Local damage identification method using finite element model updating based on a new wavelet damage function

Abstract

As the number of unknown parameters in a finite element model updating problem increases, the challenges in finding reliable and meaningful updating result surges. Although traditional damage functions have illustrated an excellent ability in reducing unknown parameters, they are imprecise for identifying local damages. To solve this problem, a new type of damage function termed the wavelet damage function, which specializes in local damage identification, is proposed in this article. It utilizes the properties of the Haar wavelet and wavelet multi-resolution analysis. During the wavelet damage function–based finite element model updating procedure, unknown parameters in finite element models are not directly adjusted. Instead, wavelet coefficients of the parameters are estimated stepwise and then the final updating results are reconstructed through the inverse discrete wavelet transform. Numerical simulations and experimental verifications are conducted, and the corresponding results show that wavelet damage function can offer better accuracy as well as higher computational efficiency in the identification of local damages than the traditional damage functions.

Keywords

finite element model updating genetic algorithm Haar wavelet local damage identification wavelet damage function wavelet multi-resolution analysis

Introduction

Damage identification through finite element model updating (FEMU) is drawing increasing attention in the literature, particularly in structural health monitoring (SHM), since it can conveniently detect structural damages by comparing the baseline model and the updated model. FEMU aims to minimize the discrepancies—quantified by the cost function or objective function—between the simulated and observed system behaviors by means of adjusting unknown parameters in finite element (FE) models (Simoen et al., 2015; Teughels and De Roeck, 2005). In cost functions, structural eigenfrequencies and mode shapes are frequently used features. More recently, derivatives of mode shapes including mode shape curvatures, modal strain energy, and modal flexibility were also developed and introduced in FEMU (Pandey and Biswas, 1994; Pandey et al., 1991; Shi et al., 2000).

The minimization of the cost function is a constrained optimization problem, for which probability-based search algorithms have the advantages, such as the genetic algorithm (GA), simulated annealing (SA), and particle swarm optimization (PSO). Many researchers successfully implemented these algorithms in the damage identification of simple structures (Friswell et al., 1998; Levin and Lieven, 1998; Marwala, 2007). However, their application in large and complicated structures is still limited. As the number of unknown parameters increases, the difficulty in using probability-based search algorithms climbs, meaning that neither reliability nor efficiency can be guaranteed. Consequently, multiple acceptable but different models may exist in the final updated results (Berman, 1995; Caicedo and Yun, 2011).

Therefore, the number of unknown parameters in the optimization procedure should be strictly controlled. A possible solution is to eliminate the less important ones through engineering experience. However, this process is subjective, and any ill-conceived decisions can lead to erroneous results. Alternatively, similar parameters can be described by a set of pre-defined damage functions. Damage functions are derived based on interpolation functions, which can approximate the unknown parameters with fewer coefficients. Thus, the unknown parameters are remarkably reduced (Abdel Wahab et al., 1999; Perera and Ruiz, 2008; Teughels et al., 2002). But the main disadvantage of these damage functions is also substantial—the parameter distribution between each interpolating node is always continuous and smooth. This characteristic is suitable for certain damage patterns, such as overall degradation or widely distributed cracks; however, it may not be capable of reflecting local but vital damages.

New types of damage functions are essential to overcome the aforementioned limitations. This article presents a new damage function termed the wavelet damage function (WDF). By utilizing the wavelet techniques, the WDF has the special ability to identify local damages. Although wavelet techniques are widely used in damage identification (Douka et al., 2003; Sohn et al., 2004; Taha et al., 2006), their combination with damage functions has seldom been reported. In the proposed WDF, the Haar wavelet (Haar, 1910) is used for fitting the parameter distributions, while wavelet multi-resolution analysis (WMRA) (Mallat, 2009) is introduced to reduce the number of unknown parameters.

Similar to traditional damage functions, parameters in FE models are not adjusted directly in WDF. They are determined by multiple levels of wavelet coefficients with a reduced number of coefficients. The corresponding updating procedure starts from the lowest level of WMRA, which contains the least number of coefficients. Then, the estimation proceeds step by step until reaches the highest level, during which a filter is introduced to automatically control the parameter number and eliminate insignificant coefficients. At last, both the locations and severities of the damages are reconstructed by implementing inverse discrete wavelet transform (IDWT) on the final wavelet coefficients.

