Artificial Neural Network-Based Automated Finite Element Model Updating with an Integrated Graphical User Interface for Operational Modal Analysis of Structures

Abstract

This paper presents an artificial neural network-based graphical user interface, designed to automate finite element model updating using data from operational modal analysis. The approach aims to reduce the uncertainties inherent in both the experimental data and the computational model. A key feature of this method is the application of a discrete wavelet transform-based approach for denoising OMA data. The graphical interface streamlines the FEMU process by employing neural networks to automatically optimize FEM inputs, allowing for real-time adjustments and continuous structural health monitoring under varying environmental and operational conditions. This approach was validated with OMA results, demonstrating its effectiveness in enhancing model accuracy and reliability. Additionally, the adaptability of this method makes it suitable for a wide range of structural types, and its potential integration with emerging technologies such as the Internet of Things further amplifies its relevance.

1. Introduction

Numerical models can be valuable tools for the continuous monitoring of structural integrity, enabling damage detection, estimating the service life, and formulating optimal maintenance strategies. In structural analysis, numerical modeling is commonly carried out through the use of finite element models (FEMs) [1,2]. The increasing emergence of novel numerical modeling techniques, along with advancements in existing methods, necessitates achieving high standards of accuracy and reliability. It is crucial to quantify the uncertainties and errors caused by model assumptions, as they frequently result in inaccuracies. This need has led to the emergence of finite element model update (FEMU) methods, which seek to adjust the numerical model according to the real-world performance of the structure as witnessed during static and/or dynamic testing. When updating FEMs, two main sources of uncertainty may arise: one related to the experimental data collected and the other concerning the predictions made by the FEM [3].

The experimental methods utilized for a FEMU include static and dynamic structural tests, along with the results obtained from structural health monitoring (SHM) approaches. In SHM, dynamic approaches are more commonly used, as they provide a more accurate evaluation of the structure’s performance under real-world conditions [4]. Among dynamic approaches, operational modal analysis (OMA) [5] is a key technique that enables the identification of modal parameters while the structure is in operation and experiencing natural excitation. OMA techniques are based in time and frequency domains, with their most advanced and refined methods being stochastic subspace identification covariance-based (SSI-Cov) in time, and frequency spatial domain decomposition (FSDD) in the frequency domain. In this field, errors frequently arise from imperfections in measuring equipment, random noise in measurements, inconsistencies in signal processing, and challenges during the post-processing of measurement data [6]. Among these, mitigating noise in data through advanced signal processing remains a critical challenge and an ongoing research focus. Denoising entails the removal of unwanted frequencies that may obscure the frequencies of interest. Wavelet transformations [7] have become a powerful tool for this purpose, particularly the discrete wavelet transform (DWT), which effectively reduces high-frequency noise by strategically selecting wavelet thresholds that are adapted to the unique characteristics of the signal being analyzed [8,9].

The uncertainty in FEM arises from inaccurate assumptions about various parameters, including material properties, section characteristics, and the thickness of shell or plate elements, as well as the physical behavior of the structure [10]. These uncertainties are further compounded by factors such as structural idealization and simplification, imprecise mass distribution assumptions, incorrect modeling of mesh connections, and the inherent uncertainties in boundary conditions and joints [11,12]. Although these errors can be mitigated in practice, they cannot be entirely eliminated [3].

To address the disparity between experimentally obtained structural dynamic properties and their numerical equivalents, a sensitivity-based function that incorporates multiple variables is utilized in conjunction with an iterative optimization process to identify optimal structural parameters, which enhances the calibration of large structural models beyond the capabilities suggested by traditional techniques. Model updating can be categorized into manual and automated approaches, with the most effective strategies often employing a hybrid model. Manual techniques, which depend on trial and error, are suitable for optimizing a limited number of parameters; however, they typically lack comprehensive physical interpretations and may yield inefficient outcomes. In contrast, automated methods are beneficial for managing larger parameter sets, effectively reducing idealization errors and improving the overall efficiency of the model updating process [3].

As structural behavior becomes increasingly complex and the demand for precise modeling grows, machine learning (ML) methods have emerged as powerful tools in this field [13,14]. Among these, artificial neural networks (ANNs) are particularly effective at learning intricate patterns and relationships within data [15,16]. By leveraging ML capabilities, ANNs can efficiently process large datasets, adapt to varying conditions, and optimize structural parameters with minimal manual intervention, making them valuable for advanced model updating and precise parameter estimation.

Numerous innovative research studies on the FEMU have significantly advanced structural health monitoring. Kim et al. [17] applied neural networks to update boundary conditions in long-span bridges using static data, enhancing stiffness identification. Sabamehr et al. [18] introduced a hybrid of ANNs and genetic algorithms to correlate structural frequencies with property changes in bridge segments, addressing computational challenges associated with a complex matrix-based FEMU. Naranjo-Pérez et al. [19] developed a collaborative algorithm combining various FEMU techniques with machine learning, improving efficiency. Probabilistic approaches, such as those by [20,21], have been employed to manage data uncertainties, though practical engineering complexities remain. Padil et al. [22] explored non-probabilistic ANN methods, offering potential solutions for uncertainty in damage detection.

Building on these findings, the present study addresses a gap in the FEMU field by proposing a fully automated and user-friendly approach that simplifies the model updating process while maintaining a high level of accuracy. This method incorporates a comprehensive range of dynamic parameters, such as natural frequencies and mode shapes, enabling it to replicate the complex dynamic behavior of real-world structures accurately. Unlike previous methods that require detailed numerical formulations or manual intervention, this approach significantly reduces computational complexity by directly providing FEM inputs without the need for additional analysis. The proposed automation allows for rapid, continuous FEM updates, facilitating real-time adaptation to environmental and operational effects. This is crucial for effective structural health monitoring, ensuring the model remains responsive to dynamic variations. Additionally, the method’s versatility allows its application across a wide range of structural types, from linear to nonlinear systems, ensuring precision through advanced modal identification techniques. By utilizing ML, the approach efficiently captures complex patterns in large datasets, which would be infeasible with traditional methods. Moreover, uncertainty reduction in OMA enhances the model’s reliability and robustness under various operational scenarios.

Section 3 of the paper details the experimental setup, including the selected laboratory structure and its instrumentation used for OMA. A key component of this section is the proposed denoising approach, which employs DWT as part of the pre-processing steps discussed in Section 3.3. Section 4 presents the FEM and its modal identification results. Section 5 provides an overview of the FEMU process, beginning with a brief introduction and sensitivity analyses for parameterization. It also outlines the creation of the dataset required for training the ANN model. Section 6 describes the process of identifying the optimal configuration and the development of the ANN model, along with its graphical user interface (GUI), designed to enhance the efficiency, accuracy, and speed of FEM input prediction. This minimizes discrepancies between the computational model and the real structure. Lastly, Section 6.6 demonstrates the effectiveness of the resulting GUI for FEMU, utilizing OMA results from the structure.

2. Research Significance

As enabling a FEM to more accurately replicate the behavior of the actual structure requires reducing the uncertainties in both experimental setups and FEMs, this paper starting from a denoising method for data from the OMA approach employing an ANN-based GUI, which enables the user to achieve an easy, fast, and precise prediction of the FEM inputs, to capture the closest behavior of the computational model to real-world observations, while considering the dynamic behavior of the real structure, such as its natural frequencies and mode shapes. Unlike traditional FEMU methods that require the definition of specific objective functions, this approach leverages neural network architecture to automatically identify the optimal structural properties, enabling the FEM to function more accurately.

