A fraction function regularization model for the structural damage identification

Abstract

The conventional model updating based on sensitivity analysis generally employs l₁-norm regularizer to characterize the sparsity of the structural damage. However, the l₁-norm regularizer inevitably excessively penalizes the larger components in the damage parameter, which certainly causes the extra estimation bias of the damage parameter and reduces the damage identification accuracy. A fraction function regularizer not only well characterizes the sparsity, but also overcomes the excessive penalty drawback of the l₁-norm regularizer. Based on this, a fraction function regularization model is proposed to improve the damage identification accuracy. Numerical and experimental results illustrate that the damage identification accuracy of the proposed model is averagely improved 4.96% and 3.68% than that of the l₁ regularization one, the iteratively reweighted l₁ regularization one and the elastic net one, respectively.

Keywords

structural damage identification fraction function regularization model updating

Introduction

The structural damage unavoidably arises because of various factors, such as aging, stress concentration, and strong winds. At early stage, most of the damages are difficultly identified by visual inspection. Nevertheless, the disastrous destruction of structures may occur if these damages continue for a long period. Therefore, it is meaningful to study the structural damage identification.

The structural damage identification methods have been extensively researched in recent years. Among them, vibration-based methods are intensively interested because of their easy operation and non-destructive nature (Hou and Xia, 2020; Debnath and Das, 2021; Chen et al., 2019; Wang et al., 2021; Zhu et al., 2014) The basic idea of these methods is that the vibration characteristics (such as modal frequencies and mode shapes) of the structure will change when the damage occurs (Cui et al., 2018a; Wang et al., 2021). Model updating is a widely utilized way in vibration-based damage identification methods. It uses stiffness parameter space as its search space, and the potential values/distributions of the stiffness parameter in an ideal finite element (FE) model are updated by minimizing the discrepancies between the measured and the simulated vibration characteristics (Mottershead et al., 2011). The updated stiffness parameter is utilized for locating and quantifying the structural damage. Among many techniques for model updating (e.g., heuristic algorithm (Sun et al., 2013), least squares based approaches (Xu et al., 2012), and filtering techniques (Chatzi and Smyth, 2002)), the sensitivity analysis is the most outstanding one because of its efficient computational capability and prominent sensitivity to small parameter changes (Mottershead et al., 2011; Zhao and Sun, 2020). However, the mathematical model of the sensitivity analysis-based model updating is generally underdetermined due to the fact that the number of available structural modal parameters is far less than that of structural elements.

Generally, the regularization model has been employed to resolve this underdetermined problem. The earliest sensitivity analysis methods are based on Tikhonov regularization model (Li and Law, 2010; Weber et al., 2011), which employs l₂-norm regularizer to promote the smooth solutions (Jiang et al., 2020; Zhang and Xu, 2016). They are effective for large area surface corrosion or significant destruction. In practice, the structural damage normally happens in a few sections at the early stage. Compared with all the elements of the whole structure, a few sections are spatially sparse (Huang and Beck, 2015; Hou et al., 2017). Therefore, Tikhonov regularization model can’t accurately characterize the sparse damage scenario. In fact, the sparse solution requires that most of items are zero except several nonzero items, which well characterizes the practical spatially sparse damage scenario. Based on this, the l₁ regularization model that based on the l₁-norm regularizer is widely employed to characterize the sparsity of the structural damage. Concretely, the l₁ regularization model only uses natural frequency for damage identification (see, e.g., Hernandez (2014); Wang and Hao (2015); Cao et al. (2018); Zhou et al. (2015); Zhang and Xu (2016)). Although frequencies can be obtained handily and accurately, they are ordinarily insensitive to local damage, which usually causes the false damage localization. Based on the fact that mode shapes involve spatial information and are more sensitive to local damage, many studies employ both natural frequencies and mode shapes for damage identification (Hou et al., 2017, 2021; Chen and Yu, 2019; Li et al., 2022). Moreover, the l₁ regularization method is introduced to accurately identify sparse coefficient solution in Chen et al. (2022), and the paper Hou et al. (2018) studies the parameter selection method of the l₁ regularization model to further improve the damage identification accuracy. In addition, the iteratively reweighted l₁ regularization (IRLR) model is presented for the damage identification in Zhou et al. (2018). More recently, the probabilistic models based on the regularization technique are proposed for the structural damage identification (Hou et al., 2019; Wang et al., 2020a, 2020b). For example, two-stage sensitivity analysis-based damage identification frameworks based on Bayesian l₁ learning are proposed for the reliable damage identification in Zhao and Sun (2020) and Zhao et al. (2020).

Although it has been widely used to promote the sparse solutions for the underdetermined sparsity problem, the l₁-norm regularizer theoretically inevitably overly penalizes the larger components of the damage parameter because it minimizes the sum of absolute values of all components of the damage parameter (Cao et al., 2016; Zhang, 2010). This certainly causes the extra estimation bias of the damage parameter and reduces the accuracy of the damage identification. In fact, l₁-norm is merely the convex relaxation of the l₀-norm that is the optimal sparsity promoting regularizer. Therefore, it is significant to further research other approximations of the l₀-norm that are better than l₁-norm.

