A combination of damage locating vector method (DLV) and differential evolution algorithm (DE) for structural damage assessment

Abstract

In this study, a two-stage method is presented for identifying multiple damage scenarios. In the first stage, the damage locating vector (DLV) method using normalized cumulative energy (nce) is employed for damage localization in structures. In the second stage, the differential evolution algorithm (DE) is used for damage severity of the structures. In addition, in the second stage, a modification of an available objective function is made for handing the issue of symmetric structures. To verify the effectiveness of the present technique, numerical examples of a 72-bar space truss and a one-span steel portal frame are considered. In addition, the effect of noise on the performance of the identification results is also investigated. The numerical results show that the proposed combination gives good assessment of damage location and extent for multiple structural damage cases.

Keywords

damage assessment / damage locating vector method (DLV) / differential evolution (DE) / multiple damage location assurance criterion (MDLAC) / mode shape error function

Introduction

Structural health monitoring (SHM) has become an important aspect in engineering fields such as aerospace, off-shore construction, civil engineering and mechanical industry. SHM helps assess the integrity of structures to ensure the correctness and safety throughout their service life. It is common that the existence of damage in structure leads to change in modal parameters including frequency, mode shape and modal damping. Based on this nature of structure, many vibration-based methods have been proposed to localize and quantify damage.

In early stage, damage localization in SHM took huge interest from researchers all over the world. Many methods have been proposed and developed for various structures. Some pioneering works relating to simple structures can be mentioned as follows. Yuen [¹] used change in both frequency and mode shape to identify damage in cantilever beam. Pandey et al. [²] proposed mode shape curvature method to locate damage in a finite element beam structure. Salawu and Williams [³] employed modal assurance criterion (MAC) value for structural damage detection. Pandey and Biswas [⁴,⁵] proposed change in flexibility matrix to identify damage location for beam like structures. Stubbs et al. [⁶] presented a modal strain energy-based method. Subsequently, Bernal [⁷] proposed the damage locating vector method (DLV) for linear elastic structures. Besides, many recent works dealing with localization in complex structures were studied in Refs. [⁸–²³]. Among various damage identification methods, DLV method has high practical potential and the feasibility of the method has been illustrated numerically and experimentally using truss structures [²⁴].

In parallel to developing damage localization strategies, severity assessment has gained much concern from many researchers. For example, Cawley and Adams [²⁵] used the difference between measured and analytical frequencies to detect, locate and quantify damage. Messina et al. [²⁶] defined the multiple damage location assurance criterion (MDLAC) to predict both location and extent of one or more damage sites. Mares and Surace [²⁷] used genetic algorithm (GA) to minimize the generalized residual force function. The optimal solution provides information of damage site and extent. Koh and Dyke [²⁸] proposed a method that assesses both damage location and extent by maximizing the MDLAC function. Mohan et al. [²⁹] used the particle swarm optimization (PSO) to minimize the frequency response function (FRF)-based objective function. Recently, Kaveh et al. [³⁰] used a mixed particle swarm-ray optimization together with harmony search (HRPSO) for identifying the location and extent of frame and truss structures. Kaveh and Maniat [³¹] and Kaveh and Zolghadr [³²] proposed two improved versions of a charged system search (CSS) algorithm [³³] to locate and quantify the damages in structures using natural frequency and mode shape of structures. Mostly, the severity estimation of damage is modified to an optimization problem in which damage variables are usually the elemental damage extents and the objective function is commonly the difference between measured and analytical data. However, the challenge in solving optimization problem is that the solution may be trapped in a local optimum and the computational cost is usually expensive especially for the case of large numbers of damage variables.