This approach is important for OMA, as the precision of the updated model directly influences the efficacy of continuous SHM. By facilitating real-time adjustments and enhancing the model’s responsiveness to varying environmental and operational conditions, the developed GUI presents a robust solution. Furthermore, the adaptability of this approach allows for its application across diverse structural types, and its potential integration with emerging technologies such as the Internet of Things (IoT) further amplifies its relevance. Such integration could facilitate continuous data input into the model, thereby supporting real-time health monitoring and automated decision-making processes.

Figure 1 is a schematic representation of the proposed FEMU approach in this paper. It is divided into two main phases: Phase 1 combines OMA techniques with FEM, and Phase 2 focuses on FEMU and ANN implementation. Phase 1 deals with the experimental results by applying pre-processing steps, including synchronization, detrending, and DWT to reduce uncertainties in OMA, followed by modal identification in the time and frequency domains. Concurrently, FEM analysis is performed by segmenting and assigning material properties, followed by modal identification analysis. The results from both sides are compared, and if there is considerable discrepancy, the FEM should be reconsidered and remodeled to align the modal parameters more closely with reality. Phase 2 aims to minimize the differences between the results of the FEM and the experimental evidence by implementing a sensitivity analysis based on the eigenvalues and eigenvectors of the FEM with different perturbation values to the material inputs. This step is followed by the creation of a database to train the ANN, iterating over all dynamic analysis combinations and using the optimal FEM input parameters identified by the sensitivity analysis. The ANN is then constructed with the optimal structure, and a GUI is developed to make the updating process efficient.

3. Experimental Setup

3.1. Lab Structure and Instrumentation

In this study, a four-story, shear-type aluminum structure was selected for OMA. The structure and its dimensions are detailed in Figure 2. Each floor was equipped with a wireless accelerometer with an ADXL 355 sensor (from Analog Devices Inc., Cambridge, MA, USA). The functionality and precision of the employed data wireless acquisition system were validated and documented in [5].

3.2. Data Acquisition Process

For this study, acceleration data were recorded on the X, Y, and Z axes at a sampling frequency of 250 Hz over a 30 min duration under controlled environmental conditions at room temperature. Data acquisition was conducted in the absence of any applied external forces, and the structure was only subjected to natural excitations. Figure 3 presents the recorded acceleration data from all four sensors in the Y direction. The recorded peak acceleration values for each floor, from the first to the fourth, are 0.00207 g, 0.00208 g, 0.00245 g, and 0.00298 g, respectively.

3.3. Pre-Processing

Subsequent to the recording, the data were synchronized, with an inaccuracy below the sample time of around 2 milliseconds, in accordance with the methodology outlined in [5], and subsequently detrended. Furthermore, to reduce the uncertainties associated with the influence of noise on the data, a novel approach utilizing DWT was implemented, as discussed in the following section.

3.3.1. Discrete Wavelet Transform Approach for Denoising

Unlike traditional Fourier transforms, which provide only frequency domain information, the DWT decomposes a signal into both frequency and time-localized components. This multi-resolution analysis is particularly effective at isolating and eliminating noise, allowing the preservation of critical features such as abrupt changes or discontinuities within the signal [23,24].

Mathematically, the DWT of a discrete signal $x\left[n\right]$ can be expressed as follows:

$$X_{j,k}={\sum }_{n}x[n]{\psi }_{j,k}[n]$$

(1)

where ${\psi }_{j,k}[n]$ are the wavelet functions obtained by scaling and translating a mother wavelet $\psi \left[n\right]$:

$${\psi }_{j,k}\left[n\right]={\displaystyle \frac{1}{\sqrt{2^j}}}\psi \left({\displaystyle \frac{n-2^{j}k}{2^j}}\right)$$

(2)

Here, $j$ and $k$ represent the scale and translation parameters, respectively.

DWT enables the representation of a signal within a hierarchical tree structure. Each level of this structure contains both wavelet details $D_i\left(t\right)$ and approximations $A_i\left(t\right)$ (Equation (3)).

$$x[n]=\sum _{i=1}^j\left(D_i\left(t\right)+A_i\left(t\right)\right)$$

(3)

A thresholding technique is subsequently applied to the detail coefficients, which represent the high-frequency components of the signal. This thresholding can be executed through hard or soft methods [25]. In this study, soft thresholding was employed, characterized as follows:

$$\widehat{c}=sign\left(c\right).max\left(\left|c\right|-\lambda ,0\right)$$

(4)

where $\widehat{c}$ represents the threshold wavelet coefficient, and $\lambda$ denotes the threshold value.

This threshold is typically determined through universal thresholding, as proposed by Donoho and Johnstone in 1994 [26]. The noise level $\sigma$ and the threshold $\lambda$ can be estimated using the median absolute deviation (MAD) of the wavelet coefficients, calculated as follows:

$$\sigma ={\displaystyle \frac{\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(\left|c-\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(c\right)\right|\right)}{0.6745}}, \lambda =\sigma \sqrt{2\mathrm{l}\mathrm{o}\mathrm{g}N}$$

(5)

Wavelet reconstruction is the last step in the denoising process, when the updated wavelet coefficients are used to recreate the denoised signal. This phase integrates the approximation coefficients, which capture the low-frequency components, with the threshold detail coefficients, resulting in a signal with reduced noise levels.

3.3.2. Wavelet Types and Decomposition Levels for the Optimal Denoising Performance

Various configurations of wavelet types and decomposition levels were systematically tested to identify the optimal settings for denoising. The effectiveness of the denoised signal was assessed using several performance metrics, including mean squared error (MSE), signal-to-noise ratio (SNR), peak signal-to-noise ratio (PSNR), mean absolute error (MAE), maximum error (ME), and structural similarity index (SSIM).

Peak Signal-to-Noise Ratio (PSNR)

The PSNR is a metric used to evaluate the quality of a reconstructed signal in relation to the original signal. It is computed by dividing the highest power of the signal by the MSE. Higher PSNR values indicate a better reconstruction quality, with less distortion from the original signal. The calculation for the PSNR is given in Equation (6) [27]:

$$\mathrm{P}\mathrm{S}\mathrm{N}\mathrm{R}=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({\displaystyle \frac{max{\left(x\right)}^2}{\mathrm{M}\mathrm{S}\mathrm{E}=\frac{1}{N}{\sum }_{i=1}^N{\left(x_i-{\widehat{x}}_i\right)}^2}}\right)$$

(6)

where $x_i$ represents the original signal, ${\widehat{x}}_i$ represents the denoised signal, and $N$ is the number of samples.

