Baseline-free damage identification strategy based on FRF-WPPE-LOF framework for multi-source information fusion and sparse field inversion

Abstract

The structural health monitoring (SHM) of large steel box girders often lacks baseline data, making traditional damage detection methods unsuitable for structures in long-term service. To overcome this problem, this study proposes a baseline-free frequency response function (FRF)-wavelet packet permutation entropy (WPPE)-local outlier factor (LOF) damage identification framework that integrates multi-source information fusion theory and sparse field inversion. Firstly, a composite damage indicator was constructed by integrating FRF, WPPE, and LOF, which can highlight non-stationary, frequency sensitive, and edge localization damage characteristics. In order to improve engineering interpretability and spatial robustness, an engineering prior weighting scheme based on stress distribution was introduced in the damage mapping stage, especially for the bending dominant region. Subsequently, a sparse field inversion method was developed by linking the indicators of the sensor domain with the stiffness attenuation at the unit level through weighted optimization. This makes the damage vector interpretable, thereby further deriving quantitative damage depth and severity. The proposed method was validated using a steel box girder finite element model and a triangular impact load induced broadband vibration test under healthy and three types of damage conditions. The results show that this method achieves accurate identification of damage locations, enhances sensitivity to slight and boundary damage, has strong robustness to noise and uncertainty of excitations, and does not require any baseline measurements. Due to these advantages, the proposed framework has great potential for application in large-span bridges or other large civil structures where baselines are difficult to obtain.

Introduction

Structural health monitoring (SHM) is a key technology to ensure the safety and durability of large-span steel structures, such as steel box girder bridges, during their service life. ¹ Large steel structures are prone to cracking or stiffness degradation due to various factors such as fatigue, corrosion, temperature changes, and traffic loads during long-term operation. ² Timely and accurate detection, ³ localization, ⁴ and quantification ⁵ of damage are of great significance for preventing catastrophic failures and extending structural life (such as bridge maintenance, wing structures, ship hulls, etc.).

At present, widely used vibration-based SHM methods, such as frequency response function (FRF), ⁶ modal parameters, ⁷ energy spectrum, ⁸ wavelet analysis, ⁹ and so on, have been extensively studied due to their non-destructive nature and global sensitivity to structural changes. These methods greatly promote the development of damage identification technology. However, most classical methods rely on healthy baseline data—that is, the response of the structure in its intact (undamaged) state prior to any stiffness degradation—for comparative analysis. ¹⁰ This baseline dependent method has many limitations: in many engineering practices, it is difficult to determine or reconstruct the initial health state of the structure; the environment, incentive conditions, and even boundary conditions may change during the service period, which can amplify the errors in the difference calculation, thereby affecting the robustness and repeatability of recognition.^11–13 This limitation has stimulated growing interest in baseline-free damage identification strategies.

Numerous studies have reported different vibration-sensitive features, including frequency-domain indicators, ¹⁴ wavelet or wavelet-packet energy distributions, ¹⁵ entropy-based complexity measures, ¹⁶ and statistical anomaly indices (AIs),^17–20 to enhance damage sensitivity under noisy or non-stationary conditions. However, most existing works focus primarily on feature-level diagnosis, where damage is inferred qualitatively from feature variation or anomaly intensity, without explicitly linking fused indicators to physical damage parameters such as stiffness attenuation or damage severity.

In recent years, multi-source information fusion ²¹ and sparse field inversion techniques ²² have shown great potential in SHM. Multi source fusion can integrate different features such as frequency-domain information (such as FRF ²³ ), time-frequency domain information (such as wavelet/wavelet packet energy^24,25), and statistical complexity (such as entropy ⁴ ), improving sensitivity and robustness to small, complex, or edge damage. Meanwhile, sparse field inversion methods allow for the reconstruction of structural damage distribution (such as thermal maps or anomaly maps) with minimal sensor data, enabling high-resolution localization without the need for large-scale sensor deployment.^13,22

Based on the above background, the main motivation of this study is to propose an FRF-wavelet packet permutation entropy (WPPE)-local outlier factor (LOF) fusion framework that does not require healthy baseline data, fully mining multi-source damage sensitive information, and using sparse field inversion technology to map it to space for accurate positioning and distribution reconstruction. Specifically, by providing frequency domain stability through FRF, extracting sub-band energy through wavelet packet decomposition (WPD), characterizing complexity through Permutation Entropy (PE) or weighted entropy, and detecting local anomalies through Local Outlier Factor (LOF), a fusion index that is sensitive and stable to damage is constructed.

Using sparse field inversion techniques such as inverse distance weighting (IDW) or sparse reconstruction algorithms, the fusion index is mapped into a continuous damage distribution heatmap, achieving high-resolution visualization of the “degree, direction and spatial distribution” of structural damage. By introducing engineering prior information derived from structural stress distribution, the proposed framework provides a physically interpretable mapping from sensor-level indicators to spatial damage distribution. This formulation allows damage localization and severity assessment to be performed within a unified modeling process, while avoiding reliance on healthy baseline measurements.

Rather than introducing new vibration-sensitive features, the contribution of this work lies in:

1. To design and propose a baseline-free FRF-WPPE-LOF damage identification method. The quantitative interpretation of damage severity can be realized without baseline measurement.

2. Implemented multi-source information fusion (combining FRF, acceleration time-frequency sub-band energy, PE/complexity index, and LOF local outlier factor). The non-uniform frequency-domain, time-frequency and statistical characteristics are organized into a unified damage index (DI).

3. Embedding engineering knowledge and sparse constraints into the inversion model with physical significance. And introducing sparse field inversion technology to generate structural damage heat maps, thereby providing qualitative-quantitative-spatial distribution identification capabilities.

The structure of the article is arranged as follows: the second section elaborates on the theoretical basis and methods used, including FRF, WPD, PE, LOF, and sparse field inversion; the third section introduces numerical simulation analysis, including finite element model, excitation and damage settings, simulation results and identification performance; The fourth part conducts sensitivity analysis, including the influence of different excitation loads and material parameters on identification performance; the fifth section conducts physical experiment verification, describing the experimental setup, data processing, application framework, and result comparison; the sixth section is a comparative discussion that conducts large-scale comparative analysis by expanding the sample; the seventh section summarizes the research results and looks forward to future expansion directions.

Methodology

Theoretical background

The baseline-free damage identification approach used in this study is based on several complementary and non-redundant frequency-domain, time-frequency/statistical features, and spatial reconstruction methods, mainly including FRF, WPD, PE, local outlier factor (LOF), and sparse field inversion. In recent years, methodological reviews have demonstrated the effectiveness of these techniques in SHM, providing a solid theoretical and practical basis for multi-source fusion when their respective information domains are appropriately distinguished and integrated.^26–28

FRF can be used to describe the input-output relationship of structures in the frequency domain. For a given excitation-response, the definition of FRF is $H\left(\omega \right)=\frac{X\left(\omega \right)}{F\left(\omega \right)}$ . Where, $\omega$ represents angular frequency. $X\left(\omega \right)$ and $F\left(\omega \right)$ represent the Fourier spectra of structural response and excitation, respectively. ²⁹

Due to its inherent robustness to unknown or unstable excitations by indirectly normalizing the excitation effects through frequency-domain transfer characteristics, it is widely used in monitoring scenarios without baseline or uncontrollable field excitations.

In this study, FRF is not directly used as a single damage indicator, but rather as a dynamic transfer representation with practical significance, which establishes a connection between the measured vibration signal and subsequent feature extraction. Recent comments and technological advancements on the FRF method have systematically summarized its advantages, estimation methods, and application limitations in complex operating conditions.^14,29 However, if the damage only results in slight or highly localized stiffness reduction, the amplitude or phase changes of the original FRF may be subtle, especially in the high frequency range. Therefore, relying solely on FRFs may be difficult to effectively distinguish damage situations, and further transformation into more sensitive feature representations is needed.

WPD can provide tree-based multi-resolution decomposition, which can obtain the energy distribution of signal sub-bands on uniform sub-bands. ²⁴ Given a response signal $x(t)$ , WPD decomposes it into $B=2^J$ sub-band signals $x_b(t)$ at decomposition level $J$ , where $J$ is selected based on frequency resolution requirements and sampling frequency. The energy of each sub-band is calculated as $E_b=\sum _{n=1}^{N_b}{|x_b(n)|}^2$ , where, $N_b$ represents the number of samples in the $b$ th sub-band.

WPD has been widely used as a multi-scale damage description index and has demonstrated effectiveness in various SHM and fault diagnosis applications.^16,26 However, purely energy-based sub-band features may exhibit sensitivity limitations under strong non-stationary excitation or extremely weak damage scenarios. Therefore, a complexity-based complementarity measurement method is introduced to address this issue.