Recently, two-stage approach in damage assessment has proved to be an efficient method for localizing and quantifying damage in structures. In this approach, the damage is identified through two stages where potential damaged elements are located in the first stage and then the severity of the damage is identified in the second stage. Usually, the above-mentioned damage localization methods are employed in the first stage and the above-mentioned severity assessment methods are utilized in the other stage. Some previous studies on this approach can be mentioned as follows. Moslem and Nafaspour [³⁴] first used the residual force method to locate damage sites and then applying the GA to estimate damage extent of identified sites. Au et al. [³⁵] employed the elemental energy quotient difference in the first stage and a micro-genetic algorithm in the second one to assess structural damage. Guo and Li [³⁶] combined an evidence theory and GA to identify damage location and extent. Seyedpoor [³⁷] proposed a two-stage method using modal strain energy based index (MSEBI) and PSO for evaluating damage in structures, and so forth. In the literature review, the two-stage methods have demonstrated the remarkable features in comparison to severity assessment methods and damage identification methods. For example, the computational cost is reduced significantly since only suspected damaged elements are considered in the optimization scheme. And, the optimal solution might eliminate misidentified elements (false alarms) and provides exact extents of damaged elements. However, there are still some issues as follows: 1) almost previous studies were limited on optimization techniques like GA and PSO which still remain restrictions in searching the global solution and computational cost; 2) the non-unique solution of symmetric structures as mentioned in Refs. [³⁸,³⁹] has not addressed yet. Therefore, finding an efficient algorithm as well as an effective objective function for two-stage approach is really necessary. Meanwhile, among various optimization methods, the differential evolution algorithm (DE) [⁴⁰,⁴¹] has been proven to be superior to other popular optimization algorithms (e.g., GA, PSO, Artificial Bee Colony Algorithms (ABC) and Cuckoo-Search (CS)) [⁴²,⁴³]. Nevertheless, the application of DE to two-stage approach hasn't been performed yet.

Based on the above considerations, in this paper, a new combination of the DLV method and DE for structural damage assessment is proposed. In the first stage, the DLV method using the normalized cumulative energy (nce) instead of the normalized cumulative stress (ncs) is employed to account for variations of internal forces along the length of each element [²⁴]. In the second stage, a combined objective function of MDLAC function and a mode shape error function is proposed to overcome the drawbacks of MDLAC function like the non-uniqueness of solution and the issue of symmetric structures. It is observed that the DLV and DE are model-based methods that require modal information of both intact and damaged stages. The proposed method is hence suitable for damage assessment of structures that is modeled numerically in design procedure or modal information is collected at the intact stage. The performance of the proposed procedure is examined by two numerical examples of a 72-bar space truss and a one-span steel portal frame with the presence of noise.

A brief introduction to DLV method and the formulation of normalized cumulative energy index

The DLV method, first proposed by Bernal [⁷] for damage localization, has been demonstrated to be an effective method through many studies including both numerical simulation analysis and experimental verifications [²⁴,⁴⁴–⁴⁸]. In this method, some load vectors designed as damage locating vectors (DLVs) are sought and treated as a static force vector onto the reference structural model. The crucial feature of these loads is that they cause zero stress in damaged elements of structures. Therefore, a so-called normalized cumulative stress (ncs) index [⁷], computed from stress of each element, was proposed to identify the present of damage in structures. The usage of ncs provided good results for damage identification in structures. However, the computation of this index is somewhat complex when it has been adjusted to various structures. As a result, Quek et al. [²⁴] proposed a normalized cumulative energy (nce) index which is adapted to account for the different types and variations of internal forces and capacities along the length of each element [²⁴]. Based on these considerations, in this study, we also used the nce for localizing damage in structures. The DLV method using the nce is briefly described below.

DLV method and the formulation of normalized cumulative energy (nce) index are based on two well-known concepts in damage detection methods namely: (1) flexibility matrix and (2) strain energy. From modal parameters of structure, the flexibility matrix is computed as follows [⁴]

(1)F={\displaystyle \sum _{i=\mathrm{1}}^{sdof}\frac{\mathrm{1}}{{\omega }_i^{\mathrm{2}}}}\hspace{0.17em}{\varphi }_i\hspace{0.17em}{\varphi }_i^T,

where F is the flexibility matrix; {\omega }_i is the ith frequency; {\varphi }_i is the ith mass-normalized mode shape; sdof is the number of degrees of freedom. In the above equation, flexibility matrix can be well approximated by using a few low modes as follows

(2)\tilde{F}={\displaystyle \sum _{i=\mathrm{1}}^{nmod}{\displaystyle \frac{\mathrm{1}}{{\omega }_i^{\mathrm{2}}}}\hspace{0.17em}}{\varphi }_i\hspace{0.17em}{\varphi }_i^T,

where nmod is the number of considered low modes. When a structure is damaged, the mode shapes, the frequencies and then the flexibility matrix are changed. Therefore, the change in flexibility matrix has been used as an indicator to detect damage locations. This change can be archived by

(3){\tilde{F}}_{\mathrm{\Delta }}={\tilde{F}}_{UD}-{\tilde{F}}_D,

where the indexes UD and D mean respectively undamaged and damaged states.

