 (which is the first iteration shown). Dashed red lines trace the five points considered in iteration #5 and the blue line indicates the true value of the Young’s modulus. This process eventually leads to estimating the global minimum of the objective function.

Inverse problem

The inverse problem can be formulated by rearranging Equation 5 to express the Young’s modulus as a function of other mechanical properties and the spatial derivatives of the measured holographic displacement field (Equation 6). The formulation presented in Equation 6 must be satisfied for all of the nodes inside the vibrating domain which results in an overdetermined system of equations.

$$\begin{gathered} E=\frac{12\left(1-v^2\right)}{h^3}{\left(\frac{{\partial }^{4}W(x,y,\Omega )}{\partial x^4}+2\frac{{\partial }^{4}W(x,y,\Omega )}{\partial x^2\partial y^2}+\frac{{\partial }^{4}W(x,y,\Omega )}{\partial y^4}\right)}^{-1}(P(x,y,\Omega ) \\ +\rho h{\text{Ω}}^{2}W(x,y,\Omega )) \end{gathered}$$ 6

This method can provide an accurate estimate when the data are noise-free [32]. However, the presence of even a very low level of noise in the vibration measurement data will result in large errors in the estimated Young’s modulus due to the noise amplification in the differentiation process of Equation 5 that is of order four in both spatial directions [33,34]. We verified that in parameter estimations using noisy data, these large errors indeed occur as expected.

To avoid noise amplification due to differentiation, we proposed the method which is presented in Figure 1. First, some initial guesses for the Young’s modulus were made, then Equation 5 was directly solved to calculate the displacement field. The calculated and measured displacement fields were then compared and the values of the Young’s modulus were iteratively modified (Section 0) to reduce the error between the calculated and measured displacement fields. This process led to an optimization problem in which the objective function, the Euclidean distance between measured and calculated fields, described by Equation 7 had to be minimized.

$$obj\left(E_i\right)=\sqrt{\sum _{j=1}^n{\left(w_j^C-w_j^{\text{m}}\right)}^2}$$ 7

where $w_j^C$ and $w_j^m$ respectively represent the calculated and measured displacements of the node j and n indicates the number of nodes in the vibrating domain. It should be noted that the calculated displacement is a function of the Young’s modulus ($E_i$) considered in each iteration of the optimization process.

Optimization process

We used Bayesian optimization in our proposed material characterization method to minimize the objective function (Equation 7). In our implementation, we used the BayesOpt algorithm in MATLAB¹.

The Bayesian optimization works with a Gaussian process model of the objective function as a response surface and trains this model by strategically exploring the parameter space and evaluating the objective function (Equation 7). In the present work, we used the Gaussian process regression $\mathcal{G}\mathcal{P}$ [35,36] to define the Gaussian model ${\widetilde{ob}}_J\left(E_i\right)$ of the objective function as:

$${\widetilde{ob}}_j\left(E_i\right)=\mathcal{G}\mathcal{P}\left[\mu \left(E_i,\theta \right);k\left(E_i,E_j;\theta \right)\right]$$ 8

where E_i and E_j are two Young’s moduli points in the parameter space, μ is the expected value function in the Gaussian process, k is a covariance kernel function, and θ is the vector of kernel parameters that are defined based on standard deviation and characteristic length scale. Standard deviation specifies the variation of the function values from the expected value and the characteristic length scale determines the amount of smoothness of the Gaussian process regression [37,38]. A small length scale means that the function becomes quickly uncertain when we get far from known positions. In this work, we used automatic relevance determination Matern 5/2 covariance kernel function [37,39]:

$$k\left(E_i,E_j;\theta \right)={\sigma }_f^2\left(1+\sqrt{5}r+\frac{5}{3}{r}^2\right)exp(-\sqrt{5}r)$$ 9

where r is:

$$r=\sqrt{\sum _{m=1}^N\frac{{\left(E_i-E_j\right)}^2}{{\sigma }_m^2}}$$ 10

and θ is defined as:

$${\theta }_m=log\left({\sigma }_m\right)form=1,2,\dots ,N\text{\&}{\theta }_{N+1}=log\left({\sigma }_f\right)$$ 11

where σ_m represent the characteristic length scale for each parameter and σ_f shows the standard deviation of the response surface. Kernel parameters are found by maximizing their marginal likelihood in each iteration [36,38,40]. Also, N is the number of parameters that we want to estimate (here N=1) [37].

In fact, Equation 8 uses the Gaussian process regression to create a posterior distribution based on our prior knowledge, and this posterior distribution can be later used to find the expected value and standard deviation of the objective function for each unexplored value of the Young’s modulus. The standard deviation of ${\widetilde{ob}}_J\left(E_i\right)$ then shows how likely it is for the expected value of the Gaussian objective function (${\widetilde{ob}}_J\left(E_i\right)$) to be close to its actual value (Obj(E_i)) at each E_i point [37].

Before the main optimization process starts, the objective function (Equation 7) is evaluated at four random Young’s moduli points and a Gaussian process regression (Equation 8) is created from them (Figure 1). The method then updates the Gaussian process regression in each iteration step through evaluating the objective function and learning more about it. In fact, in each iteration step we calculate the value of the objective function for a specific point in the Young’s modulus space and we use this knowledge to update the Gaussian process regression for the current step, we then proceed to the next iteration and calculate the value of the objective function for a new Young’s modulus point and update the Gaussian process again and this process continues until we reach the stopping criterion.