PE measures complexity by analyzing the order patterns of local amplitude arrangements in time series, which is computationally simple and robust to noise. For a given sub-band signal $x_b(t)$ , PE is computed by embedding the signal with dimension m and time delay $\tau$ , where m and $\tau$ are selected following common SHM practice (typically m = 4 – 6, τ = 1). ¹⁶ The PE of the $b$ th sub-band is defined as ${\mathrm{PE}}_b=-\sum _{k=1}^{m!}{p}_k\ln \left(p_k\right)$ . Where $p_k$ is the probability of the $k$ th sequential mode. ⁴ PE has been shown to be highly sensitive to small interference and early damage while maintaining computational efficiency and robustness to noise. In order to enhance spectral selectivity, PE is calculated on the WPD sub-band rather than on the original signal, allowing local frequency sensitive complexity characterization. ¹⁶

Based on the quantification of sample “isolation degree” based on local density differences, LOF exhibits considerable ability in uneven sensing layout, edge damage, and online anomaly detection. In recent years, comparisons and engineering applications of LOF with other unsupervised or semi-supervised anomaly detection techniques have also shown that LOF is particularly sensitive to local/edge damage when features are separable.^27,28 However, the discriminative effect of LOF is highly dependent on the quality of input features. This is also one of the reasons for coupling LOF with high-sensitivity WPD–PE-based descriptors.

Sparse field inversion and reconstruction methods include interpolation, Kriging models, and sparse regularization. It allows for approximate reconstruction of the damage field under conditions of sparse or irregular arrangement of measurement points. In recent years, there have been various studies combining sparse priors or multi strategy mixed sparse reconstruction to improve imaging resolution and robustness.^28,30

In summary, these methods are complementary in the information dimension, so they have great potential to integrate to form a non-redundant, multi-source fusion framework, which is suitable for damage identification without baseline and sparse measurements.

Integrated FRF-WPPE framework

To achieve baseline-free structural damage identification, this study proposes an integrated FRF-WPPE-LOF framework. The core motivation of this framework is to fully utilize the complementary advantages of various methods:

FRF provides stable and robust frequency-domain features against excitation uncertainty. WPD can achieve multi-scale frequency band sensitivity analysis. PE is highly sensitive to nonlinear complexity and small disturbances. In addition, the spatial variation of the reconstructed anomaly field is used to provide qualitative information about the spatial distribution trend of anomalies related to damage.

Through multi-source information fusion, the “degree” of damage can be quantitatively evaluated, while eliminating dependence on the health baseline, thus constructing the weighted WPPE (WWPPE) index^23–26 to achieve sensitive detection of minor injuries. LOF is used for local anomaly detection, which enhances the recognition ability of edge damage by analyzing the spatial density differences of WEE indicators.^4,27 The local anomaly factors output by LOF can be directly combined with sparse field inversion to generate thermal maps of continuous structural damage.

Based on the weighted AI (WWPPE × LOF), a sparse damage vector is constructed, and the sparse field inversion method is used to reconstruct the continuous damage distribution on the entire structural grid. By interpolation and spatial sparse reconstruction, high-resolution thermal maps of damage can be generated, providing a basis for subsequent analysis of damage direction and degree.^28,29

The specific implementation process of this framework is shown in Figure 1, which can be mainly divided into the following seven steps:

• Step 1: Obtain structural response and calculate FRF. Each measurement point was excited by repeated low-energy instrumented hammer impacts (5–10 repetitions) to improve signal repeatability and reduce random noise.

• Step 2: Perform WPD decomposition on FRF signals.

• Step 3: Calculate the energy and PE of each sub-band.

• Step 4: Construct a weighted energy entropy (WEE) index by multiplying the energy of the wavelet packet with the PE ( ${\mathrm{WEE}}_b=E_b\times {\mathrm{PE}}_b$ ).

• Step 5: Damage “direction” discrimination: abnormal field gradient direction (based on thermal map).

Figure 1.

Flow chart of framework.

After interpolating the final $\mathrm{AI}(x,y)$ into the grid, calculate the gradient vector $\nabla \mathrm{AI}$ as shown in Equation (1). ³¹

$$\begin{array}{l}\nabla \mathrm{AI}=\left[\frac{\partial \mathrm{AI}}{\partial x},\frac{\partial \mathrm{AI}}{\partial y}\right]=\left[\frac{\partial I\left({\mathrm{WEE}}_b\right)}{\partial x},\frac{\partial I\left({\mathrm{WEE}}_b\right)}{\partial y}\right]\\ =\lfloor \frac{\partial \frac{{\sum }_{b=1}^M\frac{{\mathrm{WEE}}_b}{{\left(r_b\right)}^p}}{{\sum }_{b=1}^M\frac{1}{{\left(r_b\right)}^p}}}{\partial x},\frac{\frac{{\sum }_{b=1}^M\frac{{\mathrm{WEE}}_b}{{\left(r_b\right)}^p}}{{\sum }_{b=1}^M\frac{1}{{\left(r_b\right)}^p}}}{\partial y}\rfloor \end{array}$$

(1)

where, $I(.)$ represents the IDW interpolation. $r_b$ is the Euler distance between the interpolation point and the $b$ th sensor, and $p$  = 2. This equation explicitly links the gradient operation to the WEE-derived anomaly field. The direction corresponding to the maximum gradient amplitude represents the direction with the fastest increase in anomaly intensity. In this study, gradient-based direction was used as a qualitative indicator to assist in explaining the damage, known as the “damage direction.”

• Step 6: Quantify the degree of damage (quantitative). The core of this step is to set the scalar “damage index” (DI) and calibration curve.

Construct a single scalar DI (the larger the indicator, the more severe it is), and calibrate it with simulation samples (different crack depths or stiffness reduction ratios) or a small number of known labels.

The four types of DI used in this article are as follows:

• (1) DI1 represents the top-k mean outlier score, calculated based on the characteristic LOF value of each measurement point. ³² Specifically, for the $s\mathrm{th}$ measurement point, the FRF signal is decomposed using WPD, and the corresponding feature vectors are constructed using the mapping operator $\mathrm{\Phi }({\mathrm{WEE}}_{s,b})$ . Then calculate the LOF values $X_s$ in the feature space of the transformation to quantify the local density deviation of each sub-band component. Subsequently, select the top-k highest LOF value and take its average as DI1, as shown in Equation (2) ³³ :

  $$\begin{array}{c}\multicolumn{1}{c}{\mathrm{DI}{1}_s=\frac{1}{k}\sum _{i=1}^k{\mathrm{LOF}}_{\left(i\right)}(X_s)=\frac{1}{k}\sum _{i=1}^k{\mathrm{LOF}}_{(i)}}\\ \multicolumn{1}{c}{(\mathrm{\Phi }({\mathrm{WEE}}_{s,b}))}\end{array}$$

  (2)

where, $\mathrm{L}{\mathrm{OF}}_{\left(i\right)}\left(X_s\right)={\mathrm{LOF}}_{\left(i\right)}(\mathrm{\Phi }({\mathrm{WEE}}_{s,b}))$ represents the $i$ th maximum LOF value within the WPP-derived feature set of the $s\mathrm{th}$ measurement point, and $k$ is the selected number of top outliers.

• (2) DI2 can capture the robust deviation of sub-band energy relative to the median across sensors, emphasizing local anomalies in the energy distribution, which can be calculated through Equation (3). ³⁴

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{DI}{2}_s=\sum _b{W}_b·\left|z_{s,b}\right|=\sum _b{W}_b·}\\ \multicolumn{1}{c}{\frac{{\mathrm{WEE}}_{s,b}-\mathrm{median}\left({\mathrm{WEE}}_b\right)}{\mathrm{MAD}\left({\mathrm{WEE}}_b\right)}}\end{array}$$

(3)

where, $W_b$ represents the weight of the sub-band b (from step 4), and $z_{s,b}$ is the robust z-score of sensor s at sub-band b. $\mathrm{MAD}(.)$ is the median absolute deviation across sensors. This indicator links the characteristics of ${\mathrm{WEE}}_b$ of multi-frequency bands with a scalar indicator, highlighting the sub-bands that contribute the most to abnormal energy patterns.

• (3) DI3 is the weighted peak/variance of phase group time delay. Let ${\tau }_{s,b}$ represent the phase group delay of sensor s in sub-band b. The weighted combination of peak and variance of ${\tau }_{s,b}$ can be calculated using Equation (4) ³⁵ :

  $$\begin{array}{c}\multicolumn{1}{c}{\mathrm{DI}{3}_s=\alpha ·\mathrm{peak}\left({\tau }_{s,b}\right)+\beta ·\mathrm{var}\left({\tau }_{s,b}\right){\tau }_{s,b}}\\ \multicolumn{1}{c}{=-\frac{\mathrm{d}}{\mathrm{d}\omega }\arg \left(H_{s,b}\left(\omega \right)\right)}\end{array}$$

  (4)

where, the coefficients $\alpha$ and $\beta$ are non-negative weighting factors that satisfy $\alpha +\beta =1$ . $\alpha$ emphasizes localized sharp phase shift, usually related to sudden stiffness reduction or crack initiation. $\beta$ reflects the overall phase instability of the entire frequency band, indicating the distributed structural degradation. $H_{s,b}\left(\omega \right)$ is the FRF of the $b\mathrm{th}$ sub-band, and $\arg \left(H_{s,b}\left(\omega \right)\right)$ represents the phase of $H_{s,b}\left(\omega \right)$ . DI3 emphasizes frequency bands with special frequency characteristics (i.e., phase characteristics), which complement DI1 and DI2 and can capture abnormal situations caused by local stiffness changes.