The DLVs defined as a basis for the null space of the change in flexibility can be calculated from a singular value decomposition (SVD) of {\tilde{F}}_{\Delta } as follows [⁷]

(4){{\displaystyle \tilde{F}}}_{\Delta }\underset{\_}{\underset{\_}{SVD}}\hspace{0.17em}{U\Sigma V}^T=[{{\displaystyle U}}_{\mathrm{1}}\hspace{1em}{{\displaystyle U}}_{\mathrm{0}}]\left[\begin{array}{ll}{{\displaystyle \Sigma }}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}\end{array}\right]{[{{\displaystyle V}}_{\mathrm{1}}\hspace{1em}{{\displaystyle V}}_{\mathrm{0}}]}^T,with\hspace{1em}DLVs={{\displaystyle V}}_{\mathrm{0}},

where U and V are orthogonal matrices; S is a diagonal matrix with diagonal terms being the singular values of {\tilde{F}}_{\Delta }; S₁ is a diagonal matrix including the nonzero singular values; U₁ is a basic for the column space, and U₀ is a basic for the left null space; V₁ is a basic for the row space, and V₀ is a basic for the null space i.e., the column of V₀ corresponding to the zero diagonal elements of S. The feature of the column vectors in V₀ is that when these vectors are treated as loads on the system, the stress of damaged elements equals to zeros. Based on this feature, the column vectors in V₀ are then used to localize the damage [⁷].

If the ith column of V₀ (the ith DLV load) is applied onto the reference structure, the strain energy of the eth element of the structure is calculated by

(5){\Xi }_i^e={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{d}_i^{eT}{K}^e{d}_i^e,

where d_i^e is the displacement of the eth element; K^e is the eth elemental stiffness matrix. The normalized cumulative energy (nce) is defined for each element by the following formula

(6){\bar{\Xi }}^e={\displaystyle \frac{{\Psi }^e}{\underset{k}{\max }\hspace{0.17em}\{{\Psi }^k\}}},

where

(7){\Psi }^e={\displaystyle \sum _{i=\mathrm{1}}^{ndlv}\frac{{\Xi }_i^e}{\underset{k}{\max }\{{\Xi }_i^k\}}},

where ndlv is the number of DLVs.

It is observed that the nce indexes of damaged elements equal to zero for all DLVs due to the feature of DLVs. Unfortunately, besides causing zero stress at damaged elements, each DLV may also cause zero stress at some undamaged elements. If only one DLV load is used, nce criteria may lead to misidentify some undamaged elements in the set of potential damaged elements. To mitigate this difficulty, all available DLVs are employed for more accurate damage localization.

Damage extent estimation using differential evolution (DE) algorithm

In this section, damage severity estimation of suspected elements located by DLV method is investigated. This is an inverse problem where the solution (damage extents of these elements) must produce the same modal parameters as those of damaged structures. This problem can be defined as

\underset{x}{Minimize}\hspace{1em}\Gamma (x),

where x is the damage variable vector that contains damage extents of suspected elements and .. is an objective function representing the above-mentioned condition.

In the following sections, an objective function that combines MDLAC and mode shape error function is presented. Then, the brief introduction of the differential evolution (DE) algorithm is presented.

Objective function

Multiple damage location assurance criterion (MDLAC) first defined by Messina et al. [²⁶] is a correlation coefficient between measured and analytical frequencies. It has been used by many authors [²⁸,³⁶,³⁷] as an efficient objective function for different damage detection methods using optimization. However, because only information of frequency is used in this function, it may not handle the non-uniqueness of solution and the issue of symmetric structures [³⁸,³⁹]. Therefore, in this study an objective function that combines MDLAC and mode shape error function is used and presented as follows.

Let us denote the measured frequency vector of the undamaged and damaged structures by f_UD and f_DM, respectively. The damage in structure can be described by a damage variable vector x = {x₁, …, x_nd} where x_i is a feasible damaged extent of the ith suspected element and nd is the number of suspected damaged elements. With respect to x, the analytical frequency vector denoted by f_DA(x) can be estimated by using the FEA. The MDLAC function is defined as follows

(8)MDLAC(x)={\displaystyle \frac{{\left|\Delta f^T\delta f(x)\right|}^{\mathrm{2}}}{(\Delta f^T\Delta f)(\delta f^T(x)\delta f^T(x))}},

where the measured frequency change vector, \Delta f, and the analytical frequency change vector, \delta f(x), are defined by [³⁷]

(9)\Delta f={\displaystyle \frac{f_{UD}-f_{DM}}{f_{UD}}},\hspace{1em}\delta f(x)={\displaystyle \frac{f_{UD}-f_{DA}(x)}{f_{UD}}}.

