Comparative Study of Adaptive l₁-Regularization for the Application of Structural Damage Diagnosis Under Seismic Excitation

Abstract

Damage identification plays a crucial role in the post-earthquake assessment and safety control of civil structures, which is usually an ill-posed inverse problem due to the presence of uncertainties and lack of measurement information. Regularization is a cutting-edge technique used to address ill-posed problems and has been developed for decades. A comprehensive review and comparison have first been conducted to identify the limitations and research gaps in the existing regularization methods for structural damage detection. Thereafter, we identified the development of the adaptive sparse regularization (ASR) method, capable of dynamically adjusting regularization parameters and sparsity according to specific damage patterns or environmental conditions, as one of the emerging research directions. Therefore, this paper systematically formulates and summarizes the theoretical framework of the ASR-based damage detection method for engineering applications to facilitate an in-depth comparative analysis. To validate the performance of the ASR method for post-earthquake structural damage diagnosis, numerical experiments are carried out on 2D and 3D models under diverse damage detection scenarios subjected to typical natural seismic excitations. These experimental investigations consider the influences of different parameter settings and uncertainties. Subsequently, the effects of damage patterns, available modal information, and solution algorithms are systematically analyzed and discussed. The results of the numerical investigation indicate that the ASR-based method is effective for damage detection, showing satisfactory accuracy and stability under complex damage scenarios and extreme conditions with a limited number of sensors and insufficient modal information. Furthermore, integrating the ASR-based damage detection method with appropriate optimization algorithms can enhance its capability to precisely identify isolated or hybrid-distributed damage.

1. Introduction

Damage detection in civil structures is essential for structural health monitoring to preserve infrastructure integrity and ensure public safety. This is particularly important for rapid disaster severity assessment during and after seismic events, as depicted in Figure 1. Our capacity to promptly identify and evaluate damage can substantially mitigate the risks associated with structural failures. Conventional inspection methods rely on visual assessments and periodic evaluations, which may not reliably detect hidden or evolving damage, or do so in a timely manner, potentially resulting in critical oversights. In contrast, vibration-based techniques analyze variations in vibration characteristics, such as natural frequencies and mode shapes, enabling the continuous monitoring and detection of internal damage within structures. Currently, the integration of advanced sensors and sophisticated data analysis techniques significantly enhances the sensitivity and robustness of vibration-based methods. This enables the detection of subtle changes and offers a more reliable assessment of a structure’s health, particularly under extreme dynamic loading conditions caused by seismic events.

Vibration-based damage detection techniques have been extensively studied and widely applied in recent decades. Recently, Hou et al. [1] presented a comprehensive review of the advancements in vibration-based damage detection for civil structures from 2010 to 2019. An et al. [2] and Rabi et al. [3] focused specifically on recent developments and future trends in vibration-based damage identification approaches for bridge structures. Furthermore, novel applications integrating artificial intelligence algorithms [4] and advanced signal processing techniques [5] for vibration feature extraction have been summarized and discussed. Beyond traditional measurement techniques, innovative vibration-based methods utilizing optical fiber sensing have attracted significant attention in recent years. Optical fiber sensing for vibration measurement has been proven effective in detecting microcracks [6,7] and localizing damage in large-scale linear structures [8,9]. However, its application in buildings remains constrained due to the challenges associated with installation difficulties and high costs, necessitating further research efforts.

From the above literature, we observe that vibration-based damage detection techniques inherently encounter several common challenges, often leading to ill-posed problems that complicate the precise evaluation of structural damage. A major challenge is the presence of unavoidable uncertainties, such as measurement noise and model errors. Measurement noise, which arises from environmental factors and sensor inaccuracies, masks subtle variations in vibration patterns indicative of damage, thereby hindering the attainment of consistent and reliable diagnostic results. Similarly, inaccuracies in analytical models can result in false alarms within model-driven damage detection outcomes. Moreover, the limited number of sensors used for damage detection often leads to incomplete information regarding structural behavior. The scarcity of comprehensive data can give rise to multiple interpretations of the same observed response, thus complicating the damage assessment process. These challenges underscore the necessity for robust advanced techniques to mitigate the ill-posed nature of the problem and enhance the efficacy of vibration-based damage detection methods.

To address the aforementioned challenges associated with ill-posed problems, regularization techniques have been developed and extensively studied by introducing a penalty term to constrain the solution space. Much research work has demonstrated the potential of these techniques in enhancing both the robustness and accuracy of solutions for ill-posed inverse problems. In recent decades, regularization methods have been widely applied across various domains, particularly in structural health monitoring, evolving into several distinct categories as summarized in Figure 2. Specifically, regularization-based damage detection methods can primarily be categorized into l₂, l₁, and other innovative sparse regularization approaches, depending on the formulation of the regularization term. Furthermore, critical application details, such as strategies for determining the regularization parameter, selected damage indices, and employed solution algorithms, were also examined. To identify the limitations and research gaps in the existing regularization methods for structural damage detection, a comprehensive review will be, respectively, presented and discussed in the subsequent paragraphs.

Among the three categories of regularization techniques, l₂-regularization is predominantly utilized for structural damage detection. This technique is distinguished by its l₂-norm regularization term, which is defined as the sum of squares and possesses an analytical solution. It plays a crucial role in mitigating ill-posed problems. Notably, Tikhonov regularization (TR), a well-known l₂-regularization method, is the most extensively employed regularization approach for damage detection. A summary of studies on l₂-regularization-based damage detection methods is presented in

Table 1. Studies of l₂-regularization-based damage detection methods.

Authors (Year)	Regularization Method	Damage Index	Regularization Factor Determination	Applications		Innovation and Improvement
				Validation Approach	Structure
Prells et al. [10] 1996	TR	Modal frequencies	A threshold-based method	Numerical	2D beam	Early application of TR for damage detection
Ahmadian et al. [11] 1998	TR	Modal frequencies and mode shapes	L-curve, GCV	Numerical	Spring-mass model and frame	Comparison of L-curve, GCV, and SVD-based regularization method
Fritzen et al. [12] 2000	TR	Modal frequencies and mode shapes	-	Numerical	3D frame	Application under seismic excitation
Weber et al. [13] 2009	TR	Modal frequencies and mode shapes	GCV	Numerical and experimental	Full-scale 3D frame	Experimental and comparison study with truncated SVD
Li et al. [14] 2010	ATR	Acceleration response	L-curve	Numerical	2D truss	Proposed a new regularization term and improved performance
Wang et al. [15] 2012	MTR	Acceleration response	L-curve	Numerical	3D frame	Modified the regularization term with reasonable physical information
Li et al. [16] 2010	ATR	Covariance of covariance matrix	L-curve	Numerical	2D truss	Adopted the new damage index of CoC
Law et al. [17] 2012	ATR	Covariance of covariance matrix	L-curve	Numerical	2D truss	Adopted CoC and derived the analytical expression of the sensitivity matrix
Zhu et al. [18] 2014	TR	Acceleration response	L-curve	Numerical and experimental	Beam	Integrated with response reconstruction
Law et al. [19] 2014	ATR	Impulse response function	L-curve	Numerical	2D truss	Adopted the new damage index of the impulse response function
Lin et al. [20] 2019	ATR	Impulse response function	L-curve	Numerical	2D truss	Introduced dimensionality reductio transformation matrix
Fu et al. [21] 2016	TR	MSE and acceleration response	L-curve	Numerical	Plate	Two-stage damage detection using MSE and acceleration
Entezami et al. [22] 2017	RLSMR	Mode shapes	Hybrid GCV	Numerical	2D truss	Modified the traditional LSMR with l2 regularization
Zeinali et al. [23] 2018	IMTR	Influence line	L-curve	Numerical and experimental	Beam	Improve TR with iterative and multi-parameter
Lin et al. [24] 2018	ATR	Response covariance	L-curve	Numerical	3D frame	Integrated with response covariance and multi-objective OSP
Lin et al. [25] 2018	ATR	Response covariance	L-curve	Numerical	3D frame	Integrated with multi-type multi-objective OSP
Lin et al. [26] 2019	ATR	Response covariance	L-curve	Experimental	3D frame	Experimental validation of the method
Xu et al. [27] 2019	ATR	Response covariance	L-curve	Numerical and experimental	Transmission tower	Multi-level damage detection integrated with multi-scale FE model