• (4) DI4 represents the average anomaly amplitude of the transmission rate of probability energy density (PED) for sensor $s$ across sub-bands, which can be calculated using Equation (5). ³⁶

  $$\mathrm{DI}{4}_s=\mathrm{mean}\left(|{\mathrm{PED}}_{s,b}-\mathrm{median}\left({\mathrm{PED}}_{s,b}\right)|\right)$$

  (5)

DI4 characterizes the energy transfer changes between sensors, providing additional sensitivity for changes caused by damage in dynamic response

To ensure comparability between different DIs at different scales, normalization is used to unify all DIs into the range of [0,1], as shown in Equation (6):

$$\mathrm{DI}{i}_s^{\mathrm{norm}}=\frac{\mathrm{DI}{i}_s-\min \left(\mathrm{DI}{i}_s\right)}{\max \left(\mathrm{DI}{i}_s\right)-\min \left(\mathrm{DI}{i}_s\right)},i=1,2,3,4$$

(6)

Isotonic (order-preserving) regression is applied to fit the damage depth as a monotonic function of the normalized DIs [DI1, DI2, …]. This calibration is performed on a set of simulated or labeled damage cases, without relying on baseline or undamaged state measurements of the monitored structure. This regression emphasizes that without assuming a specific linear relationship, the estimated depth of damage will not decrease with an increase in damage indicators. By bootstrap resampling of the calibration data, confidence intervals for the fitted monotonic relationship could be obtained to quantify the uncertainty of the regression fit. ³⁷

It should be noted that monotonic regression is only used for calibrating the severity of damage and interpreting the results, and does not participate in subsequent damage detection, fusion, or localization processes.

Subsequently, as shown in Equation (7), the normalized DI was directly subjected to multi-source damage fusion using a weighted average strategy. ²¹ The purpose of determining fusion weights offline from simulated datasets through cross validation is to enhance the monotonic consistency and stability of fusion indicators in ranking damage severity, rather than fitting absolute damage values.

This weight calibration is performed at the method level and does not rely on baseline or undamaged state data of the monitored structure. Once determined, the weights are fixed and uniformly applied to all test cases without further adjustment.

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{DI}}_s^{\mathrm{multi}}=q_1·\mathrm{DI}{1}_s^{\mathrm{norm}}+q_2·{\mathrm{DI}2}_s^{\mathrm{norm}}+q_3·\mathrm{DI}{3}_s^{\mathrm{norm}}}\\ \multicolumn{1}{c}{+q_4·\mathrm{DI}{4}_s^{\mathrm{norm}},\sum _i{q}_i=1}\end{array}$$

(7)

To reconstruct a continuous damage indicator field from discrete sensor-based values, IDW interpolation is adopted. For any spatial point i, the interpolated DI $d_i^{\mathrm{DI}}$ is calculated by weighted average of multi-source DI values measured at all m sensor positions. The weight is inversely proportional to the distance between the interpolation point and each sensor. Interpolation is represented as Equation (8), where the allocation to the $s\mathrm{th}$ sensor depends on the Euclidean distance $r_{\mathrm{is}}$ and the distance attenuation parameter $p$ . In this study, the distance attenuation parameter was set to p = 2. The normalization process is shown in Equation (9). ²¹

$$d_i^{\mathrm{DI}}=\frac{\sum _{s=1}^m{q}_{\mathrm{is}}{\mathrm{DI}}_s^{\mathrm{multi}}}{\sum _{s=1}^m{q}_{\mathrm{is}}},{\omega }_{\mathrm{is}}=\frac{1}{{\left(r_{\mathrm{is}}\right)}^p}$$

(8)

$$d_{i,\mathrm{norm}}^{\mathrm{DI}}=\frac{d_i^{\mathrm{DI}}-\min \left(d_i^{\mathrm{DI}}\right)}{\max \left(d_i^{\mathrm{DI}}\right)-\min \left(d_i^{\mathrm{DI}}\right)}$$

(9)

Due to the clear stress distribution characteristics of beam structures under bending loads (the maximum bending moment region is more prone to crack formation and stiffness reduction), prior engineering weight specifications are introduced based purely on damage inversion results from detection data. This assumption is consistent with the classical beam theory ³⁸ and has been widely adopted in SHM as a prior condition based on physical principles to improve spatial interpretability. ³⁹ To enhance the engineering rationality of the damage map, the “engineering experience vector” ${\mathit{d}}^{\mathbf{eng}}$ , which combines practical experience, is introduced. The specific calculation is shown in Equation (10):

$$d_i^{\mathrm{eng}}=\{\begin{array}{c}\multicolumn{1}{c}{0.6~0.8,\mathrm{High}\mathrm{bending}\mathrm{moment}/\mathrm{stress}\mathrm{zone}}\\ \multicolumn{1}{c}{0.3~0.5,\mathrm{Medium}\mathrm{stress}\mathrm{zone}}\\ \multicolumn{1}{c}{0~0.3,\mathrm{low}\mathrm{stress}\mathrm{zone}}\end{array}$$

(10)

The determination of zoning is based on the theoretical bending moment distribution of the beam under impact excitation, which ensures consistency with the dynamic response characteristics determined in FRF analysis (third section).

Based on prior engineering experience, the sensor-based damage inversion map is linearly weighted^40,41 as shown in Equation (11).

$${\mathit{d}}^{\mathbf{pr}\mathbf{ior}}=\gamma ·{\mathit{d}}^{\mathbf{DI}}+(1-\gamma )·{\mathit{d}}^{\mathbf{eng}},0<\gamma <1$$

(11)

Where, $\gamma$ is the weighted coefficient that controls the relative contribution of the sensor driven DI ${\mathit{d}}^{\mathbf{DI}}$ and the engineering experience index ${\mathit{d}}^{\mathbf{eng}}$ . In this study, $\gamma$ was selected based on experience to balance data-driven sensitivity and engineering rationality (unless otherwise specified, $\gamma$  = 0.7).

• Step 7: Sparse field inversion (with prior “damage vector”)

After obtaining continuous damage distribution using multi-source fusion indicators, this study further constructed a sparse damage inversion model based on structural dynamic response sensitivity to quantify the degree of damage in each region. It should be clarified that in this study, the term “baseline-free” refers to the absence of reference data corresponding to the undamaged structural state. There is no need to measure in a healthy state beforehand.

This inversion process takes into account the following three points: (1) Dynamic response variation law (model-driven ⁴² ). (2) Abnormal information inspired by sensors (data-driven ³² ). (3) Knowledge-driven engineering of structural stress laws. ⁴¹ The numerical model is only used to estimate response sensitivity and provide physical regularization for the inversion process, rather than as a baseline reference for damage comparison. These components together form the model-data-knowledge triple fusion inversion framework.

When a structure experiences local damage, the attenuation of its local stiffness can lead to changes in the measured dynamic response. The FRF at each measurement point, denoted as $H_i(\omega )$ , is first transformed into a finite dimensional FRF-based eigenvector $f_i$ by frequency-domain aggregation within a predefined frequency range.

Specifically, for each FRF Hi (ω), the signal will be decomposed into B sub-bands through WPD.^8,11 For each sub-band, calculate its energy $E_b$ and PE ${\mathrm{PE}}_b,$ ^16,25 and obtain a WEE index $f_{i,b}=$ $E_b·{\mathrm{PE}}_b$ . These indices of all sub-bands are connected to form the eigenvector of the nth measurement point: ${\mathit{f}}_i={[f_{i,1},f_{i,2},\dots ,f_{i,B}]}^T$ . Subsequently, the eigenvectors of all N measurement points are concatenated to form a eigenvector based on the global FRF: $\mathit{f}={[{\mathit{f}}_{\mathit{1}}^T,{\mathit{f}}_{\mathit{2}}^T,\dots ,{\mathit{f}}_{\mathit{B}}^T]}^T$ .

The local stiffness attenuation d will cause changes in the eigenvectors based on the FRF, which is defined as $\mathbf{\Delta }\mathit{f}$ : $\mathbf{\Delta }\mathit{f}={\mathit{f}}^d-{\mathit{f}}^0$ . Where, ${\mathit{f}}^0$ and ${\mathit{f}}^d$ represent the eigenvectors of the structure in the undamaged and damaged states, respectively.