It can be noted that the value of MDLAC is always within [0, 1] and the maximum value (equal to 1) can be attained when the analytical frequency vector is the same as the measured frequency vector of damaged structure, that mean, f_{DM}=f_{DA}(x). This condition is obtained if the damage variable vector x represents exactly the degradation of the structure.

In this study, the MDLAC function is combined with a modes shape error function to give a new objective function

(10)\Gamma (x)=\mathrm{1}-MDLAC(x)+{\displaystyle \sum _{i=\mathrm{1}}^{nmod}\frac{\Vert {\varphi }_{DM,i}-{\varphi }_{DA,i}(x)\Vert }{\Vert {\varphi }_{DM,i}\Vert }},

where {\varphi }_{DM,i} and {\varphi }_{DA,i}(x) are the ith measured and analytical mode shape, respectively. It should be noted that the minimum value of \Gamma (x) is zero if the damage variables are estimated exactly. In comparison to the MDLAC, the proposed objective function is more restricted since the solution must satisfy both frequency and mode shape conditions.

Differential evolution algorithm

Differential evolution (DE) algorithm which is first introduced by Storn and Price [⁴⁰] is one of most popular population-based algorithm. It is not only simple to implement and easy to use with a few control parameters, but its convergence properties are better compared to many other well-known methods [⁴⁰,⁴²]. The implementation of the algorithm is briefly described as follows.

At the start of the algorithm, the initial population of NP individuals is created by mean of randomly sampling from the D-dimensional feasible domain S. These individuals are then evolved through three operators: mutation, crossover and selection as discussed below.

For each individual (target vector) x_{i,G}=\{x_{\mathrm{1},i,G}, x_{\mathrm{2},i,G}, \dots , x_{D,i,G}\},\hspace{0.17em}\hspace{0.17em}i=\mathrm{1}, \mathrm{2}, \dots ,NP of a G generation, the mutation operator is used to generate the mutation vector v_{\mathrm{i,G}} by one of the five strategies as follows [⁴⁹]:

\begin{array}{ll}DE/rand/\mathrm{1}& v_{i,G}=x_{r_{\mathrm{1}},G}+F(x_{r_{\mathrm{2}},G}-x_{r_{\mathrm{3}},G}),\\ DE/rand/\mathrm{2}& v_{i,G}=x_{r_{\mathrm{1}},G}+F(x_{r_{\mathrm{2}},G}-x_{r_{\mathrm{3}},G})+F(x_{r_{\mathrm{4}},G}-x_{r_{\mathrm{5}},G}),\\ DE/best/\mathrm{1}& v_{i,G}=x_{best,G}+F(x_{r_{\mathrm{1}},G}-x_{r_{\mathrm{2}},G}),\\ DE/best/\mathrm{2}& v_{i,G}=x_{best,G}+F(x_{r_{\mathrm{1}},G}-x_{r_{\mathrm{2}},G})+F(x_{r_{\mathrm{3}},G}-x_{r_{\mathrm{4}},G}),\\ DE/current\mathrm{2}best/\mathrm{1}& v_{i,G}=x_{i,G}+F(x_{best,G}-x_{i,G})+F(x_{r_{\mathrm{3}},G}-x_{r_{\mathrm{4}},G}),\end{array}

where {\mathbf{X}}_{best,G} is the best individual of the current generation; F is a mutation constant which falls within [0, 2], and mutually different individuals x_{r_{\mathrm{1}},G},\hspace{0.17em}{x}_{r_{\mathrm{2}},G},\hspace{0.17em}{x}_{r_{\mathrm{3}},G},\hspace{0.17em}{x}_{r_{\mathrm{4}},G},\hspace{0.17em}{x}_{r_{\mathrm{5}},G} are selected from the set of population except the individual x_{i,G}.

Afterward, the crossover operator is applied to the population in which some components of target vector, x_{i,G}, is substituted by the corresponding {\nu }_{i,G}\text{'}s components to create a trial vector u_{i,G} by using either a binomial or an exponential scheme. Binomial scheme is commonly used in DE and is presented by

u_{ji,G}=\{\begin{array}{l}{v}_{ji,G}\hspace{1em}\mathrm{if}\hspace{0.17em}(randb(j)\le CR)\hspace{0.17em}\mathrm{or}\hspace{0.17em}j=rnbr(i)\\ x_{ji,G}\hspace{1em}\mathrm{if}\hspace{0.17em}(randb(j)>CR)\hspace{0.17em}\mathrm{and}\hspace{0.17em}j\ne rnbr(i)\end{array},\hspace{1em}j=\mathrm{1}, \mathrm{2}, \mathrm{...}, D,

where CR in [0, 1] is a crossover constant; randb(j) is a uniform random number in [0, 1] corresponding to the jth component; rnbr(i) is a randomly chosen index in {1, 2, …, NP} to ensure that u_{i,G} contains at least one component from v_{i,G}.