. Prells et al. [10] were the pioneering researchers to propose TR for damage identification, validating its performance through a numerical study on a beam structure. Ahmadian et al. [11] conducted a comparative analysis of various regularization factor determination methods, demonstrating the noise robustness and reliability of generalized cross-validation (GCV) and the L-curve method. Fritzen et al. [12] applied the TR-based damage detection method under seismic excitation. Subsequently, Weber et al. [13] refined TR to enhance its consistency, accuracy, and robustness. Li et al. [14] introduced adaptive Tikhonov regularization (ATR) with an improved regularization term to minimize unexpected negative solutions and significantly enhance damage detection performance. Similarly, Wang et al. [15] proposed a modified Tikhonov regularization (MTR) method that incorporated reasonable physical information into the regularization term. Furthermore, Li et al. [16] adopted the covariance of covariance (CoC) matrix as a damage index to increase the sensitivity of the ATR-based method, while Law et al. [17] further derived the analytical expression of the sensitivity matrix of CoC to improve the computation accuracy and efficiency of ATR-based damage detection. Zhu et al. [18] utilized the TR to enhance the stability of response reconstruction, thereby enabling damage detection without requiring excitation information. Law et al. [19] and Lin et al. [20] employed ATR to mitigate the ill-posedness issue for impulse response function-based damage detection under multiple excitations. Fu et al. [21] proposed a two-stage TR-based damage detection method using mean squared error (MSE) and acceleration to weaken the “vicinity effect” and minimize false alarms. Entezami et al. [22] modified the traditional least squares minimal residual method (LSMR) with l₂-regularization, demonstrating superior damage detection performance compared to TR-based methods. Zeinali et al. [23] developed an iterative multi-parameter Tikhonov regularization (IMTR) for bridge damage detection, showing improvements in detecting sharp and deep damage. To address the uncertainty caused by measurement noise, and to improve damage detection performance with a limited number of accelerometers, an ATR-based damage detection approach integrated with response covariance and multi-objective optimal sensor placement (OSP) was developed and validated through numerical studies [24]. This approach was later extended and numerically investigated by incorporating multi-objective OSP with multi-type sensors to significantly enhance the quality of damage identification [25]. Thereafter, the effectiveness of this comprehensive approach has also been thoroughly examined and validated via laboratory experiments [26]. To facilitate practical application, Xu et al. [27] further integrated ATR with a multi-scale finite element (FE) model for conducting numerical and experimental studies on loose bolt detection in a complex scaled model derived from a real transmission tower structure. Note that l₂-regularization is convenient to apply in various inverse problems with analytical solutions. According to the above review, the effectiveness of l₂-regularization has been verified in many numerical and experimental studies on different types of structures. It has been noticed that studies on l₂-regularization have investigated various constructions of the regularization term and damage index to mitigate the impact of uncertainties. Nevertheless, many researchers have observed that l₂-regularization inherently tends to distribute the solution across multiple components. This characteristic contradicts the typically sparse distribution of damage patterns, thereby representing one of its primary limitations.

To achieve some breakthroughs for the limitations associated with l₂-regularization, some researchers have concentrated on developing damage detection methods based on l₁-regularization. In contrast, the penalty term in l₁-regularization corresponds to the l₁-norm constraint, which is defined as the sum of the absolute values of the coefficients. This characteristic of l₁-regularization promotes sparsity in the solution, actively encouraging certain coefficients to become exactly zero, thereby aligning more closely with practical damage patterns. However, solving the l₁-regularized problem is non-analytical, complicating the determination of the regularization parameter and posing significant challenges for the optimization process. These aspects necessitate further investigation and much more research efforts. A summary of the current studies on l₁-regularization-based damage detection methods is provided in

Table 2. Studies of l₁-regularization-based damage detection methods.

Authors (Year)	Regularization Method	Damage Index	Regularization Factor Determination	Solution Method	Applications		Innovation and Improvement
					Validation Approach	Structure
Hernandez [28] 2014	l1-regularization	Modal frequencies	-	Primal-dual interior point method	Numerical	Beam and plate	Early application of l1-regularization for damage detection
Zhou et al. [29] 2015	l1-regularization	Modal frequencies	L-curve	Active-set	Experimental	Beam	Comprehensive case study on measurement number, damage severity, number of damages, and noise level
Wu et al. [30] 2018	l1-regularization	Modal frequencies and mode shapes	Turning point of the residual norm and solution norm curves	Active-set	Experimental	Beam	Combination of modal frequencies and mode shapes
Hou et al. [31] 2018	l1-regularization	Modal frequencies and mode shapes	Turning point of the residual norm and solution norm curves	-	Numerical and experimental	Truss and frame	Experimental study on a more complicated structure
Hou et al. [32] 2021	l1-regularization	Modal frequencies and mode shapes	Turning point of the residual norm and solution norm curves and DP	-	Numerical and experimental	Beam and frame	Comprehensive study on regularization factor determination and compared with DP
Smith et al. [33] 2018	l1-regularization	Impulse response	Cross-validation	Active-set	Numerical	Beam	Adopted impulse response as damage index
Wang et al. [34] 2019	l1-regularization	Modal frequencies and mode shapes	Threshold setting method	Alternating minimization approach	Numerical and experimental	Beam and truss	Novel method for regularization factor determination
Lai et al. [35] 2018	l1-regularization	Pseudo force	AIC	-	Experimental	Frame	Data-driven semi-supervised method
Ding et al. [36] 2019	l1-regularization	Modal frequencies and MAC	DP	I-JAYA	Numerical and experimental	Beam and truss	Integrated with I-JAYA
Fan et al. [37] 2018	l1-regularization	Resonance frequency shifts	-	Primal-dual interior point method	Numerical	PZT structure	Application on PZT damage detection
Zhang et al. [38] 2017	l1-regularization	Composed state vector	L-curve	EKF	Numerical and experimental	Beam and frame	Integrated with EKF
Zhang et al. [39] 2016	l1-regularization	Strain, displacement, acceleration	-	Primal-dual interior point method	Numerical and experimental	Beam	Utilizing multitype dynamic responses
Zhang et al. [40] 2017	l1-regularization	Strain, displacement, acceleration	-	Primal-dual interior point method	Numerical and experimental	Beam	Utilizing multitype dynamic responses and substructures
Lin et al. [41] 2023	l1-regularization	Response covariance	Equally weighting-based adaptive method	Primal-dual interior point method	Numerical	Beam	Proposed adaptive method for regularization factor determination and adopted response covariance

. Hernandez [28] was among the first to apply l₁-regularization for damage detection, solving the regularized problem via convex optimization. Zhou et al. [29] highlighted the advantages of l₁-regularization and systematically analyzed the effects of measurement number, damage severity, number of damages, and noise level on damage detection outcomes. Subsequently, Wu et al. [30] proposed an innovative method for selecting the l₁-regularization factor by combining modal frequencies and mode shapes as damage indices. Hou et al. [31,32] experimentally validated this approach on complex truss and frame structures, conducting a comprehensive comparison between the new regularization factor selection method and the traditional discrepancy principle (DP). Smith et al. [33] explored the application of LASSO regularization in impulse response sensitivity-based damage detection for structural systems. Wang et al. [34] developed a novel threshold-setting method for regularization factor determination and employed an alternating minimization approach to solve the optimization problem. Lai et al. [35] introduced a data-driven semi-supervised damage detection approach utilizing l₁-regularization for identifying a baseline model. Ding et al. [36] combined l₁-regularization with the I-JAYA algorithm, demonstrating its effectiveness in scenarios involving minor damages, significant noise interference, and limited modal data. Fan et al. [37] employed l₁-regularization to detect damage in piezoelectric ceramic lead zirconate titanate (PZT) by leveraging the damage index of resonance frequency shifts. Zhang et al. [38] integrated l₁-regularization with the extended Kalman filter to address ill-posed problems and prevent over-smooth solutions. Furthermore, Zhang and Xu [39,40] explored the application of l₁-regularization and Kalman filter-based damage detection methods using multi-type dynamic responses and multi-level damage identification. Most recently, Lin et al. [41] developed an adaptive sparse regularization (ASR) technique for precise damage identification, where the newly proposed regularization parameter determination scheme can automatically and effectively adapt to various damage scenarios and noise contamination levels. It is well known that l₁-regularization approaches lack closed-form analytical solutions, and deriving the corresponding regularization factor analytically is also challenging. Therefore, a variety of strategies have been proposed to determine the l₁-regularization factor as shown in Table 2, which significantly impacts the effectiveness and efficiency of damage detection results. Nevertheless, existing strategies still exhibit certain limitations. Some methods necessitate manual intervention, while others require prior knowledge of noise characteristics or involve computationally intensive processes with multiple candidate parameters. Furthermore, the application of l₁-regularization has predominantly focused on numerical simulations and laboratory experiments involving simple 2D structures under ideal excitation conditions, a relatively high sampling ratio, and an extensive number of sensors. Consequently, the performance of the l₁-regularization method for more complex structures with limited sensors and lower sampling ratios under ambient or ground excitations requires further investigation and validation.

With the rapid advancement of l₁-regularization-based damage detection methods, the benefits of sparsity have been substantiated. Some researchers have endeavored to further refine the regularization term to enhance sparsity or attain other desirable properties of the solution for improved performance. In the realm of structural damage detection, innovative regularization techniques, such as reweighted l₁-regularization and l_p-regularization, have been explored and applied to upgrade the accuracy of damage identification. It is worth noting that the regularized optimal problems in this category are non-analytic or non-convex, which even complicates their solution and parameter determination, demanding significant research efforts. Representative studies on enhanced sparse regularization-based damage detection studies are summarized in

Table 3. Studies of enhanced sparse regularization-based damage detection methods.