The sensitivity of characteristic vectors based on FRFs to local stiffness attenuation can be represented by the matrix ${\mathit{S}}_{\mathit{f}}$ , which is calculated through finite element numerical perturbation: ${\mathit{S}}_{\mathit{f}}=\frac{\partial \mathit{f}}{\partial \mathit{d}}$ . And the first-order approximate relationship between local stiffness attenuation and feature changes is expressed as shown in Equation (12). ⁴²

$$\mathbf{\Delta }\mathit{f}\approx {\mathit{S}}_{\mathit{f}}\mathit{d}$$

(12)

Where, $\mathit{d}$ is the local stiffness attenuation vector of the structural unit. Due to the local sparsity of structural damage, significant stiffness attenuation only occurs in a small number of elements, therefore sparse inversion is applied to solve the damage vector $\mathit{d}$ using the observed $\mathbf{\Delta }\mathit{f}$ and sensitivity matrix ${\mathit{S}}_{\mathit{f}}$ .^13,28

In addition, anomalous patterns in the feature vectors (such as high LOF scores from DI1, DI2, etc.) help guide the inversion process by highlighting potential damaged areas to ensure that both model-driven sensitivity and data-driven anomalous information are considered.

To avoid the influence of noise, non-uniqueness, and pathological problems on inversion,^40,41 the engineering prior ${\mathit{d}}^{\mathbf{prior}}$ is introduced as the initial and weight constraints for inversion. The Local stiffness attention vector in the inversion model is shown in Equation (13), using ${\mathit{d}}^{\mathbf{prior}}$ as initial values, as shown in Equation (14).

$${\mathit{d}}^{\ast }=\arg \underset{d\ge 0}{\min }{\Vert \mathit{\Delta }\mathit{f}-\mathit{S}\mathit{d}\Vert }_2^2+\lambda \sum _{i=1}^n(1-d_i^{\mathrm{prior}})\left|d_i\right|$$

(13)

$${\mathit{d}}^{(0)}={\mathit{d}}^{\mathbf{prior}}$$

(14)

Where, $\Vert \mathbf{\Delta }\mathit{f}-\mathit{Sd}{\Vert }_2^2$ ensures that the inversion results can approximate the real dynamic changes (model-driven ⁴² ). $(1-d_i^{\mathrm{prior}})\left|d_i\right|$ represents sparse regularization and prior weighting. When the engineering prior considers that a region with high stress and damage probability will result in a large ${\mathit{d}}^{\mathbf{prior}}$ , the corresponding penalty term is small, making the abnormal signal information in that region easier to identify as damage. Conversely, when the engineering prior considers that a region with low stress and damage probability will result in a large ${\mathit{d}}^{\mathbf{prior}}$ , the penalty term is large, thus avoiding false damage caused by noise. The regularization parameter $\lambda$ guides the trade-off between control data fidelity and sparsity through engineering priors. A too small $\lambda$ may lead to noise driven false damage, while a too large $\lambda$ may excessively suppress weak but real damage. In this study, $\lambda$ was selected through a grid-based cross-validation program that uses a limited set of numerical simulation samples with known damage locations and severity. ⁴³ The candidate values for $\lambda$ are selected within the range of $[{10}^{-3},{10}^{-1}]$ . For each candidate object, the inversion performance is evaluated based on damage localization accuracy, sparse consistency, and robustness to noise. The subsequent analysis will use the $\lambda$ value that provides the best overall balance. Non-negative constraint d ≥ 0 ensures that the damage can only attenuate stiffness and cannot “enhance” stiffness.

This is an L1 sparse inversion model with structural prior, which is more robust and in line with engineering practice than traditional inversion methods that use FRF sensitivity.

To enhance the sparsity and interpretability of damage identification results in engineering sense, this article further maps the obtained dimensionless DI $d_i^{*}$ (between 0 and 1) to more physically meaningful damage quantification values. This process includes two aspects: damage depth estimation and damage level classification.

• (1) Mapping to damage depth:

Since $d_i^{*}$ represents the normalization degree of relative damage in the ith region, it does not directly have an engineering unit. Therefore, in order to convert the recognition results into dimensional parameters that can be used for structural evaluation, an empirical mapping relationship, as shown in Equation (15) is adopted:

$${\mathrm{depth}}_i\approx k·d_i^{*}$$

(15)

Where, ${\mathrm{depth}}_i$ represents the estimated depth of damage in the ith region (such as the local weakening ratio of the steel plate, the effective stiffness reduction ratio of the section, etc.). The coefficient $k$ is a structure-related scaling factor determined offline through parameter finite element sensitivity analysis, ⁴² without the need for any baseline or reference measurements of the monitored structure. It should be noted that $k$ is only used as a post-processing step for interpreting the results and does not affect the damage identification or sparse inversion process.

This mapping enables abstract entropy class indicators to be transformed into quantifiable parameters with physical meaning. At the same time, due to the sparsity of $d_i^{*}$ itself (only the damaged area approaches 1, while the rest of the area approaches 0), the resulting damage depth matrix also has interpretability and local significance.

• (2) Mapping to level:

To further improve the engineering readability of recognition results and facilitate manual inspection and practical operation and maintenance decisions, this article divides $d_i^{*}$ into the following four levels based on its distribution pattern and sensitivity to structural performance degradation, as shown in Table 1.

Table 1.

Mapping level of damage.

Damage scales	Healthy	Slight	Medium	Serious
$d_i^{*}$	0–0.25	0.25–0.50	0.50–0.75	0.75–1.0

This threshold based classification is commonly used for state assessment and visualization in SHM practices. ⁴⁴ Discretization enhances the interpretability of generating damage heat maps based on IDW, highlighting key regions while preserving the sparse nature of inversion results.

Numerical simulation

Finite element model

This study uses ABAQUS to establish a finite element model of steel box girder to simulate the dynamic response characteristics under different damage states. The geometric dimensions of the steel box girder are shown in Figure 2, with a thin-walled enclosed rectangular section of 100 × 50 mm and a total length of 3 m. To ensure the reliability of end constraints, a fixed section of 157.5 mm is reserved on both sides of the beam end. The material is made of structural steel S355, with an elastic modulus of E = 210 GPa, a density of 7.84 g/cm³, and a Poisson’s ratio of 0.3. To improve the comparability between numerical simulation and laboratory beam testing, the virtual model adopts a modal damping ratio $\xi$ of 1.6%, which is consistent with the damping level identified in the subsequent experimental study.

Figure 2.

Finite element model of steel box girder.

The virtual model is built using three-dimensional solid elements (C3D8R, eight-node linear brick elements with simplified integration) in ABAQUS. Although the steel box girder studied is a thin-walled structure, solid elements are intentionally used to better capture the local stiffness degradation and strain concentration effects related to damage, as well as the high-frequency dynamic response involved in FRF-based analysis. Discretizing the cross-sectional thickness using three-layer solid elements was found to be sufficient to represent equivalent stiffness attenuation and dynamic response changes while maintaining computational efficiency. Along the length of the beam, the mesh was appropriately refined in potential damage areas to enhance sensitivity to local stiffness changes. Apply completely fixed boundary conditions at the constraint end.

As shown in Figure 3, three damage conditions (DC1–DC3) were set up to evaluate the impact of different degrees and combinations of damage on the structural dynamic response:

1. DC1: Slight single damage with dimensions of 20 × 2 × 1 mm, belonging to non-penetrating local weakening.

2. DC2: Severe single injury with dimensions of 20 × 2 × 2 mm, resulting in local penetrating weakening.

3. DC3: Double damage superposition condition, with two penetrating damages located at beam coordinates (1052.5, 50, and 50 mm) and (2385, 75, and 50 mm) respectively, simulating complex scenarios with multiple damages.

Figure 3.

Damage conditions.

The coordinates in the figure correspond to the geometric center of the top surface of the virtual damage area. The coordinates of the center point are used to ensure the reproducibility of the damage location in the numerical modeling and experimental device.

Specifically, the above-mentioned damage scenarios were created with local notches in the solid model using geometric cutting operations in ABAQUS, resulting in explicit cross-sectional reduction and corresponding local stiffness degradation. It ensures that the damage is represented as a physically interpretable geometric stiffness defect. All models are equipped with virtual accelerometers to obtain time-domain response signals under impact excitation for subsequent FRF-WPD-PE fusion analysis.

Load conditions

This study uses transient impact excitation to excite the wideband dynamic response of steel box girders. As shown in Figure 4(a), the external load adopts a typical triangular pulse form, with a duration of 0.003 s and a peak force amplitude of 70 N, and is applied to the top surface of the beam shown in Figure 4(b).

Figure 4.

Triangular impact load: (a) load amplitude and (b) location of load.

Previous studies have shown that short-term shocks can excite rich mid to high frequency modal information in structures, which is beneficial for amplifying the disturbance of local damage in frequency-domain characteristics, thereby improving the sensitivity of FRF and wavelet entropy indicators (such as the high-frequency sub bands obtained from WPD decomposition being more sensitive to subtle damage). In addition, compared with continuous harmonic excitation or sweep frequency signals, impulse loading is easier to implement, more stable, and can avoid boundary nonlinear errors caused by long-term excitation. Therefore, it is very suitable as an input excitation for baseline-free damage identification frameworks.