The fraction function

F (σ, x) = \frac{| x |}{| x | + σ}

(1)

attracts lots of theoretical studies in recent years (Li et al., 2017; Li and Hamamura, 2014; Cui et al., 2018b, 2019), where x ∈ R, and σ > 0 is the parameter to control the approximate to the l₀-norm. It is easy to verify that F(σ, x) = 0 if x = 0, and

\lim_{σ \to 0} F (σ, x) = 1

if x ≠ 0. The Figure 1 intuitively shows that the fraction function can better approximate the l₀-norm than l₁-norm by choosing a small enough σ.

Figure 1.

Comparison of the l₀-norm, l₁-norm, and fraction function F(σ, p_i). Obviously, fraction function closely matches l₀-norm with a small σ.

Motivated by this, a fraction function regularization model based on the fraction function (1) is proposed for the damage identification to avoid excessively penalizing the larger components of the damage parameters and to improve the damage identification accuracy. The iteratively reweighted least squares (IRLS) optimization algorithm is introduced to solve the consequent optimization problem, wherein the L-curve criterion is employed to choose the regularization parameter. In addition, the continuation scheme as in Malek-Mohammadi et al. (2016) is utilized to adjust the parameter σ. Numerical and experimental results illustrate that the proposed model is more accurate than the l₁ regularization one, the IRLR one and the elastic net one.

The rest of this paper is organized as follows: Section “Method” introduces the background knowledge, and proposes the fraction function regularization model. The performance of the proposed method is evaluated through both numerical and experimental studies in Section “Numerical study” and Section “Experimental study”, respectively. Finally, conclusions are presented in section “Conclusions”.

Method

Background

Model updating via sensitivity analysis

The free vibration of an undamped structure with N degree-of-freedom can be expressed by the eigenvalue equation as

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

(2)

where

[M] \in R^{N \times N}

and

[K] \in R^{N \times N}

are the mass matrix and the stiffness matrix, respectively, λ_r and

{ϕ_{r}}

are the rth order eigenvalue and mass-normalized mode shape, respectively. When the structure changes (e.g., the structural damage), the physical parameters (i.e.,

[M]

and

[K]

) of the structure will change, thereby altering the modal parameters (i.e., λ_r and

{ϕ_{r}}

). Model updating is an effective way to find the change of physical parameters according to the observed change of modal parameters.

In the existing damage identification methods, it is generally assumed that there will be a identifiable change of the stiffness with the mass remaining unchanged when damage occurs. In view of this, only the change of the stiffness is considered to indicate the structural damage in this study.

To begin with, we parameterize the structural stiffness matrix using the stiffness parameters at substructuring level. In the undamaged state, the structural stiffness matrix in FE model can be expressed as follows:

[K] = \sum_{i = 1}^{n} α_{i} [K_{i}],

(3)

where [K_i] ∈ R^N×N is the ith substructure stiffness matrix, α_i is the ith substructure stiffness parameter in the undamaged state, and n is the number of substructures in the FE model.

In the damaged state, we suppose that the stiffness matrix is reduced to

[\tilde{K}] = \sum_{i = 1}^{n} {\tilde{α}}_{i} [K_{i}],

(4)

where

[\tilde{K}] \in R^{N \times N}

, and

{\tilde{α}}_{i}

denotes the substructure stiffness parameter in the damage state. Then,

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

(5)

is defined as the ith substructure’s Stiffness Reduction Factor (SRF), where p_i ∈ [−1, 0]. Obviously, p_i = 0 indicates that the ith substructure is undamaged, and p_i = −1 indicates that the ith substructure is thoroughly damaged. The SRF {P} = {p₁, …, p_n} is defined as the damage parameter of the structure, which signifies both damage location and damage severity of the structure.

According to Taylor series, the relationship between the damage parameter and modal residue can be established. The modal residue describes the discrepancy of structural modal parameters between damaged and undamaged states, and is written as follows:

{R} = {[r_{λ}^{1} \dots r_{λ}^{j} \dots r_{λ}^{m} {r_{ϕ}^{1}}^{T} \dots {r_{ϕ}^{j}}^{T} \dots {r_{ϕ}^{m}}^{T}]}^{T},

(6)

where the superscript “T” symbolizes the transpose,

r_{λ}^{j}

is the residue from jth order eigenvalues,

r_{ϕ}^{j}

is the residue form the jth order mass-normalized mode shapes, and m is the observed mode orders. Here,

r_{λ}^{j}

is further expressed as

r_{λ}^{j} = (λ_{j}^{E} - λ_{j}^{0}) / λ_{j}^{E},

(7)

where

λ_{j}^{E}

and

λ_{j}^{0}

are jth order measured and analytical eigenvalues, respectively. Analogously,

{r_{ϕ}^{j}}

is written as

{r_{ϕ}^{j}} = {ϕ_{M}^{j}} - Γ {ϕ_{F}^{j}},

(8)

where

{ϕ_{M}^{j}} \in R^{N_{p}}

is the jth order measured mode shape at N_p measurement points,

{ϕ_{F}^{j}} \in R^{N}

is the jth order analytical mode shape, and

Γ \in R^{N_{p} \times N}

is the selection matrix with “1” and “0” to select the observed degrees-of-freedom from analytical mode shape.