Authors (Year)	Regularization Method	Damage Index	Regularization Factor Determination	Solution Method	Applications		Innovation and Improvement
					Validation Approach	Structure
Zhang et al. [42] 2015	Reweighted l1-regularization (l0)	Acceleration Response	-	Primal-dual interior point method	Numerical and experimental	Truss and beam	Early comparison study of reweighted l1-regularization
Xu et al. [43] 2018	Reweighted l1-regularization (l0)	Strain, displacement, acceleration	-	Primal-dual interior point method	Numerical and experimental	Bridge	Adopted multi-type responses
Zhou et al. [44] 2019	Reweighted l1-regularization (l0)	Modal frequencies and mode shapes	Turning point of the residual norm and solution norm curves	-	Numerical and experimental	3D frame	Adopted innovative method for regularization factor determination
Bu et al. [45] 2022	Reweighted l1-regularization (l1/2)	The fourth-order statistical moment	-	-	Numerical and experimental	Beam and frame	Reweighted to l1/2-norm constraint
Pan et al. [46] 2019	lp-regularization (0 < p < 1)	Bending moment response	BIC	Active-set	Numerical	Beam	Application for bridge structures
Yue et al. [47] 2019	lp-regularization (0 < p < 1)	Modal frequencies and mode shapes	L-curve	PSO	Numerical and experimental	Beam	Comparison study of lp-regularization (0 < p < 1)
Luo et al. [48] 2020	lp-regularization (0 < p < 1)	Modal frequencies and mode shapes	-	PSO	Numerical	2D truss	Comparison study of l2-, l1-, l1/2-regularization
Ding et al. [49] 2022	l1/2-regularization	Modal frequencies and MAC	DP	Modified JAYA	Numerical and experimental	Truss and frame	l1/2-regularization integrated with modified JAYA
Lin et al. [50] 2017	Elastic net method and ATR	Response covariance	Cross-validation	-	Numerical	Beam	Two-stage method using ATR and elastic net method
Hou et al. [51] 2021	Elastic net method	Modal frequencies and mode shapes	Cross-validation	Active-set	Numerical and experimental	Beam and frame	Experimental and comparison study of elastic net method
Chen et al. [52] 2019	Trace LASSO	Modal frequencies and mode shapes	Manual result comparison	ALO algorithm,	Numerical and experimental	Beam	Integrated trace lasso and ALO algorithm
Chen et al. [53] 2019	Trace LASSO	Modal frequencies and mode shapes	Manual result comparison	ALO-IMN algorithm,	Numerical and experimental	Truss and beam	Integrated with ALO-IMN algorithm

. Zhang et al. [42] utilized reweighted l₁-regularization for damage detection by introducing specifically determined weighting factors. This approach approximates the effect of l₀-norm, thereby enhancing sparsity while maintaining the convexity of the optimization problem. Subsequently, Xu et al. [43] extended the method by incorporating multiple types of dynamic responses into a reweighted l₁-regularization framework. Zhou et al. [44] proposed an innovative strategy for determining the regularization factor and conducted experimental validation. Bu et al. [45] explored an advanced reweighted l₁-regularization technique with distinct weighting factors that approximate l_1/2-regularization rather than l₀-regularization. Additionally, some researchers directly employed l_p-regularization (0 < p < 1), solving the resulting nonconvex l_p-regularized problem using heuristic algorithms. Pan et al. [46] applied l_p-regularization (0 < p < 1) for damage detection in bridge structures. Yue et al. [47] and Luo et al. [48] conducted a comparative study on l_p-regularization (0 < p < 1) to explore the selection of the order p and its performance in damage detection under various damage patterns. Ding et al. [49] combined l_1/2-regularization with a modified JAYA algorithm, considering the coupling effects of multiple types of uncertainties. Moreover, several alternative sparse regularization techniques have been proposed to combine the advantages of both l₂ and l₁ regularization. Lin et al. [50] proposed a two-stage damage detection approach utilizing adaptive thresholding regression and the elastic net method, which incorporates both l₂ and l₁-norm constraints to obtain refined damage detection results progressively. Hou et al. [51] performed a comparative analysis between the elastic net method and l₁-regularization, demonstrating that the elastic net method is advantageous for detecting grouped damages. Chen et al. [52,53] developed a damage detection method based on trace LASSO, where the solution was obtained through the integration of an ant-lion optimizer (ALO) algorithm and an improved Nelder–Mead (INM) algorithm. The aforementioned novel sparse regularization techniques demonstrate distinct advantages compared to traditional l₁-regularization methods. However, determining the appropriate regularization parameter and resolving the associated inverse problems continue to pose significant challenges. In some cases, these tasks may become even more complex. Moreover, akin to traditional l₁-regularization, the validation of these advanced techniques in complex application scenarios remains underreported in the literature.

To sum up, we note that l₂-regularization offers a robust mechanism for improving model generalization in damage detection applications by mitigating overfitting through the introduction of a penalty term proportional to the square of the model weights. This penalization encourages weight decay, resulting in a smoother model that can better adapt to noise within the data, thus enhancing predictive accuracy and interpretability. However, its limitations include potential underfitting, especially in scenarios where the underlying damage patterns are complex and necessitate a more nuanced model representation. Specifically, l₂- regularization uniformly shrinks all weights, which may not be optimal when certain features have inherently greater importance, leading to loss of critical information pertinent to damage characterization. In contrast, l₁-regularization offers several advantages for damage detection in various applications. The primary benefit of l₁-regularization lies in its ability to induce sparsity in the solution, effectively selecting a subset of relevant features while discarding irrelevant ones. This can enhance interpretability and significantly reduce computational complexity, making it easier to identify damage indicators among numerous potential features. Moreover, its robustness in the presence of noise allows for more reliable identification of damage without overfitting the model. However, l₁-regularization also has limitations, such as its tendency to arbitrarily select one feature over another when they are correlated, which could result in suboptimal feature selection. Addressing these research gaps could significantly advance the understanding and application of l₁-regularization in damage detection frameworks. Consequently, there is a growing demand for systematic evaluations of l₁-regularization’s effectiveness across diverse damage detection scenarios, taking into account the actual practical conditions in structural health monitoring systems, to enhance our understanding of its practical applicability and limitations in real-world applications. Based on the comprehensive review and summary presented above, it is evident that the development of adaptive l₁-regularization methods, which are capable of dynamically adjusting regularization parameters and sparsity for solutions in accordance with specific damage patterns or environmental conditions, constitutes an emerging research direction. Consequently, this paper focuses on a comparative analysis of adaptive l₁-regularization techniques for the application of structural damage diagnosis. The rest of this paper is organized as follows. In Section 2, the theoretical framework of the adaptive sparse l₁-regularization for engineering application is systematically formulated and summarized. Section 3 conducts numerical experiments for parameters influence analysis and performance examination on 2D and 3D models under diverse damage detection scenarios subject to two different natural seismic excitations. Finally, conclusions are drawn in Section 4.

2. Formulations of Adaptive l₁-Regularization for Structural Damage Diagnosis

2.1. Inverse Problem in Damage Detection

The relationship between the input excitations and the structural responses can be modeled as a linear dynamic system for practical application as follows:

$$M\ddot{d}\left(t\right)+C\dot{d}\left(t\right)+Kd\left(t\right)=-LM{\ddot{d}}_s\left(t\right)$$

(1)

where $M$, $C$, and $K$ are, respectively, the global mass, damping, and stiffness matrix; $\ddot{d}\left(t\right)$, $\dot{d}\left(t\right)$, and $d\left(t\right)$ are the acceleration, velocity, and displacement response; ${\ddot{d}}_s\left(t\right)$ is the ground excitation vector; and $L$ is the mapping matrix. By assuming that the elemental stiffness is linearly decreased where the local structural damage occurs, the global stiffness matrix $K^{\mathrm{d}}$ of the damaged structure can be formulated as the weighted summation of each element’s stiffness matrices $K_i$ as shown in Equation (2).

$$K^{\mathrm{d}}={\displaystyle \sum _{i=1}^{\mathrm{ne}}(1+\Delta {\alpha }_i)}{K}_i;\hspace{0.33em}\left(-1\le \Delta {\alpha }_i\le 0\right)$$

(2)

where $\Delta {\alpha }_i\in \Delta \alpha$ is the stiffness variation factor in the $i^{\mathrm{th}}$ element; $\Delta \alpha$ denotes the vector of stiffness variation in the elements; and $\mathrm{ne}$ denotes the total number of the elements in the structure.

According to the aforementioned linear assumption, the measured damage indicator $V^{\mathrm{m}}$ calculated by using the recorded response can be expanded as follows based on the first-order Taylor series expansion.

$$V^{\mathrm{m}}=V^{\mathrm{c}}+{\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial \alpha }}\Delta \alpha +\epsilon$$

(3)

where $V^{\mathrm{c}}$ is the computed damage indicator, and its superscript c denotes that the damage indicator is computed with the responses obtained from the FE model; and $\epsilon$ denotes the effect of measure noise or model error. Equation (3) can be subsequently reformulated as a linear sensitivity-based damage detection equation and incorporates the iterative Gaussian–Newton algorithm for model updating as follows.

$$\Delta V^k=S^k\Delta {\alpha }^{k+1}+\epsilon ;\left(k=0,1,2\dots \right)$$

(4)

with

$$\Delta V^k=V^{\mathrm{m}}-V^{\mathrm{c}}$$

(5)

and

$$S={\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial \alpha }}=\left[{\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial {\alpha }_1}},{\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial {\alpha }_2}},\dots ,{\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial {\alpha }_i}},\dots ,{\displaystyle \frac{\partial V^{\mathrm{c}}}{\partial {\alpha }_{\mathrm{ne}}}}\right]$$

(6)

where $\Delta V$ is the magnitude of deviation between $V^{\mathrm{m}}$ and $V^{\mathrm{c}}$; and $S$ is the Jacobian matrix that denotes the structure’s sensitivity to the stiffness variation, and it is computed by using the finite difference approach from the analytical model [19]. The superscript $k$ in Equations (4) and (5) represents the corresponding number of the iteration; and $\Delta {\alpha }^{k+1}$ is the vector of fractional stiffness variation to be solved at iteration ${(k+1)}^{\mathrm{th}}$, with ${\displaystyle \sum \Delta {\alpha }^{k+1}}=\Delta \alpha$.