Deciding the next Young’s modulus point to evaluate the objective function (Equation 7), is critical. In order to strategically choose the next Young’s modulus to evaluate the objective function (Equation 7) for each iteration step, we found the Young’s modulus point that maximized an acquisition function [41]. The acquisition function was evaluated using the Gaussian process regression (Equation 8) of the objective function rather than the objective function itself (Equation 7), so its extrema can be found cheaply and quickly at each iteration step [42].

The most commonly used acquisition function is the expected improvement function defined as [43]:

$$EI=\mathbb{E}\left[\max \left(0,{\widetilde{ob}}_J\left(E^{\ast }\right)-{\widetilde{ob}}_J(E)\right)\right]$$ 12

where $\mathbb{E},E^{\ast }$ and $\widetilde{o{b}_j}\left(E^{\ast }\right)$ are the expected value, best Young’s modulus and best observed value of the objective function so far, respectively. For the acquisition function, we used a modified version of the expected improvement function, known as expected improvement plus, that escapes local extrema and therefore helps better estimate the global extrema of the objective function. The expected improvement plus is similar to the typical expected improvement function (Equation 12) but it benefits from a method to stop the optimization from overexploiting a specific region by defining an exploration ratio: for each step if the algorithm finds that overexploiting is happening, it multiplies the vector of kernel parameters (θ) by the number of iterations [44]. It then generates a new point based on the new kernel function. We set the exploration ratio to be 0.5 and our verifications, which will be presented in the following sections, show that this value results in finding the global optimum for all cases considered in the current work. The same value of exploration ratio was also used in several works in the literature [45,46].

In the Bayesian optimization, it is common to consider a large enough number of evaluations of the objective function as the stopping criterion [47–49]. This is due to the complexity of defining convergence of errors in the Bayesian optimization [47]. For all of the parameter estimations in this work, therefore, the number of iterations was fixed to 180 evaluations of the objective function (as will be discussed in Section 0). This value may need to be changed for other applications or if higher accuracies than what is targeted in this paper are desired. Moreover, the range of the solution space for the Young’s modulus was chosen to be from 1 GPa to 400 GPa for the benchmarks investigated in the present study.

For providing a visual explanation of the optimization process, Figure 2-b demonstrates a few iterations (iterations 5, 20, 60, and 80) of the evolution of our Bayesian parameter estimation procedure for the simply-supported plate which will be discussed in Section 0. At first 4 random initial guesses for the Young’s modulus were made and the objective function was evaluated and then at each step, the Young’s modulus points with the highest expected improvement were chosen to be searched as potential minimizers of the objective function. In fact, from the fifth iteration (the first curve shown in Figure 2-b), the real process begins and in each step, the next evaluation point is identified based on the acquisition function. Figure 2-b shows the values of the objective function evaluated at the candidate points as well as the confidence interval region (shaded gray). For better visualization, the dashed red lines trace the five points identified in the fifth iteration while the new guesses are added in each iteration step of the optimization and the solid blue line shows the true value of Young’s modulus. In each iteration, the new evaluation points are chosen to be near the local and global minima of the function and after several iterations, the error regions become smaller. This process eventually estimates the global minimum after several iterations. It should be noted that in this figure the results of iterations 5, 20, 60, and 80 are presented while the optimization procedure continues up to iteration 180.

Results

Validation using analytical data

We analytically calculated the forced harmonic response of a square plate with two types of boundary conditions (simply-supported and clamped) via mode superposition. We then found the optimum Young’s modulus that could best replicate the analytically calculated harmonic vibration patterns using our proposed material characterization method.

Simply-supported thin square plate:

The natural frequencies ω_mn and mode shapes W_mn of a simply-supported thin rectangular plate can be expressed as [50]:

$${\omega }_{mn}={\pi }^2\sqrt{\frac{D}{\rho h}}\left[{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2\right]$$ 13

$$W_{mn}(x,y)=A_{mn}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}$$ 14

where both m and n are integers and the subscript mn represents the mode number mn of the plate. Additionally, a and b are the length and width of the rectangular plate and h is the thickness of the plate, respectively. Also, A_mn represents the amplitude coefficient.

The response of the plate to a harmonic pressure with the frequency of Ω and amplitude of f(x, y) can be written as [51]:

$$w(x,y,\text{Ω},\text{t})={\int }_0^a{\int }_0^{b}f(\overline{x},\overline{y})W(x,y,\overline{x},\overline{y},\text{Ω})d\overline{x}d\overline{y}cos(\text{Ωt})$$ 15

where:

$$W(x,y,\overline{x},\overline{y},\text{Ω})=\sum _{m,n=1}^{\infty }\frac{W_{mn}(\overline{x},\overline{y})}{\left({\omega }_{mn}^2-{\text{Ω}}^2\right){\int }_0^a{\int }_0^b\rho h{W}_{mn}^{2}dxdy}{W}_{mn}(x,y)$$ 16

In Equations 15 and 16, $\overline{x},\overline{y}$represent the coordinates at which the pressure is applied. In this section, it was assumed that the pressure was uniformly applied all over the plate.