The presentation of this work is organized as follows: the theoretical basis of traditional damage functions, the Haar wavelet, and WMRA are briefly reviewed, followed by an introduction of the proposed WDF in basic concepts and updating procedure; in numerical simulations, comparisons among WDF and two traditional methods are made and discussed, which mainly focus on their identification accuracy and computational efficiency; in addition, a laboratory test of a reinforced concrete (RC) beam is conducted to verify WDF under real situations; several conclusions of this study are briefly summarized at the end.

Theoretical basis

Traditional damage function

To detect local damages in a structure, the parameterization of the FE model should be element-wise, so that both accurate locations and severities of the damages can be guaranteed. Consequently, the unknown parameters (e.g. the bending stiffness, Young’s modulus, and mass of each element) in the optimization process will be numerous, resulting in unacceptable burdens for probability-based search algorithms. Therefore, the starting point of the traditional damage functions is to approximate these numerous unknown parameters by a set of interpolation functions, which can be characterized by a reduced number of coefficients (Teughels et al., 2002). Equation (1) shows the relationship between the unknown parameters in the FE model and the traditional damage functions

a_{i}^{e} = \sum_{j = 1}^{N} p_{j} N_{j} (x_{i}^{e}), (i = 1, 2, 3, \dots, n)

(1)

where $a_{i}^{e}$ is the unknown parameter of a certain element to update, $n$ is the total element number, $N_{j}$ is the jth damage function, $N$ is the number of damage functions, $p_{j}$ are their multiplication factors (coefficients of the damage functions), and $x_{i}^{e}$ is the central coordinate of Element $i$ .

By introducing damage functions, the coefficients in an FEMU problem are changed to $p_{j}$ instead of $a_{i}^{e}$ . Since the number of damage functions $N$ is usually controlled by the sensor number and is much smaller than the element number n, the difficulty in the optimization step is remarkably reduced. The final values of $a_{i}^{e}$ can be easily reconstructed after all the $p_{j}$ are obtained.

Traditional damage functions are originally derived from the shape functions in FE theory, such as splines, polynomials, or linear functions. But only linear damage functions (LDFs) are used in practices, because more complicated functions will also require more undetermined coefficients. Furthermore, traditional damage functions are continuous and should be allocated according to the pre-defined damage elements (a set of neighboring elements in the FE model), so that long descending slopes are inevitable in the updating results. It may be useful for representing the parameter distribution due to overall degradation or widely distributed cracks. But as for local damages, their properties cannot be precisely captured. A new type of damage function that specializes in local damage identification is essential.

Haar wavelet

The Haar wavelet, presented by Haar (1910), is the simplest and earliest type of all the wavelet functions. It is a sequence of rescaled square-shaped functions that form an orthonormal function basis, where the scaling function or father wavelet can be expressed as

h_{0} (x) = {\begin{matrix} 1, & x \in [0, 1) \\ 0, & o t h e r w i s e \end{matrix}

(2)

Its counterpart is called the mother wavelet, whose definition is

h_{1} (x) = {\begin{array}{l} 1, & x \in [0, 1 / 2) \\ - 1, & x \in [1 / 2, 1) \\ 0, & o t h e r w i s e \end{array}

(3)

All other functions in the wavelet group can be generated via translating and dilating $h_{1} (x)$ . In general, the relationship between $h_{1} (x)$ and $h_{n} (x)$ can be defined as follows

h_{n} (x) = h_{1} (2^{j} x - \frac{k}{2^{j}}), n = 2^{j} + k, j > 0, 0 < k \leq 2^{j}, n, j, k \in Z

(4)

Note that the Haar functions are mutually orthogonal, indicating that the Haar functions can be used as a transform basis. Furthermore, Haar functions are neither continuous nor differentiable. This property may be a disadvantage in certain applications, but it can be an advantage for the analysis of signals with sudden transitions.

WMRA

WMRA is used to implement the discrete wavelet transform (DWT) with filters, which was first proposed by Mallat (2009). This process decomposes the original signal into two different resolutions: coarse and fine (i.e. approximation and detail). The data with coarse resolution contain low-frequency information and provide the main features of the original signal, while the fine part contains high-frequency information and corresponds to the feature details. A down-sampling process is introduced at each decomposition stage. By ignoring the second part of each sampling pair, approximation and detail represent only half of the previous samples.