Structural Similarity Index (SSIM)

The SSIM quantifies the resemblance between two signals based on their mean, variance, and correlation structure. It is widely used to assess the quality of processed signals. In the following equation, ${\mu }_x$ and ${\mu }_{\widehat{x}}$ are the means, ${\sigma }_x$ and ${\sigma }_{\widehat{x}}$ are the variances, and ${\sigma }_{x\widehat{x}}$ is the covariance of the original and denoised signals. $C_1$ and $C_2$ are constants to stabilize the division (Equation (7) [28]).

$$\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\left(x,\widehat{x}\right)={\displaystyle \frac{\left(2{\mu }_x{\mu }_{\widehat{x}}+C_1\right)\left(2{\sigma }_{x\widehat{x}}+C_2\right)}{\left({\mu }_x^2+{\mu }_{\widehat{x}}^2+C_1\right)\left({\sigma }_x^2+{\sigma }_{\widehat{x}}^2+C_2\right)}}$$

(7)

Objective Function Determination

The objective function used to identify the optimal denoising configuration is a weighted combination of various performance metrics, emphasizing their relative importance in denoising performance. The most effective combination of wavelet type and decomposition level was determined by minimizing an objective function, which indicates the highest noise reduction while preserving the signal’s structural integrity. The primary goal is to minimize error metrics (MSE, MAE, ME) and maximize quality metrics (SNR, PSNR, SSIM). The proposed objective function (OF) for determining the optimal conditions is given by Equation (8). The coefficients were determined through a process of trial and error.

$$OF=\left(0.35\times \mathrm{M}\mathrm{S}\mathrm{E}\right)+\left(0.2\times \mathrm{M}\mathrm{A}\mathrm{E}\right)+\left(0.1\times \mathrm{M}\mathrm{E}\right) -\left(0.2\times \mathrm{S}\mathrm{N}\mathrm{R}\right)-\left(0.15\times \mathrm{P}\mathrm{S}\mathrm{N}\mathrm{R}\right)-\left(0.35\times \mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\right)$$

(8)

Consequently, various types of DWTs, as detailed in

Table 1. Tested DWT types.

Wavelet Family	Wavelet Identifiers
Haar	haar
Daubechies	db1, db2, …, db20
Symlets	sym2, sym3, …, sym20
Coiflets	coif1, coif2, …, coif5
Biorthogonal	bior1.1, bior1.3, bior1.5, bior2.2, …, bior6.8
Reverse Biorthogonal	rbio1.1, rbio1.3, rbio1.5, rbio2.2, …, rbio6.8

, each with 2 to 5 decomposition levels, were assessed, with the findings summarized in

Table 2. Iterations over all DWT types with levels ranging from 2 to 5.

Wavelet	Level	MSE	SNR	PSNR	MAE	ME	SSIM	Objective
bior1.1	2	1.07 × 10−8	5.379477	28.16202	0.000082	0.000652	0.750457	−5.56278
bior1.1	3	1.54 × 10−8	3.801784	26.58433	0.000097	0.000842	0.668469	−4.98187
bior1.1	4	1.99 × 10−8	2.685388	25.46793	0.000111	0.000977	0.648481	−4.58412
bior1.1	5	2.19 × 10−8	2.263825	25.04637	0.000117	0.000882	0.645208	−4.43543
bior1.3	2	1.10 × 10−8	5.255395	28.03794	0.000083	0.000697	0.741835	−5.51633
bior1.3	3	1.57 × 10−8	3.707644	26.49019	0.000098	0.000936	0.6571	−4.94493
bior1.3	4	1.92 × 10−8	2.841703	25.62425	0.000109	0.000892	0.644098	−4.6373
bior1.3	5	2.09 × 10−8	2.465651	25.2482	0.000114	0.000832	0.645276	−4.5061
…	…	…	…	…	…	…	…	…
sym19	2	9.15 × 10−9	6.057419	28.83996	0.000076	0.000459	0.766889	−5.80583
sym19	3	1.20 × 10−8	4.88689	27.66943	0.000087	0.000491	0.704214	−5.3742
sym19	4	1.73 × 10−8	3.288775	26.07132	0.000105	0.000604	0.661944	−4.80005
sym19	5	1.92 × 10−8	2.834529	25.61707	0.00011	0.000658	0.657882	−4.63964
sym20	2	9.15 × 10−9	6.059545	28.84209	0.000076	0.00044	0.766811	−5.80655
sym20	3	1.20 × 10−8	4.884842	27.66739	0.000087	0.000496	0.703416	−5.37321
sym20	4	1.73 × 10−8	3.287674	26.07022	0.000105	0.000625	0.660057	−4.799
sym20	5	1.92 × 10−8	2.836781	25.61933	0.00011	0.000718	0.654244	−4.63915

.

The optimal DWT type utilized in this study was ‘bior3.9′ at two levels, yielding the following performance metrics: MSE of 8.99 × 10⁻⁹, SNR of 6.133648, PSNR of 28.916192, MAE of 0.000076, ME of 0.000442, SSIM of 0.771145, and an objective value of −5.834. Figure 4 illustrates the acceleration data from the fourth floor in Y direction, both before and after the application of DWT.

Figure 4 presents a comparative analysis of the Y-direction acceleration data obtained from a sensor located on the fourth floor, highlighting the effects of DWT filtering. Figure 4a illustrates the time-domain comparison between the raw and filtered signals, demonstrating the effectiveness of DWT in reducing noise. Figure 4b focuses on a randomly selected zoomed-in region, further underscoring the improved alignment and smoothing of the filtered signal relative to the raw data. Figure 4c depicts the Power Spectral Density (PSD) in the frequency domain, showcasing the significant attenuation of noise across a broad frequency range due to DWT filtering, while preserving crucial spectral components, particularly at lower frequencies.

3.4. Modal Identification Process

To estimate the dynamic parameters of the structure, two OMA methods—SSI-Cov in the time and FSDD in the frequency domain—were applied to the pre-processed data. The PSD for FSDD was computed using the Welch method, with a resolution of 1024 and a 50% overlap. For the SSI-Cov method, a stabilization diagram was generated using 40 block rows (time lags) and a maximum model order of 60. The combination of the PSD and the stabilization diagram is illustrated in Figure 5, and

Table 3. Resulted natural frequencies and mode shapes from OMA.

Natural Frequencies [Hz]			FSDD Mode Shapes			SSI-Cov Mode Shapes
			Mode 1	Mode 2	Mode 3	Mode 1	Mode 2	Mode 3
FSDD	SSI-Cov	Average	0.252213	−0.96047	1	0.264583	−0.960617	1
Mode 1: 2.04	Mode 1: 2.01	Mode 1: 2.025	0.638551	−0.943739	−0.40772	0.65159	−0.978302	−0.381035
Mode 2: 6.16	Mode 2: 6.13	Mode 2: 6.145	0.88866	0.147326	−0.586475	0.895134	0.131195	−0.602195
Mode 3: 9.65	Mode 3: 10.01	Mode 3: 9.83	1	1	0.652448	1	1	0.642897

reports the resulting natural frequencies and mode shapes for the first three modes identified by the OMA.

Figure 6 presents the modal assurance criterion (MAC) plot, which compares the mode shapes obtained from two different methodologies. Their MAC values indicate a significant correlation, with values approaching one along the diagonal and near zero in off-diagonal elements. This pattern suggests a high degree of similarity between the mode shapes derived from both methods. The average values of the dynamic parameters from both the time- and frequency-domain analyses were taken as the output for OMA.

4. Finite Element Model (FEM)

The computational model was constructed using ABAQUS software (2022) [29], with braces and columns defined as beam elements and floors modeled in shell. To accurately represent material properties—such as elasticity modulus, mass density, and Poisson’s ratio—the structural elements were categorized into distinct groups, including the columns and braces of each floor (first to fourth) and each floor plates.

Table 4. Modal mass participation in the translational and rotational directions.

Mode	TRAN-X (%)	TRAN-Y (%)	TRAN-Z (%)	ROTN-X (%)	ROTN-Y (%)	ROTN-Z (%)
1	0	0	84.05	13.67	0	0
2	0	0	8.66	55.33	0	0
3	0	0	2.32	3.56	0	0
4	0	0	0.49	2.48	0	0

presents the modal mass participation, resulted from the computational model, in both the translational and rotational directions for the first four modes.

