A joint fraction function regularization model for the structural damage identification

Abstract

The sparse regularization (SReg) model has been prove to be an effective tool for the structural damage identification. However, the SReg model overly penalizes the larger components in the damage parameter leading to extra estimation bias, and it ignores the similarity information among different measurements that is useful for improving the performance of damage identification. To further improve the accuracy of damage identification, this study proposes a joint fraction function regularization model by jointing multiple fraction function regularization models, where fraction function regularizers are utilized to overcome the excessive penalty drawback, and the similarity of different measurements is employed through a data fusion technique. The numerical and experimental study show that, compared with the previous SReg models, the damage identification errors of the proposed model are reduced by 4.15% and 2.12% on average, respectively.

Keywords

Structural damage identification sparse regularization model fraction function regularizer

Introduction

Structural health monitoring (SHM) is a powerful and increasingly prevalent technology for long-term monitoring of structural safety (Chen, et al., 2023a; Wang et al., 2021). The essential part of SHM is structural damage identification, which identifies the structural damage and evaluates the state of the structure using acquirable information (Chen et al., 2023c; Song et al., 2023).

At present, there are many commonly used damage identification methods, such as manual visual damage identification methods (Kim et al., 2017), vibration-based damage identification methods, and image-based damage identification methods (Gao and Mosalam, 2018). Among them, vibration-based methods have been extensively studied because of their advantages of identifying internal structural damages without destruction (Hou and Xia, 2020; Debnath and Das, 2021; Goyal and Pabla, 2016; Chen, et al., 2023b). The vibration-based methods include model updating method (Mottershead and Friswell, 1993) and dynamic fingerprint method (Sun et al., 2019), among which, the model updating method (Sun et al., 2013; Xu et al., 2012; Chatzi and Smyth, 2009) has become one of the main methods for structural damage identification due to its strong interpretability and operability. Model updating method iteratively updates the stiffness parameters in a finite element (FE) model to minimize the discrepancy between simulated and measured vibration characteristics, which improves the accuracy of the FE model in reflecting the actual structural performance. And then the updated stiffness parameters are utilized for damage identification (Li et al., 2023). Moreover, among model updating methods, the model updating method based on sensitivity analysis is particularly important because of its sensitivity to small parameters changes (Mottershead et al., 2011; Marwala, 2010; Chen and Sun, 2021). Unfortunately, since the number of mode orders that can be effectively used is much smaller than the number of elements, the mathematical equation between the modal information and the damage information established through the sensitivity is an underdetermined equation.

Typically, regularization models are employed to solve the above underdetermined equations. For the structures in the early stage of damages, the damage locations, relative to the whole structure, should be spatially local and sparse (Huang and Beck, 2015; Hou et al., 2018b). Considering the spatial sparsity of structural damages, the l₀-norm regularization model is applied to obtain the sparse solution. But the l₀-norm regularization model need to be convex relaxed to the l₁-norm regularization model since the solving of the l₀-norm regularization model is a NP-hard problem. So far, the l₁-norm regularization model has been widely applied in the structural damage identification (Zhang and Xu, 2016; Hernandez, 2014; Wang and Hao, 2015; Cao et al., 2018; Zhou et al., 2015). In addition, the regularization parameter selection method in the l₁-norm regularization model and the Iterative Reweighted Least Squares (IRLS) optimization algorithm are proposed in paper Hou et al. (2018a) and (Bruckstein et al., 2009) to solve the above models, respectively, which makes the solving of the l₁-norm regularization model more complete.

Although the l₁-norm regularization model could characterize the sparsity of structural damages, as stated in paper, it overly penalizes the larger components in the damage parameter leading to extra estimation bias. In fact, the fraction function regularizer can mitigate the drawback of the l₁-norm regularizer by controlling the severity of the penalties and characterizes the sparsity of the structural damages better, which is more consistent with the spatial sparsity of actual structural damages. Moreover, the similarity among different measurements of the same modal parameter of the structure in a short time has been proved to be useful in the paper Li et al. (2023c), where the similarity among different measurements is utilized through a data fusion technique to improve the accuracy of damage identification.

Inspired by the above theories, this paper propose a joint fraction function regularization model for structural damage identification. In the proposed model, the fraction function regularizer are considered to characterize the sparsity of structural damages, and the similarity among different measurements is utilized in damage identification. Compared with the previous models, the drawbacks of the similarity among different measurements being ignored and l₁-norm regularizer excessively punishing are overcome, simultaneously, which further improves the performance of damage identification. The proposed model is solved using the IRLS optimization algorithm. Numerical and experimental study results show that, compared with the previous models, the proposed model effectively reduces the structural damage identification errors.

Method

Background

Sensitivity analysis

The free vibration of an undamped structure with N degrees of freedom is expressed by the following differential equation:

(- λ_{j} M + K) ϕ_{j} = 0, j = 1, \dots, N,

(1)

where M ∈ R^N×N and K ∈ R^N×N represent the mass and stiffness matrices of the structure, respectively; λ_j ∈ R and ϕ_j ∈ R^N are the jth order natural frequency and the mass-normalized mode shape of the structure, respectively. The physical parameters (i.e., M and K) of the structure change when the structure is damaged, and then the modal parameters (i.e., λ_j and ϕ_j) of the structure change. Therefore, the structural damages can be identified by detecting whether the modal parameters of the structure are changed.

In the FM model, the stiffness matrix of the structure in the intact state can be expressed as follows:

K = \sum_{i = 1}^{N_{e l e}} α_{i} K_{i},

(2)

where K_i ∈ R^N×N represents the ith element’s stiffness matrix, α_i represents the ith element’s stiffness parameter, and N_ele represents the number of elements.