The relationship between the damage parameter {P} and the modal residue ${R}$ can be approximately expressed with the following sensitivity equation

[S] {P} = {R},

(9)

where

[S] \in R^{(N_{P} + 1) m \times n}

is the sensitivity matrix (or named Jocabian matrix) of modal parameters with respect to stiffness parameter. 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}]

can be calculated by taking the partial derivatives with respect to the each entries of stiffness parameters on both sides of eigenvalue equation (Weng et al., 2011, 2013; Nelson, 1976). In detail,

S_{λ}^{j} = [\partial λ_{j} / \partial p_{1} \partial λ_{j} / \partial p_{2} \dots \partial λ_{j} / \partial p_{n}] \in R^{n}

and

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

Based on the above analysis, the structural damage can be identified by solving the equation (9). But in mathematics, the equation (9) is an underdetermined problem and has infinite solutions due to the number of the available modal parameters is far less than that of structural elements.

Sparse regularization model for the structural damage identification

The structural damage generally occurs at a few substructures only, which indicates that {P} is a sparse vector where only a few components are non-zero. According to the sparse recovery theory, {P} can be obtained by solving the so-called l₀-norm minimization model:

\min_{{P} \in R^{n}} {‖ {P} ‖}_{0} s . t . {‖ [S] {P} - {R} ‖}_{2}^{2} < ε

(10)

where

{‖ {P} ‖}_{0}

is the l₀-norm of vector {P}, and ɛ is subjected to the error. The model (10) is the original model in the sparse recovery theory. In order to balance the sparsity and the error, the model (10) can be equivalently transformed into the following so-called l₀ regularization model:

\min_{{P} \in R^{n}} {‖ [S] {P} - {R} ‖}_{2}^{2} + β {‖ {P} ‖}_{0}

(11)

where β > 0 is the regularization parameter, representing a tradeoff between sparsity and error. Although it is the optimal one for solving the underdetermined sparsity problem, the model (11) is NP-hard because of the discontinuous and discrete nature of the l₀-norm. The general means is to replace the l₀-norm with a continuous sparsity promoting regularizer, and then the corresponding optimization problem is solved using the relevant minimization algorithm (e.g., steepest descent method).

Generally, the l₁-norm is the most widely used continuous sparsity promoting regularizer in damage identification. Using the l₁-norm regularizer, the l₁ regularization model is expressed as follows:

\min_{{P} \in R^{n}} {‖ [S] {P} - {R} ‖}_{2}^{2} + β {‖ {P} ‖}_{1}

(12)

where

{‖ {P} ‖}_{1}

is the l₁-norm of the vector {P}.

Although it has been widely used to promote the sparsity of the damage parameter, the l₁-norm regularizer excessively penalizes the larger components of the damage parameter, which causes the extra estimation bias of the damage parameter and reduces the accuracy of the damage identification. Next, the fraction function regularization model is proposed to improve the damage identification accuracy.

The fraction function regularization model for the damage identification

The proposed model

Based on the fraction function (1), we formulate the fraction function regularizer $\sum_{i = 1}^{n} F (σ, p_{i})$ . Obviously, with σ → +0, $\sum_{i = 1}^{n} F (σ, p_{i})$ can approximate ${‖ P ‖}_{0}$ well. Thus, the fraction function regularizer can be used to characterize the sparsity of the structural damage.

Moreover, with a small σ, F(σ, p_i) ≈ 1 for ∀p_i ≠ 0. Therefore, the fraction function regularizer almost only involves the number of non-zero components of the damage parameter, which effectively avoids to over-penalize the large components in damage parameter. Consequently, the fraction function regularizer not only can characterize the sparsity, but also overcomes the excessive penalty drawback of the l₁-norm regularizer.

Substituting the l₁-norm regularizer with the fraction function regularizer in the damage identification, the fraction function regularization model is given by:

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

(13)

The initial value of σ is set to 10⁻², and the continuation scheme that presented in section “The optimization algorithm” is employed to adjust the parameter σ.

The optimization algorithm

In this section, the IRLS algorithm (Bruckstein et al., 2009) is introduced to calculate the optimization problem (13), where the gradient of the damage parameter ${P}$ is given by:

\nabla_{{P}} L ({P}, β) = {[S]}^{T} ([S] {P} - {R}) + β {P}_{F} {P},

(14)

In equation (14)

{P}_{F} ≜ d i a g (\frac{F^{'} (σ, p_{i})}{\sqrt{p_{i}^{2} + σ}}),

(15)

where F′(σ, p_i) denotes the derivative of F(σ, p_i). Based on this, the equation (14) can be further expressed as

\nabla_{{P}} L ({P}, β) = H_{F} {P} - {[S]}^{T} {R}

(16)

where H_F = [S]^T[S] + β{P}_F.

It should be noted that the derivative of $| p_{i} |$ does not exist if p_i = 0. Here, an approximation skill is adopted to overcome this obstacle by adding a slightly positive constant c to $p_{i}^{2}$ . That is,

| p_{i} | \approx \sqrt{p_{i}^{2} + c},

(17)

where c is the slightly positive constant. Obviously, the equation (17) approaches

| p_{i} |

when c → +0. Based on this approximation skill, the derivative of

| p_{i} |

can be approximated as

\frac{d | p_{i} |}{d p_{i}} \approx \frac{p_{i}}{\sqrt{p_{i}^{2} + c}},

(18)

According to the quasi-Newton (Bruckstein et al., 2009), the iteration formula can be expressed by

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

(19)

where h is the step size, and h is set to 0.8 in this study. Obviously, the formula (19) can be simplified as

{P}^{[k + 1]} = H_{F}^{- 1} {[S]}^{T} {R}

by fixing h = 1.