We used Equations 13 to 16, with mode superposition over the first 20 modes, to calculate the harmonic responses of a thin square plate as inputs to our material characterization method. Figure 3-a, shows the analytically calculated harmonic response of a plate with geometrical and material parameters listed in Table 1, excited at 7.9 kHz which is close to the 13^th natural frequency (8.1 kHz) of the plate. The magnitude of the applied uniform harmonic pressure was 10 kPa and the plate was discretized on a 1000×1000 grid. The value of the Young’s modulus was estimated to be 200.03 GPa (error of 0.01%).

Figure 3 -.

(a) The analytically calculated harmonic response of a simply-supported square plate with known material properties (given in Table 1) excited with a harmonic pressure with a magnitude of 10 kPa and frequency of Ω=7.95 kHz. This displacement field was used as the input to our material characterization method and the error of the estimated Young’s modulus was 0.01%. (b) The same analytically calculated harmonic response as in Panel (a) after application of white Gaussian noise with a signal to noise ratio of 10 dB. This noisy displacement field was also used as the input to the proposed material characterization method to study the estimation error in presence of the noise. The error of the estimated Young’s modulus was 0.68%. (c) The minimum value of the objective function with respect to the number of evaluations of the objective function for material characterization of the simply-supported square plate: using noiseless data shown in panel (a) (blue line) and using noisy data shown in panel (b) (red line). The error remained almost constant after about 20 iterations for both noiseless and noisy data. This shows that the maximum number of iterations used in this work (180 iterations) is enough for obtaining convergence.

Table 1 -.

Material properties used in analytical and some of the FEM simulations to validate the proposed holography-based material characterizing method.

Property	Value	Unit
E	200	GPa
ρ	7850	Kg/m 3
Y	0.3	-
H	0.01	m
a, b	1	m

As mentioned in Section 0, we used the Euclidian distance for our objective function in Equation 7 and the above estimation was based on that distance. We studied the effect of the distance definition in the objective function (Equation 7). When we replaced the Euclidian distance with Manhattan and Chebyshev distances, the estimated Young’s moduli were 200.75 GPa (error of 0.37%) and 199.35 GPa (error of 0.32%), respectively. Because the definition of distance did not have substantial effects on the estimated Young’s modulus, in the current work, we chose Euclidian distance for all our calculations as explained for Equation 7.

In order to assess the robustness and accuracy of the proposed material characterization method when the measured displacement magnitude field is affected by experimental noise, we applied white Gaussian noise to the magnitude of the analytical displacement field of Figure 3-a. Figure 3-b shows the magnitude of the noisy displacement field (1000×1000) with a signal to noise ratio (SNR) of 10 dB. The value of the estimated Young’s modulus using these noisy data was 201.37 GPa (error of 0.68%).

The convergence of the minimum value of the objective function with respect to the number of iterations is presented in Figure 3-c for both displacement fields shown in Figure 3. After almost 20 evaluations, the error remained almost constant. This demonstrates that 180 iterations of the objective function are enough for the studies reported in this paper.

Clamped thin square plate

The clamped boundary condition is important for our future application of the method for hearing diagnosis as the tympanic membrane is clamped around its annulus. In this section, a clamped thin square plate with the geometrical and material properties provided in Table 1, was used to further validate the proposed method. The same formulation as for the simply-supported plates (Equations 15 and 16) can be used to find the harmonic response [18], while the mode shapes and natural frequencies of a clamped plate must be used in those formulations. The mode shapes of a clamped square plate can be expressed as [50]:

$$\begin{gathered} W(x,y)=\sum _{m=1}^p\sum _{n=1}^q\{A_{mn}[cosh\frac{{ϵ}_{m}x}{a}-\text{cos}\frac{{ϵ}_{m}x}{a} \\ -{\alpha }_m\left(sinh\frac{{ϵ}_{m}x}{a}-sin\frac{{ϵ}_{m}x}{a}\right)]cosh\frac{{ϵ}_{n}y}{a}-cos\frac{{ϵ}_{n}y}{a} \\ -{\alpha }_n\left(sinh\frac{{ϵ}_{n}y}{a}-sin\frac{{ϵ}_{n}y}{a}\right)\} \end{gathered}$$ 17

where the coefficients A_mn, ϵ_m, α_m, ϵ_n and α_n and the natural frequencies of the square plate can be found in [50]. The first 6 mode shapes were used in the present study.

We calculated the magnitude of the displacement field of the clamped plate at four frequencies near the first four natural frequencies of the system (Figure 4). The proposed material characterization method was then employed to retrieve the Young’s modulus from these vibration patterns. As Table 2 shows, the estimated Young’s moduli for the resolution of 400×400 had errors of less than 0.18% for all investigated frequencies. The reason that we used a lower grid size here than what was used for the simply-supported plate in the previous section is that calculating the mode shapes in the forward analytical formulations for the clamped plate is computationally more expensive than the simply-supported plate.

Figure 4 -.

The analytically calculated harmonic response of a clamped plate with known material properties (given in Table 1) excited with a harmonic pressure with a magnitude of 10 kPa. Each of these harmonic responses was used as the input to our proposed material characterization method to validate it. The excitation frequencies were: (a) Ω=79.57 Hz; (b) Ω=159.23Hz; (c) Ω=238.85Hz; and (d) Ω=318.30Hz. The results of the estimated Young’s modulus are presented in Table 2. We observed no specific trends between the estimation errors and the excitation frequencies. The maximum estimation error was 0.18% using the displacement field at Ω=238.85Hz of panel (c).