When the structure is damaged, the stiffness matrix reduces to

\tilde{K} = \sum_{i = 1}^{N_{e l e}} {\tilde{α}}_{i} K_{i},

(3)

where

{\tilde{α}}_{i}

represents the ith element’s stiffness parameter in the damage state. In the field of damage identification, it is ordinarily assumed that structural damage typically leads to changes in stiffness with negligible changes in mass. Hence, only the changes in stiffness are considered as indicators for identifying and assessing structural damage.

The Stiffness Reduction Factor (SRF) is defined by

θ_{i} = \frac{{\tilde{α}}_{i} - α_{i}}{α_{i}},

(4)

Evidently, θ_i = 0 denotes that the ith element is not damaged and θ_i = −1 denotes that the ith element is completely damaged.

θ = [θ_{1}, \dots, θ_{n}] \in R^{n}

is defined as the damage parameter of the structure, which reflects the locations and severities of the structural damages.

The sensitivity-based model updating method establishes the relationship between damage parameter θ and modal residual R:

S \times θ = R,

(5)

where S is the sensitivity matrix of the modal parameters with respect to the stiffness parameters α_i (i = 1, …, n), and R represent the discrepancy in the structural modal parameters between the intact and damaged states of the structure. The sensitivity matrix

S = [S_{λ}^{1}; \dots; S_{λ}^{j}; \dots; S_{λ}^{m}; S_{ϕ}^{1}; \dots; S_{ϕ}^{j}; \dots; S_{ϕ}^{m}] \in R^{(N_{s} + 1) m \times n}

can be obtained from the partial derivation of each natural frequency λ_j and mode shape ϕ_j to the stiffness parameter α_i of each element, where m is the observed mode orders, and N_s is the number of the measured points (Weng et al., 2011, 2013; Nelson, 1976). In detail,

S_{λ}^{j} = [\frac{\partial λ_{j}}{\partial α_{1}} \frac{\partial λ_{j}}{\partial α_{2}} \dots \frac{\partial λ_{j}}{\partial α_{n}}] \in R^{1 \times n}

, and

S_{ϕ}^{j} = Γ [\frac{\partial ϕ_{j}}{\partial α_{1}} \frac{\partial ϕ_{j}}{\partial α_{2}} \dots \frac{\partial ϕ_{j}}{\partial α_{n}}] \in R^{N_{s} \times n} (j = 1, \dots, m)

, where

Γ \in R^{N_{s} \times N_{s}}

picks the measured degrees of freedom from the calculated mode shapes for the jth order mode shape.

By considering the analysis presented above, the damage parameter θ can be determined by solving equation (5). However, since the number of available mode orders is much smaller than the number of elements, the solving of equation (5) is an underdetermined problem with an infinite number of solutions in mathematics, which needs to be described by a regularization model. The regularization model is expanded in detail in the next subsection.

l₁-norm regularization model for structural damage identification

Based on the sparsity of the structural damage, the sensitivity analysis for damage identification can be described by the following constrained optimization problem:

\min_{θ \in R^{n}} {‖ θ ‖}_{0} s . t . {‖ S \times θ - R ‖}_{2}^{2} < ε,

(6)

where

{‖ θ ‖}_{0}

is the l₀-norm of the damage parameter θ, and ɛ is a positive number approximately equal to 0 so that S × θ ≈ R. The sparsity of θ is controlled by

\min_{θ \in R^{n}} {‖ θ ‖}_{0}

and the accuracy of the damage parameter θ is controlled by

{‖ S \times θ - R ‖}_{2}^{2} < ε

. In order to balance the two effects, model (6) is rewritten as follows:

\min_{θ \in R^{n}} {‖ S \times θ - R ‖}_{2}^{2} + β {‖ θ ‖}_{0},

(7)

where β represents the trade-off between sparsity and accuracy, and Model (7) is called the l₀-norm regularization model. Nevertheless, solving Model (7) becomes a computationally challenging task due to the discrete nature of the l₀-norm. To simplify the solving process, a common approach is to replace the l₀-norm with thel₁-norm, as the l₁-norm is a widely adopted continuous regularizer for promoting sparsity in damage identification. As a result, the l₁-norm regularization model is derived:

\min_{θ \in R^{n}} {‖ S \times θ - R ‖}_{2}^{2} + β {‖ θ ‖}_{1},

(8)

where

{‖ θ ‖}_{1}

is the l₁-norm of the damage parameter θ.

Joint fraction function regularization model for structural damage identification

The proposed model

The structural damages induces alterations in physical parameters, consequently leading to variations of modal parameters (i.e., measurements), hence the similarity among different measurements can be expressed by the similarity of sparseness profile of different damage parameters. In this paper, $\sum_{l < l^{'}} {‖ θ^{l} - θ^{l^{'}} ‖}_{1} (1 \leq l, l^{'} \leq L)$ is used to describe the similarity of sparseness profile of different damage parameters, and then describe the similarity among different measurements, where θ^l denotes the damage parameter obtained from the lth measurement, and L denotes the total number of measurements.

On the other hand, the fraction function regularizer not only provides a more accurate representation of the sparsity of structural damages compared to the l₁-norm regularizer, but also effectively addresses the issue of excessive penalties associated with the l₁-norm regularizer.

The form of the fraction function is shown as follows:

F (σ, x) = \frac{| x |}{| x | + σ},

(9)

where x belongs to the set of real numbers, and σ (where σ > 0) is a parameter utilized to control the degree of approximation to the l₀-norm. In this paper, the fraction function regularizer is used to replace l₁-norm regularizer to avoid the excessive penalty drawback for the damage parameters. That is,

\sum_{i = 1}^{n} F (σ, θ_{i}^{l})

is used instead of

{‖ θ ‖}_{1}

in l₁-norm regularization model (8), where

θ^{l} = [θ_{1}^{l}, \dots, θ_{i}^{l}, \dots, θ_{n}^{l}] \in R^{n \times 1}

denotes the damage parameter obtained from the lth measurement, and

θ_{i}^{l}

denotes the ith component of θ^l.