Considering that most of the elements in damage parameter are 0 or close to 0, we only iteratively calculate the key elements in the corresponding variables, which reduces the complexity and increases the stability of the algorithm. In detail, we set

M = {t : | p_{t} | \geq α \times \max_{i} | p_{i} |},

(20)

where α ∈ [0, 1) is the threshold parameter (α is set to 10 ^{- 5} in this study). With the above process,

{P}

and

[S]

can be simplified as

{P^{M}}

and [S^M], respectively, and they are presented by

{P^{M}} = {p_{i} | i \in M},

(21)

and

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

(22)

where p_i is the ith element of

{P}

, and s_i is the ith column of matrix [S]. The corresponding iteration steps are replaced as

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

(23)

and

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

(24)

where

\bar{M}

is the complement of M.

The parameter σ > 0 controls the approximation to the l₀-norm. Obviously, when σ is small, the fraction function F(σ, p_i) can well approximate the l₀-norm, but this results the strenuous task to solve the model (13). Conversely, with a comparatively large σ, the approximate accuracy is not good. In view of this, a continuation scheme as in Malek-Mohammadi et al. (2016) is employed to obtain the optimal solution of the model (13). In this continuation scheme, σ as a adjustable variable is iteratively adjusted in numerical algorithm to achieve a depth approach to the l₀-norm. In detail, the model (13) is initially solved with a large σ, and the obtain solution is moved as an initial value of the next iteration in which the model (13) is solved with a smaller σ. These iterations continue, until a desired accuracy is achieved. Combining this continuation scheme with the IRLS algorithm, we summarize the pseudo-codes of the numerical algorithm in Algorithm 1.

Algorithm 1 Iterative algorithm for fraction function regularization model

1. Input: sensitivity matrix [S], modal residue

{R}

, the regularization parameter β, the parameter σ,
α, and h, and the termination threshold ɛ₀ and σ_min

2. Initial: the initial solution

{P_{0}} = {[S]}^{T} {([S] {[S]}^{T})}^{- 1} {R}

3. While convergence criterion is not met

4. Update:

M = {t : | p_{t} | \geq α \times \max_{i} | p_{i} |}

5. Update:

{P^{M}} = {p_{i} | i \in M}

and

[S^{M}] = {s_{i} | i \in M}

6. Update:

H_{F} = {[S^{M}]}^{T} [S^{M}] + β d i a g (F^{'} (σ, p_{i}) / \sqrt{p_{i}^{2} + c}) (i \in M)

7. Update:

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

8. Update σ: σ = σ × 0.5

9. Stopping rule: if

{‖ {P}^{(k + 1)} - {P}^{(k)} ‖}_{2}^{2} \leq ε_{0}

or σ ≤ σ_min

10. Until Convergence

11. Output: damage parameter {P}

Numerical study

This section implements a numerical study to verify the effectiveness of the proposed model. The structural model in this numerical study is firstly introduced in this section. Next, the damage scenarios are presented, followed by presentation of the damage identification results. The proposed model and all the experiments are implemented in Matlab on a computer with Inter Core i5-4590 CPU (3.30 GHz) and 4 GB memory.

Model description

A six-bay planar truss structure with 31 bars elements and 25 degrees-of-freedom, as shown in Figure 2, is considered to verify the performance of the proposed fraction function regularization model. The material and geometry properties are shown as follows: mass density and Young’s modulus are $2.77 \times 10^{3} k g / m^{3}$ and $7 \times 10^{10} N / m^{2}$ , respectively, and the length of diagonal bar and the section dimension of each bar are 2.12 m and 0.05 × 0.05 m², respectively. Both the first six-order natural frequencies and the first three-order mode shapes are employed for damage identification in numerical study. For the truss model, 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.

Figure 2.

The geometric configuration of the truss structure.

Damage scenario simulation

As presented in Table 1, two kinds of damage scenarios are simulated by the reduction of Young’s modulus of the elements. Damage Scenario 1 (DS1 in brief) is a multiple slight damage scenario, in which four elements (element 4, 11, 20, and 17) are damaged, and the damage severity of each damaged element is randomly set between 1% and 1.5%. The l₁ regularization model (SReg model in brief) and the IRLR model are taken into account for comparative study. Damage Scenario 2 (DS2 in brief) is a grouped damage scenario. The Elastic net model (Hou et al., 2021), an effective sparse regularization model for identifying the grouped structural damage, is employed for comparative study in the damage identification of damage Scenario 2.

Table 1.

The damage scenarios in numerical study.

Damage scenario	Element no.	Damage intensity (%)	SRF
DS1	4	1.3	SRF(4) = −0.013
	11	1	SRF(11) = −0.01
	20	1.5	SRF(20) = −0.015
	27	1.2	SRF(27) = −0.012
DS2	4, 5	1.1	SRF(4) = SRF(5) = −0.011
DS2	19, 20, 21	1	SRF(19) = SRF(20) = SRF(21) = −0.01

The damage identification error δ is defined for comparing the damage identification accuracy of different models. That is

δ = \frac{{‖ {\hat{P}} - {P} ‖}_{2}^{2}}{{‖ {P} ‖}_{2}^{2}},

(25)

where {P} represents the real SRF, and

{\hat{P}}

represents the estimated SRF. The value of δ reveals the discrepancy between the real SRF and the estimated SRF. A smaller value of δ represents higher damage identification accuracy.