Finally, the selection operator takes place by comparing values of fitness function (f) at the trial vector u_{i,G} and the target vector x_{i,G} and the better one having lower value is chosen as the individual of the next generation:

x_{i,G+\mathrm{1}}=\{\begin{array}{l}{u}_{i,G}\hspace{1em}if\hspace{0.17em}f(u_{i,G})\le f(x_{i,G});\\ x_{i,G}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}.\end{array}

The three operators of evolution process are repeated until either the stop criterion is met or the maximum iteration is reached. In this paper, the stop criterion is reached when the deviation between the best and the mean of objective function (f_best and f_mean) of the current generation is less than a tolerance. To reduce the total computational effort of the algorithm, a parallel technique will be applied to the process of the fitness evaluation and evolution. The DE algorithm is summarized briefly in Fig. 1.

Numerical examples

To demonstrate the reliability and effectiveness of the present approach, two numerical examples namely: 1) a 72-bar space truss, and 2) a one-span steel portal frame are considered in this section. For each example, two cases of damage are examined. The first one is assumed to be damaged at two elements while the second one is assumed to be damaged at three elements. The damaged structures are simulated by reducing the Young’s modulus of affected elements. The effect of noise on the accuracy of the method is also investigated. For the identification of damage location, after applying the SVD algorithm to the change in flexibility, all extracted DLV loads are employed to evaluate the normalized cumulative energy (nce) index. For the assessment of the damage severity, the obtained results of the DE are compared with those of the PSO to verify the robustness of the DE. The number of modes for computing the objective function is chosen to be 5 for all examples given in this section. The control parameters, population size and stopping criterion of the DE and PSO for all examples are listed in Table 1. It is to be noted that in this study, the parameters of the DE are chosen based on recommendation in [⁴⁰]. More specifically, for the case of F, F = 0.5 is usually a good initial choice. For the case of CR, it is recommended to try CR = 0.9 or CR = 1.0 which are appropriate in order to check whether or not a quick solution is possible.

A 72-bar space truss

The first example is a 72-bar space truss (see Fig. 2), as designed in [⁵⁰]. This truss has four non-structural masses attached at nodes 1–4. Material properties of the truss and the value of added masses are provided in Table 2. The cross section of each element group shown in Table 3 is taken from the optimal result in [⁵⁰]. The details of two damage cases with multiple damage scenarios are given in Table 4. The first five frequencies of undamaged and damaged structure are presented in Table 5. In case 1, a comparison between the proposed objective function (modified MDLAC function) and the MDLAC function is conducted to demonstrate the efficiency of the proposed objective function.

Case 1: Two damaged elements

The influence of number of modes (3, 4, 5 and 6) for approximating the flexibility matrices on the nce values of all elements is depicted in Fig. 3. It can be observed that the nce of elements 7 and 9 is significantly smaller than that of the others, and therefore be classified as suspected elements. Moreover, the increment of number of modes leads to the decrease of the nce values of the elements 7 and 9. This feature helps locate damaged elements more accurately when a large number of modes is used.

After the implementation of the DLV method, the optimization process using the DE and PSO is then performed to estimate damage severities of the suspected elements. The damage ratios of these elements are considered as design variables. The results of the damage severity evaluation in ten independent runs of the DE and PSO are provided in Table 6. It can be seen from the table that both algorithms identified the damage ratios of elements 7 and 9 correctly. However, it can also be realized that the number of iterations in the DE is smaller than that of the PSO. The convergence history of the run having smallest number of iterations of the DE and PSO is shown in Fig. 4 for comparison.

Besides using the proposed objective function, the MDLAC function approximated using the first five modes is also exploited as the objective function in this case. The results of the DE and PSO for the MDLAC function in 10 independent runs are given in Table 7. As shown in Table 7, the damage ratios of elements 7 and 9 alternate between the approximate values 10% and 30%. This phenomenon which is caused by the symmetry of the structure may lead to wrong results for damage extent evaluation if the MDLAC function is used. On the other hand, if the proposed objective function is utilized, this issue is tackled. The result shows that the proposed objective function is more effective than traditional MDLAC function.