To sum up, on the basis of l₁-norm regularization model (8), a joint fraction function regularization model (Joint-Fra-Reg) is proposed as follows:

\begin{array}{l} \min_{[θ^{1}, \dots, θ^{l}, \dots, θ^{L}] \in R^{n \times L}} L (θ^{l}, β, λ) \\ = \min_{[θ^{1}, \dots, θ^{l}, \dots, θ^{L}] \in R^{n \times L}} \sum_{l = 1}^{L} (\frac{1}{2} {‖ S \times θ^{l} - R^{l} ‖}_{2}^{2} + β \sum_{i = 1}^{n} F (σ, θ_{i}^{l})) + λ \sum_{l < l^{'}} {‖ θ^{l} - θ^{l^{'}} ‖}_{1} \end{array}

(10)

where R^l denotes the modal residual used in the lth measurement, and β and λ represent the sparsity regularization parameter and the similarity regularization parameter, respectively.

In the model (10), minimization of ${‖ S \times θ^{l} - R^{l} ‖}_{2}^{2}$ ensures the accuracy of the damage parameter θ^l, minimization of $\sum_{i = 1}^{n} F (σ, θ_{i}^{l})$ ensures the sparsity of the damage parameter θ^l, minimization of $\sum_{l < l^{'}} {‖ θ^{l} - θ^{l^{'}} ‖}_{1}$ ensures the similarity of sparseness profile of different damage parameters, and thus ensures the similarity among different measurements.

For the regularizer $β \sum_{i = 1}^{n} F (σ, θ_{i}^{l})$ , on the one hand, as shown in equation (9), σ is smaller, the fraction function could approximate the l₀-norm better, which is more conducive to characterize the sparsity of structural damage parameter θ^l. On the other hand, $F (σ, θ_{i}^{l}) \approx 1$ when $θ_{i}^{l} \neq 0$ . Therefore, same as the l₀-norm, the fraction function primarily captures the count of non-zero components in the damage parameter θ^l, thereby effectively mitigating the drawback that the l₁-norm disproportionately penalizes larger components of θ^l. This enhancement in regularization helps improve the accuracy of damage identification.

For the regularizer $λ \sum_{l < l^{'}} {‖ θ^{l} - θ^{l^{'}} ‖}_{1}$ , different measurements are the same as each other when λ → +∞; the similarity among different measurements is not considered when λ = 0, and the model (10) degenerates into a simple summation of L fraction function regularization models. When L = 1, there are no more θ^l or θ^l′ but only θ, $λ \sum_{l < l^{'}} {‖ θ^{l} - θ^{l^{'}} ‖}_{1}$ no longer exists, and the model (10) degenerates into a fraction function regularization model.

In this paper, the initial value of the hyperparameter σ in the fraction function is set to 0.01. To ensure convergence of the algorithm, a continuation scheme is introduced in the “Optimization algorithm” subsection, which dynamically adjusts the hyperparameter σ during the optimization process.

Optimization algorithm

Joint-Fra-Reg model is a special case of the fused lasso model (Hoefling, 2010). A efficient algorithm for the solution is available in Hocking et al. (2011). Specifically, Joint-Fra-Reg model can be solved in two steps: a sparsification step and a fusion step. And the details as follows:

Firstly, in the sparsification step, λ = 0 and the similarity among different measurements is not considered, provisionally. That is, the following optimization problems is solved:

\min_{[θ^{1}, \dots, θ^{l}, \dots, θ^{L}] \in R^{n \times L}} L (θ^{l}, β) = \min_{[θ^{1}, \dots, θ^{l}, \dots, θ^{L}] \in R^{n \times L}} \sum_{l = 1}^{L} (\frac{1}{2} {‖ S \times θ^{l} - R^{l} ‖}_{2}^{2} + β \sum_{i = 1}^{n} F (σ, θ_{i}^{l}))

(11)

IRLS algorithm (Bruckstein et al., 2023b) is employed to solve the model (11). The gradient of the damage parameter θ^l obtained from the lth measurement is given by the following equation:

\nabla_{θ^{l}} L (θ^{l}, β) = S^{T} \times (S \times θ^{l} - R^{l}) + β \times {(θ^{l})}_{F} \times θ^{l},

(12)

where

{(θ^{l})}_{F} ≜ d i a g (\frac{F^{'} (σ, θ_{i}^{l})}{\sqrt{{(θ_{i}^{l})}^{2} + c}}),

(13)

Specifically,

F^{'} (σ, θ_{i}^{l})

denotes the derivative of

F (σ, θ_{i}^{l})

with respect to

θ_{i}^{l}

, and c is a positive constant approximately equal to zero. Equation (12) can be further expressed as

\nabla_{θ^{l}} L (θ^{l}, β) = H_{F}^{l} \times θ^{l} - S^{T} \times R^{l},

(14)

where

H_{F}^{l} = S^{T} \times S + β \times {(θ^{l})}_{F}

It is worth noting that the derivative of $| θ_{i}^{l} |$ does not exist when $θ_{i}^{l} = 0$ . To solve this problem, an approximation skill is employed in this paper by adding a positive constant approximately equal to zero to ${(θ_{i}^{l})}^{2}$ i.e.,

| θ_{i}^{l} | \approx \sqrt{{(θ_{i}^{l})}^{2} + c},

(15)

where

\lim_{c \to 0^{+}} \sqrt{{(θ_{i}^{l})}^{2} + c} = | θ_{i}^{l} |

. By this approximation skill, the derivative of

| θ_{i}^{l} |

with respect to

θ_{i}^{l}

can be approximated as

\frac{d | θ_{i}^{l} |}{d θ_{i}^{l}} \approx \frac{θ_{i}^{l}}{\sqrt{{(θ_{i}^{l})}^{2} + c}} .