In order to obtain the spatial dynamic response of the structure under impact conditions, 14 virtual accelerometers were arranged along the length direction on both sides of the beam, as shown in Figure 5. This type of layout covers the mid span area and the transition sections at both ends, and can simultaneously capture the overall mode changes and response disturbances caused by local damage. The symmetrical arrangement of accelerometers on both sides of the beam helps to improve the robustness of anomaly identification and provides bilateral gradient information for subsequent sparse field inversion, making the reconstruction of damage heat maps more stable. This layout strategy has been verified to have high repeatability and accuracy in previous research on impact load damage identification.

Figure 5.

Position of virtual accelerometers.

All measurement points record the full time-domain acceleration response after the impact load is applied, which is used for subsequent calculation of FRF, execution of WPD decomposition, and PE feature extraction, providing basic data for the FRF-WPPE-LOF multi-source fusion framework.

Simulation results and analysis

To verify the applicability of the proposed FRF-WPPE-LOF fusion framework under different DCs and its robustness to noise and excitation uncertainties, this section conducts simulation analysis based on three types of DCs: DC1, DC2, and DC3.

To simulate harsh detection scenarios, environmental noise with a signal-to-noise ratio (SNR) of 10 dB was added to the original time-domain response of the virtual accelerometer, as shown in Figure 6. Subsequently, FRF was extracted from the noisy signal to obtain the frequency-domain response as shown in Figure 7. The corresponding structural modal shape is shown in Figure 8. It can be observed that although noise significantly increases background fluctuations, the main modal peaks are still clear and recognizable, laying the foundation for subsequent WPD and PE feature extraction.

Figure 6.

Acceleration response in time domain (sensor 7): (a) DC2 and (b) DC3. DC: damage condition.

Figure 7.

FRF of sensor 4 (SNR = 10 db): (a) original FRF, (b) DC2, and (c) DC3. FRF: frequency response function; DC: damage condition.

Figure 8.

Model shape and frequency: (a) model 1 (50.05 Hz), (b) model 2 (135.49 Hz), (c) model 3 (260.02 Hz), and (d) model 4 (418.27 Hz).

Based on the integrated framework proposed in “Integrated FRF-WPPE framework” section, multi-source feature indicators corresponding to three types of DCs were fused, and sparse field inversion was further performed to generate a damage heatmap (Figure 9). Specifically, the responses obtained from FRF-based frequency-domain interference analysis, WPD sub-band energy variation, WEE, and LOF based anomaly detection all exhibit low amplitude but distinguishable local mutations under slight damage (DC1). For more severe damage cases (DC2 and DC3), these indicators show more significant changes, and a high degree of consistency is observed between these single indicator-based analysis methods, indicating that the proposed fusion strategy has strong feature complementarity.

Figure 9.

Thermal map of damage: (a) DC1, (b) DC2, and (c) DC3. DC: damage condition.

Figure 9(a) shows that when the damage is mild, non-penetrating weakening (DC1), the peak value of the inverted DI falls within the range of 0.25–0.50, with a maximum value of about 0.35, indicating mild damage. The peak position is located about 1 m away from the end of the beam, which is consistent with the height of the preset damage location. The abnormal amplitudes obtained by sensors 5 and 6 along the width direction (Y direction) of the beam are close, which accurately locates the damage near the midpoint of the section, verifying the spatial sensitivity of the frame in mild damage scenarios.

In the severe penetration weakening (DC2) shown in Figure 9(b), the peak value of the reconstructed heatmap exceeds 0.75, significantly higher than the minor damage benchmark, indicating a clear determination of severe damage. The positioning results also accurately correspond to the preset damage coordinates, proving the high-resolution ability of the WEE index and LOF anomaly factor in strong damage feature extraction.

For the double damage superposition condition (DC3), as shown in Figure 9(c), this method accurately identified two independent peak positions of damage, and both peaks were close to 0.8, successfully distinguishing the spatial distribution and mutual independence of damage. This result indicates that sparse field inversion has good discriminability in dealing with complex scenes with multiple damages, without the occurrence of hotspot merging or misjudgment.

In addition, as shown in Figure 9, the LOF anomaly score exhibits significant edge sensitivity, especially under DC3 (Figure 9(c)), which highlights local outliers near the damage boundary. And these outliers are not obvious in the WEE response. ¹⁶ When combined with entropy-based indicators, this complementary behavior enhances the detectability of edge damage. Overall, the FRF-WPPE-LOF fusion framework maintained stable damage localization performance under noisy conditions, and different indicators showed consistent spatial damage trends.

It should be pointed out that the analysis in this section is based on a single excitation condition and a limited damage sample size. Although the preliminary results have verified the robustness of the framework under noise and excitation uncertainty, its statistical stability still needs to be further evaluated in the fifth section through more expanded samples, including different noise intensities, random excitation distributions, and more complex damage combinations, to comprehensively verify the generalization ability and practical application potential of the method.

Sensitivity analysis

To evaluate the robustness and generalization ability of the proposed framework under real-world uncertainty conditions, sensitivity analysis considering changes in incentive conditions and material properties was adopted. These changes simulate the possible degradation situations that may occur in practical applications.

Localization error (LE), damage magnitude error (DME), and sparsity consistency index (SCI) are used to quantitatively evaluate the stability of identification under parameter changes. The specific calculation process is shown in Equations (16)–(18).

$$\mathrm{LE}=\frac{|N_{\mathrm{pred}}-N_{\mathrm{true}}|}{N}$$

(16)

$$\mathrm{DME}=\frac{\Vert D^{*}-D^{\mathrm{true}}{\Vert }_2}{\Vert D^{\mathrm{true}}{\Vert }_2}$$

(17)

$$\mathrm{SCI}=\frac{N_{\mathrm{ci}}}{N_{\mathrm{true}}}$$

(18)

where, $N_{\mathrm{true}}$ is the actual number of damaged elements, $N_{\mathrm{pred}}$ is the predicted number of damaged elements, $N$ is the total number of elements, and $N_{\mathrm{ci}}$ is the correctly identified number of damaged elements. $D^{*}$ is the estimated damage magnitude vector, which is obtained from a sparse inversion model. $D^{\mathrm{true}}$ is the ground truth damage vector, which is manually set in numerical simulations and is only non-zero at the actual damaged elements.

Different impulse forces

To evaluate the robustness of the proposed FRF-WPPE-LOF framework under changes in excitation characteristics, sensitivity analysis was conducted by varying the impact duration while maintaining a constant peak force.

The DC of case baseline $I_0$ is consistent with DC2 shown in Figure 3, and the excitation load and its location are shown in Figure 4. The reason for using this case as a control group is that it represents a moderate shock duration that provides sufficient broadband frequency excitation. Used for the main numerical simulation results (“Simulation results and analysis” section) to ensure consistency. It also provides stable recognition performance, thus providing a reliable reference for robustness comparison.

Variation in the duration of the impact

Sensitivity analysis was conducted by changing the duration of the impact while maintaining a constant peak force. In practical engineering scenarios, the duration of impact loads may vary depending on the collision object, contact stiffness, or boundary constraints. These changes will affect the frequency components of the excitation and may affect the damage sensitive features extracted from the acceleration response. The adopted cases are shown in Table 2.

Table 2.

Different duration of the impact.

Case	Duration of the impact	Force peak (N)
$I_0$	$2\times {10}^{-3}$	70
$I_1$	$2\times {10}^{-2}$	70
$I_2$	$2\times {10}^{-4}$	70

Figure 10 shows the changes in LE, DME, and SCI under different impact durations.

Figure 10.

Changes in LE, DME, and SCI values under impact loads with different durations. LE: localization error; DME: damage magnitude error; SCI: sparsity consistency index.

The LE values of $I_1$ and $I_2$ are both close to $I_0$ . Due to the decrease in high-frequency content, $I_1$ slightly increases, which slightly weakens the sensitivity of the high-resolution sub-band. However, this growth rate is still very small, indicating strong localization robustness. In these three cases, the fluctuation amplitude of DME is extremely small. Although $I_2$ introduces richer high-frequency excitation, which may slightly improve the accuracy of amplitude estimation, the magnitude of this improvement is not significant. This framework maintains consistent reconstruction performance, indicating the stability of the sparse inversion model. SCI remained at a level close to 1.0 in all cases, confirming that correctly identified damaged components remained unchanged under excitation changes. This indicates that the proposed method does not rely on specific excitation durations.

Offset of the impact position

In practical engineering applications, due to operational uncertainty, environmental interference, or measurement deviations, it is not always possible to ensure accurate impact positions. Therefore, it is necessary to investigate whether the proposed FRF-WPD-PE framework can still maintain stable recognition performance when the excitation position deviates from the reference situation.

In the case $I_0$ , the amplitude of impact load is 70 N and the duration is 2 × 10⁻³ s. The impact position is located in a relatively high modal participation area, which can ensure sufficient overall structural vibration and serve as a benchmark case for comparison. To evaluate spatial sensitivity, two additional cases were designed as shown in Table 3:

Table 3.