2.2. Response Covariance-Based Damage Index

A response covariance-based damage indicator has been developed, and its sensitivity to local damage, as well as its insensitivity to noise contamination in measurements, has been validated [50]. Consequently, the response covariance-based damage index will be utilized in the subsequent sections of this paper for damage detection with a limited number of accelerometers. The response covariance-based damage indicator $V_{pq}$ is formulated as

$$\begin{array}{l}{V}_{pq}=[C_{p_1{q}_1}\left({\tau }_0\right),C_{p_1{q}_2}\left({\tau }_0\right),\dots ,C_{p_i{q}_j}\left({\tau }_0\right),\dots ,C_{p_s{q}_s}\left({\tau }_0\right),C_{p_1{q}_1}\left({\tau }_1\right),C_{p_1{q}_2}\left({\tau }_1\right),\dots ,C_{p_i{q}_j}\left({\tau }_1\right),\dots ,\\ C_{p_s{q}_s}\left({\tau }_1\right),\dots ,C_{p_1{q}_1}\left({\tau }_{nt-1}\right),C_{p_1{q}_2}\left({\tau }_{nt-1}\right),\dots ,C_{p_i{q}_j}\left({\tau }_{nt-1}\right),\dots ,C_{p_s{q}_s}\left({\tau }_{nt-1}\right){]}^{\mathrm{T}}\end{array}$$

(7)

with

$$C_{pq}\left(\tau \right)=\mathrm{E}\left[{\widehat{\ddot{d}}}_p(t){\widehat{\ddot{d}}}_q(t+\tau )\right]=\mathrm{E}\left[{\displaystyle \frac{{\ddot{d}}_p(t)}{{\sigma }_p^0}}{\displaystyle \frac{{\ddot{d}}_q(t+\tau )}{{\sigma }_q^0}}\right]$$

(8)

where $C_{pq}\left(\tau \right)$ represents the normalized cross-covariance function that composes the damage indicator; the subscript $pq$ indicates that the corresponding $C_{pq}\left(\tau \right)$ is calculated with the acceleration data recorded by the sensors p and q, respectively; the subscript i or j denotes the number of the corresponding sensor where $i,j\in \left[1,\mathrm{s}\right]$, and s is the total number of the sensors; ${\ddot{d}}_p(t)$ and ${\ddot{d}}_q(t)$ are the acceleration time history measured by using the sensors p and q; ${\widehat{\ddot{d}}}_p(t)$ and ${\widehat{\ddot{d}}}_q(t)$ are the normalized acceleration time history; ${\sigma }_p^0$ and ${\sigma }_q^0$ are, respectively, the normalization coefficients equal to the standard deviations of ${\ddot{d}}_p$ and ${\ddot{d}}_q$, and their superscript 0 means the data are obtained from the intact structure; $\tau$ and $t$, respectively, represent the time lag and the time variable; nt represents the total number of time lags; and the superscript T is the transposition operation and $\mathrm{E}\left[·\right]$ denotes the expectation. Since responses measured from different sensor locations result in acceleration with varying orders of magnitude, the acceleration responses are normalized in Equation (8) to alleviate the ill-posedness in the following damage identification problem. Computing the damage index $V_{pq}^{\mathrm{m}}$ and $V_{pq}^{\mathrm{c}}$ using the Equations (7) and (8), and then substituting into Equation (4), the damage detection equation cooperating with the response covariance-based damage indicator is finally written as follows.

$$\Delta V_{pq}^k=S^k\Delta {\alpha }^{k+1}+\epsilon ;\left(k=0,1,2\dots \right)$$

(9)

By solving $\Delta {\alpha }^{k+1}$ from Equation (9) and updating $\Delta V_{pq}^k$ and $S^k$ from the FE model in each iteration, the stiffness reduction of each component can be gradually accumulated by $\Delta \alpha ={\displaystyle \sum \Delta {\alpha }^{k+1}}$ after the iteration convergence.

2.3. EfI-Based Optimal Sensor Placement

As outlined in the preceding literature, the configuration of sensors plays a critical role in determining the quality of the measured response data and has a substantial impact on the performance of damage detection. The well-known effective independence (EfI) method [54] is utilized for optimal sensor placement in this paper to ensure high-quality data acquisition and improve the effectiveness of damage detection. The EfI method assesses the effective independence of sensor measurement, ensuring that sensors are strategically positioned to capture diverse and non-redundant information. This approach is particularly advantageous for enhancing the efficiency of sensor networks. For structural damage detection, the EfI method focuses on selecting a specific number of sensor locations that contribute maximally to the independence of the measured mode shapes. Specifically, sensor locations are iteratively eliminated from a candidate list based on their contribution to independence, as expressed by the following vector.

$$EI=\mathrm{diag}\left[\Phi {\left({\Phi }^{\mathrm{T}}\Phi \right)}^{-1}{\Phi }^{\mathrm{T}}\right]$$

(10)

where $\Phi$ is the mode shape matrix corresponding to the current sensor locations, and $\mathrm{diag}\left[·\right]$ denotes the diagonal elements of the matrix. Equation (10) is computed from the current candidate list in each iteration; then, the location corresponding to the smallest elements in $EI$ is removed from the candidate list. The aforementioned computation process is iteratively repeated until the number of remaining sensor locations is reduced to the predetermined threshold. By employing this backward sequential sensor placement strategy, the EfI method is capable of efficiently determining the optimal sensor configuration for complex structures, thereby rendering it highly effective for practical engineering applications.

2.4. Adaptive l₁-Regularization Applied for Damage Diagnosis

Solving Equation (9) directly is often infeasible owing to the ill-posed nature of the associated damage detection problem, which is further exacerbated by uncertainties and the limited number of sensors. Regularization methods can provide an effective approach to address this challenge. The optimization problem of damage detection corresponding to Equation (9) can typically be formulated by using l₂- or l₁-regularization as

$$\underset{\Delta {\alpha }^{k+1}}{\mathrm{Minimize}}\left[{‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2+\lambda {‖\Delta {\alpha }^{k+1}‖}_2\right]$$

(11)

and

$$\underset{\Delta {\alpha }^{k+1}}{\mathrm{Minimize}}\left[{‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2+\lambda {‖\Delta {\alpha }^{k+1}‖}_1\right]$$

(12)

The primary distinction between the two cost functions presented in Equations (11) and (12) lie in their respective adoption of the l₂-norm and l₁-norm as regularization terms. The l₂-norm and l₁-norm constraints exhibit distinct characteristics, which can be elucidated through an example of a two-dimensional programming problem, as illustrated in Figure 3. In the given context, the range indicated by the red line represents the possible solution set, while the range represented by the blue dotted line corresponds to the l₂- or l₁-norm constraint. The intersection point of these two ranges signifies the optimal solution. It is evident that the l₂-norm constraint forms a circular shape, whereas the l₁-norm constraint forms a square. Due to the narrower and sharper range of the l₁-norm constraint, the intersection point is more likely to occur on an axis, implying that some components are zero, thereby resulting in a sparse solution.

Based on the advantages of l₁-regularization, the authors recently developed an adaptive sparse regularization (ASR) method [41]. This method is employed in this paper for structural damage identification. Specifically, the aforementioned regularized problem presented in Equation (12) is specifically formulated as the following optimization problem.

$$\begin{array}{l}\underset{\Delta {\alpha }^{k+1}}{\mathrm{Minimize}}\left[{‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2+{\lambda }^{\ast k+1}{‖{\displaystyle \sum \Delta {\alpha }^{k+1}}‖}_1\right]\\ \mathrm{s}.\mathrm{t}.\hspace{1em}-1\le \Delta {\alpha }_i^{k+1}+{\displaystyle \sum \Delta {\alpha }_i^k}\le 0;\hspace{0.33em}{\lambda }^{\ast k+1}\ge 0\end{array}$$

(13)

where ${‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2$ is the residue norm and represents the data fidelity, while ${‖{\displaystyle \sum \Delta {\alpha }^{k+1}}‖}_1$ is the norm of the solution and constrains its sparsity, both of which make a significant contribution to the correctness and robustness of damage identification. Therefore, to effectively balance the l₂ residue norm and the l₁ solution norm, selecting an appropriate adaptive regularization factor is an essential issue. If the regularization factor is too small, the residue norm will be overemphasized potentially leading to an over-fitting problem for the cases with contamination of the measurement noise or model error. Conversely, if the regularization factor is too large, the solution norm will dominate excessively and lead to an over-smooth solution. As discussed in Section 1, the regularization factor for l₁-regularization is more difficult to determine compared to l₂-regularization methods, and the l₁-regularization factor is often subjectively chosen or set according to empirical strategies without adaptive expressions.

