An efficient approach for optimal sensor placement and damage identification in laminated composite structures

Advances in Engineering Software 119 (2018) 48–59

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Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

Research paper

An efficient approach for optimal sensor placement and damage identification in laminated composite structures D. Dinh-Conga,c, H. Dang-Trungb,c, T. Nguyen-Thoib,c,

T

⁎

a

Division of Construction Computation, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam c Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam b

A R T I C L E I N F O

A B S T R A C T

Keywords: Optimal sensor placement Damage identification Laminated composite structures Jaya algorithm Model reduction Limited sensors

This paper proposed an efficient approach for optimal sensor placement and damage identification in laminated composite structures. This approach first utilized a model reduction technique, namely iterated improved reduced system (IIRS) method, to develop a reduced order model for optimal sensor placement (OSP), and then the OSP strategy using Jaya algorithm is conducted by formulating and solving an optimization problem for finding the best sensor locations. The objective function of the optimization problem is defined based on the correlation between the flexibility matrix obtained from an original finite element model and the corresponding one calculated from IIRS method. Next, the approach uses the measured incomplete modal data from optimized sensor locations for detecting and assessing any stiffness reduction induced by damage. In order to do this, the damage identification problem is formulated as an optimization problem where the damage extent of elements and the modal flexibility change are taken as the continuous design variables and the objective function, respectively. The Jaya algorithm is again adopted to solve the optimization problem for determining the actual damage sites and extents. Numerical simulations of a three cross-ply (0°/90°/0°) beam and a four-layer (0°/90°/90°/0°) laminated composite plate are carried out to demonstrate the applicability and efficiency of the proposed approach.

1. Introduction With significantly increasing applications of composite materials in mechanics, aerospace, marine, civil and many other industries, structural health monitoring (SHM) for composite materials have gained much attention from the scientific and engineering communities. A reliable and effective damage diagnose method is extremely important to ensure the conditions of integrity and safety of structures made of composite materials. Over the last few decades, many works and studies have been focused on vibration-based global SHM techniques for solving damage identification problems in composite structures. The basic idea of these techniques is that the change of either modal parameters (natural frequencies [1], mode shapes [2]) or their variations (frequency-response function [3], curvature mode shapes [4], flexibility matrix [5], etc.) can be used as signals to detect and locate damages in the structures. For more detailed information on these techniques, the reader can refer to good review articles [6,7]. Besides, several recent studies related to flaw detection problems in smart composite structures were presented in Refs. [8–10]. In essence, the problem of damage identification in composite ⁎

structures can be formulated as an optimization problem, where the location and degree of damage are found by minimizing the objective function which is commonly defined in terms of the difference between the vibration data measured by modal testing and those calculated from analytical model. For generating data from the analytical model used in health monitoring studies, the damage can be simulated by either reducing the stiffness of elements in damaged areas [5,11,12] or using crack models [13–15] or delamination models [16–18]. As a result, the traditional finite element (FE) analysis or isogeometric analysis [19–22] can be employed as a tool for damage diagnosis through the model updating process which typically requires an optimization algorithm. In order to meet this requirement, several meta-heuristic optimization algorithms have been applied as intelligent searching techniques to deal with the problem. For example, Su et al. [16] used genetic algorithms (GAs) and artificial neural networks (ANNs) for quantitative assessment of delamination in glass fiber-reinforced epoxy (GF/EP) composite laminates. In their study, the efficiency of the GA was compared with ANNs in term of both the prediction precision and computational cost. Qian et al. [17] proposed a hybrid optimization algorithm featuring cooperative particle swarm optimization (PSO) with simplex method

Corresponding author at: Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail addresses: [email protected] (D. Dinh-Cong), [email protected] (H. Dang-Trung), [email protected] (T. Nguyen-Thoi).

https://doi.org/10.1016/j.advengsoft.2018.02.005 Received 12 December 2017; Received in revised form 6 February 2018; Accepted 18 February 2018 0965-9978/ © 2018 Elsevier Ltd. All rights reserved.

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the optimal location for a given set of sensors. The correlation between the flexibility matrix obtained from an original FE model and the corresponding one calculated from IIRS method is considered as the objective function, where the sensor positions are defined to be the discrete optimization variables. We also present a comparison between the proposed objective function with an objective function based root mean square (RMS) of modal assurance criterion (MAC) [33,34] to validate the superiority of the proposed objective function. The second one is to locate and quantify structural damage using incomplete modal data obtained from optimized sensor locations. In this second part, the optimization-based damage identification problem is first formulated by defining the damage extent of elements as the continuous design variables and the modal flexibility change as the objective function, and then solved by using again the Jaya algorithm. To investigate the applicability and efficiency of the proposed damage diagnosis approach, two numerical examples including a three cross-ply (0°/90°/0°) beam and a four-layer (0°/90°/90°/0°) laminated composite plate are conducted. In addition, the influence of noise in the measured incomplete modal data on the accuracy of proposed approach is also examined in the examples. The remaining parts of the article are organized as follows. Initially, the IIRS method for OSP is presented in Section 2. Then, Section 3 generally provides the formulation of optimization-based damage detection problem. In Section 4, the description of the Jaya algorithm for discrete and continuous design variables is introduced. Section 5 shows the performance of the proposed approach through numerical examples. Finally, the concluding remarks are given in Section 6.