Table 2 -.

Results of the proposed material characterization method estimated from noiseless (Figure 4) and noisy data (SNR=10 dB, Figure 5) of a clamped square plate with the true Young’s modulus of 200 GPa.

	Noiseless data		Noisy data – SNR=10 dB
Ω (Hz)	Estimated Young’s Modulus (GPa)	Error (%)	Estimated Young’s Modulus (GPa)	Error (%)
9.57	200.19	0.09	200.11	0.05
159.23	199.69	0.15	201.80	0.9
238.85	200.36	0.18	199.88	0.06
318.3	200.12	0.06	192.15	3.92

Like the simply-supported plate, we added white Gaussian noise (10 dB SNR) to the analytically calculated response to assess the accuracy and robustness of the proposed method (Figure 5). The estimated Young’s moduli from these noisy displacement fields (Table 2) had less than 3.92% error with even quite noisy displacement fields.

Figure 5 -.

White Gaussian noise with a signal to noise ratio of 10 dB applied to the analytically calculated harmonic response of a clamped plate (Figure 4) with known material properties (given in Table 1) excited with a harmonic pressure with a magnitude of P=10 kPa. Each of these noisy displacement fields was used as the input to our proposed material characterization method to validate it in the presence of noise. The excitation frequencies were (a) Ω=79.57 Hz; (b) Ω=159.23 Hz; (c) Ω=238.85 Hz; and (d) Ω=318.30 Hz. The results of the estimated Young’s modulus are presented in Table 2. We observed no specific trends between the errors and the excitation frequencies. The maximum estimation error was 3.92% using the displacement field at Ω=318.30 Hz of panel (d).

Validation using finite-element data

To more rigorously validate the proposed material characterization method by using more complicated plate geometries and to investigate the sensitivity of the method to various experimental errors, we used finite-element analyses to generate complex ground-truth vibration patterns more similar to the data obtained using holographic measurements. In our analyses, we used COMSOL (Version 5.4, MA, USA) and Code_Aster (Version 14.4.0, France) finite-element packages to calculate the steady-state forced harmonic vibration patterns.

Elliptical and triangular plates

To further validate the accuracy of the proposed method for plates with more complicated geometries, an elliptical plate and a triangular plate were studied in this section. The mesh convergence was checked for the finite-element calculated displacement fields. Besides, to consider the accuracy of the proposed material characterization method in the presence of noise, white Gaussian noise and non-Gaussian noise were applied to the displacement fields obtained from finite-element calculation.

We considered both domains to be inside a 10 cm×10 cm square. For the elliptical plate, the radii were considered to be 4.52 cm and 1.59 cm and for the triangular plate, two edges were considered to be equal and 8.94 cm long and the other edge was 8.00 cm long. The domains in the field of view outside the elliptical and triangular plates were fixed. The thickness of both plates was assumed to be 100 μm and the stimulus pressures of 1 kPa with various frequencies were uniformly applied on the plates. Besides, both plates were considered to have the material properties listed in Table 1 and the resolution was considered to be 1000×1000 for both cases. Figure 6 and Figure 7 show the displacement magnitude of the finite-element calculated harmonic vibration patterns of the elliptical and triangular plates, respectively. Table 3 shows the values of the estimated Young’s moduli and estimation errors for both elliptical and triangular domains for all the cases shown in Figure 6 and Figure 7. The maximum error of the estimated Young’s modulus was 1.49% for the elliptical plate and 0.23% for the triangular plate. Figure 8 shows the displacement fields of both plates after adding white Gaussian noise of 10 dB SNR. The maximum error of the estimation was 1.43% for the elliptical and 0.76% for the triangular plates with these noisy displacement fields. Besides, we studied the effects of different values of SNR in the range of 0 to 15 with intervals of 0.5 for both plates and for all 4 frequencies that were discussed before. The maximum error was 3.21% in the estimated Young’s modulus even with small values of SNR.

Figure 6 -.

Finite-element calculated harmonic displacement magnitude response of an elliptical plate inside a 10 cm × 10 cm square field of view (top: 3-D and bottom: 2-D projections). The material properties are given in Table 1. These displacement magnitude responses were used to validate the proposed material characterization method. Excitation amplitude was P=1 kPa and the excitation frequencies were: (a) Ω=79.57 Hz; (b) Ω=1.59 kHz; (c) Ω=3.18 kHz; and (d) Ω=7.96 kHz. The results of the estimated Young’s modulus are presented in Table 3. We observed no specific trends between the errors and the excitation frequencies. The maximum estimation error was 1.49% using the displacement field at Ω=79.57 Hz of Panel (a).

Figure 7 -.

Finite-element calculated harmonic displacement magnitude response of a triangular domain inside a 10 cm × 10 cm square field of view (top: 3-D and bottom: 2-D projections). The material properties are given in Table 1. These displacement magnitude responses were used in the material characterization method to validate it. The excitation amplitude was P = 1 kPa and excitation frequencies were: (a) Ω=79.57 Hz; (b) Ω=1.59 kHz; (c) Ω=3.18 kHz; and (d) Ω=7.96 kHz. The results of the estimated Young’s modulus are presented in Table 3. We observed no specific trends between the errors and the excitation frequencies. The maximum estimation error was 0.23% using the displacement field at Ω=3.18 kHz of Panel (c).