To overcome the difficulty in l₁-regularization factor determination, the ASR method provides an adaptive strategy, with the mathematical expression for selecting the regularization factor ${\lambda }^{\ast k+1}$ being the following.

$${\lambda }^{\ast k+1}={\displaystyle \frac{{‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2}{{‖{\displaystyle \sum \Delta {\alpha }^{k+1}}‖}_1}}\approx {\displaystyle \frac{{‖S^k\Delta {\alpha }_{\mathrm{r}}^{k+1}-\Delta V_{pq}^k‖}_2}{{‖{\displaystyle \sum \left(\Delta {\alpha }^k\right)}+\Delta {\alpha }_{\mathrm{r}}^{k+1}‖}_1}}$$

(14)

The above adaptive regularization factor ${\lambda }^{\ast k+1}$ in Equation (14) is derived under the assumption that the norms of residue and solution are of equal importance for effective damage diagnosis, namely, ${‖S^k\Delta {\alpha }^{k+1}-\Delta V_{pq}^k‖}_2={\lambda }^{\ast k+1}{‖{\displaystyle \sum \Delta {\alpha }^{k+1}}‖}_1$. ${\lambda }^{\ast k+1}$ need to be determined by Equation (14) before solving the damage detection problem in Equation (13), but $\Delta {\alpha }^{k+1}$ remains unknown before Equation (13) is solved. Thus, the terms $\Delta {\alpha }^{k+1}$ and ${\displaystyle \sum \Delta {\alpha }^{k+1}}$ are replaced by their approximate value $\Delta {\alpha }_{\mathrm{r}}^{k+1}$ and ${\displaystyle \sum \left(\Delta {\alpha }^k\right)}+\Delta {\alpha }_{\mathrm{r}}^{k+1}$. Specifically, ${\displaystyle \sum \left(\Delta {\alpha }^k\right)}$ is the vector of accumulated stiffness reduction in the previous k iterations, and $\Delta {\alpha }_{\mathrm{r}}^{k+1}$ is the estimated reference value of the stiffness reduction variation at the ${(k+1)}^{th}$ iteration obtained by

$$\underset{\Delta {\alpha }_{\mathrm{r}}^{k+1}}{\mathrm{Minimize}}\left[{‖S^k\Delta {\alpha }_{\mathrm{r}}^{k+1}-\Delta V_{pq}^k‖}_2\right]$$

(15)

Consequently, an appropriate ${\lambda }^{\ast k+1}$ can be automatically determined in Equation (14) by substituting $\Delta {\alpha }_{\mathrm{r}}^{k+1}$ calculated through Equation (15) at each iteration. The ${\lambda }^{\ast k+1}$ is then substituted into Equation (13) to calculate the ${(k+1)}^{th}$ iterative solution with appreciable data fidelity and sparsity. The above optimal problems Equation (13) and Equation (15) are effectively solved by using the SDPT3 algorithm [55]. The implementation of the ASR-based damage detection method is illustrated in Figure 4, and the computational procedure is detailed in Algorithm 1.

It should be pointed out that the incorporation of adaptive l₁-regularization for diagnosing structural damage within the context of seismic excitation in this paper does not inherently restrict its applicability to environments characterized by significant ground motions alone. This method can be effectively utilized for general damage detection under a wide range of dynamic loads, including operational forces, wind loads, and vibrations induced by machinery. In routine operations, adaptive l₁-regularization can be utilized in structural health monitoring to analyze data collected under operational conditions. This enables the early detection of damage prior to significant deterioration. Specifically, this method is predominantly employed to improve the estimation of structural parameters while promoting sparsity in the damage detection solution. By minimizing the residuals between the observed and predicted responses of a structure under various loading conditions, it enhances the accuracy of damage identification.

Algorithm 1. The implementation of the ASR-based damage detection method

① Establish the FE model, set $k=0$ and $\Delta {\alpha }^k=0$. ② Conduct the optimal sensor placement. ③ Measure the structural responses from optimally placed sensors, and compute the measured damage indicator $V^{\mathrm{m}}$.

for $k=1:k_{\max }$

④ Compute the structural response from the FE model;

⑤ Construct the computed damage indicator $V^{\mathrm{c}}$, compute the sensitivity $S^k$ and the deviation $\Delta V_{pq}^k$;

⑥ Solve Equation (15) and obtain the reference value $\Delta {\alpha }_{\mathrm{r}}^{k+1}$;

⑦ Determine the adaptive regularization factor ${\lambda }^{\ast k+1}$ through Equation (14);

⑧ Solve Equation (13) and obtain $\Delta {\alpha }^{k+1}$, and compute the total stiffness reduction ${\displaystyle \sum \left(\Delta {\alpha }^{k+1}\right)}$;

⑨ Update the FE model according to the total stiffness reduction until achieving the stopping criterion.

end

⑩ Output the damage detection result.

2.5. Simulation of Measurement Noise and Model Error

Structural damage identification inherently represents a challenging inverse problem, largely due to its sensitivity to various uncertainties, particularly measurement noise and model discrepancies. In practical engineering applications, measurement noise arising from sensor inaccuracies and environmental factors, combined with model inaccuracies caused by idealized assumptions, is inevitably present. These uncertainties further complicate the damage detection problem, often resulting in non-unique or unstable solutions. To ensure the robustness and applicability of damage identification methods, it is crucial to systematically incorporate noise and model errors into the subsequent numerical experiment. The measurement noise and model error are simulated by adding a Gaussian random component to the unpolluted structural responses and the material elasticity modulus vector, respectively. Specifically, the polluted structural responses [19] and the material elasticity modulus with model error [41] can be simulated as follows:

$${\ddot{d}}^{\mathrm{m}}={\ddot{d}}^{\mathrm{c}}+{\mathrm{N}}_{\mathrm{p}}\times \left[\mathrm{std}({\ddot{d}}^{\mathrm{c}})\times N_{\mathrm{oise}}\right]$$

(16)

and

$$E_{\mathrm{err}}=\mathrm{diag}({1+\mathrm{E}}_{\mathrm{p}}{\epsilon }_e)\times E_0$$

(17)

where ${\ddot{d}}^{\mathrm{m}}$ and ${\ddot{d}}^{\mathrm{c}}$ denote the polluted and unpolluted acceleration response, ${\mathrm{N}}_{\mathrm{p}}$ is the noise level, $\mathrm{std}(·)$ denotes the standard deviation, and $N_{\mathrm{oise}}$ is the standard Gaussian random noise times series. $E_{\mathrm{err}}$ and $E_0$ denote the material elasticity modulus vector with and without model error, ${\mathrm{E}}_{\mathrm{p}}$ is the model error level, $\mathrm{diag}(·)$ denotes the diagonal matrix, and ${\epsilon }_e$ is the standard Gaussian random model error vector. Such simulations not only validate the method’s resilience under realistic conditions but also provide insights into parameter selection and regularization strategies to mitigate adverse effects caused by these uncertainties. For all the case studies in the subsequent numerical experiment, the noise level ${\mathrm{N}}_{\mathrm{p}}$ is set to 10%, and the model error level ${\mathrm{E}}_{\mathrm{p}}$ is set to 1%.

3. Numerical Experiment for Comparative Analysis and Performance Evaluation

3.1. Numerical Study on a 2D Beam Structure

3.1.1. Description of FE Model for the 2D Beam Structure and Seismic Excitation

To illustrate the characteristics and effectiveness of the proposed ASR method under various damage scenarios subjected to ground excitation with differing amounts of modal information, a 2D beam structure was utilized in the subsequent numerical experiment. The FE model of the beam is constructed as shown in Figure 5, comprising 41 nodes and 40 equal-length Euler–Bernoulli beam elements. Additionally, eight accelerometers, represented by blue circles on the model, are optimally positioned using the previously mentioned EfI method. Furthermore, the slashed elements are selected as the preset damage locations for subsequent study, based on the theoretical experience that maximum stress and strain values predominantly occur near the supports of the beam. More detailed structural information is provided in

Table 4. The properties of the FE model of the 2D beam structure.

Properties	Specifications
Element length	100 mm
Element width	50 mm
Element height	9.5 mm
Elastic modulus	$2.05\times 10^{11}$ Pa
Damping ratios (${\xi }_1={\xi }_2$)	0.02
Damage scenarios	DS1: E12(20%) DS2: E12(20%), E13(30%) DS3: E11(15%), E12(20%), E13(15%)

, while the first eight natural frequencies of the model are listed in

Table 5. The first eight natural frequencies of the 2D beam structure.

Mode No.	Frequency (Hz)
1	3.06
2	5.06
3	10.51
4	26.01
5	39.01
6	43.97
7	61.15
8	95.65

. Given that the performance of the ASR-based damage detection method under wide-band concentrated white noise excitation with a fixed sampling rate of 200 Hz has been investigated in a prior study [41], this research further examines damage detection under narrow-band ground excitation and evaluates the impact of incorporating varying amounts of modal information on the accuracy of damage detection results. In particular, the beam model is subjected to seismic forces derived from the 2011 Tohoku earthquake [56,57], a catastrophic event with a recorded magnitude of 9.0 that caused extensive destruction along Japan’s northeastern coast. The peak ground acceleration (PGA) of this seismic excitation is scaled down to 1 m/s² for the subsequent study, as illustrated in Figure 6. It is known that the inclusion of different levels of modal information may significantly affect the performance of structural damage detection. By adjusting the sampling rates to control the incorporated modal information, the sampling rates are set at 200 Hz, 100 Hz, and 50 Hz, respectively, covering 8, 6, and 3 natural frequencies of the structure in the subsequent comparative study.