Different impact position.

Case	Impact position
$I_0$	1/3 span
$I_3$	1/4 span
$I_4$	1/12 span

Case $I_3$ represents a slight deviation from the reference excitation point, while case $I_4$ corresponds to the excitation point closer to the boundary, where the modal contribution and vibration energy distribution differ significantly from the excitation conditions at the mid span position. The test results are shown in Figure 11.

Figure 11.

Changes in LE, DME, and SCI values under impact loads with different positions. LE: localization error; DME: damage magnitude error; SCI: sparsity consistency index.

Compared with the baseline data, there was a slight increase in LE in both cases $I_3$ and $I_4$ . This is because changes in the excitation position can alter the modal participation factor and local vibration energy distribution, which may slightly affect the clarity of the concentration of damage-related features. However, since the normalization based on FRF reduces excitation dependence, and WPPE extracts frequency complexity features rather than relying solely on amplitude, the variation of LE should remain within a small range.

DME may experience some degree of fluctuation, especially in case $I_4$ (1/12 span), where approaching boundaries may weaken the excitation energy transmitted to certain damaged components. The reduction of signal noise characteristics in specific sub-bands may slightly affect the estimation accuracy of damage size. However, the sparse inversion mechanism limits the solution space and prevents large-scale deviations in the estimated damage size.

In all cases, SCI will remain relatively stable. Even with minor deviations, the framework should still be able to correctly identify most of the truly damaged components. This indicates that the method can still preserve the sparse characteristics of the structure in the presence of uncertainty in the excitation position.

Different material properties

Baseline configuration (case M₀) is defined as: $E_0=210\mathrm{GPa}$ , ${\rho }_0=7840\mathrm{kg}/{\mathrm{m}}^3$ , ${\xi }_0=1.6\%$ . This is consistent with the model parameters in “Finite element model” section. The following sections will analyze the recognition stability under independent and joint disturbances.

Elastic modulus variation

To simulate the effects of temperature changes and material aging, the elastic modulus was adjusted as shown in Table 4, while maintaining constant density and damping.

Table 4.

Different elastic modulus.

Case	Elastic modulus	Changed area
M0	$E_0$	Whole beam
M1	0.9 $E_0$	Between sensors 3, 4, 5, 6
M2	1.1 $E_0$	Between sensors 3, 4, 5, 6
M3	0.9 $E_0$	Between sensors 7, 8, 9, 10
M4	1.1 $E_0$	Between sensors 7, 8, 9, 10

The LE, DME, and SCI are computed for each case, as shown in Figure 12.

Figure 12.

Changes in LE, DME, and SCI values under different elastic modulus. LE: localization error; DME: damage magnitude error; SCI: sparsity consistency index.

The results indicate that the LE value remains almost unchanged under changes in stiffness. As the modulus deviation increases, DME will slightly rise, but still within an acceptable range. The SCI value is still close to 1, indicating that the damaged elements have been continuously identified.

Although stiffness disturbances can alter the overall frequency distribution, the energy characteristics of sub bands based on FRFs will be standardized and aggregated among sensors. Therefore, the relative spatial anomaly pattern remains stable. The weighted sparse inversion technique further eliminates the proportional effect of uniform stiffness, ensuring stable positioning even when the stiffness variation amplitude reaches ±10%.

Density variation

To reflect the quality changes caused by corrosion products, water accumulation, or surface attachments, the change in density is shown in Table 5, while keeping the stiffness and damping unchanged. The result is shown in the Figure 13.

Table 5.

Different density.

Case	Density	Changed area
M0	${\rho }_0$	Whole beam
M5	0.95 ${\rho }_0$	Between sensors 3, 4, 5, 6
M6	1.05 ${\rho }_0$	Between sensors 3, 4, 5, 6
M7	0.95 ${\rho }_0$	Between sensors 7, 8, 9, 10
M8	1.05 ${\rho }_0$	Between sensors 7, 8, 9, 10

Figure 13.

Changes in LE, DME, and SCI values under different density conditions. LE: localization error; DME: damage magnitude error; SCI: sparsity consistency index.

In the case of density disturbance: LE fluctuates slightly around the value of $I_0$ . DME has hardly changed either. SCI is still close to 1.

The change in density mainly has a uniform impact on the modal frequency. Due to the proposed framework being based on relative sub-band energy distribution rather than absolute frequency values, disturbances in quality will only cause minimal spatial deformation. This confirms that the method does not rely on precise quality calibration.

Damping ratio variation

To simulate the changes in energy dissipation caused by aging, the changes in damping ratio are shown in Table 6. Simultaneously maintaining the same stiffness and density. The analysis results are shown in Figure 14.

Table 6.

Different damping ratio.

Case	Damping ratio (%)	Changed area
M0	1.6	Whole beam
M9	1	Between sensors 3, 4, 5, 6
M10	3	Between sensors 3, 4, 5, 6
M11	1	Between sensors 7, 8, 9, 10
M12	3	Between sensors 7, 8, 9, 10

Figure 14.

Changes in LE, DME, and SCI values under different damping ratio conditions. LE: localization error; DME: damage magnitude error; SCI: sparsity consistency index.

The results indicate that damping disturbance will cause slight changes in the peak amplitude of the FRF; However, LE remains almost at zero level. DME exhibits some volatility, but remains within a controllable range. The SCI index remains high.

Although damping can alter the amplitude level of the FRF, the proposed DI index is constructed based on weighted sub-band energy and entropy measurements, which are less sensitive to overall amplitude attenuation. Sparse regularization can prevent erroneous damage areas caused by noise amplification due to damping effects.

Experimental validation

Although numerical studies can provide controlled modeling of different levels of damage scenarios, thereby facilitating mechanism level analysis and sensitivity evaluation, experimental testing focuses on verifying the effectiveness of the proposed method under various practical factors such as real measurement noise, boundary uncertainty, and inevitable modeling differences.

Experimental setup

This section conducts solid tests on steel box girders to verify the applicability and robustness of the proposed FRF-WPPE-LOF fusion framework in practical environments. The geometric and material parameters of the steel box girder specimens used in the experiment are consistent with the numerical model, as shown in Figure 15. The test piece has a thin-walled enclosed rectangular section of 100 × 50 mm, made of S355 steel, with a plate thickness of 2 mm and a total length of 3 m. The beam end is rigidly fixed using piers, end cover plates, and high-strength bolts, with the end cover plate covering a length of 157.5 mm, which is consistent with the fixed section length set in the numerical simulation model to ensure comparability of boundary conditions.

Figure 15.

Experimental steel box girder specimen: (a) view 1, (b) view 2, and (c) details of cross-section.

In order to obtain the time-domain response of the structure under impact excitation, 14 acceleration sensors (PCB 352C33, sensitivity: 100 mV/g) were arranged on both sides along the beam length direction, and their layout was completely consistent with the arrangement of the virtual sensors mentioned above (Figure 16(a)). This symmetrical arrangement is not only beneficial for obtaining response information across the entire span, but also for verifying the accuracy of gradient field construction required for damage directionality judgment. The sensor signal is synchronously collected through a multi-channel data acquisition system, and the sampling frequency = 12.8 kHz is set according to the width of the impact frequency band to ensure the capture of mid to high frequency dynamic features.

Figure 16.

Detail and layout of sensors: (a) detail of sensors and (b) layout of sensors.

The incentive adopts a handheld instrumented impact hammer (PCB 086D05, sensitivity = 1 mV/lbf ≈ 0.23  mV/N), equipped with a medium stiffness hammer tip, and the hammering position is consistent with the excitation point in the numerical simulation to ensure the spatial consistency of the input excitation (Figure 4). Shock excitation can stimulate rich frequency band content and has high repeatability, highly matching the FRF-WPD-PE analysis method. In the experiment, repeated tapping is used to offset random errors and ensure the stability of subsequent FRF calculations.

To compare the consistency of damage identification performance between numerical simulation and solid structure, three types of specimen-DCs (SDC) were set for the beam, which were the same as those in the simulation (Figure 17). Among them, SDC1 represents minor local weakening damage, SDC2 represents penetrating damage (Figure 17(a)), and SDC3 represents dual penetrating DCs (Figure 17(b)). Damage is introduced through cutting or local slotting to ensure that the size, location, and shape of the damage are consistent with the numerical model.

Figure 17.

DCs on the actual structure: (a) SDC1 and SDC2 and (b) SDC3. DC: damage condition.

The overall experimental plan aims to verify:

1. The robustness of the proposed method under conditions of real noise, non-ideal boundaries, and excitation randomness.

2. The consistency between the FRF, WPD sub-band energy changes and WEE characteristics of the physical structure and the numerical samples.

3. The correspondence between the experimental damage heat map and the numerical sparse field inversion results.

Data processing and FRF-WPPE implementation

The experiment used an instrumented hammer to repeatedly strike the beam 10 times, obtaining 10 sets of acceleration response signals (as shown in Figure 18). The FRF was calculated by input-output ratio, and the experimental FRF spectrum is shown in Figure 19. The reconstructed mode is shown in Figure 20. It can be seen that the peak positions of each frequency order are highly consistent with the numerical simulation, indicating that the experimental data quality is reliable and the excitation and sensor arrangement meet the requirements.