Table 3 -.

Estimated Young’s modulus for elliptical and triangular plates with the true Young’s moduli of 200 GPa using the noiseless displacement fields shown in Figure 6 and Figure 7.

	Elliptical plate		Triangular plate
Ω (Hz)	Estimated Young’s Modulus (GPa)	Error (%)	Estimated Young’s Modulus (GPa)	Error (%)
79.57	202.99	1.49	200.17	0.08
1592	200.87	0.43	200.34	0.17
3184	201.07	0.53	200.47	0.23
7961	197.75	1.12	200.16	0.08

Figure 8 -.

White Gaussian noise with a signal to noise ratio of 10 dB applied to the finite-element calculated harmonic displacement magnitude responses of the elliptical plate of Figure 6 and triangular plate of Figure 7. These responses were used to validate the material characterization method in the presence of noise. The excitation amplitude was P=1kPa. The excitation frequencies and the error in estimating the Young’s modulus for each panel are: (a) Ω=79.57 Hz & error=1.43%; (b) Ω=1.59 kHz & error=0.61%; (c) Ω=3.18 kHz & error=1.20%; (d) Ω=7.96 kHz & error=0.51%; (d) Ω=79.57 Hz & error=0.33%; (e) Ω=1.59 kHz & error=0.14%; (f) Ω=3.18 kHz & error=0.76%; and (g) Ω=7.96 kHz & error=0.03%. The results reveal that there are no specific trends between the errors and the excitation frequencies. The maximum estimation errors for the elliptical and triangular plates were related to the excitation frequencies of Ω=79.57 Hz and 3.18 kHz, respectively.

Non-Gaussian noise

In practice, the experimental data may be contaminated with non-Gaussian noise such as speckle noise and any other probable hardware noise. In order to investigate the robustness of the proposed material characterization method in presence of non-Gaussian noise, the vibration patterns of the elliptical and triangular plates with white Gaussian noise with SNR of 5 dB were converted to non-Gaussian noisy data using the method presented in [52] and [53] with skewness of larger than 1 and kurtosis of larger than 3 [54,55]. Table 4 shows that the estimated Young’s modulus with these noisy data had less than 4.2% of error for both plates at all excitation frequencies studied in this section.

Table 4 -.

Estimated Young’s modulus for elliptical and triangular plates with the true Young’s moduli of 200 GPa using displacement fields obtained by applying non-Gaussian noise to the displacement fields shown in Figure 6 and Figure 7.

	Elliptical plate		Triangular plate
Ω (Hz)	Estimated Young’s Modulus (GPa)	Error (%)	Estimated Young’s Modulus (GPa)	Error (%)
79.57	202.20	0.1	191.63	4.18
1592	200.79	0.39	201.17	0.58
3184	196.07	1.96	202.58	1.29
7961	202.27	1.13	204.46	2.23

Effect of uncertainty in thickness, Poisson’s ratio, and boundary conditions

Our proposed material characterization method estimates the Young’s modulus assuming no uncertainty in the Poisson’s ratio, thickness, and boundary conditions of the plate. In this section, the goal is to determine the effects of uncertainty in these assumptions on the accuracy of the proposed method. We used the finite-element models of elliptical and triangular plates which were benchmarked in previous sections.

Figure 9-a shows the error in the estimated Young’s moduli due to the error in the thickness assumption for the elliptical domain. These data were obtained by calculating the error in the estimated Young’s modulus by varying the thickness from the true value while all other geometrical parameters and the Poisson’s ratio and boundary conditions, were kept at the true value in the estimation process. We performed similar investigations on the triangular plate but did not show the results in Figure 9 as the results of both domains followed similar trends. In this and all other figures, the dots indicate the evaluation points and the lines indicate regressions for each dataset which was fitted for better visibility. In both domains and for all investigated excitation frequencies, assuming a thickness less than the true value caused overestimating the Young’s modulus because the material characterization method compensates the smaller thickness with higher stiffness to obtain the same displacement field. Although the error is different for each domain and each excitation frequency, the method is more sensitive to errors in the thickness assumption at lower frequencies. This suggests that when uncertainty in the thickness is expected, characterization at higher frequencies can yield more accurate results. In addition, for both domains, at all investigated frequencies, there is an asymmetry for the estimated Young’s modulus with respect to the vertical line of zero error in the thickness. Except at the very low frequency (~80 Hz), 3% error in thickness caused 10% or smaller errors in the estimated Young’s modulus.

Figure 9 -.

Errors in the assumed value of thickness and Poisson’s ratio used in the proposed material characterization method. (a) Effects of errors in the assumed thickness for the elliptical plate of Figure 6 on the estimated Young’s modulus. The error in the estimation of the Young’s modulus increases as the error in accurately specifying the thickness increases. Using a low frequency displacement field causes higher errors in the estimation of the Young’s modulus than using higher frequency displacement fields does (e.g., with 10% underestimation of the thickness: 40% error using displacement data at 79.61 Hz vs 24% error using displacement data at 1.59 kHz or higher frequencies).When the thickness is uncertain, measuring the displacement field at higher frequencies can lead to a more accurate estimation of the Young’s modulus. (b) Effects of errors in the assumed Poisson’s ratio for the elliptical plate of Figure 6 on the estimated Young’s modulus. The error in the estimation of the Young’s modulus increases as the error in accurately specifying the Poisson’s ratio increases and the increase is not sensitive to the frequency of excitation. (c) To study the error due to uniform thickness assumption, the effect of variable thickness was studied by revolving this variation of thickness along one radius of the circular plate and calculating the displacement field using the finite element method. Zero error is at the mean thickness. The displacement fields using the variable thickness displacement field were then used in the material characterizing method assuming constant mean thickness to calculate the error in the estimation of the Young’s modulus due to the uniform thickness assumption.