As depicted in Figure 5, three distinct damage scenarios were designed to investigate the impact of varying damage patterns on the proposed ASR method. The first damage scenario (DS1) comprises a single isolated damage element (E12 with a 20% stiffness reduction). The second damage scenario (DS2) involves two grouped damage elements (E12 with a 20% stiffness reduction and E13 with a 30% stiffness reduction). The third scenario (DS3) encompasses three concentrated damage elements (E11 and E13 with a 15% stiffness reduction, and E12 with a 20% stiffness reduction). In these scenarios, both the number of damaged elements and the severity of damage progressively increase, encompassing both isolated and multiple-grouped damage cases. Furthermore, 10% measurement noise and 1% model error were introduced and simulated as described in Section 2.5. Consequently, the effectiveness and robustness of the proposed method can be validated across a variety of complex damage scenarios.

3.1.2. Comparison Study Between the l₁-Regularization and l₂-Regularization

A comparative investigation with Tikhonov regularization (TR) [13] is conducted to validate the effectiveness of the proposed ASR-based damage detection method [41] in scenarios involving various damage patterns. The damage detection results are illustrated in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and summarized in

Table 6. The damage detection results of the three damage scenarios by using the TR- and ASR-based methods.

Damage Scenario	Element No.	Stiffness Reduction (%)
		Preset Value with Model Error	Detected Damage
			TR	ASR
DS1	E12	20.74	10.96	22.07
DS2	E12	20.74	22.78	22.28
	E13	30.55	24.60	30.05
DS3	E11	16.97	10.71	0.00
	E12	20.74	13.92	36.43
	E13	15.55	11.84	0.00

.

• CASE 1-1: Damage scenario with an isolated damage

The evolution of the accumulated stiffness reduction in the damaged elements and the primary false alarms in DS1 is illustrated in Figure 7. By employing the TR-based damage detection method, the approximate location of the damage can be identified; however, the severity of the damage is underestimated and erroneously distributed to undamaged elements. The curve representing the stiffness reduction in the damaged element E12 increases rapidly during the initial iterations and gradually approaches the red dashed line, which denotes the preset stiffness reduction as shown in Figure 7a. Additionally, false alarms are observed in adjacent undamaged elements E11 and E13, persisting throughout the iterative process. In contrast, when utilizing the ASR-based damage detection method, the curve of the stiffness reduction in the damaged element converges quickly toward the red dashed line as depicted in Figure 7b, with only one significant false alarm occurring in E20 throughout the iterative process. Figure 7 demonstrates the superior performance of ASR compared to the alternative method, emphasizing its enhanced accuracy and robustness in detecting isolated damage under conditions of 10% noise and 1% model error contamination.

The final damage detection results are obtained in the last iteration and presented in Figure 8 and Table 6. By utilizing the TR-based method, the detected damage corresponds to a 10.96% stiffness reduction in element E12. Although the preset damage location is accurately identified, the severity of the damage is significantly underestimated by nearly 10%. Additionally, false alarms are observed in the adjacent intact elements E11 and E13, with respective stiffness reductions of 5.36% and 9.41%. These findings indicate that the TR-based method tends to produce non-sparse solutions, which may lead to false alarms in isolated damage detection scenario. In contrast, the proposed ASR-based method demonstrates a tendency to provide sparse solutions, thereby enabling more accurate identification of isolated damage. Using the ASR-based method, the detected sparse damage corresponds to a 22.07% stiffness reduction in element E12. A single notable false alarm is observed in element E20, with a stiffness reduction of 5.26%, primarily attributed to the cumulative effect of model errors in other elements. The comparison of damage identification results shown in Figure 8 highlights the superior performance of the proposed ASR-based method, which provides a more accurate and sparse solution compared to the traditional TR-based method in scenarios involving isolated damage detection under significant model error and noise contamination.

• CASE 1-2: Damage scenario with two grouped damages

The evolution of the accumulated stiffness reduction in the damaged elements and the primary false alarms in DS2 is illustrated in Figure 9. By employing the TR-based damage detection method, both the damage location and severity are successfully identified in this case. The curves representing the stiffness reduction in the damaged elements E12 and E13 gradually approach the red dashed line, which indicates the preset stiffness reduction as shown in Figure 9a. However, a significant false alarm occurs in the adjacent undamaged element E14 and persists until the final iteration. Additionally, all these curves converge to the true value at a relatively slow rate. In contrast, when using the ASR-based damage detection method, the curves of stiffness reduction in the damaged elements E12 and E13 are closer to the red dashed lines, as depicted in Figure 9b. Furthermore, false alarms are small, and all curves stabilize after convergence. The comparison of the evolution process presented in Figure 9 highlights the superior accuracy and stability of the ASR method.

The final damage detection results are obtained in the last iteration and presented in Figure 10 and Table 6. By utilizing the TR-based method, the detected damage includes a 22.78% stiffness reduction in E12 and a 24.60% stiffness reduction in E13. A significant false alarm is observed in E14, with an 8.46% stiffness reduction. In contrast, by adopting the proposed ASR-based method, the detected damage includes a 22.28% stiffness reduction in E12 and a 30.05% stiffness reduction in E13. The maximum false alarms occur in E20, with a 2.82% stiffness reduction. Damage scenarios involving multiple grouped damages are more advantageous for the TR-based method. Compared to DS1, it can be observed from the results that the performance of the TR-based method has improved in this case, and its drawback of providing non-sparse solutions is concealed. Despite being under the grouped damage condition, the ASR-based method still outperforms the TR-based method in terms of accuracy and robustness. The effectiveness of the proposed method in detecting grouped damage under significant noise and model error contamination is thus validated.

• CASE 1-3: Damage scenario with three concentrated damages

The evolution of the accumulated stiffness reduction and the final damage identification results for DS3 are, respectively, illustrated in Figure 11 and Figure 12. By employing the TR-based damage detection method, the approximate locations of damage were accurately identified; however, the severity of the damage was not precisely quantified. As shown in Figure 11a, the stiffness reduction curves for the damaged elements E11, E12, and E13 exhibited rapid increases during the first two iterations. Subsequently, these curves gradually decreased and stabilized at a value approximately 5% lower than the preset value. Notably, at least four false alarms were observed in adjacent elements, with minimal variation throughout the iterative process. In contrast, by utilizing the ASR-based damage detection method, the approximate locations of damage were also correctly identified; however, the three concentrated damages were misidentified as a single isolated damage. According to Figure 11b, the stiffness reduction curves for the preset damaged elements increased during the first three iterations. Thereafter, the curves for E11 and E13 decreased, while the curve for E12 continued to increase and remained stable above the red dashed lines. A significant false alarm occurred in E10 and persisted until the end of the process. It is evident from Figure 11 that the TR-based and ASR-based methods exhibit distinct tendencies in detecting highly concentrated multiple damages, which differently affect the accuracy of the results.

The final damage detection results are obtained in the last iteration and presented in Figure 12 and Table 6. By utilizing the TR-based method, the detected damage includes a 10.71% stiffness reduction in E11, a 13.92% stiffness reduction in E12, and an 11.84% stiffness reduction in E13. Although all preset damages are accurately identified, the severity of the damage is underestimated by approximately 5% for each element. Additionally, non-negligible false alarms occur in adjacent elements E8, E9, E10, and E14, with up to 5% stiffness reduction misidentified in these intact elements. This indicates that the TR-based method tends to produce non-sparse solutions, distributing the detected damage across more elements. Conversely, when employing the proposed ASR-based method, the detected damage includes a 36.43% stiffness reduction in E12 and no stiffness reduction (0.00%) in E11 and E13. A single false alarm, corresponding to a 7.05% stiffness reduction, is observed in E10. These results demonstrate that concentrated multiple damages in E11, E12, and E13 are misinterpreted as an isolated damage in E12 with a higher degree of severity. The ASR algorithm failed to detect the damage in elements E11 and E13 of DS3 primarily due to the high correlation among the damages in E11, E12, and E13. Additionally, factors such as measurement noise, model errors, limited sensor numbers and sampling ratios, as well as narrow-band seismic excitations significantly increased the complexity of accurate damage detection. Under these uncertainties and constraints, the combined effect of the three damaged elements on the measured response resembled that of a single severe damage. As an l₁-regularization-based method, the ASR approach tends to yield sparse solutions. Consequently, the algorithm provided a solution with only a single damaged element (E12) instead of the true solution involving all three damaged elements. Although the ASR method classifies DS3 as a case of severe isolated damage, it nonetheless provides valuable insights into the approximate location and severity of the damage, which can facilitate the issuance of early warnings in practical structural health monitoring applications.

In the aforementioned study, the number of damage elements and the overall severity of damage progressively increase across the three scenarios. The damage patterns observed include both isolated and grouped damage. Additionally, significant measurement noise and model errors were intentionally introduced to systematically investigate their impact on the performance of damage detection. The comparative analysis indicates that the proposed ASR-based method demonstrates superior accuracy and stability in detecting isolated damage compared to the TR-based method. Detecting multiple-grouped damage remains challenging for both methods; however, the ASR-based method retains an advantage when the damage elements are not excessively grouped. In cases of highly grouped damage, both methods can roughly identify damage locations with either concentrated or dispersed distributions, though the severity of damage may be underestimated or overestimated. Overall, the ASR-based method is more effective in identifying various damage patterns under seismic excitation, even in the presence of 10% noise and 1% model error contamination, as compared to the traditional TR-based method.