Damage identification for damage scenario 1

Before damage identification, the optimal regularization parameter β is selected using the L-curve criterion to achieve the balance between the data fitting and the sparsity of the damage parameter (Hou et al., 2018; Yao et al., 2011). The curves of the residual norm and solution norm versus different β are displayed in Figure 3(a), wherein β is changed from 0.005 to 1 with the increment Δβ = 0.005. With the change of β, the curves of the residual norm and solution norm remain stable. According to the L-curve criterion, 0.005 ≤ β ≤ 1 is considered as a proper parameter range, which indicates that our model is endowed with stability with respect to regularization parameter β. Figure 3(b) displays the convergence process of the SRFs, which shows that our model obtains a stable value after five times iteration. The convergence process indicates the high efficiency of the numerical algorithm. Figure 3(c) presents the damage identification result of our model. It can be seen from Figure 3(c) that our model accurately identifies all damaged elements, and the damage identification error δ is 1.06%. Figure 3(d) presents the damage identification results of the SReg model and the IRLR model, which displays that the damage elements can be accurately identified, and the damage identification error δ of the SReg model and IRLR model are 9.63% and 7.01%, respectively.

Figure 3.

Identification of stiffness reduction factors (SRFs) for DS1 on the truss structure: (a) solution norm and residual norm of our model for different β in DS1; (b) convergence of SRFs of our model. (c) SRFs of our model in DS1 (the damage identification error δ is 1.06%); (d) SRFs of SReg model and the IRLR model in DS1 (the damage identification error δ of SReg model and IRLR model are 9.63% and 7.01%, respectively).

Considering the effect of the measurement noise, the performance of the proposed model is inspected by introducing different levels of noise into the modal parameters. For noise level 1, noises follow standard normal distribution, and the SD of noise in frequencies and mode shapes are 1% and 10% of the real data, respectively. For noise level 2, noises follow standard normal distribution as well, and the SD of noise in frequencies and mode shapes are 2% and 20% of the real data, respectively. 100 reduplicative tests are performed for each noise level, and the results are displayed in Figure 4, where the bars and the short-horizontal-lines represent mean and SD which obtain from 100 times tests. Although the minor false positives occur in Figure 4, the identified SRFs can distinctly indicate the damaged elements under these two noise levels. The damage identification errors δ for noise level 1 and noise level 2 are 11.94% and 10.28%, respectively. These results show the reliable robustness of the proposed model.

Figure 4.

SRFs of our model with different noise levels for DS1 on the truss structure. The damage identification errors δ for noise level 1 and noise level 2 are 11.94% and 10.28%, respectively.

Damage identification for damage scenario 2

The performance of the proposed damage identification model is further validated by the grouped damage scenario. According to the L-curve criterion (Figure 5(a)), 0.005–1.0 is considered as the proper range of the regularization parameter β for our model. Figure 5(b) shows the convergence process of the SRFs. After twenty-nine times iteration, our model obtains the stable SRFs, indicating the well convergence of the numerical algorithm. The performances of our model and the elastic net model are compared, and the damage identification results are presented in Figure 5(c). It can be seen from Figure 5(c) that the identified SRFs from our model and the Elastic net model well match the ground truth with minor discrepancies, and the damage identification error δ of our model and the Elastic net model are 1.14% and 1.49%, respectively.

Figure 5.

Identification of stiffness reduction factors (SRFs) for DS2 on the truss structure: (a) solution norm and residual norm of our model for different β in DS2; (b) convergence of SRFs of our model. (c) SRFs of our model and the Elastic net model in DS2 (the damage identification errors δ are 1.14% and 1.49%).

Next, the above two kinds of noise are also added into the modal parameters. Similarly, 100 reduplicative tests are also performed for each noise level, and the results are displayed in Figure 6. It is observed that the damage identification results are acceptable, although the minor false positives occur. The identified SRFs can distinctly indicate the damaged elements under these two noise levels. The damage identification errors δ for noise level 1 and noise level 2 are 12.02% and 19.66%, respectively. These results also show the reliable robustness of our model.

Figure 6.

SRFs of our model with different noise levels for DS2 on the truss structure. The damage identification errors δ for noise level 1 and noise level 2 are 12.02% and 19.66%, respectively.

In addition, we study the effect of the step size h on performance of the algorithm. In detail, turning the step size h from 0.05 to 1 with the increment Δh = 0.05, the model performance is inspected. The results show that when 0 < h < 0.55, the algorithm doesn’t converge, and our model can’t identify the damaged elements. When 0.55 < h < 1, our model obtains satisfactory damage identification results. As shown in the Figure 7, we display the damage identification error δ, where h is changed from 0.5 to 1 with the increment Δh = 0.05. The results show that when 0.55 < h < 1, the damage identification error δ maintains a small value. As a conclusion, 0.55 < h < 1 could be considered as the appropriate range of the step size h.

Figure 7.

The effect of the step size h on the model performance: (a) the varying of the σ with the increase of the h in the DS1 of the truss structure; (b) the varying of the σ with the increase of the h in the DS2 of the truss structure.