(SM) for identification of delamination in laminated composite beams. Vosoughi and Gerist [11] presented a hybrid method based on continuous genetic algorithm (CGA) and PSO for damage detection of laminated composite beams. In their work, nevertheless, the effect of measurement noise on the accuracy of the hybrid method was not investigated. More recently, a few studies have been focused on plate-like structures. Ashory et al. [23] used GA to solve the optimization problem for damage location and intensity identification in composite plates. Dinh-Cong et al. [12] introduced an efficient multi-stage optimization procedure using a modified differential evolution algorithm (MS-MDE) for damage assessment in a laminated composite plate. It is noted that the above-mentioned researches required full FE models to be able to determine accurately and completely the vibration characteristics in composite structures, which restricts applicability to real-world and large-scale implementations. Given that in practical situations it is almost impossible to determine entire experimental modal information corresponding to every node/ degree of freedom (DOF) of the FE structural model because the number of measurement sensors is typically limited, especially for large-scale structures. Therefore, one of the practical challenges to the problems of damage identification in composite structures is the use of incomplete modal data instead of complete modal data in calculation due to the limited number of measurement sensors used in practice. However, there have not been many reported studies in the literature for dealing with this challenge [1,18]. Generally, the more sensors are placed on a structure, the more information from measurement data can be obtained. However, due to cost and practicality issues, there are usually only a small number of sensors installed to a predefined set of possible locations. In fact, the quality of these data, as well as the quality of damage prediction depends much on the placement of sensors, and hence optimal sensor placement (OSP) is an important constituent in the SHM of composite structures. In the last two decades, many techniques have been proposed and developed to achieve OSP which can help collect the best identification of structural characteristics. An overview of OSP techniques was presented by Yi and Li [24]. Among them, combinatorial optimization methods have been widely employed owing to its computational efficiency for solving OSP problems of large-scale structures. Recently, some meta-heuristic optimization algorithms such as improved particle swarm optimization (IPSO) algorithm [25], niching monkey algorithm (NMA) [26], firefly algorithm (FA) [27], artificial bee colony (ABC) algorithm [28] have been successfully applied to the OSP problems. Nevertheless, there are still several issues for further improvement such as: (1) how to reduce significantly the computational cost of the optimization algorithms; (2) how to validate the optimal sensor layout obtained from the OSP strategies via a damage detection technique; and (3) how to minimize the number of sensors used for the problem of structural damage diagnosis. As an effort to fill in the above-mentioned research gaps, the current paper hence proposes an efficient approach for optimal sensor placement and damage identification in laminated composite structures. The main contributions of the paper can be addressed in three following aspects: (i) Propose a more effective OSP strategy for finding proper sensor locations installed on laminated composite structures; (ii) Conduct an optimization-based damage detection technique to validate the optimal sensor layout obtained from the proposed OSP strategy for structural damage detection, simultaneously showing its capacity in damage diagnosis and assessment by using the first several lower incomplete modes; and (iii) Apply effectively the Jaya optimization algorithm for solving both OSP and damage diagnose problems without trapping into local optima. The present work has two main parts. The first one is to determine the optimal location of a given limited number of sensors placed on a structure. For this purpose, we use a model reduction technique, namely iterated improved reduced system (IIRS) method [29], to develop a reduced order model for OSP, and then Jaya algorithm as robust optimization tool [30–32] is adopted to determine

2. Formulation of optimal sensor placement problem In this section, the mathematical formulation of optimal sensor placement (OSP) strategy is conducted by considering three following main points: (1) the OSP as an optimization problem; (2) the iterated improved reduced system (IIRS) method; and (3) the objective function for OSP problem. The details of the three points are presented in the next three sub-sections. 2.1. Optimal sensor placement as optimization problem The main goal of OSP in SHM is to determine the optimal sensor layout that can collect as much information of structural dynamic characteristics as possible. To achieve this goal, the OSP problem can be formulated as a constrained optimization problem in which the sensor positions are considered as the discrete design variables and the constraint is typically a given limited number of sensors. The objective function, usually based on the dynamic characteristics of a structure, can be maximized or minimized to determine the optimal locations for a given limited set of sensors. Thus, the mathematical model of OSP problem can be defined by the following optimization equation.

min f (S), s ∈  + s. t . g (S) = n, Slb ≤ S ≤ Sub .

(1)

where f is the objective function;S = (s1,s1, … , sn) is denoted as the candidate sensor locations placed at nodes/ DOFs of the FE structural model; n is the given limited number of sensors; Slb and Sub represent the vectors of lower and upper bound of S, respectively; and  + is the set of positive integers. In the OSP problem, a numerical model is required to identify the modal parameters of structural system such as natural frequency and mode shapes. Nevertheless, such a model will have many more nodes/ DOFs, while the optimal potential sensor locations are only chosen from a subset of the total nodes/ DOFs. Hence, we use a model reduction technique, namely IIRS method [29], to eliminate those DOFs that do not relate to the candidate sensor locations required. A brief description 49

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of IIRS method for OSP is given in the next sub-section.