(16)

According to quasi-Newton (Bruckstein et al., 2023b), the iterative formula can be represented as follows:

\begin{array}{l} {θ^{l}}^{[k + 1]} = {θ^{l}}^{[k]} - h \times {(H_{F}^{l})}^{- 1} \times (H_{F}^{l} \times {θ^{l}}^{[k]} - S^{T} \times R^{l}) \\ = (1 - h) \times {θ^{l}}^{[k]} + h \times {(H_{F}^{l})}^{- 1} \times S^{T} \times R^{l}, \end{array}

(17)

where the step size, denoted as h, is determined in this paper based on empirical knowledge, and it is set to 0.8. Apparently, when h = 1, the iterative equation (17) can be simplified as

{θ^{l}}^{[k + 1]} = {(H_{F}^{l})}^{- 1} \times S^{T} \times R^{l}

Considering that most components of the damage parameters are 0 or close to 0, to reduce complexity and enhance algorithm stability, the iterative calculations in the corresponding variables focus only on the key components. Specifically, we define a specific setting as follows:

M^{l} = {t | | θ_{t}^{l} | \geq η \times \max_{i} | θ_{i}^{l} |},

(18)

where

η \in [0,1)

, set to 0.005 in this paper, is the threshold parameter. By the above process,

{θ^{l}}^{[k]}

and S are reduced to

{θ^{M^{l}}}^{[k]}

and

S^{M^{l}}

, respectively, and are denoted by

{θ^{M^{l}}}^{[k]} = {{θ_{i}^{l}}^{[k]} | i \in M^{l}},

(19)

and

S^{M^{l}} = {s_{i} | i \in M^{l}},

(20)

where

{θ_{i}^{l}}^{[k]}

is the ith component of θ^l in the kth iteration and s_i is the ith column of S. The algorithm replaces the original iteration steps with the following procedure:

H_{F}^{l} = {(S^{M^{l}})}^{T} \times S^{M^{l}} + β d i a g (\frac{F^{'} (σ, θ_{i}^{l})}{\sqrt{{(θ_{i}^{l})}^{2} + c}}) (i \in M^{l}),

(21)

and

{\begin{cases} {θ^{M^{l}}}^{[k + 1]} = (1 - h) \times {θ^{M^{l}}}^{[k]} + h \times {(H_{F}^{l})}^{- 1} \times {(S^{M^{l}})}^{T} \times R^{l}, \\ θ^{\bar{M^{l}}} = 0, \end{cases}

(22)

where

{\bar{M}}^{l}

is the complement of M^l.

Obviously, the l₀-norm can be well approximated by the fraction function $F (σ, θ_{i})$ when σ is small, but which makes it very hard to solve Model (11). On the contrary, the approximation accuracy is not good enough when σ is large. Based on which, σ can be adjusted iteratively by a continuation scheme in the numerical algorithm as an adjustable variable to approximate the l₀-norm (Malek-Mohammadi et al., 2016). To achieve the desired accuracy, the algorithm follows a specific procedure. Initially, Model (11) is solved using a relatively large value of σ. The solution obtained from this iteration is then utilized as the initial value for the subsequent iteration. In each subsequent iteration, Model (11) is solved with a progressively smaller value of σ until the desired level of accuracy is attained.

Secondly, in the fusion step, the components of θ^l and θ^l′ without significant difference are fused. The detailed fusion progress is as follows:

{\begin{cases} {\hat{θ}}_{i}^{l} = {\hat{θ}}_{i}^{l^{'}} = \frac{1}{2} (θ_{i}^{l} + θ_{i}^{l^{'}}), when | θ_{i}^{l} - θ_{i}^{l^{'}} | \leq λ, \\ {\hat{θ}}_{i}^{l^{'}} = θ_{i}^{l^{'}} - s i g n (θ_{i}^{l} - θ_{i}^{l^{'}}) λ and {\hat{θ}}_{i}^{l^{'}} = θ_{i}^{l^{'}} + s i g n (θ_{i}^{l} - θ_{i}^{l^{'}}) λ, else, \end{cases}

(23)

where

s i g n (\cdot)

is the symbolic function,

s i g n (x) = 1

when x ≥ 1,

s i g n (x) = - 1

when x < 1. λ represents the similarity regularization parameter.

{\hat{θ}}^{l} = [{\hat{θ}}_{1}^{l}, \dots, {\hat{θ}}_{i}^{l}, \dots, {\hat{θ}}_{n}^{l}] \in R^{n \times 1}

denotes θ^l after fusion,

{\hat{θ}}_{i}^{l}

denotes the ith component of

{\hat{θ}}^{l}

θ_{i}^{l}

and

θ_{i}^{l^{'}}

are fused to

1 / 2 (θ_{i}^{l} + θ_{i}^{l^{'}})

when

| θ_{i}^{l} - θ_{i}^{l^{'}} | \leq λ

; the difference between

θ_{i}^{l}

and

θ_{i}^{l^{'}}

is large enough, after fusion, appropriate differences are allowed when

| θ_{i}^{l} - θ_{i}^{l^{'}} | > λ

The numerical algorithm, which combines the fusion step with the sparsification step, is presented in Algorithm 1.

Algorithm 1. Optimization algorithm for the joint fraction function regularization model.

2.2.3. Parameter selection

There are two important parameters β and λ in the proposed joint fraction function regularization model. β is the sparsity regularization parameter, and λ is the similarity regularization parameter. As mentioned in the subsection “The proposed model”, β controls the sparsity of the damage parameter θ^l, and λ controls the similarity of sparseness profile of different damage parameters, and thus controls the similarity among different measurements. A two-step parameter selection method is employed in this paper Fang et al. (2016) and Du et al. (2016).

In the first step, the sparsity regularization parameter β is selected. In regularization problems, the L-curve criterion is commonly applied for parameter selection. Inspired by previous researches, linear scaling and the corner of the curve is considered to be the best selection for sparsity regularization parameter β (Hou et al., 2018a). During this step, the similarity regularization parameter λ is set to zero, and only a single measurement, denoted as L = 1, is utilized.