Table 2 displays the damage identification errors δ and CPU time of the numerical study. As shown in Table 2, compared with the SReg model, IRLR model, and Elastic Net model, the damage identification error δ of the proposed model improves 8.57%, 5.95%, and 0.35%, respectively. The average improvement is 4.96%. It is apparent that the proposed fraction function regularization model yields more accurate damage identification results than the related models. In addition, the CPU time of our model is significantly less than SReg one, IRLR one, and Elastic net one, which indicates the simpler computation and optimization scheme of the numerical algorithm. In brief, the above numerical study results show the superiority of the proposed fraction function regularization model in the structural damage identification.

Table 2.

The damage identification errors δ and CPU time of numerical study.

	Model	Our model	SReg	IRLR	Noise level 1	Noise level 2
DS1	δ	1.06%	9.63%	7.01%	11.94%	10.28%
DS1	CPU Time (s)	0.0147	1.0678	1.1222	1.4773	1.4707

	Model	Our model	Elastic Net	Noise level 1	Noise level 2
DS2	δ	1.14%	1.49%	12.02%	19.66%
DS2	CPU Time (s)	0.0150	0.9771	1.4776	1.4791

Experimental study

Model description

As shown in Figure 7(a), a steel cantilever beam in laboratory is utilized to further validate the proposed model. The length and the cross-section area of the cantilever beam are 0. 9 m and 0.05 × 0.005 m², respectively. The mass density and Young’s modulus are 7850 kg/m³ and 210 Gpa, respectively. Two kinds of damage scenarios are successively introduced in the beam structure, and they are shown in Figure 7(b). In the damage scenario 1, the beam structure has only one cut(named cut 1) which moves away from the support 40 mm. In the damage scenario 2, the beam structure is supplemented two cut (named cut 2, and cut 3, respectively) which moves away from the support 250 mm and 540 mm, respectively. Each cut has the length b = 10 mm and depth d = 1.5 mm as shown in Figure 7(b).

The hammering method is used to carry out vibration test for undamaged scenario, damage scenario 1 and damage scenario 2, respectively. As shown in Figure 8(a), 11 acceleration-sensors are uniformly installed on the cantilever beam to measure the acceleration signal. The sampling frequency is 2560 HZ. Figure 8 (c)–(e) show the test instruments. A spectrum analyzer is employed to obtain the input and output signals (Figure 8(c)). Then, the signals are amplified through a dynamic-signal-amplifier (Figure 8(d)). After the vibration test, modal analysis is achieved using uTekMa software (http://www.utekl.com/) (Figure 8(e)). Because the first-order model parameter is difficult to acquire, as a alternative, the 2–7 order modal parameters are used for damage identification in experimental study, and the modal analysis results are listed in Table 3.

Figure 8.

The experimental cantilever beam: (a) Photo of the test beam; (b) Locations of Cut 1-Cut 3; (c) Signal acquisition box and force hammer; (d) dynamic signal amplifier; (e) desk computer (include: Signal acquisition system and modal analysis system).

Table 3.

Frequencies and MAC of the beam in undamaged and damaged scenarios.

		Damage scenario 1		Damage scenario 2
Mode no.	Undamaged Frequency(HZ)	Frequency(HZ)	MAC	Frequency(HZ)	MAC
2	27.50	26.75(−0.75)	95.51	26.25(−1.25)	96.49
3	75.00	74.50(−0.50)	95.57	73.75(−1.25)	95.21
4	147.50	146.25(−0.75)	92.02	145.00(−2.50)	96.90
5	245.00	242.50(−2.50)	90.29	241.25(−3.75)	88.57
6	366.25	361.25(−5.00)	88.29	358.75(−7.50)	83.55
7	510.00	503.75(−6.25)	84.14	500.00(−10.00)	90.85

Note. MAC = modal assurance criterion.

The beam is modeled with 90 identical Euler-Bernoulli beam elements (i.e., n = 90), and the length of each element is 10 mm. From support to free-end, the beam elements are numbered as Element 1 to Element 90. Cut 1, Cut 2 and Cut 3 are located at Element 5, Element 26 and Element 55, respectively. Quantified by the elemental stiffness reduction, the damage severity equals the reduction of the moment of inertia of the section. Table 4 presents the damage locations and damage severities of these two damaged scenarios.

Table 4.

Two damage scenarios for experimental study.

Scenario	Element no.	Damage length(b) and depth(d)	Damage severity (%)	SRFs
Damage scenario 1	5	b = 10 mm, d = 1.5 mm	6	SRF(5) = −0.06
Damage scenario 2	5	b = 10 mm, d = 1.5 mm	6	SRF(5) = −0.06
	26	b = 10 mm, d = 1.5 mm	6	SRF(26) = −0.06
	55	b = 10 mm, d = 1.5 mm	6	SRF(55) = −0.06

Before damage identification, the measured modal data in the undamaged state is employed to update the initial FE model to reduce the influence of modeling errors. The regularization is not utilized in this process. The updated FE model, which is closer to the undamaged structure, will be adopted for the subsequent damage identification.

Damage identification for damage scenario 1

The regularization parameter of our model is determined as β = 0.8 in experimental study. The actual damage severity of damage scenario 1 is SRF(5) = −0.06. Figure 9 shows the damage identification results for damage scenario 1. For all models, although the damage severity is slightly greater than the reference value, the damage locations can be accurately identified. The damage identification errors δ of our model, SReg model, and IRLR model are 5.11%, 5.32%, and 5.59%, respectively.