The previous coarse part can be further decomposed into the next level, resulting in an additional pair of different resolution parts. If the decomposition process is repeated M times, the original signal can be described by the combination of the Mth approximation part and all the detail parts. Its reconstruction equation is described as

S (n) = \sum_{k = - \infty}^{\infty} a_{M, k} \emptyset_{M, k} (n) + \sum_{j = 1}^{M} \sum_{k = - \infty}^{\infty} d_{j, k} ψ_{j, k} (n)

(5)

where S(n) is the original signal, M is the resolution level, $a_{M, k}$ is the Mth approximation coefficient, $d_{j, k}$ is the jth detail coefficient, $φ_{M, k} (n)$ is the scaling function, and $ψ_{j, k} (n)$ is the wavelet basis function.

WDF

Basic concepts of WDF

Suppose a structure with multiple local damages (Figure 1(a)), each element has its own parameter to update (e.g. bending stiffness $EI$ ). Before introducing the WDF, it is better to normalize the unknown parameters in the FE model, which divides the stiffness change by its initial value

d_{i}^{e} = \frac{{(EI)}_{i, d}^{e} - {(EI)}_{i}^{e}}{{(EI)}_{i}^{e}} \times 100 %, (i = 1, 2, 3, \dots, n)

(6)

in which $(EI)_{i}^{e}$ is the original bending stiffness of Element i, $(EI)_{i, d}^{e}$ is the damaged value, $d_{i}^{e}$ is the normalized parameter, and n is the total element number. $d_{i}^{e}$ is also referred to the damage index, where $d_{i}^{e} = 0$ denotes no damage and $d_{i}^{e} = - 100 %$ indicates complete damage.

Figure 1.

Basic concepts of wavelet damage function based on Haar wavelet: (a) FE model; (b) parameters to update; (c) distribution diagram of damage indexes; (d) Haar wavelet and (e)Wavelet coefficients.

Therefore, all the unknown parameters in the FEMU procedure can be grouped into a vector $S$

S = [d_{1}^{e}, d_{2}^{e}, d_{3}^{e}, \dots, d_{n}^{e}], (i = 1, 2, 3, \dots, n)

(7)

When local damages occur, $d_{i}^{e}$ of the damaged elements will suddenly change, leaving several spikes in the distribution diagram of $S$ , as plotted in Figure 1(c). It is very similar to a digital signal sequence and provides an inspiration that the wavelet techniques, which are widely used in signal processing, can also be used for presenting the parameter distribution caused by local damages. Due to the square-shaped and discontinuous property of $S$ , the Haar wavelet is consequently the most suitable choice among the various types of wavelet functions (see Figure 1(d)). The next step is to implement WMRA on $S$ . Assume the decomposition level is M, the length of the wavelet coefficients $A_{M}$ (Mth approximation part) and $D_{M}$ (Mth detail part) will be n/2^M, because the signal length is halved during every decomposition step. Figure 2 illustrates the corresponding decomposition result.

Figure 2.

Pyramidal structure of WMRA.

It should be mentioned that although other forms of functions with discontinuity and non-differentiability other than Haar wavelet functions can also be used to define the damage function, they cannot be conveniently decomposed into multiple levels. As a consequence, the reduction in parameter number is hard to achieve since all the coefficients of the damage functions have to be estimated simultaneously instead of stepwise.

The vector $S$ containing n parameters is unknown in FEMU, so that the updating is a totally inverse process of the WMRA (see Figure 2). The parameter updating may start from the lowest wavelet coefficients $A_{M}$ and $D_{M}$ containing n/2^M unknown wavelet coefficients, then the higher wavelet coefficients can be reconstructed though the IDWT. This process should be repeated for M times, until the final vector $S$ is reconstructed by $A_{1}$ and $D_{1}$ . Note that the initial number of unknown parameters is only n/2^M instead of n, hence the parameter reduction is also achieved. Therefore, the proposed procedure can be treated as a new type of damage function, which is named as WDF in this article due to its utilization of the Haar wavelet and the WMRA technique.