The second step involves the selection of the similarity regularization parameter λ. In this paper, a grid search method is employed, where the sparsity regularization parameter β, determined in the first step, remains fixed. Subsequently, the similarity regularization parameter λ is selected based on this fixed β (Du et al., 2016).

The experimental results show that there is no interaction among different $θ^{l} (l = 1, \dots, L)$ when λ < 10⁻⁴; different $θ^{l} (l = 1, \dots, L)$ are the same as each other when λ > 10⁻¹. Therefore, λ will be selected in the range of $[10^{- 4}, 10^{- 1}]$ . Specifically, the parameters are turned in the candidate set ${10^{- 4}, \frac{10^{- 3}}{2}, 10^{- 3}, \frac{10^{- 2}}{2}, 10^{- 2}, \frac{10^{- 1}}{2}, 10^{- 1}}$ in this paper, and the λ with the smallest error between the estimated damage parameter $\hat{θ}$ and the real damage parameter θ is selected as the best value of the similarity regularization parameter. In this paper, the value of λ is determined to be 0.005, which is considered the most suitable choice.

Numerical study

In numerical studies, a planer simply supported truss is utilized to illustrate the effectiveness of the proposed model. The proposed model and all numerical experiments are executed using Matlab on a computer equipped with an Intel(R) Core(TM) i7-7700 CPU operating at a frequency of 3.60 GHz, and 8 GB of memory.

Model description

The numerical study involves a planer simply supported truss structure composed of 31 elements with 25 degrees-of-freedom. The configuration of the truss is depicted in Figure 1. The material/geometry properties are given as follows: mass density is $2.77 \times 10^{3} k g / m^{3}$ , Young’s modulus is $7 \times 10^{10} N / m^{2}$ , the cross section of each bar is 0.05 × 0.05 m², and the length of the diagonal bar is 2.12 m.

Figure 1.

Configuration of the planer simply supported truss.

Damage scenario simulation

Different groups of natural frequencies and mode shapes of the truss structure can be obtained by changing the Young’s modulus of different elements. In the numerical study, the first six natural frequencies and corresponding mode shapes are utilized for the purpose of damage identification. Two damage scenarios are set to enhance the reliability of the numerical study. The two damage scenarios are listed in Table 1, which are represented as DS1 and DS2, where one damage is set in DS1 and two damages are set in DS2. In the numerical study, the mode shapes at 7 points are measured in the undamaged and damaged states, which are the horizontal displacement at 2th, 6th and 10th pin joints, and the vertical displacement at 4th, 8th, 12th and 14th pin joints.

Table 1.

The damage scenarios in the numerical study.

Damage scenario	No. of element	Damage intensity (%)	SRF
DS1	4	9.0	SRF(4) = −0.090
DS2	3	7.0	SRF(3) = −0.070
DS2	7	8.5	SRF(7) = −0.085

In this paper, the relative model error defined as $δ = \frac{{‖ \hat{θ} - θ ‖}_{2}^{2}}{{‖ θ ‖}_{2}^{2}}$ is employed to assess the accuracy of damage identification, where θ and $\hat{θ}$ represent the real and the estimated SRFs. Obviously, a smaller value of δ represents a higher accuracy of damage identification.

Damage identification for damage scenario 1

Firstly, in this numerical study, the L-curve criterion is applied to select the optimal sparsity regularization parameter β in damage scenario 1 for the joint fraction function regularization model (10), joint sparse regularization model (Li et al., 2023c) and fraction function regularization model (Li et al., 2023b), respectively Hou et al. (2018b); Yao et al. (2011). Then the selected parameter was substituted into the corresponding model. Secondly, two different noise levels are introduced into the modal parameters to inspect the performance of the proposed model: Noise Level 1 and Noise Level 2, where the standard deviation of noises in the natural frequency and the mode shape on Noise Level 1 are 1% and 10% of the real data, respectively; the standard deviation of noises in the natural frequency and the mode shape on Noise Level 2 are 2% and 20% of the real data, respectively. All the above noises follow standard normal distribution. Specifically, the noisy modal data are represented as

\hat{D} = (1 + σ μ) D,

(24)

where D and

\hat{D}

represent the noise-free and the noisy modal parameters, respectively, and σ and μ represent the standard deviation ratio and the distribution of noise, respectively. In this study, μ follows a standard normal distribution vector with zero mean and unit standard deviation. Noise levels of different measurements in the damage scenario 1 are listed in Table 2.

Table 2.

Noise levels of different measurements in different damage scenarios.

Noise level of frequency and mode shape	Damage scenario	Mea
1% and 10%	Damage scenario 1	Mea 1
		⋮
		Mea L
	Damage scenario 2	Mea 1
		⋮
		Mea L
2% and 20%	Damage scenario 1	Mea 1
		⋮
		Mea L
	Damage scenario 2	Mea 1
		⋮
		Mea L

Note: “Mea” means “Measurement”.

Figure 2(a) illustrates the variation of the residual norm and solution norm with different values of the regularization parameter β in the fraction function regularization model. The range of β is set from 0.005 to 1 with an increment of Δβ = 0.005. Based on the L-curve criterion, the parameter range $β \in [0.005,1]$ is deemed appropriate, indicating that the fraction function regularization model exhibits stability in relation to the regularization parameter β within this range. For the joint fraction function regularization model, according to the “Parameter selection” subsection, in the step of selecting the optimal sparse regularization parameter β, the similarity regularization parameter λ is set to zero and only one measurement is used i.e., L = 1. Then the joint fraction function regularization model degenerates into the fraction function regularization model, so the process of selecting the optimal sparse regularization parameters β is the same. Figure 2(a) shows that a suitable parameter range for $β \in [0.005,1]$ . On the other hand, in the joint sparse regularization model, Figure 2(b) illustrates the variations of the residual norm and solution norm for different values of the regularization parameter β, ranging from 0 to 0.05 with an increment of Δβ = 0.001. Applying the L-curve criterion, a proper parameter range for β is determined as $β \in (0,0.032]$ . Regarding the similarity regularization parameter λ in both the joint fraction function regularization model and the joint sparse regularization model, its selection process is described in the “Parameter selection” subsection. For all subsequent studies in this paper, a consistent value of λ = 0.005 is employed.