Figure 18.

Impact load and acceleration response.

Figure 19.

Experimental FRF spectrum: (a) sensor 7, (b) sensor 8, and (c) sensor 12. FRF: frequency response function.

Figure 20.

The first four modes of vibration of steel box girder reconstructed from FRF data based on impact hammer excitation: (a) model 1 (43.75 Hz), (b) model 2 (129.69 Hz), (c) model 3 (243.75 Hz), and (d) model 4 (378.13 Hz). FRF: frequency response function.

Subsequently, the FRF signal is input into the FRF-WPPE framework for 64 sub-band WPD, weighted PE calculation, and combined with LOF anomaly factor to identify local structural anomalies. Based on the comprehensive index of WPE and LOF at each sensor position, the experimental damage thermal map was obtained through IDW sparse field inversion (Figure 21).

Figure 21.

Thermal diagram of experimental damage: (a) SDC1, (b) SDC2, and (c) SDC3.

Experimental results and discussion

From Figure 21, it can be seen that:

1. The three types of DCs (SDC1–SDC3) were accurately located, and the experimental thermal map and numerical simulation results maintained good consistency, verifying the feasibility and stability of the FRF-WPPE framework in practical structures.

2. The experimental peak of SDC1 (slight damage) is slightly higher than the simulated value: it has increased from about 0.35 in the simulation to about 0.5 in the experiment. This is mainly caused by the following factors: the presence of welds, residual stresses, and minor manufacturing errors in solid beams, which make the local stiffness changes corresponding to minor damages more sensitive; Actual incentives are more difficult to achieve complete consistency, and the disturbance of micro damage to FRF is slightly amplified in the experiment; sensor noise has a more significant impact on the PE index under low DCs. This phenomenon indicates that FRF-WPPE-LOF still maintains high sensitivity to weak damage.

3. The responses of SDC2 and SDC3 exhibit gradient enhancement characteristics: SDC2 (penetrating damage) forms a significantly high-value area near the damage location, and the distribution of the heat map is almost the same as the simulation results; SDC3 (multiple penetrating damage) exhibits multimodal characteristics, with multiple high-energy zones corresponding to different crack positions. The experimental results accurately reflect the superposition effect of multiple damages.

4. The experimental error is mainly reflected in local fluctuations in the high-frequency range, but it does not affect the localization of damage. The FRF-WPPE-LOF comprehensive index shows good robustness to noise, and there is no misjudgment in the abnormal distribution of edge sensors.

Overall, the experimental results have verified the effectiveness of the FRF-WPPE-LOF framework in actual steel box girder structures, which can maintain stable identification accuracy under different degrees of damage and multiple DCs, and is highly consistent with simulation results.

To verify the performance of the new framework under baseline-free conditions, the experimental results were compared using the traditional baseline WPPE method (with healthy vibration data as a reference) and the proposed baseline-free FRF-WPPE method in this study. The damage identification results are shown in Figure 22(a) and (b), respectively.

• (1) Comparison of positioning accuracy: Consistent between baseline-free and baseline-based methods

From the results in Figure 22, it can be seen that both the baseline-free method (Figure 22(b)) and the baseline based method (Figure 22(a)) accurately identified the approximate spatial location of the injury; the three experimental DCs (SDC1–SDC3) all showed obvious local peak areas, consistent with the actual damage location; The FRF-WPPE framework demonstrates comparable damage localization capability to traditional baseline methods even without the need for health reference data.

• (2) Estimation of damage severity: Method without baseline is even more stable and consistent

Comparison of peak damage levels shows that for minor damage (SDC1), both methods can provide distinguishable mid to low amplitude responses; for both penetrating damage (SDC2) and multiple damage (SDC3), significant amplitude enhancement features can be observed in both Figure 22(a) and (b). But the important difference is that traditional baseline methods exhibit amplitude instability under multiple DCs, with a significant difference in peak values between two equally severe damages (Figure 22(a)), showing strong asymmetry. The reason is that there is a baseline method that relies on calculating the difference between health data and injury data. In multi-damage scenarios, inconsistent local excitations, boundary condition disturbances, or health data errors can be amplified, forming a “false difference” that results in one damage peak being significantly higher and the other being weakened. This instability is very common in practical engineering, making traditional methods prone to false positives or false negatives in multi-damage detection.

• (3) The advantage of baseline-free FRF-WPPE: More robust to complex scenarios

In contrast, baseline-free FRF-WPPE does not rely on health data and does not perform data subtraction, so there is no problem of difference amplification. FRF-WPD-PE-LOF forms a multi-source fusion index, balancing frequency-domain sensitivity and time-domain robustness. Two severe injuries exhibit almost equal peak intensities in Figure 22(b). The heatmap is evenly distributed with no false hotspots at the boundaries. Therefore, the baseline-free framework maintains higher consistency and robustness under multiple DCs.

Figure 22.

Comparison of results between baseline based and baseline-free methods: (a) baseline-based method and (b) baseline-free method.

The FRF-WPPE-LOF framework proposed in this study has the following advantages: it has no baseline and does not require any healthy reference samples. Directly utilize the current FRF response to construct damage indicators. Capable of frequency-domain noise resistance and multi-damage identification. In the experiment, it showed high stability (peak values remained basically consistent and there were no false peaks).

Comparative discussion

Building upon the aforementioned virtual experiments, this chapter further expands the sample size to systematically evaluate the generalization capability and robustness of the proposed baseline-free FRF-WPPE-LOF framework under multiple DCs, noise disturbances, and random operational scenarios.

An extended dataset was constructed by generating 50 damage cases at randomly distributed locations on the surface of the beam, as illustrated in Figure 23. The damage location is sampled from a two-dimensional Gaussian distribution to avoid spatial bias and ensure sufficient coverage of the structural domain. Two types of damage severity were considered: slight damage (SD1) and serious damage (SD2), consistent with “Finite element model” section. All damage scenarios are generated within the numerical model and are only used to evaluate the statistical robustness and generalization ability of the proposed framework, rather than providing reference state information for damage identification.

Figure 23.

Random generated single damage location on the beam.

Comparison of single damage scenarios

Five methods were used to detect 50 random single injury cases: (1) PE is tested separately, (2) FRF is tested separately, (3) traditional WPPE (with baseline), (4) FRF-WPPE (with baseline), and (5) FRF-WPPE-LOF (without baseline: the new framework proposed in this article). The confusion matrix is shown in Figure 24, and the detection performance indicators are shown in Table 7.

Figure 24.

Confusion matrix: (a) PE, (b) FRF, (c) WPPE, (d) FRF-WPPE, and (e) FRF-WPPE-LOF.

Table 7.

Accuracy, Precision, and Recall of different methods to identify single damage.

Method	Total correct	Overestimated	Underestimated	Accuracy (%)	Precision (%)	Recall (%)	Omission (%)
PE	33	10	7	66.0	—	—	8.0
Slight damage	14	10	—	—	66.7	—	—
Serious damage	19	—	7	—	—	73.1	—
FRF	35	9	6	70.0	—	—	6.0
Slight damage	15	9	—	—	71.4	—	—
Serious damage	20	—	6	—	—	76.9	—
WPPE	45	3	2	90.0	—	—	0.0
Slight damage	21	3	—	—	91.3	—	—
Serious damage	24	—	2	—	—	92.3	—
FRF-WPPE	47	2	1	94.0	—	—	0.0
Slight damage	24	2	—	—	96.0	—	—
Serious damage	25	—	1	—	—	96.2	—
FRF-WPPE-LOF	44	5	1	88.0	—	—	0.0
Slight damage	19	5	—	—	95.0	—	—
Serious damage	25	—	1	—	—	96.2	—

PE: permutation entropy; FRF: frequency response function; WPPE: wavelet packet permutation entropy; LOF: local outlier factor.

From the WPPE-based method onward, the “None” class remains zero, indicating that no damaged samples are misclassified as undamaged, which reflects improved detection robustness.

Testing accuracy and reliability results

The “perfect recognition rate” of using PE or FRF alone is less than 70%, and there is a significant bias in estimating the degree of damage. WPPE, FRF-WPPE, and FRF-WPPE-LOF all exceed 85% in terms of “perfect recognition.” The new framework is basically consistent with baseline FRF-WPPE in terms of accuracy (95%) and recall (96.2%), and even better than mature WPPE methods. It is particularly noteworthy that the latter three methods achieve a 100% recognition rate in the qualitative judgment of whether there is damage, with no missed detections. This indicates that FRF-WPPE-LOF still has detection accuracy close to or even exceeding traditional methods without the need for a healthy baseline.