3.1.3. Comparison Study of Damage Detection with Different Number of Modes

The performance of the ASR-based method in scenarios with varying damage patterns was comparatively analyzed in the preceding section. Additionally, a further comparative study will be carried out to evaluate the effectiveness of the ASR-based method when the sampling ratio is reduced with different model information. In this section, the sampling ratio is set at 200 Hz, 100 Hz, and 50 Hz, thereby covering 10, 6, and 3 structural frequencies, respectively. The details of the modal information can be found in Table 5. Subsequently, the damage detection is conducted, respectively, and the corresponding results are illustrated in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 and summarized in

Table 7. The damage detection result obtained by using different number of modes.

Damage Scenario	Element No.	Stiffness Reduction (%)
		Preset Damage with Model Error	Detected Damage
			200 Hz	100 Hz	50 Hz
DS1	E12	20.74	22.07	22.33	21.99
DS2	E12	20.74	22.28	12.82	25.82
	E13	30.55	30.05	34.95	26.64
DS3	E11	16.97	0.00	24.07	0.00
	E12	20.74	36.43	0.00	36.99
	E13	15.55	0.00	25.98	0.00

.

• CASE 2-1: Damage scenario with an isolated damage

The evolution of the accumulated stiffness reduction in the damaged elements and the primary false alarms in DS1 is illustrated in Figure 13. As observed, the iterative process in cases employing various sampling ratios exhibits similarity according to Figure 7b and Figure 13. The curve corresponding to the damaged element E12 quickly converges toward the red dashed line within the first five iterations and subsequently stabilizes. A solitary minor false alarm has occurred, with its location and severity remaining consistent across cases using different sampling ratios. This indicates that the reduction in modal information does not significantly affect the accuracy and stability of the ASR-based damage detection method in DS1.

The final damage detection results are presented in Figure 14 and Table 7. When the sampling frequency is set to 100 Hz and six natural frequencies are considered, a stiffness reduction of 22.33% is detected in element E12. The maximum false alarm corresponds to a stiffness reduction of 5.90% in element E21. In contrast, when the sampling frequency is reduced to 50 Hz and only three natural frequencies are utilized, the detected damage is a stiffness reduction of 21.99% in element E12, with the maximum false alarm being a stiffness reduction of 6.36% in element E20. As illustrated in Figure 14, the damage detection results obtained using different modal information sets are highly consistent. All three preset isolated damages are accurately identified even at the significantly reduced sampling frequency of 50 Hz. Furthermore, the false alarms induced by the accumulation of modeling errors remain within an acceptable range. Based on the analysis conducted in the preceding section, the proposed ASR-based method demonstrates its effectiveness in detecting isolated damage. Low-order modal information proves to be sufficient for such detection, thereby highlighting the robustness and efficacy of the ASR-based approach in this context.

• CASE 2-2: Damage scenario two grouped damages

The evolution of the accumulated stiffness reduction in the damaged elements and the primary false alarms in DS2 is illustrated in Figure 15. It can be observed that the curves corresponding to the damaged elements E12 and E13 converge toward the red dashed lines within the first few iterations and subsequently remain stable. The stability and convergence rate are unaffected by variations in the sampling ratio. However, unlike the highly accurate identification of the two damages depicted in Figure 9, the discrepancy between the curves of the damaged elements and the red dashed lines becomes more pronounced when the modal information is reduced.

The aforementioned phenomenon is evident in the final damage detection results, as illustrated and compared in Figure 16 and Table 7. When the sampling rate is set to 100 Hz and encompasses six natural frequencies, the detected damage includes a 12.82% stiffness reduction in element E12 and a 34.95% stiffness reduction in another component. A single false alarm occurs in element E10, with a 5.61% stiffness reduction. In contrast, when the sampling rate is reduced to 50 Hz and covers only three natural frequencies, the detected damage comprises a 25.82% stiffness reduction in element E12 and a 26.64% stiffness reduction in element E13. Additionally, one false alarm arises in element E19, with a 3.37% stiffness reduction. Based on these findings, the true damage pre-set in elements E12 and E13 can still be successfully identified even at lower sampling rates, and the occurrence of false alarms remains within an acceptable range. However, the loss of modal information due to the reduced sampling rate leads to diminished damage resolution, making it more challenging to accurately distinguish between the severity levels of adjacent damaged elements. Consequently, the estimated damage severity for the affected elements tends to be more significantly over- or under-estimated compared to the isolated damage scenario in DS1.

• CASE 2-3: Damage scenario with three concentrated damages

The evolution of the accumulated stiffness reduction and the final damage detection results are, respectively, illustrated in Figure 17 and Figure 18. Similar to the scenario depicted in Figure 11 where a sampling ratio of 200 Hz was utilized, only two or one curves increase during the damage detection process and remain above the red threshold lines. This indicates that the proposed method’s ability to detect concentrated multiple damages, as well as its stability and convergence speed, are not influenced by variations in the sampling ratio.

The aforementioned phenomenon is also evident in the final damage detection results, as illustrated and compared in Figure 18 and Table 7. The concentrated damages in E11, E12, and E13 are identified as 24.07%, 0.00%, and 25.98% stiffness reduction with a sampling rate of 100 Hz, respectively, and as 0.00%, 36.99%, and 0.00% stiffness reduction with a sampling rate of 50 Hz. The results presented in Figure 18 indicate that the proposed ASR-based method tends to yield sparse solutions, consolidating grouped damage into fewer locations. As previously explained, the inherent uncertainty and insufficient modal information further complicate the distinction of highly grouped damage. The above findings demonstrate that the modal information in DS3 is already insufficient, and further decreasing the sampling rate has minimal impact. Similar to the scenario with a 200 Hz sampling rate, the true damage location can still be approximately detected even when the sampling rate is reduced.

In the above comparative study, the sampling ratio and the covered mode number are progressively reduced. The lack of modal information has a distinct impact on the damage scenarios with different damage patterns. The comparison indicates that the proposed ASR-based method maintains high accuracy and robustness in detecting isolated damage. The precision of quantifying multiple-grouped damage becomes more sensitive to a reduction in the sampling ratio; however, the identification of approximate damage locations remains successful even when the sampling ratio is significantly insufficient. Overall, the ASR-based method proves to be effective and robust even under conditions with limited modal information.

3.2. Numerical Study on a 3D Frame Structure

The characteristics of the ASR-based damage detection method and the influence of modal information were elaborated in the preceding sections. As outlined in the literature review in Section 1, many popular solution algorithms have been employed to address the regularized damage detection problem, exerting a substantial impact on the method’s performance. In this section, the influence of the incorporated different solution algorithms in the ASR-based method is further investigated through an additional numerical experiment conducted on a more complex 3D frame structure.

3.2.1. Description of FE Model for the 3D Frame Structure and Seismic Excitations

To examine the robustness and effectiveness of the proposed ASR-based method under extreme conditions, a more complex asymmetric 3D model subjected to bidirectional narrowband seismic excitations is adopted for further investigation. Various types of damage elements and patterns are taken into account to create a comprehensive and complicated damage scenario. Additionally, 10% measurement noise and 1% model error are still incorporated. Moreover, the sampling ratio is set at a low level with limited modal information.

Specifically, the FE model of the frame structure is depicted in Figure 19a, featuring a five-span and three-story configuration with a total of 28 nodes and 39 Euler–Bernoulli beam elements. The optimal sensor placement is illustrated in Figure 19b, determined through the application of the EfI method and comprising 15 accelerometers arranged in two horizontal directions. With a sampling frequency set at 50 Hz, the configuration captures up to 15 structural modes. Additional details regarding the structural properties are provided in

Table 8. The properties of the FE model of the 3D frame structure.

Properties	Specifications
Cross section area ${\mathrm{A} (\mathrm{m}}^2)$	3.14 × 10−2
Inertia moment ${\mathrm{I}}_{\mathrm{y}}{(\mathrm{m}}^4)$	7.85 × 10−5
Inertia moment ${\mathrm{I}}_{\mathrm{z}}{(\mathrm{m}}^4)$	7.85 × 10−5
Torsion constant ${\mathrm{J} (\mathrm{m}}^4)$	1.57 × 10−4
Elastic modulus $\mathrm{E} \left(\mathrm{Pa}\right)$	2.10 × 1011
Shear modulus $\mathrm{G} \left(\mathrm{Pa}\right)$	E/2.6
Material density ${\mathsf{\rho } (\mathrm{kg}/\mathrm{m}}^3)$	7.8 × 103
Damping ratios (${\xi }_1={\xi }_2$)	0.02
Damage scenario	E1(30%), E5(20%), E14(20%), E16(20%), E34(20%)

, while the first 16 natural frequencies of the model are listed in

Table 9. The first 10 natural frequencies of the 3D frame structure.