Figure 2.

Parameter selection and damage identification in DS1 on the truss structure.

The results of 50 repeated tests on Noise Level 1 and Noise Level 2 are presented in Figure 2(c) and (d), respectively. In these figures, the mean values are represented by bars, while the standard deviations obtained from the 50 tests are depicted as short thin horizontal lines.

Figure 2(c) shows the comparison among the damage identification results estimated by the Joint-Fra-Reg model, the Joint-SReg model, the Fra-Reg model on Noise Level 1 and the true damage, which are represented by Joint-Fra-Reg, Joint-SReg, Fra-Reg and True Damage, respectively. The x axis represents 31 elements, and the y axis represents the SRF of each element. As can be seen from the figure, all the three models identify the damage at element 4, but there are intermittent identification errors between element 10 and element 31 in the damage identification results of the Joint-SReg model and the Fra-Reg model. There are obvious errors at element 12 and element 22 in the damage identification results of the Joint-SReg model and an obvious error at element 12 in the damage identification results of the Fra-Reg model. However, there is only one obvious identification error at elements 22 in the damage identification results of the Joint-Fra-Reg model. The damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 3.19%, 3.88% and 3.53%, respectively. So the damage identification quality of the Joint-Fra-Reg model is higher.

The comparison among damage identification results estimated by three models on Noise Level 2 and the true damage is shown in Figure 2(d). The results show that all the three models can identify the damage at element 4. But interfered by noise, there are a certain degree of identification errors between element 11 and element 22 in the damage identification results of three models. The damage severity of element 4 identified by the Joint-SReg model is less than the true damage, which brings certain security risks. At the same time, there is an obvious recognition error at element 22 in the damage identification results of the Fra-Reg model. The damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 5.65%, 6.93% and 7.31%, respectively. Therefore, the damage identification quality of the Joint-Fra-Reg model is higher.

Damage identification for damage scenario 2

Firstly, in this numerical study, the L-curve criterion is applied to select the optimal sparsity regularization parameter β in damage scenario 2 for the joint fraction function regularization model (10), joint sparse regularization model (Li et al., 2023c) and fraction function regularization model (Li et al., 2023b), respectively (Hou et al., 2018b; citealpL-curve). Then the selected parameter was substituted into the corresponding model. Secondly, two different noise levels are designed in this numerical study, and the specific data are the same as the damage scenario 1. In order to simulate the real damage identification better, the two noise levels are introduced into different measurements in the same damage scenario. Noise levels of different measurements in the damage scenario 2 are listed in Table 2.

Figure 3 shows the results of damage identification for the damage scenario 2. By the L-curve criterion, as shown in Figure 3(a) and (b), $β \in [0.005,1]$ is considered as a proper parameter rang of the joint fraction function regularization model, and $β \in (0,0.019]$ is considered as a proper parameter range for joint sparse regularization model.

Figure 3.

Parameter selection and damage identification in DS2 on the truss structure.

The results of 50 repeated tests on Noise Level 1 and Noise Level 2 are presented in Figure 3(c) and (d), respectively. In these figures, the mean values are represented by bars, while the standard deviations obtained from the 50 tests are depicted as short thin horizontal lines.

The comparison among damage identification results estimated by three models on Noise Level 1 and the true damage is shown in Figure 3(c). As can be seen from the figure, all the three models identify the damage at elements 3 and 7, but there are identification errors at elements 4, 19 and 22 in the damage identification results of the Joint-SReg model, and serious identification errors at elements 12, 22, 26 and 31 in the damage identification results of the Fra-Reg model. However, there is no obvious damage error in the damage identification results of the Joint-Fra-Reg model. The damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 1.97%, 7.60% and 6.49%, respectively. So the damage identification quality of the Joint-Fra-Reg model is higher.

The comparison among damage identification results estimated by three models on Noise Level 2 and the true damage is shown in Figure 3(d). The results show that all the three models can identify the damage of elements 3 and 7, but they are disturbed by noise. There are serious identification errors at elements 4, 16, 19 and 22 in the damage identification results of the Joint-SReg model, and several serious identification errors at elements 12 to 31 in the damage identification results of the Fra-Reg model. However, there are only slight damage errors at elements 16, 19 and 22 in the damage identification results of the Joint-Fra-Reg model. The damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 4.70%, 16.25% and 12.21%, respectively. Therefore, the damage identification quality of the Joint-Fra-Reg model is higher.

Finally, Figure 4 summarizes the damage identification errors δ estimated by different models on different noise levels in the two damage scenarios and the reduction of damage identification errors δ of the proposed model compared with the previous models.

Figure 4.

Damage identification results estimated by different models on different noise levels in two damage scenarios in the numerical study: (a) the comparison of damage identification errors δ estimated by different models on different noise levels in two damage scenarios. (b) the reduction of damage identification error δ of the proposed model on different noise levels in two damage scenarios compared with the previous models.

Figure 4 shows that, in damage scenario 1, the damage identification error δ estimated by the Joint-SReg model affected by Noise Level 1 is 3.88%, that estimated by the Fra-Reg model is 3.53%, and that estimated by the Joint-Fra-Reg model is 3.19%. The errors are reduced by 0.69% and 0.34%, respectively. The damage identification error δ estimated by the Joint-SReg model affected by Noise Level 2 is 6.93%, that estimated by the Fra-Reg model is 7.31%, and that estimated by the Joint-Fra-Reg model is 5.65%. The errors are reduced by 1.28% and 1.66%, respectively.