2.3. Objective function for OSP problem

2.2. Iterated IRS (IIRS) method

The Modal Assurance Criterion (MAC) [37] is a good tool to check the correlation between two mode vectors of equal order. The off-diagonal elements in the MAC matrix express the consistency between two modal vectors. Hence, it was recommended to use the lowest offdiagonal terms in the MAC matrix as a useful criterion to choose optimal sensor locations [26,28]. The MAC matrix can be constructed by taking into consideration the candidate sensor positions as

Iterated IRS method employed in this present study is based on Friswell's method [29,35]. Here, we define the potential sensor locations as master DOFs denoted by m, and the eliminated locations as slave DOFs denoted by s, with m + s = N (where N is total DOFs of structural system discretized by FE model). For an un-damped system with N DOFs, the generalized eigenvalue problem of a structural system with the partitioned mass, stiffness matrices and mode shapes governed by the master and slave DOFs can be written as follows

⎡ Kmm Kms ⎤ ⎧ Φmm ⎫ = ⎡ Mmm Mms ⎤ ⎧ Φmm ⎫ Λ ⎢ Ksm Kss ⎥ ⎨ Φsm ⎬ ⎢ Msm Mss ⎥ ⎨ Φsm ⎬ mm ⎣ ⎭ ⎣ ⎭ ⎦⎩ ⎦⎩

MACij =

(2)

(3)

(4)

and substituting Eq. (4) into Eq. (3) and rearranging, we obtain the following equation 1 t = TG + K−ss1(Msm + Mss t)Φmm ΛmmΦ−mm

f (S) ≡ fRMS = 1 − RMS

(5)

RMS =

⎧ Φmm ⎫ = ⎡ Imm ⎤ Φ = T Φ IIRS mm ⎢ t ⎥ mm Φ ⎨ ⎭ ⎣ ⎩ sm ⎬ ⎦ 

1 nmod (nmod − 1)

where Imm is the unit matrix of size m × m. In addition, to get a specific way for calculating the transformation matrix t or TIIRS , we conduct some following mathematics transformations: Appling Eq. (6) into Eq. (2) and per multiplication by TTIIRS , we can obtain the following reduced eigenvalue problem

f (S) ≡ fMACFLEX = 1 −

nmod

(MACij )2 , (j ≠ i)

MACFLEX = (8)

1 nc

nc

∑ MACFLEXk k

T re 2 (Funre k ) F k (S) , (k = 1, 2, ⋯, nc ) T F unre)((F re (S))T F re (S)) ((Funre ) k k k k

(9)

∑

Φi* (Λ*)−1 (Φi*)T

(16)

where Φi* and F* are the ith mode shape and the flexibility matrix, respectively; nc is the total number of columns in the flexibility matrix; the superscripts unre and re denote the unreduced model and reduced model, respectively. For finding the optimal sensor placements, an optimization technique namely Jaya algorithm is adopted to minimize this objective function. The brief review of the algorithm will be presented in Section 4.

As can be seen from Eq. (9), t is an implicit function and cannot be directly solved. Friswell et al. [29] proposed an IIRS technique to solve for Eqs. (7) and (9), in which t is updated through an iterative process given by −1

(15)

nmod

F* =

i=1

t k = TG + K−ss1(Msm + Mss t k − 1)(MkR− 1) KkR− 1

(14)

where

Substituting Eq. (8) into Eq. (5), the transformation matrix t can be rewritten as

t = TG + K−ss1(Msm + Mss t) M−R1KR

(13)

In Eq. (14), we denote: (7)

From Eq. (7), we can get 1 Φmm ΛmmΦ−mm = M−R1KR

nmod

∑i =1 ∑ j=1

With the same purpose, the MAC criterion can be utilized to estimate the correlation between the flexibility matrix obtained from an original FE model and the corresponding one calculated from IIRS method. Based on the idea, an objective function for OSP is proposed here by using the diagonal terms of the flexibility matrix as follows

(6)

KRΦmm = MR Φmm ΛmmwithKR = TTIRS KTIRS , MR = TTIRS MTIRS

(12)

where the RMS is defined as

where TG = K−ss1Ksm is defined as the Guyan transformation matrix [36]. By using the transformation matrix t, the transformation between the master DOFs and the complete set of DOFs is represented by

TIIRS

(11)

where is the ith columns of the mode shape matrix calculated from full FE model; and Φire (S) is the ith columns of the mode shape matrix calculated by the reduced order model, which is functions of the sensor location parameter S; and nmod is the number of identified mode shapes. Note that each element of the MAC matrix ranges from 0 to 1. When i = j, MAC ^ 1 if two mode shape matrices are consistent. For the case of i ≠ j, a small value of the MAC off-diagonal element indicates the less correlation between corresponding mode shape vectors; otherwise, large off-diagonal values indicate that the two mode shape matrices are fairly indistinguishable [33]. Thus, the main goal of constructing the MAC matrix is to determine sensor locations such that off-diagonal terms of the MAC matrix are as small as possible. For this purpose, the root mean square (RMS) values of the off-diagonal elements have been employed to construct the objective function for OSP [25,33]. The equation of this function can be presented as follows

By assuming a transformation matrix t between Φsm and Φmm, as

Φsm = tΦmm

(i, j = 1, 2, ...,nmod )

Φunre i

where K and M are, respectively, the global stiffness and mass matrices; Φ is the mass-normalized eigenvectors; Λ is a diagonal matrix containing corresponding eigenvalues λi (i = 1, 2, … , m); and the subscripts m and s denote the sizes of the master and slave DOFs, respectively. To eliminate the slave DOFs field, the second row of Eq. (2) is employed and rearranged to give

Φsm = K−ss1KsmΦmm + K−ss1(Msm Φmm + Mss Φsm)Λmm

(Φunre )T Φrej (S) 2 i , unre T unre ((Φi ) Φi )((Φrej (S))T Φrej (S))

(10)

where the superscript k denotes the kth iteration, with k ≥ 2. When k = 1, t(1) = TG, which leads to Guyan's method; and k = 2, it is right the standard IRS method. Further, the convergence of the IIRS technique was proved in Ref. [35]. Thereby, the modal parameters can be obtained by solving the reduced eigenvalue problem defined by MR and KR, and then they are utilized in the OSP problem.