Case 2: Three damaged elements

The nce values of all elements for various numbers of modes (from 3 to 10) are depicted in Fig. 5 and Fig. 6. It can be observed that for the case of three damaged elements, the usage of few numbers of modes may lead to false alarms. For example, some healthy elements such as 13, 14, 15, 16, 17 and 18 can be placed in the set of potential damaged elements. However, these false alarms will be eliminated when a larger number of modes such as ten modes is used. Comparison of the damage identification result with the previous case shows that in this example, the number of damage locations also influences on the accuracy of the DLV method. For the same number of modes, the DLV method yields the better results in the case of less number of damaged elements.

After identifying potential damaged locations, the DE and PSO are used to quantify the suspected damaged elements. The design variables are taken to be the damage ratios of elements 11, 22 and 52. The results of the damage severity assessment in ten independent runs of the DE and PSO are listed in Table 8. It is clearly that the damage ratios of the damaged elements are estimated correctly in all runs for both algorithms. Again, the number of iterations for the DE is less than that of the PSO. Figure 7 illustrates the convergence history of the run having smallest number of iterations of the DE and PSO.

A one-span steel portal frame

This example considers a one-span steel portal frame which was previously analyzed by Hao and Xia [³⁹]. The geometry of the frame is illustrated in Fig. 8. The material parameters are given by Young’s modulus 2.0e11 N/m² and mass density 7.67e3 kg/m³. The cross sections of the beam and the columns are 40.50×6 mm² and 50.5×6 mm², respectively. Two cases of damage are considered and their details are presented in Table 9. In the first case, the effect of refinement mesh on the damage identification result is also investigated. The first five frequencies of the undamaged and damaged frame and the corresponding results of undamaged frame in [³⁹] are listed in Table 10.

Case 1: Two damaged elements

The nce values of all elements for different number of modes (3, 4, 5 and 6) are presented in Fig. 9. It can be observed from the figure that the nce values of elements 4 and 25 are considerably smaller than those of the other elements. According to this distinguished feature, the suspected elements are chosen to be elements 4 and 25 which are also identical to the real scenario. From Fig. 9, it is also noted that the larger the number of modes utilized, the more accurate the identification of damage location is.

The influence of refinement mesh on damage identification result of DLV method is also investigated in this case. From the initial mesh as shown in Fig. 8, each element is divided into two and three elements to yield the mesh of 60 and 90 equal size elements, respectively. The numbering of elements in refined meshes is inherited from the initial mesh. Particularly, element e of the initial mesh will become elements 2e-1 and 2e for mesh size of 60, or elements 3e-2, 3e-1, 3e for mesh size of 90. Therefore, damaged elements and their damage ratios of the refined meshes are easily determined and given in Table 11. It should be noted from Table 11 that the damage of the frame is the same for both initial and refined meshes. By using the first 5 modes, the nce values of all elements for three different meshes are depicted in Fig. 10. It can be seen from the figure that the damaged elements are predicted correctly for all meshes.

The results of the damage severity assessment in ten independent runs and the convergence history of the run having the smallest number of iterations of the DE and PSO are presented in Table 12 and Fig. 11, respectively. Here, the results are given for the initial mesh. From the results in Table 12, it can be observed that the DE and PSO find exactly damage extents of elements 4 and 25 in all ten runs. Nonetheless, the DE requires smaller number of iterations than the PSO does.

Case 2: Three damaged elements

Figure 12 illustrates the nce values for all elements of the portal frame with various numbers of modes (3, 4, 5 and 6). It is easy to see that the nce values of elements 4, 19 and 25 are significantly smaller than those of the other elements. Furthermore, the difference is getting deeper as the number of utilized modes increases. From these features, it can be concluded that the suspected elements are 4, 19 and 25 which is the actual case. The damage ratios of these elements are then considered as design variables for the identification of the damage severity.

Table 13 provides the obtained results in ten independent runs of the DE and PSO for the damage severity assessment. These results show that the damaged extents of elements 4, 19 and 25 are identified exactly by using the DE and PSO. Again, the number of iterations of the DE is smaller than that of the PSO. The convergence history of the run having the smallest number of iterations of the DE and PSO is shown in Fig. 13.