Figure 9-b shows the error in the estimated Young’s moduli with respect to the error in the assumed Poisson’s ratio for the elliptical plate. These data were obtained by calculating the error in the estimated Young’s modulus by varying the Poisson’s ratio from the true value while the thickness was kept at the true value in the estimation process. We performed similar investigations on the triangular plate but did not show the results in Figure 9 as the results of both domains followed similar trends. The values of Poisson’s ratio which are smaller than the true value resulted in a larger estimated Young’s modulus and vice versa. For both plates, the general trend of the error in the estimated Young’s modulus is almost the same for all frequencies used in this study. For both domains, 20% error in the Poisson’s ratio leads to less than 10% error for all cases. Also, the whole range of 60% error leads to a maximum error of 20% in the estimated Young’s modulus.

Non-uniform thickness

In order to investigate the accuracy of the estimation procedure for structures with non-uniform thickness, we studied a circular thin plate with a radius of 3 mm and with an average thickness of 99.20 μm and a standard deviation of 9.18 μm along its radius. The thickness variation of the plate along one radius is shown in Figure 9-c and the thickness distribution of the plate was made by revolving this radial thickness distribution. The plate material was chosen to have the following properties: E=60 MPa, υ=0.33 and ρ=1100 kg/m³ and was excited with a uniform pressure with an amplitude of P=0.7 Pa and frequency of Ω=2900 Hz. The finite-element calculated displacement fields of the plate with non-uniform thickness were used to estimate the Young’s modulus. In the proposed material characterization method, however, the average thickness was used. The estimated Young’s modulus was found to be 56.27 MPa (6.21% error) showing that a standard deviation of 10% causes less error in the estimation than the error that a 10% uniform error in the thickness does (Figure 9-a).

Thick plates

The classical plate theory that was used in the characterization method is valid for thin plates only. In this section, we want to study the errors of the estimated Young’s modulus if the method is applied to thick plates. We considered the triangular plate with the same geometry as the plate in Figure 7 but with increased thicknesses in the FEM simulations (using 3D solid elements) to obtain the input displacement field. The results for the triangular plate with thicknesses of 5 and 10 mm are presented in Table 5. For the plate thicknesses of 5 and 10 mm (50 and 100 times thicker than the plate in Figure 7, respectively), the Young’s modulus was underestimated with errors of about 17% and 45%, respectively, for all investigated frequencies. These errors are substantially worse than the maximum error of less than 0.25% for all investigated frequencies for the thin plate reported in Table 3. These experiments show that the proposed material characterization method can result in large errors if it is used for characterizing thick plates and it should only be used for thin plates as intended in the development of the method.

Effects of uncertainty in boundary conditions

We also studied the effects of uncertainty in boundary conditions. In our proposed material characterization method, we assume all boundaries of the plate to be clamped. However, in practice there may be some imperfection in the boundary conditions. To study the effect of error in the boundary condition assumption on the accuracy of the proposed method, we considered the same triangular plate that we studied in Sections 0 and 0 but changed the boundary conditions of the input displacement fields obtained using FEM. We considered three different boundary condition scenarios: (A) one edge of the triangle was simply-supported while the two other edges were clamped, (B) two edges of the triangle were simply-supported while the third edge was clamped, and (C) all edges were simply-supported. In the proposed material characterization method, however, the boundary conditions, were considered to be clamped for all these scenarios. The results are provided in Table 6 at the same four frequencies investigated before for the triangular plate. For all three scenarios, the lowest accuracies are related to the lowest frequency (79.57 Hz) responses and errors of 31.26% to 74.46% with growing errors by increasing the deviation from the true boundary condition from Scenario A to Scenario C. At this frequency when the true boundary condition was assumed in the material characterization method of this triangular plate, the error was 0.08% (Table 3). This shows that if the boundary condition is uncertain the proposed method breaks at low frequencies. However, the errors decrease by increasing the excitation frequency so that the errors drop to 8%, 12.45% and 21.48% for Scenarios A, B and C respectively at the highest frequency of 7961 Hz. This suggests since the method is less sensitive to the errors in boundary conditions at high frequencies, when the boundary condition is not certain, high-frequency measurements should be considered. Also Table 6 shows that, as it is expected, for each frequency the largest errors are related to Scenario C where the maximum deviation from the true clamped boundary condition exists and all edges of the input displacement field were considered to be simply-supported.

Mesh size and runtime

To study the effect of the mesh size used in the characterization method on the estimated Young’s modulus, we used finite-element displacement data of the elliptical and triangular plates with an excitation amplitude of P=1 kPa and the excitation frequency of Ω=159.23 Hz with the dimensions and mechanical properties used in previous sections (Table 1). The finite-element mesh for calculating the input displacement field was fixed but the characterization mesh size (corresponding to the holographic resolution) was varied.