Optimization steps based on probability-based search algorithms

The updating process of vector $S$ using WDF can be expressed by a backward recurrence formula

{\begin{matrix} A_{m - 1} = IDWT (A_{m}) + IDWT (D_{m}), m = 1, 2, 3, \dots, M \\ S = A_{0} \end{matrix}

(8)

where $M$ is the decomposition level of WMRA; $A_{m}$ and $D_{m}$ are the coarse and fine wavelet coefficients at the mth level, respectively.

Then, the probability-based search algorithms are used for optimizing the wavelet coefficients in each step. The detailed procedure is presented below:

Select the decomposition level M, and let the length of $A_{M}$ remain in the proper range.

Use IDWT of $A_{M}$ for the approximation of S , get $A_{M}^{*} = [A_{M}^{1^{*}}, A_{M}^{2^{*}}, \dots, A_{M}^{l^{*}}]$ by minimizing the cost function with probability-based optimization algorithms.

Define a filter so that the small coefficients below the filter criterion will be set to zero. As a result, $A_{M}^{*}$ is modified to $A_{M}^{*} = [0, \dots, A_{M}^{p^{*}}, \dots, A_{M}^{q^{*}}, \dots, 0]$ . Thus, the number of unknown parameters is reduced $(1 \leq p \leq q \leq length (A_{M}^{*}))$ .

Since $D_{M}$ is the fine counterpart wavelet coefficient, $D_{M}$ should have a form identical to $A_{M}^{*}$ , which can be expressed as $D_{M} = [0, \dots, D_{M}^{p}, \dots, D_{M}^{q}, \dots, 0]$ .

Use $(IDWT (A_{M}) + IDWT (D_{M}))$ for the approximation of S , and introduce optimization algorithms again. This time, the unknown number is $2 \times (q - p + 1)$ , and the corresponding results are $A_{M} = [0, \dots, A_{M}^{p}, \dots, A_{M}^{q}, \dots, 0]$ and $D_{M} = [0, \dots, D_{M}^{p}, \dots, D_{M}^{q}, \dots, 0]$ .

$A_{M - 1}^{*} = I D W T (A_{M}) + I D W T (D_{M})$ ; then, exchange the $A_{M}^{*}$ in Step 3 by $A_{M - 1}^{*}$ and repeat the procedure from Step 3 to Step 5 until the highest coefficients $A_{1}$ and $D_{1}$ are calculated.

Reconstruct the damage indexes S through equation (8) to obtain the final updated damage indexes S .

The number of unknowns in a single optimization step should be approximately the same as the sensor number to provide a reliable result. When too many unknown coefficients exist in $A_{M}$ (n/2^M is too large), it is recommended to decompose it into the next level. Zero padding is essential if the length of the wavelet coefficients is odd. Due to the randomness in probability-based algorithms, the estimated values in $A_{M}^{*}$ relating to no damage will have some numerical residuals instead of exact zeros. A filter must be set to eliminate them from the following process (in Step 3), whose criteria can be adjusted from 0% to 100%. For example, the value 30% means the wavelet coefficients less than 30% of the largest one in $A_{M}^{*}$ will be ignored. a higher criterion will ensure fewer unknown parameters and higher computational efficiency. However, note that not only numerical residuals but also small damages (less than 30% of the primary damage) may be filtered out simultaneously.

FEMU procedure using WDF

Figure 3 shows a complete flowchart of the damage identification using the proposed WDF, where the FE updating part is the same as the procedure of classic FEMU (Teughels and De Roeck, 2005), except for the parameter estimation (highlighted by the gray box in Figure 3). WDF should be introduced before FE analyses, hence the following optimization steps are implemented on the wavelet coefficients which have fewer unknown parameters. The local damages can then be easily detected, localized, and quantified from the updated parameters at the final step.

Figure 3.

Flowchart of local damage identification procedure using WDF.

Numerical study

Numerical simulations are conducted for testing WDF under various situations, including different damage cases, different numbers of unknown parameters and noise influence. The testbed structure is a simply supported beam with 16 elements, see Figure 4.

Figure 4.

Testbed beam in the numerical simulations.