In damage scenario 2, the damage identification error δ estimated by the Joint-SReg model affected by Noise Level 1 is 7.60%, that estimated by the Fra-Reg model is 6.49%, and that estimated by the Joint-Fra-Reg model is 1.97%. The errors are reduced by 5.63% and 4.52%, respectively. The damage identification error δ estimated by the Joint-SReg model affected by Noise Level 2 is 16.25%, that estimated by the Fra-Reg model is 12.21%, and that estimated by the Joint-Fra-Reg model is 4.70%. The errors are reduced by 11.55% and 7.51%, respectively. The numerical study results presented above demonstrate the superior performance of the proposed model in structural damage identification.

Experimental study

Model description

In this section, a cantilever beam as shown in Figure 5(a) is conducted to further validate the effectiveness of the proposed model. The cantilever beam structure used here is presented in Li et al. (2023c). The material/geometry properties of the fixed end beam are given as follows: the length is 0.6 m, the cross-section has a shape of 0.055 × 0.005 m², the Young’s modulus is estimated as 2.1 × 10¹¹ N/m², and the density of the material is 7850 kg/m³.

Figure 5.

The experimental cantilever beam and the experimental equipment: (a) Photo of the cantilever beam; (b) Signal acquisition instrument; (c) dynamic signal amplifier; (d) modal analysis system.

As shown in Figure 5(a), two damage scenarios are prefabricated into the cantilever beam, which are sample 1 and sample 2. All the damage are prefabricated in the form of cuts on the beam structure. As shown in Figure 5(a), it is assumed that the width of the cut is b, the depth is d, and the width of each cut is the same, which is 0.01 m, but the depths are different. Different depths simulate different damage severities, so the damage severity of each cut is reflected by the reduction in the moment of inertia of the cross-section.

The beam is modeled as an Euler-Bernoulli beam with 60 elements (i.e., N_ele = 60), and each element length is 10 mm. As shown in Figure 6, the beam elements are numbered from 1 to 60 from clamped end to free end. Based on the FE model, the details of the two damage scenarios are as follows: there is only one cut on sample 1, located at the location of element 9; there are two cuts on sample 2, located at the locations of element 9 and element 25. The above two damage scenarios are represented by DS1 and DS2, respectively, specific situations are presented in Table 3.

Figure 6.

Configuration of the planer simply supported truss.

Table 3.

The damage scenarios in the experimental study.

Damage scenario	No. of element	Damage depth/m	Damage severity (%)	SRF
DS1	9	0.0075	27.3	SRF(9) = −0.273
DS2	9	0.0075	27.3	SRF(9) = −0.273
DS2	25	0.0075	27.3	SRF(25) = −0.273

In this experiment, the hammering method is adopted for the modal tests. A small hammer with a rubber cover (as shown in Figure 5(b)) is employed to strike the cantilever beam to make it vibrate. The acceleration signals of the cantilever beam vibration are measured by 7 acceleration sensors (the weight of each sensor is 6 g) installed on the cantilever beam using the glue (as shown in Figure 5(a)), and the sampling frequency is 2560 Hz. The testing instruments are shown in Figure 5(b) to (d). A spectrum analyzer (as shown in Figure 5(b)) performs spectrum analysis on acceleration signals after receiving them. Then, the signals are amplified through a dynamic signal amplifier (as shown in Figure 5(c)). After the modal tests, as shown in Figure 5(d), the modal analysis are performed using the uTekMa software (https://www.utekl.com/). The modal test results are listed in Table 4, where “MAC”, “Freq.”, and “Ud freq.” mean “modal assurance criterion”, “Frequency”, and “Undamaged frequency”, respectively. Measurement 1 and measurement 2 are the modal parameters obtained from two serially modal tests, respectively.

Table 4.

Frequencies and MAC of the beam in undamaged state, DS1 and DS2.

Mode	Ud freq. (Hz)	DS1				DS2
		Measurement 1		Measurement 2		Measurement 1		Measurement 2
		Freq. (Hz)	MAC	Freq. (Hz)	MAC	Freq. (Hz)	MAC	Freq. (Hz)	MAC
1	10.03	9.86	96.18	9.91	92.85	9.86	93.95	9.84	98.90
2	62.70	62.57	99.78	62.60	99.65	61.83	99.78	61.79	98.42
3	177.16	177.14	99.94	177.05	99.92	176.49	99.85	176.60	93.61
4	348.09	346.78	99.81	347.48	99.86	344.53	78.09	345.13	98.95
5	573.12	566.78	99.17	569.81	99.41	561.53	99.03	563.56	98.20
6	855.08	845.03	89.23	849.04	91.36	844.93	92.97	845.96	97.96

In order to reduce the error of finite element modeling, the initial finite element model is updated by using the modal data in the undamaged state before damage identification, and regularization is not used in this process. The updated finite element model provides a more accurate reference model for the undamaged structure, which is applied to subsequent damage identification. The natural frequencies and MAC of the updated finite element model and experimental model are listed in Table 5, respectively.

Table 5.

Modal data of the experimental model and initial FE model.

Mode	FE model Freq. (Hz)	Experimental Freq. (Hz)	Freq. difference (%)	MAC
1	10.71	10.03	−6.35	96.56
2	66.07	62.70	−5.10	95.53
3	189.43	177.16	−6.48	96.06
4	368.37	348.09	−5.51	96.21
5	603.59	573.12	−5.05	93.47
6	895.32	855.08	−4.49	91.69

Note: “Freq.” means “Frequency”.