Figure 10-a presents the error of the estimated Young’s modulus for different mesh sizes for the elliptical plate. The horizontal axis in these figures is the number of nodes considered in each of the edges of the square field of view per centimeter. We performed similar investigations on the triangular plate but did not show the results in Figure 10 as the trends of the results for both plates were similar. For a coarse mesh with a size of 2.5×2.5 per cm, errors of more than 70% in both elliptical and triangular plates were obtained. However, upon refining the mesh size, the error rapidly dropped to less than 10% with 10×10 per cm grids for both plates. This figure shows that the method gives less than 2.5% error with 55×55 per cm or finer. For both domains, the errors decreased to less than 1.5% for finer mesh sizes.

Figure 10 -.

(a) Effect of grid resolution on the estimated Young’s modulus for the elliptical plate of Figure 6. This study was performed to find the sensitivity of the estimation error to the mesh size. It can be observed that the estimation error is large for coarse meshes (for instance for a mesh with a size of 2.5×2.5 per cm, the estimation error is larger than 70%). However, the error rapidly drops by refining the mesh. (b) Runtime with respect to total number of nodes for the elliptical plate shown in Figure 6. Increasing the number of nodes results in an approximately linear increase in the runtime. (c) Error of the estimated Young’s modulus with respect to the error in the viewing angle of the camera (α) for the elliptical plate of Figure 6. For all frequencies except the lowest frequency, the estimation error remains within 3% for angles errors of up to 45°. However, for the lowest frequency (79.6 Hz), the error is less than 3% for angles up to 10° and increases to about 40% for α = 45°. This suggests that low frequency responses are more sensitive to the errors in the viewing angle.

Figure 10-b shows the runtime with respect to the total number of nodes on the square domain for the elliptical domain. The calculations were performed with one central processing unit (Intel CPU, core i7–8700K, 3.70 GHz). The operating system was Windows 10 and MATLAB R2019a was used for all calculations. We performed the runtime analysis for the triangular plate as well however in Figure 10 only the results of the elliptical plate are presented. As this figure shows, increasing the resolution (or the number of calculation nodes), causes an approximately linear increase in the calculation time. A similar linear trend was observed for the triangular plate.

Effects of camera angle

Our proposed material characterization method assumes orthogonality of the sensitivity vector of the holography system to the vibrating plane. We investigated the errors that deviations from this assumption may introduce in the estimated Young’s modulus by using finite-element calculated displacement fields viewed with an angle α. For this aim, the displacement filed was projected on the orthogonal direction for different values of α and the results were used as inputs to the material characterization method.

Figure 10-c shows the errors in the estimated Young’s modulus with respect to the error in the orthogonality assumption quantified by the angle α for the elliptical plate. We did the same analysis for the triangular plate but only the results of the elliptical plate are presented in Figure 10 as the trend of the errors for both plates were similar. The plates were modeled as described in Section 0 with the same material properties. For both plates, the error in the Young’s modulus estimation remains within 3% for up to 45° angle deviation for all frequencies except the lowest frequency investigated. At the lowest frequency, the method is sensitive to errors in the angle as the error in the estimated Young’s modulus remains within 3% only for angle errors of less than 10° and the estimation error grows to about 40% for a 45° error in the angle. These results suggest that for non-ideal experimental setups, it is safer to use higher frequencies stimuli to benefit from the insensitivity of the method to angle errors.

Validation using experimental data

In order to assess the accuracy of the proposed method for material characterization of real structures, we measured the harmonic vibration response of a thin circular copper plate using a stroboscopic holography system that is being developed for hearing diagnosis. The diameter and thickness of the plate were 6.25 mm and 0.02 mm, respectively. The tolerance of these measurements was ±2.54 μm. The experimental setup (Figure 11-a) consists of a calibrated microphone, a speaker, and an otoscope head which includes a camera and an object illumination component. More details of the experimental setup can be found in [56].

Figure 11 -.

(a) The digital holography system used in this work. The experimental setup consists of a speaker, a microphone, and an otoscope head which includes a camera and an object illumination component. (b) Experimental displacement magnitude normalized by the stimulus sound pressure at 4350 Hz of a thin copper plate obtained using the holography setup. This displacement field was used in the material characterization method and the estimated Young’s modulus was in agreement with published material properties of copper [58].

Copper has a low level of damping relative to many other metal and non-metals [57]. The plate was stimulated with a harmonic pressure of 4350 Hz and the displacement field and sound pressure were measured. Figure 11-b shows the displacement magnitude field obtained using holography at 13 stroboscopic phases with a resolution of 800×800. The proposed method was used to estimate the Young’s modulus using the measured displacement magnitude field assuming the Poisson’s ratio and density of 0.34 and 8960$\frac{Kg}{m^3}$, respectively [58,59]. The estimated Young’s modulus using our method was 142.48 GPa which is in the reported range of Young’s modulus for copper alloys [58].

We found that it is crucial that the displacement domain including the entire plate and its non-moving boundary is measured and used as the input to the proposed material characterization method. Failing to provide this domain in its entirety can result in large errors in the material characterization.

Discussions

As mentioned in Section 0, the main aim of this work is to use full-field 2-D single frequency harmonic vibration data for material characterization where there is no access to high-quality modal data. Our ultimate goal is to develop this method to estimate the Young’s modulus of the human tympanic membrane and use it as a diagnostic tool for hearing pathologies. We used the Bayesian optimization to make this a clinically viable method that can find the global optimum value with the fewest possible number of iterations.