3. Optimization-based damage detection problem In optimization-based damage detection problem, the damage detection process is generally accomplished by minimizing an objective function based on modal parameters. With this scheme, a correct choice of the objective function plays a critical role in the successful 50

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Fig. 1. Node and element numbering of the discretized composite beam.

In this research, the numerical results illustrated clearly its better performance in comparison with other well-known optimization algorithms such as differential evolution (DE) and cuckoo search (CS). In the present research, the Jaya algorithm is further extended to solve for both the optimal sensor placement and damage detection problems, in which the sensor locations and the degree of damage elements are discrete and continuous design variables, respectively. The detail of Jaya algorithm for the problems with continuous or discrete variables is briefly described via four phases as follows. Initially, an initial population consisting NP individuals is generated randomly in the search space. Each individual in NP is a vector comprising n design variables xj = (x1,x2, … , xn) and is initialized as

Table 1 Sensor placements of the composite beam obtained by Jaya algorithm using two objective functions. Objective function

fRMS fMACFLEX fRMS fMACFLEX fRMS fMACFLEX

Number of sensors

6 7 8

Sensor no.

f

1

2

3

4

5

6

7

8

2 3 2 3 2 2

4 4 3 4 4 3

7 8 6 7 6 5

12 10 9 9 10 7

15 13 12 11 11 9

16 15 15 14 12 11

– – 16 15 15 14

– – – – 16 15

7.67 × 10 − 1 3.38 × 10 − 2 7.29 × 10 − 1 5.33 × 10 − 5 7.50 × 10 − 1 2.91 × 10 − 5

implementation of the process. Modal frequencies and mode shapes, the basic dynamic parameters, have been widely used for the problem. In practical applications, it is usually challenging to achieve the complete measured modes of a structure from a given limited number of sensors and the high-order mode shapes are typically quite difficult to measure accurately. To cope with the real issue, the modal flexibility is considered a suitable parameter because it can be approximately constructed by using only the measured first few frequencies and mode shapes. In addition, it was proved to be more sensitive to damage than either the natural frequency or the mode shape [38]. Due to these advantages, an objective function based on modal flexibility is defined herein by using the norm of the discrepancies between the flexibility matrix obtained from an experimental modal and the corresponding one calculated from a reduced analytical model, which can be formulated as

Γ(x) =

F exp − F re (x) F exp Fro

Fro

, x = (x1, ...,x ne ) ∈ [0, 1]ne

if xj is a continuous variable xj, i = x lj, i + rand [0, 1] × (x uj, i − x lj, i), (i = 1, 2, ...,NP; j = 1, 2, ...,n)

xj, i = x lj, i + round [rand [0, 1] × (x uj, i − x lj, i)], (i = 1, 2, ...,NP; j = 1, 2, ...,n)(20)

end if where x uj and x jl denote the upper and lower bounds of x j , respectively; rand[0, 1] is a uniformly distributed between 0 and 1; round is the function to round real values of integer design variables; NP is the population size. It should be noted that for OSP problem in Eq. (20), the sensor locations are selected from a candidate set of nodes of a model and a single sensor could only be placed on a single node. Next, at any Gth iteration, assume that x j, i, G is the value of the jth variable for the ith candidate during the Gth iteration, then this value is ′ utilized to generate a vector x i, G as

(17) if xj,i is a continuous variable x′j,i, G = xj,i, G + r1,j, G × (xj,best, G − |xj,i, G|) − r2,j, G × (xj,worst, (21) G − |xj,i, G|) else if xj,i is a discrete variable x′j,i, G = xj,i, G + round[r1,j, G × (xj,best, G − |xj,i, G|) − r2,j, (22) G × (xj,worst, G − |xj,i, G|)] end if

where the components of vector x are the degree of damage of ne elements, and they are considered as continuous design variables; F exp is the flexibility matrix obtained from the damaged FE model, which is taken as the flexibility matrix from experiments by adding noise at the modal parameters, while F re is the flexibility matrix obtained from reduced analytical model; and ‖ • ‖Fro presents the Frobenius norm of a matrix. To find a set of damage variables, a robust optimization solver is applied for minimizing the objective function Γ(x), which is expressed as follows

finding x = {x1, x2, ⋯, x ne} MinimizeΓ(x) S. t . 0 ≤ x i ≤ 1, (i = 1, 2, ⋯, ne )

(19)

else if xj is a discrete variable

where xj,worst, G and xj,best, G are the value of the jth variable for the worst candidate and the best candidate, respectively; r1,j, G and r2,j, G are the random numbers in the range [0, 1]; the term ′′r2,j, G × (xj,worst, G − |xj,i, G|)′′ points out the tendency of the solution avoiding the worst solution; and the term ′′r1,j, G × (xj,best, G − |xj,i, G|)′′ points out the tendency of the solution toward the best solution. The absolute value of the candidate solution |xj,i, G| helps enhance the exploration ability of the algorithm. Subsequently, the values of components x′j,i, G are checked to reflect back to the allowable region if its values exceed the corresponding upper and lower bounds. This operation is executed as

(18)