In this paper, the first three modes were selected for analysis because, as Table 4 reveals, they collectively contributed 95.03% to the mass participation in the TRAN-Z direction and 72.55% in the ROTN-X direction. Meanwhile, mode 4 contributed only 0.49% in TRAN-Z and 2.48% in ROTN-X, rendering its influence on the overall dynamic behavior negligible. According to Chopra [30], modal analysis can effectively capture the dynamic response with a relatively small number of modes, depending on the specific load and response parameters. Fragiadakis [31] supports this, stating that in some building codes, it is recommended to consider so many modes as to achieve an effective modal mass of at least 90% of the structure’s mass. Since the effective modal mass has a relationship with the base shear, this rule implies that the error in the base shear estimation should be less than 10%. In addition, Eurocode 8 (EC8 2004) [32] specifies that all modes with effective modal masses greater than 5% should be considered in the analysis. Furthermore, Chung et al. [33] highlight that the effective mass and inertia would equal the physical mass and inertia in an ideal linear structure if all modes were included in the analysis. However, in practice, including modes that contribute to a total of 90% of the effective mass is considered sufficient to capture the significant dynamic responses of the structure. Moreover, from the practical point of view, including each additional mode would add five inputs (a natural frequency and its mode shape with four degrees of freedom) to the defined input for the proposed machine learning model in this paper, making the process more complicated. Therefore, it was decided to exclude mode 4 and higher modes from this analysis, as this approach aligns with established guidelines and ensures a streamlined yet accurate representation of the structure’s dynamic behavior.

Figure 7 presents the results of the modal analysis of the FEM for the first three modes, including the corresponding natural frequencies and mode shapes.

Table 5. Comparison of OMA and FEM results for natural frequencies and mode shapes.

Loc.	OMA Mode Shapes			FEM Mode Shapes			Natural Frequencies
	Mode 1	Mode 2	Mode 3	Mode 1	Mode 2	Mode 3
Floor 1	0.26	−0.96	1.00	0.32	−0.89	1.00	OMA [Hz]	FEM [Hz]	Discrepancies %
Floor 2	0.65	−0.96	−0.39	0.65	−0.90	−0.40	Mode 1: 2.025	Mode 1: 2.117	Mode 1: 4.54
Floor 3	0.89	0.14	−0.59	0.89	0.11	−0.79	Mode 2: 6.145	Mode 2: 6.334	Mode 2: 3.08
Floor 4	1.00	1.00	0.65	1.00	1.00	0.82	Mode 3: 9.83	Mode 3: 10.343	Mode 3: 5.21

provides a comparison of the natural frequencies and mode shapes derived from the FEM and OMA, and Figure 8 shows the resulting MAC between their mode shapes.

Table 5 and Figure 8 show discrepancies between the predictions made by the computational model and the observed behavior of the structure through OMA. These differences are to be expected and can be attributed to the inherent limitations and assumptions within the computational model. These include idealized boundary conditions, material properties, and geometric simplifications, which fail to fully account for the complexities and uncertainties inherent in the real-world structure. Therefore, to achieve a more accurate correlation between the model predictions and the observed structural behavior, the implementation of FEM updating techniques becomes important.

5. Finite Element Modeling Update (FEMU)

5.1. Model and Model Class

In the context of a FEMU, a model refers to the mathematical representation of a physical system’s input–output relationship. The “model class” encompasses a group of models that share similar structural characteristics. These models are defined by input variables, such as structural parameters $\theta$, and output variables $z$, which may include natural frequencies, mode shapes, and other related factors [3]. The relationship between inputs and outputs is mathematically expressed as follows, where $M$ denotes the model operator:

$$z=M\left(x,\theta \right)$$

(9)

In numerous FEMU applications, the output results are often treated as independent of the input vector $x$. This assumption simplifies the relationship [3], allowing it to be expressed as follows:

$$z=M\left(\theta \right)$$

(10)

5.2. Experimental Datasets

Experimental datasets, denoted as $\tilde{M}$, consist of vectors that can be classified as either homogeneous, comprising a single type of data, or heterogeneous, containing multiple data types. These vectors include output quantities such as natural frequencies, mode shapes, strains, and displacements, either individually or in combination. Such datasets play a critical role in refining finite element models to ensure their accuracy in representing the physical behavior of structures. In this article, these vectors are referred to as the desired parameters from the eigenvalues and eigenvectors obtained from OMA.

5.3. Model Updating

Model updating is an iterative process wherein model parameters are refined to reduce discrepancies between model predictions and experimental observations. This is achieved by defining and minimizing an objective function. The process involves parameter estimation within a predetermined model framework, taking into account uncertainties inherent in both the model and the experimental data. The measured dataset can be represented as a vector [3].

$$\tilde{M}=M_m\left(\theta \right)+ϵ+\mu$$

(11)

where $ϵ$ represents model uncertainty, and $\mu$ represents measurement uncertainty. The objective is to determine the optimal model parameters ${\theta }_{opt}$ such that the numerical model outputs $M\left({\theta }_{opt}\right)$ align closely with the experimental data $\tilde{M}$ [3]. This can be formulated as follows:

$$M_m\left({\theta }_{opt}\right)\approx \tilde{M}$$

(12)

5.4. Sensitivity Analysis

To optimize the number of parameters for FEM updating, it is crucial to identify which properties of each structural segment most significantly influence the results of modal analysis, while disregarding less impactful parameters. Sensitivity analysis is integral to FEM updating as it evaluates how variations in model parameters affect output responses.

In this study, a sensitivity analysis was conducted to observe the influence of material properties (elastic modulus, mass density, and Poisson’s ratio) on the dynamic characteristics of the structure (natural frequencies and mode shapes). A similar approach to sensitivity-based finite element model updating, which focuses on ambient vibration test data and the modal parameters of bridge structures, was employed by Saidin et al. [34].

To accurately assess the structural behavior during the modal identification process, the material properties of the columns, floors, and braces for each floor were assigned separately. An iterative process was established to calculate sensitivity values for the FEM analyses. During each iteration, one material property of each segment was perturbed, as outlined in

Table 6. Levels of perturbations for each of the material properties of the structural elements.

Material Properties	Structural Elements	Perturbations [%]
Elasticity module $[\mathrm{G}\mathrm{P}\mathrm{a}$]	Col (1–4)-Brace (1–4)-Floor (1–4)	1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Mass density $[\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$]	Col (1–4)-Brace (1–4)-Floor (1–4)	1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Poisson ratio	Col (1–4)-Brace (1–4)-Floor (1–4)	1, 2, 3, 4, 5, 6, 7, 8, 9, 10

.

It should be noted that, for more complex structures, an ANN-based sensitivity analysis can be employed to capture the relationships between design parameters and structural responses. After training, the ANN can estimate output sensitivity to input changes by calculating gradients, offering a more efficient analysis, particularly for models where theoretical sensitivity calculations are computationally expensive.

This paper employed Equation (13) as the natural frequency sensitivity factor $S_f$, by taking the absolute difference between each pair of corresponding frequencies, normalizing by the baseline frequency, and summing these values. Mathematically, the sensitivity $S_f$ can be expressed as follows:

$$S_f={\sum }_{p_i}^{n=3}\left({\displaystyle \frac{\left|f_{p_i}-f_b\right|}{f_b}}\right)$$

(13)

where

• $f_p$ is the $i^{th}$ frequency resulting from the perturbation to the input;
• $f_b$ is the $i^{th}$ frequency from the list of baseline frequencies;
• $n$ is the total number of the considered modes.