Key advantages of baseline-free methods

The results of 50 sets of data show that the new framework only resulted in one underestimation (severe injury misclassified as minor injury). There are five overestimates (acceptable, consistent with the principle of no missed detections). For baseline-free methods, recall rate (no missed detections) is more important than slightly higher accuracy, as missed detections in engineering inspections may lead to structural safety hazards. Therefore, this feature actually reflects the engineering applicability of the new framework.

Comparison of multiple DCs

Further use random combination method to randomly combine the previous 50 injuries into groups of two to four, generating 50 sets of multi-damage cases (as shown in Figure 25). The same five aforementioned methods were still used to detect the damages.

Figure 25.

Random generated multi-damage group location on the beam (three examples from 50 groups): (a) sample 1, (b) sample 2, and (c) sample 3.

Qualitative identification of the existence of damage

When making qualitative judgments on whether damage has occurred, the confusion matrix of the recognition results is shown in Figure 26, the Receiver Operating Characteristic (ROC) curve is shown in Figure 27, and the performance indicators are shown in Table 8. The recognition rate of “whether there is damage” for WPPE, FRF-WPPE, and FRF-WPPE-LOF methods all reached 100%. However, the single module methods of PE and FRF are still significantly inadequate, with high rates of false positives and false negatives. The new framework maintains good qualitative robustness under complex operating conditions.

Figure 26.

Confusion matrix (qualitative judgment): (a) PE, (b) FRF, (c) WPPE, (d) FRF-WPPE, and (e) FRF-WPPE-LOF.

Figure 27.

ROC curve (qualitative judgment).

Table 8.

Accuracy, Precision, and Recall of different methods to identify multi-damage (qualitative judgment).

Method	TP	FP	FN	Accuracy (%)	Precision (%)	Recall (%)	Omission (%)
PE	40	6	4	80.0	87.0	90.9	8.0
FRF	41	4	5	82.0	91.1	89.1	10.0
WPPE	50	0	0	100.0	100.0	100.0	0.0
FRF-WPPE	50	0	0	100.0	100.0	100.0	0.0
FRF-WPPE-LOF	50	0	0	100.0	100.0	100.0	0.0

PE: permutation entropy; FRF: frequency response function; WPPE: wavelet packet permutation entropy; LOF: local outlier factor.

Determination of damage degree (quantitative identification)

For multi-damage degree recognition, the confusion matrix is shown in Figure 28, the performance indicators are shown in Table 9, and the ROC curves of each method are shown in Figure 29. The results indicate that the accuracy of the new framework is between WPPE and FRF-WPPE. The recognition accuracy of the three fusion methods is about 80%. The recall rate of the new framework still reaches 96%, which is completely consistent with the baseline FRF-WPPE. Similarly, the baseline-free method prioritizes ensuring a higher recall rate, making it more reliable in engineering tasks.

Figure 28.

Confusion matrix (quantitative identification): (a) PE, (b) FRF, (c) WPPE, (d) FRF-WPPE, and (e) FRF-WPPE-LOF. PE: permutation entropy; FRF: frequency response function; WPPE: wavelet packet permutation entropy; LOF: local outlier factor.

Table 9.

Accuracy, Precision, and Recall of different methods to identify multi-damages (quantitative identification).

Method	TP	FP	FN	Accuracy (%)	Precision (%)	Recall (%)	Omission
PE	29	12	9	58.0	70.7	76	8.0
FRF	31	8	11	62.0	79.4	74	10.0
WPPE	37	8	2	74.0	82.2	88	0.0
FRF-WPPE	40	8	2	80.0	83.3	96	0.0
FRF-WPPE-LOF	38	10	2	76.0	79.2	96	0.0

PE: permutation entropy; FRF: frequency response function; WPPE: wavelet packet permutation entropy; LOF: local outlier factor.

Figure 29.

ROC curve of damage detection system (quantitative identification): (a) PE, (b) FRF, (c) WPPE, (d) FRF-WPPE, and (e) FRF-WPPE-LOF. PE: permutation entropy; FRF: frequency response function; WPPE: wavelet packet permutation entropy; LOF: local outlier factor.

Analysis of abnormal cases and potential engineering limitations

Further analysis was conducted on cases with incorrect classification. Representative examples are shown in Figures 26(a) 30, 31, and 32. In this case, only one of the four damages is the slight damage, while the rest are serious damages. But the final test results showed that all four damages were judged as serious (Figure 32).

Figure 30.

Finite element model of the typical case.

Figure 31.

FRF for the typical case: (a) FRF magnitude overview (sensors 3–12) (unit: m/s²/N) and (b) FRF magnitude comparison. FRF: frequency response function.

Figure 32.

Thermal map distribution of the typical case.

From sparse field reconstruction and thermal map distribution, it can be seen that due to the close proximity of multiple damage locations, damage coupling and superposition effects occur in the FRF mode distribution and WPD sub-band energy structure. The contribution of serious damage in the dominant mode is greater, leading to the “drowning” of slight damage features and resulting in overestimation of partial damage. This phenomenon belongs to the typical “serious damage dominant effect” and is also commonly present in practical structures.

Therefore, damage that is too close in spatial distance may still result in feature superposition, affecting the distinguishability of slight damage. At present, there is still room for improvement in the independent discrimination of extremely dense damage points by LOF. At extremely high noise levels, low order modes may still interfere with anomaly detection to some extent. However, the new framework still has significant advantages in recall rate, baseline detection capability, and robustness to multiple damage scenarios.

The proposed framework is mainly designed for those cases where the damage is sparsely or evenly distributed in space, which is consistent with the common assumptions in vibration based SHM. In the case of extremely dense or closely distributed damage, feature overlap may occur, and local abnormal indicators may be affected by major severe damage modes. In this study, the subsequent sparse field inversion and spatial interpolation partially alleviate this effect by forcing spatial continuity. However, the introduction of explicit spatial correlation modeling is a valuable direction for future research.

Conclusion and future work

This study proposes a baseline-free structural damage identification framework based on FRF-WPPE-LOF, and systematically verifies its effectiveness in numerical simulations and solid steel box girder experiments. The research results indicate that the framework can still achieve high accuracy in identifying the location and degree of damage without the need for health status reference data, providing a practical and feasible technical path for solving the long-term problem of “difficulty in obtaining health baselines” in engineering structures.

Firstly, by deeply integrating the excitation robustness of FRF, the multi-resolution characteristics of WPD, the complexity sensitivity of PE, and the local anomaly detection capability of LOF, this study constructs a multi-source fusion index system that can simultaneously handle noise, excitation uncertainty, and multiple damage superposition. The numerical simulation results show that the framework maintains stable sensitivity to slight, serious, and multiple damages, with accurate damage localization and no missed detections. In the real steel box girder experiment, the new framework also demonstrated recognition ability consistent with the simulated height, further verifying the feasibility and engineering applicability of the method.

Secondly, by expanding the sample size and conducting systematic comparisons, the new framework exhibits recognition accuracy that is close to or even partially surpasses its baseline dependent methods in both single damage and multi-damage scenarios. Especially in terms of recall rate, noise robustness, and edge damage detection, baseline-free FRF-WPPE-LOF is significantly better than traditional methods, and is suitable for engineering scenarios such as bridges, steel structure buildings, and large equipment where healthy baselines cannot be obtained in advance. It has significant engineering significance.

Although this method has demonstrated good performance, there is still room for further improvement in the future. In the practical application of SHM, environmental and operational changes such as temperature, humidity and long-term material degradation will significantly affect the vibration response. Although these factors have not been clearly changed in this experimental activity, the proposed framework relies on no baseline feature fusion and local anomaly detection, which reduces the sensitivity to the overall response changes caused by environmental changes. However, systematic research under different environmental conditions and the gradual deterioration of materials are an important direction of future research. Subsequent research can focus on the following directions:

1. Long term monitoring application of physical bridges: Future research may apply the proposed framework to the existing bridge structure to study its long-term stability, adaptability and robustness under the changes of actual environment and operating conditions, so as to promote its transition from laboratory verification stage to field application stage.

2. Integrating machine learning/deep learning models: For key processes such as sparse field inversion and damage heat map feature extraction, lightweight deep neural networks can be introduced to enhance automation and real-time recognition capabilities.

3. Multimodal sensing data extension: The framework can be further extended by incorporating heterogeneous sensing modalities, such as strain, temperature, and vision-based measurements, enabling cross-modal information fusion and enhancing damage discrimination capability under complex environmental conditions.

4. Optimization of large-scale structural adaptability: Further improvement of sparse field inversion algorithm and sensor layout strategy can enhance the scalability of the method for large-span bridges and spatial steel frame structures. In addition, for real-time monitoring applications, lightweight algorithms can be used to simplify the WPD process and optimize sensor layout through topology analysis, thereby achieving high efficiency in on-site data processing while ensuring detection accuracy.

In summary, the FRF-WPPE-LOF baseline-free damage identification framework proposed in this article provides a highly robust, accurate, and engineering friendly solution for SHM, with broad application prospects. With the integration of multimodal data and intelligent algorithms in the future, this framework is expected to develop into a high-performance intelligent monitoring system suitable for more complex engineering structures.