In general, optimization algorithms can be categorized into derivative-based and derivative-free algorithms [60]. The derivative-based methods use derivatives to find the optimal value of the objective function. However, sometimes it is difficult to evaluate the derivative of the objective function and it becomes more challenging when the objective function is noisy [61]. Also, the derivative-based methods are usually considered as local optimization techniques [61]. Derivative-free methods include but not limited to evolutionary algorithms (such as genetic algorithm [62]) pattern search [63], simulated annealing algorithm [63], particle swarm algorithm [63], and Bayesian optimization method.

In order to compare the performance of the Bayesian optimization method with another global optimization technique, we used the genetic algorithm to find the Young’s modulus of one of the high-frequency cases with high noise intensity. We chose the elliptical plate with the excitation frequency of 7961 Hz with SNR of 0 dB (presented in Section 0) and used MATLAB genetic algorithm toolbox and considered two evolution scenarios. In Scenario 1, we considered the population of 20 with 9 generations while in Scenario 2, we set the population to be 60 with 3 generations. Both scenarios result in overall 180 times evaluations of the objective function which is the same as the number of evaluations we used with the Bayesian optimization algorithm. The crossover fraction was set to be 0.8. We found that for Scenario 1, the accuracy can be significantly different each time we run the optimization. We repeated the optimization four times for this scenario and the range of error of the estimated Young’s modulus was from about 6.6% to 84% while it was 0.63% with our proposed method using Bayesian optimization for this noisy vibration data. This can be due to the differences in the initial populations each time we run the optimization as it is known that in genetic algorithm the accuracy depends on the initial population [64]. We also ran the optimization for Scenario 2 four times and the range of the error of the estimated Young’s modulus was from 0.46% to 8.70%. One may be able to get better results by using different parameters in the genetic algorithm making the material characterization process operator dependent [26]. Our proposed method, on the other hand, is not operator dependent because the Bayesian optimization algorithm does not need parameters to be set by the operator.

Among other derivative-free optimization methods, pattern search methods search in arbitrary directions and modify the motion of the points to get as close as possible to the optimum values. However, these methods may be attracted by local optima [65]. The simulated annealing algorithm is a probabilistic technique that uses a temperature-like variable to control the degree of stochasticity in the random search and finds the optimum value by iteratively going down from a high temperature to a low temperature. In addition to be stochastic, another disadvantage of this method is its slow convergence [64]. The particle swarm method is a derivative-free method that is also highly sensitive to the initial particles chosen [26] and is known to be trapped in local optima [64]. Bayesian optimization method, which is used in the current work, is not affected by the initialization and is known to be able to find the global optimum with a fewer number of iterations in comparison with many other optimization algorithms such as genetic algorithm, particle swarm, and pattern search.

We also presented some unidentifiable cases, conditions under which the method has a poor performance and remedies for such situations. For instance, as mentioned in Section 0, our estimation method does not provide accurate results for thick plates. Also, as discussed in Section 0, the displacement field of the entire object and its non-moving boundary should be captured in measurements to be able to accurately estimate the Young’s modulus. Obtaining such a view, to enable such captures, might be challenging for some measurement scenarios. Moreover, the method was developed based on a negligible damping assumption that can be justified for many low damped engineering structures. However, this limits the application of the method to damped structures. The developments are underway to generalize the method to damped structures.

Conclusions

In this paper, a novel method was introduced to estimate the Young’s modulus of 2-D thin structures using harmonic vibration data from holography measurements. We validated the method using several analytical, numerical, and experimental benchmarks. For all noise-free benchmark vibration data, which was obtained using analytical and FEM calculations, the highest value of the estimation error was 3.92%. We also studied the effects of adding Gaussian and non-Gaussian noise to the input vibration data and found that the proposed method is robust to high intensity noise and observed less than 5% error in the estimated Young’s modulus for all benchmarks. For each case study, estimations at different frequencies revealed that when uncertainties in the geometrical and material parameters are negligible, the effects of the excitation frequency are not considerable on the estimation outcome. Hence, the estimated values of the Young’s modulus at different frequencies are very close to each other for when we do not have considerable uncertainties (the maximum difference is less than 4%). We studied the effects of uncertainties in the material and geometrical parameters as well as boundary conditions. As expected from basic plate equations, we observed that the accuracy of our proposed method can be compromised if the boundary condition is incorrectly specified. We also studied the effects of error in the viewing angle of the experimental holography setup and found that for the cases studied here, the error in the estimated Young’s modulus remains within 3% for camera angle errors of up to 10° for all frequencies. Our investigations show that when uncertainty in the thickness of the object exists or an ideal perpendicular viewing angle is hard to obtain in the experimental setup, material characterization at higher frequencies is recommended. The method is promising especially for material characterization applications in which access to high-quality free vibration or modal data is limited.

Highlights.

• We propose a non-modal vibration-based method for material characterization.

• The method is developed for estimating the elasticity of thin planar structures.

• Our method works based on single frequency harmonic vibration data of whole domain.

• The method is robust to noise typical in holography measurements of soft tissue.

• Sensitivity of the method to uncertainties and measurement artefacts was studied.