Figure 9 presents sensitivity plot of the natural frequencies in response to the defined perturbations in the elasticity modulus, mass density, and Poisson’s ratio, for each individual group of elements. Figure 9a demonstrates that perturbations in the elasticity modulus of columns 1, 4, 2, and 3 exert the most significant influence on the resulting natural frequencies. In Figure 9b, it is evident that the greatest impact, in terms of mass density, is attributed to floors 4 and 1. Additionally, the sensitivity plot in Figure 9c indicates that the effects of perturbations on the Poisson ratio of each group of structural elements are negligible.

The mode shape sensitivity $S_{Fi}$ is also employed in this study by first extracting the mode shape vectors corresponding to the specified perturbation. Subsequently, the MAC values for each pair of mode shape vectors are calculated. The mode shapes’ difference is then obtained by subtracting the MAC values from one.

$$S_{Fi}=1-\mathrm{M}\mathrm{A}\mathrm{C}\left({Fi}_{pi}^{vec}, {Fi}_b^{vec}\right)=1- {\displaystyle \frac{\left|{\left({Fi}_{pi}^{vec}·{Fi}_b^{vec}\right)}^2\right|}{\left({Fi}_{pi}^{vec}·{Fi}_{pi}^{vec}\right)\left({Fi}_b^{vec}·{Fi}_b^{vec}\right)}}$$

(14)

where

• ${Fi}_{pi}^{vec}$ is the $i^{th}$ mode shape vector resulting from the perturbation to the input;
• ${Fi}_b^{vec}$ is the $i^{th}$ mode shape vector from the list of the baseline.

Figure 10 depicts the sensitivity plot of the mode shapes in relation to the specified perturbations, focusing on the elasticity modulus, and the mass density for each individual group of elements. Figure 10a shows that the perturbations in columns 1, 2, 3, and 4 resulted in significant deviations in the mode shapes. As depicted in Figure 10b, the mass densities of the floors 4 and 1 are the most effective. Due to the minor effect of the Poisson ratio on the results of the natural frequencies, it was decided to skip the sensitivity analysis for this material property on the mode shapes.

5.5. Design of Experiment (DOE)

DOE is a systematic approach used to determine the relationship between factors influencing a process and its output [35]. It is employed to plan, conduct, analyze, and interpret controlled tests, evaluating the factors that affect one or more parameters [35,36]. DOE is particularly important in FEM updating, as it reduces the number of simulations required, conserves computational resources, and enhances understanding of the model’s behavior [37].

The most straightforward DOE method is full factorial design, which explores all possible combinations of design variables [38,39]. While it provides a comprehensive view of factor interactions, it becomes impractical with a large number of factors due to the exponential increase in experiments. Specifically, if there are $k$ factors each at $n$ levels, the total number of experiments is $n^k$.

In this paper, the full factorial design method is applied due to the relatively low complexity of the model compared to real-world structural cases, and to generate a robust dataset for training a machine learning model, capturing the complete interactions between parameters. Figure 9 and Figure 10 highlight the most influential parameters on the modal results: the elasticity modulus of columns 1, 2, 3, and 4, as well as the mass density of floors 1 and 4.

The elasticity modulus and mass density of aluminum are commonly cited as having average values of approximately 70 $\mathrm{G}\mathrm{P}\mathrm{a}$ and 2700 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, respectively [40,41]. These values serve as useful benchmarks when analyzing the material behavior of structures. In this study, the bounds for these design variables were established based on aluminum’s known physical properties, ensuring that the updating process was rooted in realistic conditions. To account for natural variability, the elasticity modulus was set within a range of 68 $\mathrm{G}\mathrm{P}\mathrm{a}$ (minimum) to 72 $\mathrm{G}\mathrm{P}\mathrm{a}$ (maximum), while the mass density was considered between 2600 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ (minimum) and 2800 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ (maximum). The selection of these bounds takes into consideration typical fluctuations in material properties due to various factors, such as differences in alloy composition, variations in manufacturing processes, and the influence of environmental conditions like temperature or humidity. These factors can lead to small but significant changes in mechanical properties, which need to be accounted for to enhance the accuracy of the analysis [42]. By defining these realistic bounds, the updating process remains valid for practical applications, ensuring that the results are both precise and applicable to real-world scenarios. It should also be noted that, during each iteration of the analysis, the material properties of other structural elements were maintained at their average values.

Using the full factorial design, the elasticity modulus of columns 1 to 4 involved $6^4$ combinations, and the mass density of floors 1 and 4 involved $3^2$ combinations, resulting in a total of $\mathrm{11,664}$ iterations. This number of iterations was selected to ensure an accurate structural behavior representation, as fewer iterations would underestimate the system, and more would make the approach computationally infeasible.

For more complex structures, such as bridges or large buildings, fractional factorial designs become more practical. For instance, a full factorial design with five factors at two levels would require 32 experiments, but a half-fractional design reduces this to 16, allowing for the screening of significant factors while maintaining manageable computational costs [43]. Other methods, such as the response surface methodology (RSM) [36,44], are suitable for nonlinear relationships between inputs and outputs. Additionally, techniques like central composite design (CCD) [45], Box–Behnken design [46], and Taguchi methods [47] can be employed depending on the specific requirements [48].

5.6. FEMU Dataset

Table 7. Iterative modal analysis process for the defined combination for the elasticity and mass density of the elements.