4. Jaya algorithm for discrete and continuous design variables Jaya algorithm, a novel population-based search method, recently proposed by Venkata Rao [30]. This algorithm is quite simple and easy to use with only common controlling parameters (population size and a number of generations), which escape the burden of tuning of algorithmic-specific control parameters [39]. It has recently been introduced for dealing with different engineering optimization problems [31,32,40], and its results have shown the better performance than those of other population-based algorithms. The main idea of this algorithm is that it always tries to move toward the best optimum solution and to avoid worst solution. In the previous our work [40], the Jaya algorithm is successfully applied in structural damage assessment.

x ′j, i, G

u u ⎧ 2x j − x ′ j, i, G if x ′ j, i, G > x j ⎪ l = 2x j − x ′ j, i, G if x ′ j, i, G < x jl ⎨ ⎪ x ′ j , i, G otherwise ⎩

(23)

Finally, the vector x′i,G is compared to its counterpart target individual xi,G. If the vector x′i,G has lower functional value, it will survive to the next generation, otherwise, the target vector xi,G will be saved in the population. 51

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Table 2 The first five frequencies of the laminated composite beam calculated by various models. Method

Modal frequency (Hz)

Unreduced model (FOBT) [42] Unreduced model (present) IIRS method (k = 2), using 6 optimal sensor locations from fRMS IIRS method (k = 2), using 6 optimal sensor locations from fMACFLEX

1

2

3

4

5

19.051 19.125 19.125

– 38.983 38.985

– 61.861 62.472

– 85.374 86.191

– 109.741 127.987

(0.00%) 19.125

(0.01%) 38.987

(0.99%) 61.873

(0.96%) 85.740

(16.63%) 113.447

(0.00%)

(0.01%)

(0.02%)

(0.43%)

(3.38%)

Fig. 2. The MAC values of the first eight modes for the composite beam obtained by Jaya algorithm using two objective functions.

52

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Fig. 3. Convergence process of the Jaya algorithm for OSP (8 sensor positions) of the composite beam using two objective functions: (a) fMACFLEX; (b) fRMS.

x′i, G if f (x′i, G) ≤ f (x i, G) x i, G + 1 = ⎧ ⎨ ⎩ x i, G otherwise

5.1.1. Optimal sensor placement First, we investigate the capability of the present OSP strategy for better data acquisition. For this purpose, we assume that the strategy is performed by installing 6, 7 and 8 sensors on the composite beam, respectively. In addition, it is also assumed that only the translational DOFs of the beam are considered for possible sensor installation. Consequently, a total of 15 nodes is available for installing the sensors, and the nodal number varies from 2 to 16. For conducting OSP of the composite beam, the IIRS technique is applied to obtain a reduced model consisting the translational DOFs only. Two cost functions, as provided in Eqs. (12) and (14), are employed to determine optimal locations for the given limited number of sensors. The first eight modes are selected for calculating these functions. The OSP results of the composite beam obtained from Jaya algorithm using two different objective functions are summarized in Table 1. The results show that when the objective functions employed for OSP are different, the corresponding optimal sensor layouts will have discrepancies. That means that the OSP problem may depend on the different objective functions used in the optimization process. The capability of capturing the vibration behavior of the beam model with two sets of sensor nodes (6 measured points) from two objective functions are reported in Table 2. The identified modal frequencies results from IIRS method are compared with those calculated by the unreduced model in the second row. It is obvious that the use of 6 measurement points from the function fMACFLEX help identify modal frequencies with more accuracy than that from the function fRMS. Fig. 2 shows the MAC values between the modes identified from different sets of sensors and the corresponding modes calculated from all DOFs of the FE model. From the figure, it can be seen that: (1) the number of identified modes will increase when more sensors are properly added; (2) with 7 measurement points from the function fMACFLEX, the modes from 1 to 8 are well identified, while with 7 measurement points from the function fRMS, only the modes from 1 to 5 are identified. This indicates that the OSP strategy based on the function fMACFLEX is found to be superior to that based on the function fRMS in determining the optimal sensor positions. The typical convergence process of the Jaya algorithm for OSP of

(24)

where f is the cost function which is given as either Eq. (14) for continuous variable or Eq. (17) for discrete variable.

5. Numerical examples In this part, two numerical examples including a three cross-ply (0°/ 90°/0°) beam and a four-layer (0°/90°/90°/0°) laminated composite plate are presented to verify the applicability and efficiency of the proposed approach for structural damage assessment using incomplete modal data identified from optimized sensors placement. In the following examples, damage is simulated by reducing the stiffness of senele lected elements to a certain level, i.e. K = ∑e = 1 (1 − ae ) ke , where K and e k are the global stiffness matrix of damage structures and the stiffness matrix of the eth element, respectively; and ae represents the damage ratio of the eth element. For each of these examples, three different damage scenarios are considered with and without the effect of measurement noise on identifying single and multiple structural damage sites. The added noise level in the measured modal data is ± 0.15% noise in natural frequencies and ± 3% noise in mode shapes [5]. The common control parameters of the Jaya algorithm in all examples are given by population size (NP) = 30, maximum integration = 900, random number (r) = [0, 1] and stop criterion = 10−6. Each optimization problem of OSP in all examples is repeated 10 times, and the best optimal solution with the minimum objective function and a good MAC index is considered as the representative OSP result. For each damage scenario, the average and statistical results of 10 independent runs are reported in figures and tables, respectively. All the computations are carried out in MATLAB environment.