Mode No.	Frequency (Hz)	Mode No.	Frequency (Hz)
1	3.37	9	13.65
2	3.62	10	16.14
3	4.93	11	16.94
4	9.25	12	18.03
5	9.98	13	19.96
6	10.42	14	22.37
7	11.82	15	23.69
8	12.91	16	27.43

. The frame model is subjected to El Centro seismic excitation, with the peak ground acceleration (PGA) rescaled to 1 m/s², as depicted in Figure 20. In subsequent analyses, a comprehensive damage scenario is considered, encompassing both isolated and grouped damaged elements across various directions and stories, as shown in Figure 19a. Specifically, the damaged elements include column E1 with a 30% stiffness reduction on the first story, column E34 with a 20% stiffness reduction on the third story, and grouped beams and columns (E5, E14, and E16) with a 20% stiffness reduction on the first story.

To investigate the influence of solution algorithms, four popular algorithms are selected and integrated with the proposed ASR-based method. These include the originally adopted SDPT3 algorithm (a convex optimization algorithm) in the reference [41], the active set algorithm (another convex optimization algorithm utilized in references [29,30,33,46,51]), the particle swarm optimization (PSO) algorithm (a heuristic algorithm employed in references [47,48]), and the I-JAYA algorithm (a heuristic algorithm applied in references [36,49]). The effectiveness of these algorithms is compared and observed in the following study.

3.2.2. Comparison Study of Damage Detection Using Different Solution Algorithms

The evolution of the accumulated stiffness reduction achieved by the four algorithms for the 3D frame structure is, respectively, illustrated in Figure 21. As shown in Figure 21a, the detected stiffness reduction rapidly increases to a plateau within the initial few iterations, after which the five curves gradually converge toward the red reference lines while other curves progressively diminish. Despite minor fluctuations during the iteration process, the original ASR-based method utilizing the SDPT3 algorithm successfully identifies all five damaged elements with high accuracy and stability. In Figure 21b, when employing another convex optimization algorithm based on the active set approach, the evolution of the detected stiffness reduction exhibits a pattern similar to that in Figure 21a, achieving equally satisfactory precision and stability. By contrast, the evolution of the detected damage obtained using heuristic algorithms, as depicted in Figure 21c,d, is less predictable. Specifically, when using the PSO algorithm, the iterative process remains relatively stable, with only one instance of an unpredictable sudden change observed in Figure 21c. However, this phenomenon becomes more pronounced in Figure 21d, where the I-JAYA algorithm is employed. Although the true damage is roughly identified, unpredictable sudden changes occur throughout the entire iterative process.

As outlined in Section 3.2.1, the current numerical experiment poses significant complexity and challenges for damage detection under extreme conditions. Fluctuations in the damage detection results are expected to occur throughout the iterative identification process. Consequently, the final damage detection results in this case are presented statistically with uncertainty quantification. Given that substantial changes in the iterative process of the four algorithms primarily occur within the first 30 iterations, the mean value of the detected stiffness reduction during the last 20 iterations is calculated and adopted as the final damage detection outcome. In the context of uncertainty quantification for damage detection, the error bar serves as a critical tool for conveying the confidence interval associated with results derived from various solution algorithms. Error bars illustrate the variability or uncertainty in the estimation of damage detection parameters, providing a visual representation of the range within which the true value is anticipated to reside. When utilizing error bars to describe confidence intervals, a conventional approach involves defining them based on a specified confidence level, typically set at 95%. This indicates that the error bars extend from the lower bound to the upper bound of the confidence interval, within which 95% of the solutions are expected to fall if the experiment were replicated multiple times.

Specifically, the final damage detection results are presented and compared in Figure 22 and

Table 10. The detected stiffness reduction obtained from different algorithms and the corresponding computation time.

Method	Stiffness Reduction (%)						Computation Time (Second)
	E1	E5	E14	E16	E34	Maximum False Alarm
ASR + SDPT3	28.19	20.14	20.52	18.76	14.98	2.36 (E9)	399
ASR + Active set	25.05	17.79	21.25	19.86	17.78	4.24 (E4)	139
ASR + PSO	24.99	17.90	21.22	18.97	15.94	4.19 (E4)	25,530
ASR + I-JAYA	29.56	22.05	20.13	20.95	13.36	1.28 (E33)	99,481
Ture damage	29.14	20.44	20.82	20.62	21.30	-	-

. In these visualizations, blue solid dots represent the mean values, while the error bars indicate the confidence intervals. When employing the SDPT3, active set, PSO, and I-JAYA algorithms, the detected stiffness reductions are as follows: 28.19%, 25.05%, 24.99%, and 29.56% for E1; 20.14%, 17.79%, 17.90%, and 22.05% for E5; 20.52%, 21.25%, 21.22%, and 20.13% for E14; 18.76%, 19.86%, 18.97%, and 20.95% for E16; and 14.98%, 17.78%, 15.94%, and 13.36% for E34, respectively. Additionally, the maximum false alarms observed for the respective algorithms are 2.36% in E9, 4.24% in E4, 4.19% in E4, and 1.28% in E33. It is evident that the ASR-based method generally succeeds in detecting all five damaged elements with minimal false alarms. As shown in

Table 11. The uncertainty corresponding to the damaged elements obtained from different algorithms.

Method	Interval Half-Width for 95% Confidence (%)
	E1	E5	E14	E16	E34
ASR + SDPT3	1.61	2.40	2.27	1.66	2.25
ASR + Active set	2.32	1.56	1.88	1.00	2.71
ASR + PSO	3.22	2.31	1.54	1.81	2.59
ASR + I-JAYA	4.29	5.02	7.64	2.98	6.24

, the confidence interval half-widths for the damaged elements, when using the SDPT3, active set, PSO, and I-JAYA algorithms are, respectively, as follows: stiffness reductions of 1.61%, 2.32%, 3.22%, and 4.29% in E1; 2.40%, 1.56%, 2.31%, and 5.02% in E5; 2.27%, 1.88%, 1.54%, and 7.64% in E14; 1.66%, 1.00%, 1.81%, and 2.98% in E16; and 2.25%, 2.71%, 2.59%, and 6.24% in E34. Overall, the SDPT3 algorithm demonstrates superior performance due to its high accuracy, particularly in detecting damage on the first story with minimal false alarms and relatively low uncertainty. The remaining algorithms also provide accurate results with acceptable false alarm rates in general. Notably, the statistical outcomes from the active set and PSO algorithms exhibit significant similarity, the I-JAYA algorithm demonstrates less accurate identification of damage severity in certain elements. Moreover, it is observed that the uncertainty in the damage detection results obtained via the I-JAYA algorithm is substantially higher than those of the other three algorithms, as evidenced by significant error bars in both detected damage elements and some intact elements, indicating reduced reliability. Furthermore, the computational times for different solution algorithms are compared and presented in Table 10. It is noteworthy that heuristic algorithms (PSO and I-JAYA) demand significantly longer computational times, specifically 25,530 s and 99,481 s, respectively, compared to convex optimization algorithms (SDPT3 and active set), which require only 339 s and 139 s, respectively, to achieve comparable damage detection precision. Additionally, solutions derived from heuristic algorithms exhibit randomness, leading to potentially significant variations across multiple runs unless substantial computational resources are available, which is often impractical in engineering applications.

The above supplementary numerical study on the 3D frame structure verified that the proposed ASR-based method remains effective for a more complex structure, even under limited modal information and in the presence of significant noise and model errors. Additionally, the comparative analysis revealed that the ASR-based method is capable of detecting both isolated and grouped damage with satisfactory accuracy when combined with various solution algorithms. Specifically, the results suggest that integrating ASR with convex optimization algorithms, such as SDPT3 and the active set algorithm, offers greater advantages in terms of stability and efficiency compared to integrating it with heuristic algorithms, such as PSO and I-JAYA.

4. Conclusions

The development of the ASR-based method for structural damage detection is acknowledged as one of the pivotal research directions following a thorough review and comparative analysis of existing regularization techniques. Consequently, this paper systematically formulates and summarizes the theoretical framework of the ASR-based method for in-depth comparative studies, validating its feasibility and capability for post-earthquake structural damage diagnosis through extensive and comprehensive numerical experiments. The identification results demonstrate that the ASR-based method outperforms the TR-based method, which exhibits high accuracy and reliability in detecting damage in both simple and complex structures. This holds true for the cases under various damage patterns with significant uncertainties, even under extreme conditions characterized by a limited number of sensors and insufficient modal information. Regarding the application of ASR, comparisons reveal that the proposed ASR-based method maintains high accuracy and robustness in detecting isolated damage, while the precision of quantifying multiple-grouped damage becomes more sensitive to reductions in the sampling ratio. It is evident that providing sufficient modal information can enhance the precision of identifying grouped or highly concentrated damage using the ASR-based method. Furthermore, comparative analysis confirms that the ASR-based method, when integrated with appropriate solution algorithms, is capable of detecting both isolated and grouped damage with satisfactory accuracy and efficacy. Future studies could explore the integration of the proposed method with the Bayesian framework to enhance uncertainty quantification capabilities or achieve complex nonlinear damage detection for post-earthquake structural damage diagnosis. Additionally, adopting a multi-scale finite element model or a surrogate model could facilitate multi-scale damage detection and improve computational efficiency. Moreover, the performance of the proposed method could be further validated by applying it to practical and significant civil structures such as long-span bridges and high-rise buildings.