A simple single-objective cost function is defined using only the eigenfrequencies and mode shapes of the structure

F (S) = \frac{1}{2} \sum_{r = 1}^{R} \frac{{({\bar{λ}}_{r} - λ_{r} (S))}^{2}}{{\bar{λ}}_{r}^{2}} + \frac{1}{2} \sum_{r = 1}^{R} \frac{‖ {\bar{ϕ}}_{r} - ϕ_{r} (S) ‖_{2}^{2}}{‖ {\bar{ϕ}}_{r} ‖_{2}^{2}}

(9)

where $S$ is the vector of unknown parameters; ${\bar{λ}}_{r}$ and $λ_{r}$ are measured and simulated eigenfrequencies, respectively, with ${\bar{ϕ}}_{r}$ and $ϕ_{r}$ as their corresponding mode shapes and R is the number of interested modes.

In the simulation cases, the first four vertical eigenfrequencies (R = 4) and corresponding mode shapes (seven points in total) are assumed. The updating results of WDF are compared with two traditional methods, especially focusing on the identification accuracy and computational efficiency. One is the multistage updating procedure using linear damage function (LDF) (Perera and Ruiz, 2008), and the other is the direct updating (DU) method achieved by adjusting all parameters independently.

The GA tool in the MATLAB toolbox is used for optimization of the cost function. Notably, its settings in different cases and methods should be kept consistent for better comparison: population size = 30; crossover fraction = 0.8; the range of damage indexes is [–30%, 10%], and the stopping criterion is either the value of the cost function reaching $10^{- 6}$ or the number of stall generations exceeding 10. To speed up the convergence rate and increase the accuracy within local ranges, a hybrid function (fmincon) is introduced at the end of each optimization step, which aims to find a constrained minimum of the cost function starting from the previous GA estimates. Since GA is a probability-based technique, the final results can vary slightly each time. To make the results more conclusive, all the three methods are repeated 1000 times in each damage case, respectively, so that the results can be discussed from a statistical point of view.

Single damage identification

The local damage is assumed at Element 8 with the stiffness reduction of 20%, and one-level decomposition is sufficient for WDF. The updating procedure is a two-step approach as a result, where Step 1 aims to estimate $A_{1}^{*}$ , and Step 2 calculates $A_{3}^{*}$ and $D_{1}$ simultaneously. The filter criterion is set to 10% for this noise-free case. As for the LDF method, the updating procedure is also divided into two steps: first, every two adjacent elements are grouped into one damage element; and second, the potential elements are selected for DU. Meanwhile, the DU method just adjusts all the 16 bending stiffness values independently and simultaneously.

The final updated damage indexes are plotted in Figure 5(a) to (c) by box plot. Obviously, all the three methods have obtained the precise location and severity of the damage. The median, 25th percentile, 75th percentile, and whiskers of each box are almost coincident. The total generations of these methods in each repetition are also extracted and compared. For better illustration, their histograms are fitted into probability density functions (PDFs), as shown in Figure 5(d). It indicates that FEMU procedures using damage functions are more efficient, because both WDF and LDF converged faster than DU. Among them, the WDF method has the smallest generation expectation.

Figure 5.

Updating results of a single damage: (a) WDF method, (b) LDF method, (c) DU method, and (d) fitted PDFs.

Multiple damage identification

Local damages may also occur in multiple locations. Therefore, damage functions should have the ability to identify all the possible damages. Two extra local damages are introduced in Element 3 and Element 10 with severities of 10% and 15%, respectively.

The updating procedures and settings stay unchanged as those in the previous case, and the corresponding results are plotted in Figure 6(a) to (c). The damage identifications by WDF and LDF are still relatively precise, and their box plots are essentially identical. Although the accuracy of the DU method decreases slightly, the error ranges are negligible compared with the damage severities, but it still has the highest expectation of generations. Since multiple damages exist in the beam, the first step of the WDF method will reserve more wavelet coefficients to the following step, the computational efficiency of WDF and LDF are comparable (see Figure 6(d)).

Figure 6.

Updating results of multiple damages: (a) WDF method, (b) LDF method, (c) DU method, and (d) fitted PDFs.

Damage identification with a large number of unknown parameters

A finer mesh is applied to the FE model in Figure 4, where each element is further divided into four smaller elements. The total element number increases from 16 to 64, but the number of measured nodes is still remained the consistent (seven points). The local damage occurs in Element 30, and its severity is set to 20%.When 64 unknown parameters are adjusted independently, meaningful results cannot be obtained due to severe ill conditioning. Therefore, the DU method is excluded in this case. Only WDF and LDF are implemented.