5.1. A three cross-ply (0°/90°/0°) beam The first example is a cross-ply (0°/90°/0°) rectangular beam with clamped boundary conditions at both ends, as depicted in Fig. 1. The geometrical parameters of the beam are given by the length L = 0.2 m, the width b = 0.02 m and the thickness t = 0.02 m. The physical material properties are given by E1 = 40 N/m2, E2 = 1 N/m2, G12 = G13 = 0.6E2, G23 = 0.5E2, v12 = 0.25, and the thickness of each layer is t/3. The composite beam is discretized into 16 beam elements using first-order shear deformation theory (FSDT) as presented in Ref. [41], and thus each element has two nodes with three DOFs (two translational displacements in the horizontal and vertical directions, and one rotation) per node.

Table 3 Three different damage scenarios in the laminated composite beam.

53

Scenario

1

2

Element no. Damage ratio

1 0.30

5 0.20

3 16 0.30

1 0.20

2 0.3

9 0.30

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Fig. 4. Comparison of obtained damage identification results from Jaya algorithms for the laminated composite beam without noise and with noise from various optimal sensor placement: (a) case 1; (b) case 2; (c) case 3.

5.1.2. Damage detection using incomplete modal data obtained from sensors Second, we focus on damage detection in the composite beam using incomplete modal data obtained from optimized sensor locations. In order to examine the effect of the proposed damage identification

the composite beam is shown in Fig. 3, in which the mean and best convergence lines of the two objective functions in 8 measured points’ case are depicted. It can be seen that the average fitness has the tendency to steadily approach to best fitness with the increase of the number of generations, which shows a good stability of convergence.

Table 4 The statistical results of damage identification for three scenarios of the laminated composite beam without noise and with noise from various optimal sensor locations. Scenario

1 2

3

Noise level

Actual location

Jaya algorithm, using 6 optimal sensor locations

Jaya algorithm, using 7 optimal sensor locations

Jaya algorithm, using 8 optimal sensor locations

Avg. value

Std. dev.

Avg. NSA

Avg. value

Std. dev.

Avg. NSA

Avg. value

Std. dev.

Avg. NSA –

0%

α1

0.2951

0.0004

5,346

–

–

–

–

–

3%

α1

0.2322

0.0930

7,155

0.2564

0.0414

6,450

0.2790

0.0238

5,877

0%

α5

0.1992

0.0005

5,952

–

–

–

–

–

–

α16 α5

0.2736

0.0007

–

–

–

–

3%

0.1629

0.1348

0.1582

0.0571

α16

0.2004

0.0815

0.2483

0.0483

–

–

–

–

–

–

0.1391

0.0667

0%

3%

7,752

α1

0.2012

0.0007

α2 α9

0.3063

0.0005

6,249

0.2993

0.0003

α1

0.1296

0.0786

α2 α9

0.3328

0.0625

0.3216

0.2793

0.0773

0.2816

7,311

6,387 –

6,546

0.1951

0.0723

0.2449

0.0647

–

–

–

–

–

–

0.1876

0.0123

0.0678

0.3157

0.0223

0.0413

0.3033

0.0286

Avg. value = average value of damage ratio with respect to Γ; Std. dev. = standard deviation with respect to Γ; Avg. NSA = an average number of structural analyses.

54

6,459 –

6,195

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D. Dinh-Cong et al.

Fig. 5. Convergence process of the Jaya algorithm for damage detection of the composite beam: (a) without noisy data from 6 measurement points; (b) with noisy data from 8 measurement points.

Fig. 6. (a) A four-layer (0°/90°/90°/0°) square composite plate; (b) element numbering of the plate.

Table 5 Optimal sensor placements of the composite plate obtained by Jaya algorithm using the objective function fMACFLEX. Number of sensors

10

Sensor no.

of noise, the proposed method can accurately detect the actual damaged locations with only 6 sensors placed on the composite beam. However, in noise-contaminated condition and with 6 measurement points, the proposed method has some false alarms elements (element 2 in scenario 1; elements 12 and 15 in scenario 2) appeared in its predictions, whereas with 8 measurement points the proposed method can successfully determine the actual site of damage with negligible false alarms. Also from the figures, it is worth mentioning that the better damage assessment results are achieved when the more sensors are utilized on the structure. For further investigation, the statistical results of damage detection including the average, standard deviation, and the average number of structural analyses for three scenarios of the composite beam without and with noisy data from a different number of sensors are provided in Table 4. The results show clearly that for all three scenarios, the standard deviation of predicted results with noise-free data from 6 measurement points is much lower than those with corresponding noisecorrupted data. However, the standard deviation of predicted results with noise-corrupted data can be decreased by adding more sensors.

f

1

2

3

4

5

6

7

8

9

10

7

10

17

19

27

30

31

36

40

45

3.19 × 10 − 9

method with the different number of sensors (6, 7 and 8 sensors), three damage scenarios are studied in this example. The details of damaged elements and their damage ratios of these scenarios are summarized in Table 3. The first five natural frequencies and corresponding mode shapes are used for approximating the objective function as in Eq. (17). To determine the damage extent and its locations, Jaya algorithm is adopted for solving the inverse problem. The average results of damage ratio of all elements obtained by the Jaya algorithm for scenario 1 to 3 are shown in Fig. 4(a), (b) and (c), respectively. It can be seen that when the measured modal data is free Table 6 The first six frequencies of laminated composite plate calculated by IIRS method. Method

Unreduced model (present) IIRS method (k = 2), using 10 optimal sensor locations

Modal frequency (Hz) 1

2

3

4

5

6

0.022 0.022 (0.00%)

0.056 0.056 (0.00%)

0.138 0.138 (0.00%)

0.204 0.204 (0.40%)

0.402 0.402 (0.00%)