Damage identification for damage scenario 1

Damage scenario 1 is a single damage scenario, and 27.3% damage is introduced in element 9 (i.e., SRF(9) = −0.273). According to the conclusions obtained in the numerical study in this paper, the Fra-Reg model and the Joint-Fra-Reg model are endowed with stability with respect to the sparse regularization parameter β when $β \in [0.005,1]$ . Therefore, in the experimental study, the sparse regularization parameter β of the above two models is the same as the numerical study in this paper (i.e., $β \in [0.005,1]$ ). For the Joint-SReg model, since the experimental data is the same as the paper Li et al. (2023c), the sparse regularization parameter β of the model can be selected by referencing the paper Li et al. (2023c) (i.e., β = 0.0085).

Figure 7 shows the comparison among the damage identification results estimated by the Joint-Fra-Reg model, the Joint-SReg model, the Fra-Reg model and the true damage, which are represented by Joint-Fra-Reg, Joint-SReg, Fra-Reg and True Damage, respectively. The x axis represents 60 elements. The y axis represents the SRF of each element. Figure 7 shows that, for the Joint-SReg model, although the model identifies the damage at element 9, there are different degrees of false damage identification results at elements 1, 8 and 53, so the damage identification error is large. For the Fra-Reg model, although the model identifies the damage at element 9, there are different degrees of false damage identification results at elements 4 and 8, and the damage identification error is large. For the Joint-Fra-Reg model, although there is a small damage identification result at element 8, the model accurately identifies the damage at element 9, which reflects the superiority of fraction function in structural damage identification.

Figure 7.

SRFs of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model in DS1 (the damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 3.67%, 4.24% and 4.83%, respectively).

Damage identification for damage scenario 2

Damage scenario 2 is a multiple damage scenario, and 27.3% damage is introduced into elements 9 and 25 (i.e., SRF(9) = SRF(25) = −0.273). The sparse regularization parameter β of the Fra-Reg model and the Joint-Fra-Reg model is the same as that in damage scenario 1, and the sparse regularization parameter β of the Joint-SReg model is the same as that in the paper Li et al. (2023c) (i.e., β = 0.0045).

Figure 8 shows the comparison among the damage identification results estimated by three models and the true damage. The x axis represents 60 elements. The y axis represents the SRF for each element. Figure 8 shows that, for the Joint-SReg model, although the model identifies the damage of elements 9 and 25, the damage severity identified at element 25 is far less than the true damage severity, which presents a huge security risk for the practical application. In addition, there are large degree of false damage identification results in elements 1 and 10, so the damage identification error is large. For the Fra-Reg model, although the model identifies the damages at elements 9 and 25, there is an obvious false damage identification result at element 8, and the damage identification error still needs to be reduced. For the Joint-Fra-Reg model, there is no obvious false damage identification result and the model accurately identifies the damages at elements 9 and 25, which reflects the superiority of Joint-Fra-Reg model in structural damage identification, especially in multiple damage scenarios.

Figure 8.

SRFs of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model in DS2 (the damage identification error δ of Joint-Fra-Reg model, Joint-SReg model and Fra-Reg model are 12.59%, 17.12% and 14.81%, respectively).

Figure 9 shows the δ estimated by different models in two damage scenarios and the reduction of the damage identification errors δ of the proposed model compared with the previous models. Figure 9(a) and (b) show that, in damage scenario 1, the δ estimated by the Joint-SReg model is 4.24%, that estimated by the Fra-Reg model is 4.83%, and that estimated by the Joint-Fra-Reg model is 3.67%, which reduces the error by 0.57% and 1.16%, respectively. In damage scenario 2, the δ estimated by the Joint-SReg model is 17.12%, that estimated by the Fra-Reg model is 14.81%, and that estimated by the Joint-Fra-Reg model is 12.59%, which reduces the error by 4.53% and 2.22%, respectively. In general, the δ of the proposed model is reduced by 2.55% and 1.69% on average compared with the previous models, respectively.

Figure 9.

Damage identification results estimated by different models in two damage scenarios in the experimental study: (a) the comparison of damage identification errors δ estimated by different models in two damage scenarios; (b) the reduction of damage identification error δ of the proposed model compared with the previous models in two damage scenarios.

Conclusions and discussions

A joint fraction function regularization model for structural damage identification is proposed in this paper. The similarity among the different measurements and the fraction function regularizer are considered in the proposed model. Compared with the previous models, the drawbacks of the similarity among different measurements being ignored and l₁-norm regularizer excessively punishing are overcome, simultaneously, which further improves the structural damage identification performance. The numerical study on a six-span planar truss structure and the experimental study on a cantilever beam structure show that, compared with the previous sparse regularization models, the damage identification errors of the proposed model are reduced by 4.15% and 2.12% on average, respectively.

Although the proposed model further improves the accuracy of damage identification compared with the previous models, there are still some problems that can be optimized in practical applications. Firstly, the sensitivity matrix S is related to the damage state because of the nonlinear relationship between the changes in modal parameters and the damage parameter. Therefore, it is beneficial to update the sensitivity matrix S during the solution process. Future researches should consider the updating of the sensitivity matrix S. Secondly, the influence of temperature on the modal parameters of the structure could be considered. Finally, damage identification methods without considering undamaged structures could be further studied. Future researches could be carried out from the above aspects.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partly supported by the Open Fund of China Highway Engineering Consulting Group limited company (zzyhyfzx-2022-03), the Scientific Research Fund of China Highway Engineering Consulting Group limited company (zzkj-2022-02).

ORCID iD

Rongpeng Li

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A joint fraction function regularization model for the structural damage identification

Abstract

Keywords

Introduction

Method

Background

Sensitivity analysis

l1-norm regularization model for structural damage identification

Joint fraction function regularization model for structural damage identification

The proposed model

Optimization algorithm

2.2.3. Parameter selection

Numerical study

Model description

Damage scenario simulation

Damage identification for damage scenario 1

Damage identification for damage scenario 2

Experimental study

Model description

Damage identification for damage scenario 1

Damage identification for damage scenario 2

Conclusions and discussions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

l₁-norm regularization model for structural damage identification