Figure 9.

Identification of stiffness reduction factors (SRFs) for DS1 on the beam structure of our model, SReg model, and the IRLR model in DS1 (the damage identification errors δ of these three models are 5.11%, 5.32%, and 5.59%, respectively).

Damage identification for damage scenario 2

Damage scenario 2 is a multiple damage scenario, where the element 5, 26, and 55 are damaged 6%, respectively, that is, SRF(5) = SRF(26) = SRF(55) = −0.06. Figure 10(a) shows the damage identification results of our model. Although elements 4 and 25 are false identified, the SRFs of the damaged elements are larger than that of the undamaged elements. The damage identification error δ of our model is 27.95%. Figure 10(b) shows the damage identification results of the SReg model and the IRLR model. For the SReg model, although all the damaged elements can be identified, false identifications occur in elements 4 and 60. The damage identification error δ of the SReg model is 35.15%. For IRLR model, the identified SRF is less than the real SRF, and the false identifications occur in element 64. The damage identification error δ of the IRLR model is 34.78%. This comparison verifies that our model is superior to the SReg model and the IRLR model in damage identification.

Figure 10.

Identification of stiffness reduction factors (SRFs) for DS2 on the beam structure: (a) SRFs of our model in DS2 (the damage identification error δ is 27.95%); (b) SRFs of SReg model and IRLR model in DS2 (the damage identification errors δ are 35.15% and 34.78%, respectively).

Figure 11 shows the convergence process of the damage parameters. Our model almost approaches a stable value after 27 and 32 times iteration in damage scenario 1 and damage scenario 2, respectively, which shows the high efficiency of the proposed numerical algorithm.

Figure 11.

(a) Convergence of SRFs of our model in DS1; (b) Convergence of SRFs of our model in DS2.

The damage identification errors and CPU time of the experimental study are reported in Table 5. It is find that almost the same identification errors are obtained in the identification of damage scenario 1. However, in the identification of damage scenario 2, the proposed model has obvious superiority, which potently indicates that the proposed fraction function regularization model has more advantages than the relative models. Compared with the SReg model and IRLR model, the damage identification errors δ of the proposed model improves 0.21% and 0.48% in damage scenario 1, respectively, and they improves 7.20% and 6.83% in damage scenario 2, respectively. The averaged improvement is 3.68%. In addition, the CPU time of the proposed model is significantly less than SReg model and IRLR model. From the CPU time, we also find the efficiency of the used numerical algorithm under the given convergence tolerance.

Table 5.

The damage identification errors δ and CPU time of experimental study.

	Damage scenario 1			Damage scenario 2
Model	Our model	SReg	IRLR	Our model	SReg	IRLR
δ	5.11%	5.32%	5.59%	27.95%	35.15%	34.78%
CPU Time (s)	0.0106	0.9240	0.8958	0.0112	0.9048	0.9174

Conclusions

In this paper, we have proposed a fraction function regularization model for damage identification, where the fraction function regularizer is used. The fraction function regularizer not only well describes the sparsity, but also overcomes the excessive penalty drawback of the l₁ regularizer, which effectively improves the accuracy of the damage identification. Numerical and experimental results have illustrated that the damage identification accuracy of the proposed model is averagely improved 4.96% and 3.68% than that of the l₁ regularization one, the iteratively reweighted l₁ regularization one and the elastic net one, respectively.

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 was supported by the Special Fund for Basic Scientific Research of Central Colleges in Chang’an University (300102129202 and 310812163504).

ORCID iDs

Rongpeng Li

Xueli Song

References

Bruckstein

Donoho

Elad

(2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Review 51: 34–81.

Cao

Cai

Tan

, et al. (2016) Image super-resolution via adaptive l_p (0 < q ≤ 1) regularization and sparse representation. IEEE Transactions on Neural Networks and Learning Systems 27: 1550–1561.

Cao

Shuai

Tang

(2018) Structural damage identification using piezoelectric impedance measurement with sparse inverse analysis. Smart Materials and Structures 27: 035020.

Chatzi

Smyth

(2002) The unscented Kalman filter and particle filter methods for nonlinear structural system identification with noncollocated heterogeneous sensing. Mechanical Systems and Signal Processing 16: 99–123.

Chen

(2019) A hybrid ant lion optimizer with improved Nelder-Mead algorithm for structural damage detection by improving weighted trace lasso regularization. Advances in Structural Engineering 23: 468–484.

Chen

Yang

, et al. (2019) Bridge influence line identification based on adaptive B-spline basis dictionary and sparse regularization. Structural Control and Health Monitoring 26: e2355.

Chen

Sun

(2020) Sparse representation for damage identification of structural systems. Structural Health Monitoring 20: 1644–1656.

Chen

Zhang

Zheng

, et al. (2020) Sparse Bayesian learning for structural damage identification. Mechanical systems and signal processing 140: 106689.

Chen

Zhao

Zhang

, et al. (2022) Damage quantification of beam structures using deflection influence line changes and sparse regularization. Advances in Structural Engineering 24: 1997–2010.

10.

Cui

Wen

, et al. (2018a) Sparse signals recovered by non-convex penalty in quasi-linear systems. Journal of Inequalities and Applications 2018: 59.

11.

Cui

Peng

, et al. (2018b) A damage detection method based on strain modes for structures under ambient excitation. Measurement 125: 438–446.