For the application of WDF, three-level WMRA is used. In the first optimization step, the number of wavelet coefficients in $A_{3}^{*}$ is only eight. According to the decomposition results in Figure 2, the final parameters will be reconstructed through four successive optimization steps. For the LDF method, the two-step approach proposed in the literature (Perera and Ruiz, 2008) is not sufficient. An extended version containing three steps is designed in this study: first every four adjacent elements are grouped into one damage element, on which LDF is defined; second, possible damage locations are manually selected based on the previous results, and new LDFs containing every two adjacent elements are redefined accordingly, in order to ensure better identification accuracy; third, after localizing narrower possible damage range, the location and severity of the damage are then estimated by applying local DU.

Inevitably, the difficulty in optimization surges when there are numerous parameters but limited measured data. Neither WDF nor LDF are as precise as the previous cases, merely rough location and severity of the damage can be obtained, as plotted in Figure 7(a) and (b). WDF outperforms LDF since the damage location is narrower and the severity is closer to the pre-set value. The largest improvement of WDF lies in the computational efficiency. Thanks to favorable local properties of WDF, the number of unknown parameters are reduced to its full extent—despite the WDF method has one more step, its computational efficiency is still far higher than that of LDF; as illustrated in Figure 7(c).

Figure 7.

Updating results under numerous unknown parameters: (a) WDF method, (b) LDF method, and (c) fitted PDFs.

Noise influence

Noise is an inevitable factor that strongly influences the damage identification accuracy in practice. To test the proposed method against noise, the eigenfrequencies and mode shapes of the damaged beam are added by artificial random noise

{\begin{matrix} λ_{r} = λ_{r}^{0} (1 + ε_{λ} C_{r}) \\ ϕ_{r, i} = ϕ_{r, i}^{0} (1 + ε_{ϕ} C_{r, i}) \end{matrix}

(10)

where the subscript r is the mode number, i is the node number relating to a measured point, the superscript 0 denotes noise-free values, C is a random number subjected to standard normal distribution, and $ε$ is the noise level.

In this case, a noise level with $ε_{λ} = 1 %$ and $ε_{ϕ} = 2 %$ is chosen. The damage scenario and updating procedure are the same as the singe damage case, but the filter criteria is modified to 30% for better eliminating the noise-induced numerical residuals. The updating results of WDF, LDF, and DU methods are given in Figure 8(a) to (c).

Figure 8.

Updating results under noise influence: (a) WDF method, (b) LDF method, (c) DU method, and (d) fitted PDFs.

Although all the median values can reasonably reflect the damage information, the stability of each method differs significantly. The DU method is the most vulnerable to noise, as the updated damage indexes spread widely and many indexes reach the upper limit (10%). By contrast, the WDF and LDF methods provide better estimations. In the WDF results, only the damage indexes near the real damage are updated. A similar phenomenon exists in the LDF results, but the index distribution is less concentrated. Meanwhile, the WDF method is still the most efficient one among these methods (Figure 8(d)).

But compared to the former noise-free results, computational efficiency of all the methods seems improved. When noisy modal parameters are used in the cost function, they may hinder the GA in finding the real global minimum. Usually local/fake minima are found, so that the convergence rate can be increased in turn. This phenomenon is inevitable in real practice, but as long as the errors can be removed by taking averages, the identification results still show high reliability. In brief, the DU method failed to provide accurate damage identification under the influence of noise, while the results of WDF and LDF are much better. Between them, WDF shows obvious advantages over LDF in both identification accuracy and computational efficiency.

Experimental verification

Experiment overview

For further verification of the proposed WDF method, it is then implemented on the damage identification of a real RC beam in the laboratory. The beam is cast by C40 concrete, and the reinforcement ratio is 2.62%. Two special stainless steel supports, one hinged and one sliding, are designed to ensure time-invariant boundary conditions. This beam belongs to one of the specimens in a carbonation test and more details can refer to Zhang and Sun (2016). The beam should remain undamaged before accelerated carbonation, so the local damage is simulated by loading an 8.56-kg concrete block, because mass increase can also change the modal parameters without introducing any real damage to the beam. Seven vertical and one horizontal piezoelectric accelerometers are uniformly installed along the beam to measure its dynamic properties. Their technical parameters are listed in Table 1

Table 1.