Iter.	${\mathbf{E}}_{\mathit{c}1}$	${\mathbf{E}}_{\mathit{c}2}$	${\mathbf{E}}_{\mathit{c}3}$	${\mathbf{E}}_{\mathit{c}4}$	${\mathbf{M}}_{\mathit{f}1}$	${\mathbf{M}}_{\mathit{f}2}$	${\mathbf{N}\mathbf{f}}_1$	${\mathbf{N}\mathbf{f}}_2$	${\mathbf{N}\mathbf{f}}_3$	${\mathbf{M}}_1{\mathbf{F}\mathbf{i}}_1$	${\mathbf{M}}_1{\mathbf{F}\mathbf{i}}_2$	${\mathbf{M}}_1{\mathbf{F}\mathbf{i}}_3$	${\mathbf{M}}_1{\mathbf{F}\mathbf{i}}_4$	${\mathbf{M}}_2{\mathbf{F}\mathbf{i}}_1$	${\mathbf{M}}_2{\mathbf{F}\mathbf{i}}_2$	${\mathbf{M}}_2{\mathbf{F}\mathbf{i}}_3$	${\mathbf{M}}_2{\mathbf{F}\mathbf{i}}_4$	${\mathbf{M}}_3{\mathbf{F}\mathbf{i}}_1$	${\mathbf{M}}_3{\mathbf{F}\mathbf{i}}_2$	${\mathbf{M}}_3{\mathbf{F}\mathbf{i}}_3$	${\mathbf{M}}_3{\mathbf{F}\mathbf{i}}_4$
1	68	68	68	68	2600	2600	2.05	6.16	10.06	0.1679	0.3377	0.4606	0.5173	0.4422	0.4466	0.0649	0.5057	0.5344	0.2061	0.4154	0.4540
2	68	68	68	68	2600	2700	2.04	6.13	10.03	0.1667	0.3356	0.4582	0.5156	0.4399	0.4483	0.0570	0.5006	0.5342	0.1989	0.4221	0.4434
3	68	68	68	68	2600	2800	2.03	6.10	9.99	0.1656	0.3335	0.4559	0.5139	0.4376	0.4499	0.0493	0.4955	0.5340	0.1920	0.4284	0.4331
4	68	68	68	68	2700	2600	2.05	6.14	10.01	0.1681	0.3377	0.4603	0.5169	0.4434	0.4421	0.0673	0.5037	0.5289	0.2176	0.4089	0.4537
5	68	68	68	68	2700	2700	2.04	6.11	9.97	0.1669	0.3355	0.4580	0.5153	0.4411	0.4438	0.0594	0.4987	0.5288	0.2103	0.4157	0.4432
6	68	68	68	68	2700	2800	2.03	6.08	9.94	0.1657	0.3334	0.4556	0.5136	0.4389	0.4455	0.0518	0.4936	0.5288	0.2035	0.4220	0.4331
7	68	68	68	68	2800	2600	2.05	6.12	9.96	0.1682	0.3376	0.4600	0.5165	0.4446	0.4375	0.0697	0.5016	0.5232	0.2286	0.4026	0.4536
8	68	68	68	68	2800	2700	2.04	6.09	9.92	0.1670	0.3354	0.4577	0.5149	0.4423	0.4393	0.0619	0.4967	0.5232	0.2214	0.4095	0.4432
9	68	68	68	68	2800	2800	2.03	6.06	9.89	0.1659	0.3333	0.4554	0.5132	0.4401	0.4411	0.0543	0.4917	0.5233	0.2145	0.4159	0.4331
…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…
11,656	72	72	72	72	2600	2600	2.10	6.30	10.29	0.1672	0.3375	0.4607	0.5177	0.4410	0.4475	0.0638	0.5059	0.5344	0.2039	0.4161	0.4537
11,657	72	72	72	72	2600	2700	2.09	6.27	10.26	0.1660	0.3353	0.4583	0.5160	0.4387	0.4491	0.0559	0.5008	0.5342	0.1967	0.4227	0.4431
11,658	72	72	72	72	2600	2800	2.08	6.24	10.22	0.1649	0.3332	0.4560	0.5143	0.4365	0.4508	0.0483	0.4957	0.5341	0.1898	0.4289	0.4329
11,659	72	72	72	72	2700	2600	2.10	6.27	10.24	0.1674	0.3374	0.4604	0.5173	0.4422	0.4430	0.0662	0.5039	0.5290	0.2154	0.4095	0.4535
11,660	72	72	72	72	2700	2700	2.09	6.25	10.21	0.1662	0.3352	0.4581	0.5156	0.4400	0.4447	0.0584	0.4988	0.5289	0.2081	0.4163	0.4430
11,661	72	72	72	72	2700	2800	2.08	6.22	10.17	0.1650	0.3331	0.4557	0.5139	0.4378	0.4464	0.0508	0.4938	0.5288	0.2013	0.4225	0.4328
11,662	72	72	72	72	2800	2600	2.10	6.25	10.19	0.1675	0.3373	0.4601	0.5169	0.4434	0.4385	0.0686	0.5018	0.5233	0.2263	0.4033	0.4534
11,663	72	72	72	72	2800	2700	2.08	6.22	10.15	0.1663	0.3352	0.4578	0.5152	0.4411	0.4403	0.0608	0.4968	0.5233	0.2191	0.4101	0.4429
11,664	72	72	72	72	2800	2800	2.08	6.20	10.12	0.1652	0.3331	0.4555	0.5136	0.4390	0.4420	0.0532	0.4919	0.5234	0.2123	0.4165	0.4328

${\mathrm{E}}_{cn}$: elasticity module column n, ${\mathrm{M}}_{fn}$: mass density floor n, ${\mathrm{N}\mathrm{f}}_n$: natural frequency of mode n, ${\mathrm{M}}_n{\mathrm{F}\mathrm{i}}_m$: mode shape of mode n, and number of the floor m.

presents a subset of the comprehensive dataset through all iterations, illustrating the mode shapes expressed in absolute values and scaled according to the Abaqus FEM format.

6. Proposed ANN Model

6.1. Data Preparation

To improve the model performance and ensure that each feature contributed equally to the training process, data were scaled between 0.1 and 0.9. The transformation is defined as Equation (15), where $x$ is the original data, $x^{\prime }$ is the scaled data, and $x_{min}$ and $x_{max}$ are the minimum and maximum values of the original data, respectively.

$$x^{\prime }=0.1+{\displaystyle \frac{\left(x-x_{min}\right)·\left(0.9-0.1\right)}{x_{max}-x_{min}}}$$

(15)

Furthermore, subsequent to the model predictions, the scaled values will be transformed back to their original scale through the application of the inverse transformation as illustrated in Equation (16).

$$x=x_{min}+\left({\displaystyle \frac{\left(x^{\prime }-0.1\right)·\left(x_{max}-x_{min}\right)}{0.9-0.1}}\right)$$

(16)

Following scaling, the data were segmented into two groups: ${\mathrm{M}}_n{\mathrm{F}\mathrm{i}}_m$ and ${\mathrm{N}\mathrm{f}}_n$ as inputs, and ${\mathrm{E}}_{cn}$ and ${\mathrm{M}}_{fn}$ as outputs. Subsequently, the dataset was randomly partitioned into training and testing subsets, with the training set comprising 80% of the data (9332 data) and the testing set comprising 20% (2332 data). It is also noteworthy that 20% of the training set was allocated for validation purposes.

6.2. Performance Evaluation of the Proposed Model

The proposed model’s performance was evaluated using several criteria, including the coefficient of determination (${\mathrm{R}}^2$) and various error metrics: mean absolute error (MAE), normalized mean absolute error (NMAE), mean absolute percentage error (MAPE), root mean square error (RMSE), mean square error (MSE), and normalized root mean square error (NRMSE). These metrics were calculated based on the following equations, with $N$ representing all numbers of data points, $M_i$ as the actual value of the $i^{th}$ observation, $P_i$ as the predicted value, $M_{ave.}$ as the mean of the actual values, and $P_{ave.}$ as the mean of the predicted values.

$${\mathrm{R}}^2={\left({\displaystyle \frac{{\sum }_{i=1}^N\left(M_i - M_{ave.}\right)\left(P_i-P_{ave.}\right)}{\sqrt{\sum _{i=1}^N{\left(M_i-M_{ave.}\right)}^2\sum _{i=1}^N{\left(P_i-P_{ave.}\right)}^2}}}\right)}^2$$

(17)

$$\mathrm{N}\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{\frac{1}{N}\sum _{i=1}^N\left|M_i-P_i\right|}{max\left(M_i\right)-min\left(M_i\right)}}\times 100$$

(18)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|{\displaystyle \frac{M_i-P_i}{M_i}}\right|\times 100$$

(19)

$$\mathrm{N}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(M_i-P_i\right)}^2}}{max\left(M_i\right)-min\left(M_i\right)}}\times 100$$

(20)

6.3. Optimal ANN Configuration

The proposed model was developed using a feed-forward architecture and trained with the Adam optimizer via backpropagation. It consists of one input, one hidden, and one output layer, where the hidden layer employed the exponential linear unit (ELU) [49] activation function, and the output layer utilized a linear activation function. This design aimed to mitigate vanishing gradient problems associated with the ELU, while providing a linear output suitable for regression tasks. The training process involved a learning rate of 0.003 and mean squared error loss function, conducted over 250 epochs with a batch size of 64. A validation split of 0.2 was implemented to evaluate the model’s performance on unseen data.