Effect of measurement noise

In practice, the measured response data are usually contaminated with noise and the accuracy of damage identification might be impacted. Hence, the effect of measurement noise on the performance of the proposed method is investigated in this section. Here, measurement noise for both frequency and mode shape are considered. By referring to Refs. [⁵¹–⁵³] and [⁵⁴] the measurement outputs added noise of frequency and mode shape are represented by following equations

(11)f_i^{noise}=f_i^{noise-free}(\mathrm{1}+{\eta }_f(\mathrm{2}rand-\mathrm{1})),

(12){\varphi }_{ij}^{noise}={\varphi }_{ij}^{noise-free}\left(\mathrm{1}+{\eta }_{\varphi }\gamma \underset{i}{\max }\{\left|{\varphi }_{ij}^{noise-free}\right|\}\right),

where f_i^{noise} and f_i^{noise-free} are the ith natural frequency with and without noise, respectively; {\eta }_f is the noise level for frequency; rand is uniformly random number in [0, 1]; \hspace{0.17em}\hspace{0.17em}{\varphi }_{\hspace{0.17em}ji}^{noise} and {\varphi }_{ij}^{noise-free} are the jth component of the ith mode shape vector with and without noise, respectively; {\eta }_{\varphi } is the noise level for mode shape;γ is random number with zero mean and unit variance.

Numerical example given in Section 4.1 is used again for the investigation. After performing 100 independent runs of the DLV procedure, the mean values of nce of all elements with different number of modes for two cases of noise ({\eta }_f = 1%, {\eta }_{\varphi } = 3%) and ({\eta }_f = 1%, {\eta }_{\varphi } = 5%) are depicted in Fig. 14 and Fig. 15, respectively. It can be seen that the nce values of elements 7 and 9 for two cases of noise with various number of modes are significantly different with those of the other elements. Therefore, these two elements are identified as the suspected elements. In addition, it is also observed that the higher the noise level of mode shape the greater the nce values of the elements 7 and 9.

Damage severity evaluation using the DE and PSO is then applied to elements 7 and 9 for the case of noise ({\eta }_f = 1%, {\eta }_{\varphi } = 5%). The obtained results in ten independent runs of the DE and PSO are presented in Table 14. As can be seen from Table 14, the damage ratios of elements 7 and 9 are determined with minor error. For the DE, the errors between the approximated result and the exact result are 0.6% and 0.3% corresponding to elements 7 and 9, and for the PSO these errors are respectively equal to 0.6% and 0.1%. From Table 14, it is also noted that the average number of iterations of the DE is less than that of the PSO.

Conclusions

A new two-stage method for structural damage identification is proposed to identify locations and extents of damage in structures. First, damage locating vector (DLV) method using the normalized cumulative energy (nce) is applied to locate potential damaged elements. Then, the estimation of damage severity of these potential damaged elements is achieved by using the differential evolution (DE), where a combination of MDLAC and mode shape error functions is proposed as an efficient objective function. The accuracy and efficiency of the proposed combination are verified numerically for multi-damage cases namely (1) a 72-bar space truss and (2) a one-span steel portal frame. In addition, the influence of measurement noise on the performance of the proposed combination is also investigated. The results show two main points including: (1) the proposed combination can determine exactly damaged elements and their severities with the presence of measurement noise; and (2) the more number of modes utilized, the better the identification result. For future work, the influence of the uncertainties of material properties on the damage identification results should be investigated by using robust toolbox for uncertainty and sensitivity analysis methods as presented in Ref. [⁵⁵].

References

[1] Yuen M M F. A numerical study of the eigenparameters of a damaged cantilever. Journal of Sound and Vibration, 1985, 103(3): 301–310

[2] Pandey A K, Biswas M, Samman M M. Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration, 1991, 145(2): 321–332

[3] Salawu O, Williams C. Bridge Assessment Using Forced-Vibration Testing. Journal of Structural Engineering, 1995, 121(2): 161–173

[4] Pandey A K, Biswas M. Damage detection in structures using changes in flexibility. Journal of Sound and Vibration, 1994, 169(1): 3–17

[5] Pandey A K, Biswas M. Experimental verification of flexibility difference method for locating damage in structures. Journal of Sound and Vibration, 1995, 184(2): 311–328

[6] Stubbs N, Kim J T, Topole K. An efficient and robust algorithm for damage localization in offshore platforms. Proc. ASCE Tenth Struct. Congr., 1992, 543–546

[7] Bernal D. Load vectors for damage localization. Journal of Engineering Mechanics, 2002, 128(1): 7–14

[8] Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[9] Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[10] Areias P, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72(0): 50–63