Technical parameters of the piezoelectric accelerometers.

Technical parameters	Values	Technical parameters	Values
Axial sensitivity	$\approx 140 mV / (m / s^{2})$	Working temperature	$Approximately - 40 \circ C to 150 \circ C$
Transverse sensitivity	$< 5 %$	Measurement range	$0 to 600 g$
Frequency response	$0.2 to 4000 Hz$	Net weight	$38 g$

The sampling frequency of the accelerometers is set to 1000 Hz, which is enough to capture the fourth vertical bending model (326.6 Hz). Other instrumentation includes signal cables, a charge optimal modulator, a 16-channel data acquisition device, and a laptop. Detailed experimental arrangements are shown in Figure 9. During the modal identification process, the beam is excited by random taps. The first four eigenfrequencies and corresponding mode shapes are extracted by the stochastic subspace identification (SSI) technique (Peeters and De Roeck, 1999) from the acceleration time histories before and after loading respectively.

Figure 9.

Experimental layout: (a) overview, (b) data acquisition instruments, (c) hinged support, (d) sliding support, and (e) piezoelectric accelerometer.

Baseline FE model

The initial FE model of the RC beam is modeled by 16 beam elements (BEAM188) in ANSYS, and rotational springs (COMBIN14) are added to the both ends of the beam, as illustrated in Figure 10. This initial FE model should be updated to the baseline model for better representing of the real structure. Here, Young’s moduli and densities of the elements are assumed to be the same along the beam, since this beam is one-time cast and do not contain any damage. Four physical quantities, which are Young’s modulus $E_{c}$ , density $ρ$ , and spring rotational stiffness values $k_{1}$ and $k_{2}$ are chosen as unknown parameters.

Figure 10.

Illustration of the FE model in ANSYS: (a) FE model of the tested beam and (b) cross section.

The entire updating process is controlled by MATLAB, while ANSYS serves as a slave software, whose role is calculating the modal properties of the FE model. It should be pointed out that the combination of these two software programs will cost much more time in FE analyses, because their data exchange is through txt files. But the combination of these commercial FE software can relieve the challenges in modeling, particularly for large and complicated structures. The DU method is used in this step, and the final updated values are $E_{c} = 36.7 GPa$ , $ρ = 2677 kg / m^{3}$ , $k_{1} = 21.2 kN m / rad$ , and $k_{2} = 52.9 kN m / rad$ .

Local damage identification

A concrete block is loaded on Element 8, which is equivalent to 88.8% density increase of that element. Therefore, the unknown parameters in this case are the densities of beam elements. Analogous to equation (6), the densities are normalized before the updating procedure

d_{i}^{e} = \frac{{(ρ)}_{i, d}^{e} - {(ρ)}_{i}^{e}}{{(ρ)}_{i}^{e}} \times 100 %, (i = 1, 2, 3, \dots, 16)

(11)

in which $(ρ)_{i}^{e}$ is the original density of Element i and $(ρ)_{i, d}^{e}$ is the value of its damaged state.

WDF, LDF, and DU are implemented to update the FE model for comparison. The parameter range is modified to [–20%, 120%] in consideration of this specific problem, where other settings are unchanged as those in the simulation case under noise influence. Due to the low computational efficiency in the combination of MATLAB and ANSYS, it is impractical to repeat the updating procedure too many times; therefore, only a typical result of each method is illustrated in Figure 11.

Figure 11.

Updating results of a real RC beam: (a) WDF method, (b) LDF method, and (c) DU method.

The WDF result indicates the density increase in the Element 8 is 80.5%, which is the closet to the true value of 88.8% (Figure 11(a)). Due to the influence of noise during measurement, some small deviations exist in the adjacent regions of Element 8. This phenomenon coincides with the previous numerical results (Figure 8). Additionally, there are some small values near both ends. A possible reason is that the boundary conditions have been slightly altered due to the static deformation after loading. The LDF result is quite similar to that of WDF (Figure 11(b)). However, the damage index is less accurate (77.5%), and the deviations around Element 8 are a bit larger. The DU method gives the poorest identification result as expected, since errors spread along the entire beam (Figure 11(c)), and the only 68.3% of the density increase is reported.