To determine the optimal number of neurons in the hidden layer, a trial-and-error method was employed, testing a range of 6 to 20 neurons. It is noteworthy that while more complex ANN configurations, involving two or more hidden layers, were tested, the single-hidden-layer configuration was preferred due to its comparable accuracy to the more complex structures, despite being less computationally demanding. Various performance metrics were calculated, including MSE, RMSE, ${\mathrm{R}}^2$, and MAPE. Figure 11 illustrates the MAPE and ${\mathrm{R}}^2$ values for different neuron configurations in the single hidden layer using the testing datasets. Furthermore, a comprehensive set of error metrics for both training and testing datasets is provided in

Table 8. Error metrics and performance measures—MSE, RMSE, MAPE, and R²—for training and testing datasets across varied neuron numbers in a single hidden layer.

Neuron Num.	Training			Testing
	MSE	RMSE	${\mathbf{R}}^2$	MSE	RMSE	${\mathbf{R}}^2$	MAPE
6	1.3065	1.1430	0.9842	1.2853	1.1337	0.9837	0.1576
7	2.0506	1.4320	0.9834	2.0690	1.4384	0.9828	0.1701
8	2.0739	1.4401	0.9840	2.0539	1.4331	0.9834	0.1622
9	1.7264	1.3139	0.9839	1.7071	1.3066	0.9833	0.1576
10	1.5119	1.2296	0.9836	1.4802	1.2166	0.9829	0.1768
11	0.5741	0.7577	0.9825	0.5644	0.7512	0.9820	0.1848
12	0.8112	0.9007	0.9847	0.8028	0.8960	0.9842	0.1568
13	0.6369	0.7981	0.9860	0.6426	0.8016	0.9855	0.1469
14	0.8566	0.9255	0.9867	0.8437	0.9185	0.9862	0.1565
15	2.1669	1.4720	0.9864	2.1065	1.4514	0.9859	0.1720
16	1.2368	1.1121	0.9868	1.2091	1.0996	0.9863	0.1625
17	0.8228	0.9071	0.9858	0.7953	0.8918	0.9853	0.1548
18	2.5116	1.5848	0.9855	2.5159	1.5861	0.9850	0.175
19	0.9237	0.9611	0.9866	0.8963	0.9467	0.9862	0.1580
20	0.8944	0.9457	0.9867	0.9015	0.9495	0.9862	0.1518

.

The results illustrated in Table 8 and Figure 11 demonstrate that the ANN-15-13-6 model outperformed the others evaluated. A detailed analysis of the model featuring 13 neurons in the hidden layer is presented in

Table 9. Performance evaluation of the model utilizing 13 neurons in the hidden layer.

Neuron Num.	Training				Testing
	MAE	NMAE	RMSE	NRMSE	MAE	NMAE	MAPE	RMSE	NRMSE
13	0.4483	0.0473	0.7981	0.0852	0.4503	0.0476	0.1469	0.8016	0.0857

. Furthermore, Figure 12 depicts the configuration of the proposed ANN model, and Figure 13 provides the training and validation loss (Figure 13a) and the mean absolute error (Figure 13b) for the model over 250 epochs.

6.4. Proposed Model Equations

The proposed model equations for estimating material properties from the structure’s modal parameters, including the associated weights, biases, and transfer functions, are outlined in this section. The ELU is utilized to compute the activation function of each neuron in the hidden layer, as defined below. For each neuron $j$ in the hidden layer,

$$h_j=\mathrm{E}\mathrm{L}\mathrm{U}\left({\sum }_{i=1}^{13}{w}_{ij}{x}_i+b_j\right)$$

(21)

where

• $w_{ij}$ are the weights connecting the $i^{th}$ input to the $j^{th}$ hidden neuron;
• $x_i$ is the $i^{th}$ input feature;
• $b_j$ is the bias term for the $j^{th}$ hidden neuron;
• $\mathrm{E}\mathrm{L}\mathrm{U}\left(z\right)=\left\{\begin{array}{c}x if z>0\\ \alpha \left(e^z-1\right) if z\le 0\end{array}\right.$, where $\alpha$ is a constant (usually set to 1) [49].

The output layer consists of six output neurons with linear activation functions. For each output $k$,

$$y_k={\sum }_j{w}_{jk}^{\left(out\right)}{h}_j+b_k^{\left(out\right)}$$

(22)

where

• $w_{jk}^{\left(out\right)}$ are the weights connecting the $j^{th}$ hidden neuron to the $k^{th}$ output neuron;
• $h_j$ is the output of the $j^{th}$ hidden neuron;
• $b_k^{\left(out\right)}$ is the bias term for the $k^{th}$ output neuron.

Combining these equations, the final equation for the output $y_k$ in terms of the input features $x_i$ is the following:

$$y_k={\sum }_j{w}_{jk}^{\left(out\right)}\mathrm{E}\mathrm{L}\mathrm{U}\left({\sum }_{i=1}^{13}{w}_{ij}{x}_i+b_j\right)+b_k^{\left(out\right)}$$

(23)

Equation (23) illustrates the transformation from the input layer to the output layer for the ANN model. The linking weights and biases for the connections between the input and hidden layers, as well as between the hidden and output layers, are reported in

Table 10. Input-to-hidden layer weights and biases of the ANN-15-13-6 model.

Neuron Num.	Weights															Bias
	${\mathit{w}}_{\mathit{i}\mathit{j}}$
	${\mathbf{N}\mathbf{f}}_1$	${\mathbf{N}\mathbf{f}}_2$	${\mathbf{N}\mathbf{f}}_3$	${\mathbf{M}}_1{\mathbf{F}\mathbf{i}}_1$	${\mathit{M}}_1{\mathit{F}\mathit{i}}_2$	${\mathit{M}}_1{\mathit{F}\mathit{i}}_3$	${\mathit{M}}_1{\mathit{F}\mathit{i}}_4$	${\mathit{M}}_2{\mathit{F}\mathit{i}}_1$	${\mathit{M}}_2{\mathit{F}\mathit{i}}_2$	${\mathit{M}}_2{\mathit{F}\mathit{i}}_3$	${\mathit{M}}_2{\mathit{F}\mathit{i}}_4$	${\mathit{M}}_3{\mathit{F}\mathit{i}}_1$	${\mathit{M}}_3{\mathit{F}\mathit{i}}_2$	${\mathit{M}}_3{\mathit{F}\mathit{i}}_3$	${\mathit{M}}_3{\mathit{F}\mathit{i}}_4$	${\mathit{b}}_{\mathit{j}}$
1	−0.0274	0.1977	0.6722	0.4107	0.2533	0.0623	−0.1863	0.0016	0.0140	−0.0324	0.1745	0.3159	−0.6603	−0.0568	−0.2386	−0.1777
2	−0.2822	0.1086	0.0994	−0.0792	0.1432	0.0236	−0.1568	−0.2866	−0.4225	−0.0921	−0.3875	0.0701	−0.9975	−0.1688	0.4594	−0.3914
3	0.3479	0.6948	−0.1440	−0.0220	−0.6140	0.4101	0.0645	−0.1211	0.4806	0.4230	−0.6743	0.0363	−0.3153	0.1790	−0.0851	0.0613
4	0.3184	0.5989	−0.0427	−0.5481	0.0819	−0.4474	0.1540	−0.5850	−0.4852	−0.1329	0.6927	−0.0234	0.6811	−0.2001	0.1149	0.1914
5	−0.0516	−0.1913	−0.2758	−0.0417	0.2068	0.7112	0.2016	0.5313	0.1342	−0.0674	−0.4149	0.7619	−0.1784	−0.2806	0.3941	−0.2774
6	−0.0210	−0.1086	−0.5050	−0.1433	0.3476	−0.5339	0.1235	−0.0811	0.5381	−0.0392	−0.3970	0.7094	0.2480	−0.1977	0.1611	0.2268
7	0.0970	0.1234	0.9031	−0.3259	0.0825	0.1497	−0.5607	0.3618	−0.0025	−0.0237	−0.4882	−0.1497	0.3826	−0.3789	−0.3585	−0.1374
8	−0.0713	0.1013	0.7304	−0.2703	−0.6110	0.1335	0.5268	−0.3832	0.0353	0.2489	0.0092	−0.7882	0.6593	−0.0416	−0.1109	0.1205
9	0.2764	0.2899	0.5052	0.6084	−0.1122	−0.6000	−0.4883	0.1669	−0.0297	0.2047	0.2370	−0.2584	0.1027	0.2385	−0.3435	−0.2139
10	0.0340	−0.3038	−0.6449	0.1469	−0.4290	−0.2123	0.2416	0.1417	0.3327	−0.1832	0.5040	0.3843	−0.1264	0.4069	0.5993	−0.2157
11	−0.2074	0.5981	0.9059	−0.4574	−0.0519	−0.4599	0.3224	−0.0825	0.3929	−0.5056	0.6014	−0.2542	−0.2932	0.1125	0.0296	0.0423