12.

Cui

Peng

, et al. (2019) Affine matrix rank minimization problem via non-convex fraction function penalty. Journal of Computational and Applied Mathematics 352: 478–485.

13.

Debnath

Das

(2021) A multi-objective framework for finite element model updating using incomplete modal measurements. Structural Control and Health Monitoring 28: e2770.

14.

Hernandez

(2014) Identification of isolated structural damage from incomplete spectrum using l₁-norm minimization. Mechanical Systems and Signal Processing 46: 59–69.

15.

Hou

Xia

(2020) Review on the new development of vibration-based damage identification for civil engineering structures: 2010-2019. Journal of Sound and Vibration 491: 115741.

16.

Hou

Xia

Zhou

(2017) Structural damage detection based on l₁ regularization using natural frequencies and mode shapes. Structural Control and Health Monitoring 25: 1–17.

17.

Hou

Xia

Bao

, et al. (2018) Selection of regularization parameter for l₁-regularized damage detection. Journal of Sound and Vibration 423: 141–160.

18.

Hou

Xia

Zhou

, et al. (2019) Sparse Bayesian learning for structural damage detection using expectation-maximization technique. Structural Control and Health Monitoring 26: e2343.

19.

Hou

Wang

Xia

(2021) Sparse damage detection via the elastic net method using modal data. Structural Health Monitoring 21: 1072–1096.

20.

Huang

Beck

(2015) Hierarchical sparse Bayesian learning for structural health monitoring with incomplete modal data. International Journal for Uncertainty Quantification 5: 139–169.

21.

Jiang

Tang

Mohamed

, et al. (2020) Augmented Tikhonov regularization method for dynamic load identification. Applied Sciences, 10: 6348.

22.

Hamamura

(2014) Smooth approximation l₀-norm constrained affine projection algorithm and its applications in sparse channel estimation. The Scientific World Journal 2014: 937252.

23.

Law

(2010) Adaptive Tikhonov regularization for damage detection based on nonlinear model updating. Mechanical Systems and Signal Processing, 24: 1646–1664.

24.

Zhang

Cui

(2017) Minimization of fraction function penalty in compressed sensing. IEEE Transactions on Neural Networks and Learning Systems, 31: 1626–1637.

25.

Wang

Deng

, et al. (2022) A novel joint sparse regularization model to structural damage identification by the generalized fused lasso penalty. Advances in Structural Engineering 25: 1959–1971.

26.

Malek-Mohammadi

Koochakzadeh

Babaie-Zadeh

, et al. (2016) Successive concave sparsity approximation for compressed sensing. IEEE Transactions on Signal Processing 64: 5657–5671.

27.

Mottershead

Link

Friswell

(2011) The sensitivity method in finite element model updating: a tutorial. Mechanical Systems and Signal Processing 25: 2275–2296.

28.

Nelson

(1976) Simplified calculation of eigenvector derivatives. AIAA Journal 14: 1201–1205.

29.

Sun

Lus

Betti

(2013) Identification of structural models using a modified artificial bee colony algorithm. Computers and Structures 116: 59–74.

30.

Wang

Hao

(2015) Damage identification scheme based on compressive sensing. Journal of Computing in Civil Engineering 29: 04014037.

31.

Wang

Hou

Xia

, et al. (2020a) Laplace approximation in sparse Bayesian learning for structural damage detection. Mechanical Systems and Signal Processing 140: 106701.

32.

Wang

Hou

Xia

, et al. (2020b) Structural damage detection based on variational Bayesian inference and delayed rejection adaptive Metropolis algorithm. Structural Health Monitoring 20: 1518–1535.

33.

Wang

Xiao

, et al. (2021) A strain modal flexibility method to multiple slight damage localization combined with a data fusion technique. Measurement 182: 109647.

34.

Weng

Xia

, et al. (2011) An iterative substructuring approach to the calculation of eigensolution and eigensensitivity. Journal of Sound and Vibration 330: 3368–3380.

35.

Weng

Zhu

Xia

, et al. (2013) Substructuring approach to the calculation of higher-order eigensensitivity. Computers and Structures 117: 23–33.

36.

Dyke

(2012) Structural parameters and dynamic loading identification from incomplete measurements: approach and validation. Mechanical Systems and Signal Processing 28: 244–257.

37.

Yao

Gerstoft

Shearer

, et al. (2011) Compressive sensing of the Tohoku-Oki Mw 9.0 earthquake: frequency-dependent rupture modes. Geophysical Research Letters 38: L20310.

38.

Zhang

(2016) Comparative studies on damage identification with Tikhonov regularization and sparse regularization. Structural Control and Health Monitoring 23: 560–579.

39.

Zhang

(2010) Nearly unbiased variable selection under minimax concave penalty. The Annals of statistics 38: 894–942.

40.

Zhou

Xia

Weng

(2015) l₁ regularization approach to structural damage detection using frequency data. Structural Health Monitoring 14: 571–582.

41.

Zhou

Hou

(2018) Structural damage detection based on iteratively reweighted l₁ regularization algorithm. Journal of Sound and Vibration 22: 1479–1487.

42.

Zhu

Chen

Cai

, et al. (2014) Locate damage in long-span bridges based on stress influence lines and information fusion technique. Advances in Structural Engineering 17: 1089–1102.