Conclusion

A new type of damage function termed the WDF is proposed in this article, which aims at identifying local damages through FEMU. Similar to traditional damage functions, WDF can also detect, localize, and quantify damages with a significantly reduced number of unknown parameters. Based on the numerical and experimental results, several brief conclusions about the proposed WDF can be drawn:

The WDF method exhibits a faster convergence rate and better anti-noise performance than traditional LDF and DU methods. Both identification accuracy and computational efficiency are improved, particularly when unknown parameters are numerous.

Possible damage ranges are automatically selected by the filter, during which noise influence and numerical residuals can be mitigated as well. But there is always a trade-off between the damage identifiability and the noise/residuals resistance, so that the filter criteria must be carefully considered.

The use of the Haar wavelet strengthens the ability of the damage functions in identifying local damages, but it losses the representativeness for global parameter changes. Therefore, the proposed WDF can only serve as a supplement to the traditional damage functions other than replacing them.

Model updating is a complex problem since many factors, for example, different measured data, cost functions, optimization algorithms, and their settings, which are not discussed in this article, may also significantly influence the performance of WDF.

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 National Key Basic Research Program of China (973 Program) (grant no. 2013CB036305) and the Ministry of Transport of People’s Republic of China (grant no. 2015318J38230).

References

Abdel Wahab

De Roeck

Peeters

(1999) Parameterization of damage in reinforced concrete structures using model updating. Journal of Sound and Vibration 228: 717–730.

Berman

(1995) Multiple acceptable solutions in structural model improvement. AIAA Journal 33: 924–927.

Caicedo

Yun

(2011) A novel evolutionary algorithm for identifying multiple alternative solutions in model updating. Structural Health Monitoring 10: 491–501.

Douka

Loutridis

Trochidis

(2003) Crack identification in beams using wavelet analysis. International Journal of Solids and Structures 40: 3557–3569.

Friswell

Penny

JET

Garvey

(1998) A combined genetic and eigensensitivity algorithm for the location of damage in structures. Computers & Structures 69: 547–556.

Haar

(1910) Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen 69: 331–371.

Levin

Lieven

NAJ

(1998) Dynamic finite element model updating using simulated annealing and genetic algorithms. Mechanical Systems and Signal Processing 12: 91–120.

Mallat

(2009) A Wavelet Tour of Signal Processing. Cambridge, MA: Academic Press.

Marwala

(2007) Dynamic Model Updating Using Particle Swarm Optimization Method (arXiv Preprint Arxiv:0705.1760). New York: Cornell University.

10.

Pandey

Biswas

(1994) Damage detection in structures using changes in flexibility. Journal of Sound and Vibration 169: 3–17.

11.

Pandey

Biswas

Samman

(1991) Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration 145: 321–332.

12.

Peeters

De Roeck

(1999) Reference-based stochastic subspace identification for output-only modal analysis. Mechanical Systems and Signal Processing 13: 855–878.

13.

Perera

Ruiz

(2008) A multistage FE updating procedure for damage identification in large-scale structures based on multiobjective evolutionary optimization. Mechanical Systems and Signal Processing 22: 970–991.

14.

Shi

Law

Zhang

(2000) Structural damage detection from modal strain energy change. American Society of Civil Engineers 126: 1216–1223.

15.

Simoen

De Roeck

Lombaert

(2015) Dealing with uncertainty in model updating for damage assessment: a review. Mechanical Systems and Signal Processing 56– 57:123–149.

16.

Sohn

Park

Wait

et al . (2004) Wavelet-based active sensing for delamination detection in composite structures. Smart Materials and Structures 13: 153.

17.

Taha

MMR

Noureldin

Lucero

et al . (2006) Wavelet transform for structural health monitoring: a compendium of uses and features. Structural Health Monitoring-an International Journal 5: 267–295.

18.

Teughels

De Roeck

(2005) Damage detection and parameter identification by finite element model updating. Archives of Computational Methods in Engineering 12: 123–164.

19.

Teughels

Maeck

De Roeck

(2002) Damage assessment by FE model updating using damage functions. Computers & Structures 80: 1869–1879.

20.

Zhang

Sun

(2016) Effect of concrete carbonation on natural frequency of reinforced concrete beams. Advances in Structural Engineering 20: 316–330.