and

Table 11. Hidden-to-output layer weights and biases of the ANN-15-13-6 model.

Neuron Num.	Weights					Bias
	${\mathit{w}}_{\mathit{j}\mathit{k}}^{\left(\mathit{o}\mathit{u}\mathit{t}\right)}$
						${\mathit{b}}_{\mathit{k}}^{\left(\mathit{o}\mathit{u}\mathit{t}\right)}$
1	−0.0699	0.4695	0.5362	0.2935	−0.6040	−0.2199	0.1767
2	−0.3030	−0.1373	0.0405	−0.2963	−0.4703	−0.5999	−0.0850
3	0.0776	0.0618	−0.5035	0.1684	−0.3281	0.2291	0.1384
4	0.5821	−0.2035	0.7514	−0.1699	0.4618	−0.4248	0.0279
5	−0.2669	−0.4222	−0.3478	0.1831	−0.5054	−0.8905	0.2283
6	0.1233	−0.4897	−0.0419	0.0958	−0.0224	0.4844	0.2391
7	0.0209	−0.0682	0.0592	0.8157	−0.0619	0.1165
8	0.6878	0.6968	−0.0675	0.2528	0.6332	−0.2486
9	−0.2971	0.5339	0.3478	0.1936	−0.1363	0.0690
10	−0.1598	0.0497	−0.2857	−0.5607	−0.3487	0.0146
11	0.3230	0.0684	0.2072	0.0383	−0.3453	0.1130

, respectively.

6.5. GUI in the Python Environment

The proposed model demonstrated high accuracy in predicting material property values, significantly reducing uncertainties in computational models and ensuring that the simulated structural behavior closely aligned with the actual structure. However, the practical implementation of the ANN, particularly in adjusting linking weights and biases, presented considerable challenges. To address these, a GUI was developed in Python, enabling continuous updates and streamlining the model’s application. As illustrated in Figure 14, the GUI facilitates the estimation of material properties, such as the modulus of elasticity and mass density, by processing the desired modal parameters as inputs. Users can input the desired natural frequencies and corresponding mode shapes for the first three vibrational modes in the “Input Parameters” section. The tool subsequently calculates and displays the predicted input for the FEM in the “Prediction Results” section.

6.6. GUI Application with OMA

In this section of the research, the constructed GUI was used to update the model from Section 4 based on the OMA results from the experimental setup in Section 3, ensuring that the model closely aligned with the actual behavior of the structure.

Table 12. Comparison of OMA and FEMU results.

Loc.	OMA Mode Shapes			FEMU Mode Shapes			Natural Frequencies
	Mode 1	Mode 2	Mode 3	Mode 1	Mode 2	Mode 3
Floor 1	0.26	−0.96	1.00	0.302	−0.944	1.00	OMA [Hz]	FEM [Hz]	Discrepancies %
Floor 2	0.65	−0.96	−0.39	0.653	−0.991	−0.467	Mode 1: 2.025	Mode 1: 2.033	Mode 1: 0.40
Floor 3	0.89	0.14	−0.59	0.890	0.154	−0.638	Mode 2: 6.145	Mode 2: 6.110	Mode 2: 0.57
Floor 4	1.00	1.00	0.65	1.00	1.00	0.728	Mode 3: 9.830	Mode 3: 9.890	Mode 3: 0.61

presents the natural frequencies and mode shapes resulting from the model after the FEMU process, and Figure 15 depicts the MAC values corresponding to the identified mode shapes of the updated model compared to the real-world behavior.

The comparison of results from OMA and FEMU, as shown in Table 12 and Figure 15, revealed a strong correlation between the identified eigenvalues and vectors. The discrepancies in natural frequencies for the first three modes were minimal. Additionally, MAC values were near to one along the diagonal and approached zero for the off-diagonal elements. These findings demonstrate the effectiveness of the developed GUI in accurately updating the structural model based on continuous OMA data, thereby enhancing the reliability of the computational model for subsequent analyses.

7. Conclusions

This paper proposes a novel approach for automating FEMU using OMA through an ANN-based GUI. It discusses a denoising method for OMA and a machine learning model aimed at minimizing uncertainties in both experimental setups and computational simulations. The ANN model achieves a prediction precision of 99.85% by utilizing both natural frequencies and mode shapes, allowing for a comprehensive evaluation of the structure’s dynamics. The integrated GUI makes the prediction process for the user efficient and accelerates it. This procedure is applicable to various types of structures, requiring only adjustments to segmentation and desired material properties. It can be scaled for more complex structures by first utilizing machine learning-based sensitivity analysis and then applying advanced DOE methods to optimize the number of iterations needed to generate the dataset for the ANN model. This approach enables its application to more intricate structures, such as large-span bridges or high-rise buildings, with numerous interacting design variables. This is particularly useful for OMA, as it facilitates real-time adjustments and enhances the model’s responsiveness to varying environmental and operational conditions; thus, the developed GUI presents a robust solution.

Looking ahead, this approach has the potential for integration with advanced technologies, such as the IoT, as it employs machine learning to identify complex patterns. This capability enables the continuous feeding of data into the machine learning model through the dynamic evaluation of structures, thereby facilitating the ongoing update of the computational simulation, health monitoring, and automated decision-making processes. Furthermore, it is possible to increase the number of variables representing the dynamic behavior of the structure within the model, enhancing the FEMU with greater detail. Additionally, a dataset could be generated that includes negative perturbations to the material properties, allowing for the simulation of damage scenarios. A machine learning method could then be trained to predict which material property of which element has been perturbed, facilitating the localization of damage. This aspect will be addressed in upcoming research. In addition, the proposed strategy will be implemented in real-world structures, including the Lamberti Bridge in the Parma province of Italy, as well as several other bridges.