0.467 0.467 (0.00%)

55

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1

Particularly, the average standard deviation over all scenarios with noise-corrupted data from 6, 7 and 8 measurement points are, respectively, 9.13%, 5.09% and 3.78%. In addition, the mean error between obtained average damage ratios and the actual damage ratios for three cases with noise-corrupted data from 6, 7 and 8 measurement points are 21.23%, 16.07% and 6.72%, respectively. These results demonstrate that the proposed method can determine both the location and severity of structural damage with high accuracy when at least 8 sensors are placed on the structure. Moreover, we also note that the increase of measurement points will lead to the decrease of the number of structural analyses. The convergence histories of the Jaya algorithm for a sample run of each damage scenario in the case without noise and with measurement noise are shown in Fig. 5. It can be found that the algorithm can converge rapidly to the optimal solution after 60 generations.

0.8 1

MAC

0.6 0.5 0.4 0 0

2

4

6

8 Mode 10 12

10

14

0

2

4

14 12

0.2

8 6 Mode

0

Fig. 7. The MAC values of the first fifteen modes for the composite plate.

2.5

Next, we consider a cantilever-laminated composite plate (0°/90°/ 90°/0°), which has the dimension Lx = 0.5 m × Ly = 1 m × t = 0.04 m. The thickness of each layer is t/4. The physical material parameters (Mpa) for each layer are given by E1 = 40, E2 = 1, G12 = G13 = 0.6E2, G23 = 0.5E2, ν12 = 0.25. The FE model of the composite plate (as shown in Fig. 6) is discretized into a mesh 4 × 8 or 32 four-node Reissner–Mindlin plate elements using first-order shear deformation theory (FSDT) as presented in Ref. [43]. Thus, the simulated model has a total of 45 nodes with five DOFs (three translations and two rotations) per node.

x 10-7

Jaya algorithm: Best value Jaya algorithm: Mean value

2

Fitness value

5.2. A four-layer (0°/90°/90°/0°) laminated composite plate

1.5 1

5.2.1. Optimal sensor placement In this example, we assume that the OSP problem in SHM is followed by installing 10 sensors on the composite plate. The selection of the number of sensors is dependent on experiences, engineering judgment and/or trial and error method. As shown in Fig. 6(b), a total of 40 nodes in the FE model, exclude the constrained nodes, are considered as the possible candidate set of sensor locations. The first ten modes of the plate are used for calculating the objective function. With this model, only the proposed function fMACFLEX is adopted to find optimal locations for a given limited number of sensors. Table 5 shows the optimal locations of sensors obtained using the Jaya algorithm with the proposed function fMACFLEX. As can be seen in Table 6, the reduced model from NSEMR-II method using 10 optimal sensor locations provides the identified modal frequencies results with very high accuracy. Fig. 7 shows the MAC values between the first fifteen modes identified from 10 optimal sensor locations and the corresponding modes calculated the original full FE model. It is shown that the MAC off-diagonal values are close to zero and all the diagonal elements are close to 1, which means a good correlation between

0.5 0 0

20

40 Iteration

60

80

Fig. 8. Convergence process of the Jaya algorithm for OSP of the composite plate.

Table 7 Three different damage scenarios in the laminated composite plate. Scenario

1

2

3

Element no. Damage ratio

3 0.25

14 0.25

19 0.40

5 0.30

23 0.30

26 0.40

0.3

Damage ratio

Damage ratio

0.3

0.2 3 0.1

0.2

3

0.1

0

0 8

7

8

6 5 4 Y 3

2 1

2 1 4 3X

(a)

7

6 5 4 3 Y 2 1

2 1 4 3X

(b)

Fig. 9. The damage identification results obtained from the Jaya algorithm for damage scenario 1 of the composite plate with 10 measurement points: (a) noise-free; (b) noise 3%.

56

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0.4 19

19

0.3

Damage ratio

Damage ratio

0.4

14

0.2 0.1

0.3 0.2

14

0.1 0

0 8

7

8 6

5 4 Y 3

7

2 1 4 3X

2 1

6 5 4 3 Y 2

(a)

1

2 1 4 3X

(b)

Fig. 10. The damage identification results obtained from the Jaya algorithm for damage scenario 2 of the composite plate with 10 measurement points: (a) noise-free; (b) noise 3%.

26

0.3

26

0.4

23

Damage ratio

Damage ratio

0.4

5 0.2 0.1 0

0.3 23 5

0.2 0.1 0

8

7

8 6 5 Y 4 3

2 1

7

6

2 1 4 3X

5 Y 4 3

(a)

2

2 1 4 3X

1

(b)

Fig. 11. The obtained damage identification results obtained from the Jaya algorithm for damage scenario 3 of the composite plate with 10 measurement points: (a) noise-free; (b) noise 3%.

related vectors as expected for MAC criterion. The typical convergence process of the Jaya algorithm for OSP of the composite plate is depicted in Fig. 8.

Table 8 The statistical results of damage identification for three scenarios of the laminated composite beam without noise and with noise from 10 measurement points.

5.2.2. Damage detection using incomplete modal data obtained from sensors After obtaining the sensor layout for the composite plate, the modal data measured from these sensors can be used for damage assessment by the proposed damage diagnosis method. In this example, three different damage scenarios consisting of one, two and three damaged elements are considered. The details of three damage scenarios are listed in Table 7. The first six natural frequencies and corresponding mode shapes are employed for calculating the objective function as in Eq. (17). Figs. 9, 10 and 11 present the average results of damage ratio of all elements obtained by the Jaya algorithm for scenario 1, 2 and 3, respectively. The graphical results show that for three scenarios, the proposed method can accurately locate all locations of damage in the plate even under noisy and incomplete modal data from 10 measurement points. Further, the details of statistical results including the average, standard deviation, and the average number of structural analyses of the Jaya algorithm for three scenarios are reported in Table 8. The results show that the extent of damaged elements is determined by the proposed method with satisfactory accuracy. Particularly, all scenarios without noise, the mean error between obtained average damage ratios and the actual damage ratios are 0.30%, while those for all scenarios with measurement noise are 9.69%. Also, the standard deviation of all predicted results is less than 5%. These results demonstrate the good ability of the proposed method in detecting and

Scenario

1 2

Noise level

Jaya algorithm, using 10 optimal sensor locations Avg. value

Std. dev.

Avg. NSA

0%

α3

0.2498

0.0001

16,212

3%

α3

0.2612

0.0639

17,115

0%

α14

0.2492

0.0295

15,675

α19

0.3983

0.0258

α14

0.2191

0.0258

α19

0.3842

0.0266

α5

0.2997

0.0002

α23

0.2976

0.0011

3% 3

Actual location

0%

3%

α26

0.4000

0.0002

α5

0.2637

0.0463

α23

0.2683

0.0471

α26

0.3862

0.0415

17,619 16,782

16,815

Avg. value = average value of damage ratio with respect to Γ; Std. dev. = standard deviation with respect to Γ; Avg. NSA = an average number of structural analyses.

quantifying the structural damage in spite of incomplete modal data and measurement inaccuracy. Fig. 12 illustrates the convergence histories of the Jaya algorithm for all damage scenarios of the composite plate with and without noise in modal data. From the figure, it is easy to see that the algorithm converges to the minimum cost after 310 generations. 57

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Fig. 12. Convergence process of the Jaya algorithm for damage detection of the composite plate: (a) without noisy data from 10 measurement points; (b) with noisy data from 10 measurement points.

Fig. 13. The damage identification results obtained from the Jaya algorithm for the composite plate with 10 measurement points and noise 5%: (a) scenario 1; (b) scenario 3.

deviation of the predicted results are much larger than the previous results presented in Table 7. From these results, it is reasonable to conclude that with this case study, the increase of measurement noise leads to the decrease of the precision of identification. Note also that when the noise level increases, the damage detection process needs more structural analyses for reaching the optimal solution.

Table 9 The statistical results of damage identification for the laminated composite beam with noise from 10 measurement points and noise 5%. Scenario

Noise level

Actual location

Jaya algorithm, using 10 optimal sensor locations Avg. value

Std. dev.

Avg. NSA

1

5%

α3

0.1835

0.1548

19,236

3

5%

α5

0.1315

0.0911

19,719

α23

0.2170

0.2890

α26

0.2550

0.1531

6. Conclusions The article proposes an efficient approach for optimal sensor placement and damage identification in laminated composite structures. The sensor placement strategy is first utilized to determine the optimal sensor layout of a given limited number of sensors placed on a structure, while damage identification method is proposed to locate and quantify structural damage using incomplete modal data collected from optimized sensor locations. The problems of optimal sensor location and damage identification are formulated as discrete and continuous optimization problems, respectively, and then solved by the Jaya algorithm. The applicability and efficiency of the proposed approach are verified through two numerical examples comprising a three cross-ply (0°/90°/0°) beam and a four-layer (0°/90°/90°/0°) laminated composite plate. In addition, various challenges such as the presence of noise in measured incomplete modal data and the impacts of the number of measurement locations are also investigated. Based on the numerical results presented in this article, some conclusions are summarized as follows

Avg. value = average value of damage ratio with respect to Γ; Std. dev = standard deviation with respect to Γ; Avg. NSA = an average number of structural analyses.

In this example, to further investigate the effect of measurement noise on the performance of the proposed algorithm, the increase of noise level in measured modal data is considered here by adding ± 1% instead of ± 0.15% noise in natural frequencies and ± 5% instead of ± 3% noise in mode shapes. Damage scenarios are assumed to be same as scenarios 1 and 3 in Table 7. For both the damage scenarios, the first six vibration modes are utilized for damage assessment in the plate. The final identified results using measured modal data from ten measurement points and the noise level are reported in Fig. 13 and Table 9. Fig. 13 shows that although the proposed damage detection method is still able to determine the damaged sites correctly, some false damage elements appear especially in triple damage case. Besides, the statistical results in Table 9 indicate that the mean error and standard

(1) The Jaya algorithm is found to be an efficient optimization tool for solving both the discrete and continuous optimization problems. 58

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(2) In comparison with the objective function fRMS, the proposed objective function fMACFLEX can capture better vibration modes and ensures a good agreement between the identified and calculated mode shapes. In addition, since more sensors are properly added in the structure, the number of identified modes will be increased. (3) When the best sensor locations are installed in the structure, the best modal information can be obtained. This will ensure the high precision of the collected data which is regarded as optimal input data for damage diagnostics in the following step. (4) The proposed damage identification method can successfully determine both the site and severity of damages in composite structures with noise in measured incomplete modal data. (5) It is worth mentioning that the proposed approach, which combines an optimal sensor placement strategy and a damage identification method, can be considered as a promising tool for practical SHM applications.

[17]

[18]

[19]

[20]

[21]

[22]

Acknowledgements

[23]

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.08.

[24]

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[26]

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