[11] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110(0): 113–137

[12] Rabczuk T, Areias P M A. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Geomechanics, 2006, 30(11): 1159–1172

[13] Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[14] Rabczuk T, Belytschko T. Application of Particle Methods to Static Fracture of Reinforced Concrete Structures. International Journal of Fracture, 2006, 137(1): 19–49

[15] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[16] Zhu H, Zhuang X, Cai Y, Ma G. High rock slope stability analysis using the enriched meshless shepard and least squares method. International Journal of Computational Methods, 2011, 08(02): 209–228

[17] Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: Framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998

[18] Zhuang X, Zhu H, Augarde C. An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function. Computational Mechanics, 2014, 53(2): 343–357

[19] Nanthakumar S S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176

[20] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of multiple flaws in piezoelectric structures using XFEM and level sets. Computer Methods in Applied Mechanics and Engineering, 2014, 275(12): 98–112

[21] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of flaws in piezoelectric structures using extended FEM. International Journal for Numerical Methods in Engineering, 2013, 96(6): 373–389

[22] Vo-Duy T, Ho-Huu V, Dang-Trung H, Nguyen-Thoi T. A two-step approach for damage detection in laminated composite structures using modal strain energy method and an improved differential evolution algorithm. Composite Structures, 2016, 147: 42–53

[23] Lie S T, Zhang Y, Wang L Q. Damage Detection in Compressed Natural Gas (CNG) Cylinders Based on Auxiliary Mass Induced Frequency Shift. Experimental Mechanics, 2015, 55(3): 487–498

[24] Quek S T, Tran V A, Hou X Y, Duan W H. Structural damage detection using enhanced damage locating vector method with limited wireless sensors. Journal of Sound and Vibration, 2009, 328(4–5): 411–427

[25] Cawley P, Adams R D. The location of defects in structures from measurements of natural frequencies. Journal of Strain Analysis for Engineering Design, 1979, 14(2): 49–57

[26] Messina A, Williams E J, Contursi T. Structural damage detection by a sensitivity and statistical-based method. Journal of Sound and Vibration, 1998, 216(5): 791–808

[27] Mares C, Surace C. An application of genetic algorithms to identify damage in elastic structures. Journal of Sound and Vibration, 1996, 195(2): 195–215

[28] Koh B H, Dyke S J. Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data. Computers & Structures, 2007, 85(3–4): 117–130

[29] Mohan S C, Maiti D K, Maity D. Structural damage assessment using FRF employing particle swarm optimization. Applied Mathematics and Computation, 2013, 219(20): 10387–10400

[30] Kaveh A, Javadi S M, Maniat M. Damage Assessment via Modal Data with a Mixed Particle Swarm Strategy, Ray Optimizer, and Harmony Search. Asian Journal of Civil Engineering, 2014, 15(1): 95–106

[31] Kaveh A, Maniat M. Damage detection based on MCSS and PSO using modal data. Smart Struct Syst. Techno, 2015, 15(5): 1253–1270

[32] Kaveh A, Zolghadr A. An improved CSS for damage detection of truss structures using changes in natural frequencies and mode shapes. Advances in Engineering Software, 2015, 80: 93–100

[33] Kaveh A, Talatahari S. A novel heuristic optimization method: charged system search. Acta Mechanica, 2010, 213(3): 267–289

[34] Moslem K, Nafaspour R. Structural damage detection by genetic algorithms. AIAA Journal, 2002, 40(7): 1395–1401

[35] Au F T K, Cheng Y S, Tham L G, Bai Z Z. Structural damage detection based on a micro-genetic algorithm using incomplete and noisy modal test data. Journal of Sound and Vibration, 2003, 259(5): 1081–1094

[36] Guo H Y, Li Z L. A two-stage method to identify structural damage sites and extents by using evidence theory and micro-search genetic algorithm. Mechanical Systems and Signal Processing, 2009, 23(3): 769–782

[37] Seyedpoor S M. A two stage method for structural damage detection using a modal strain energy based index and particle swarm optimization. International Journal of Non-linear Mechanics, 2012, 47(1): 1–8

[38] Fan W, Qiao P. Vibration-based Damage Identification Methods: A Review and Comparative Study. Structural Health Monitoring, 2011, 10(1): 83–111

[39] Hao H, Xia Y. Vibration-based Damage Detection of Structures by Genetic Algorithm. Journal of Computing in Civil Engineering, 2002, 16(3): 222–229

[40] Storn R, Price K. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 1997, 11(4): 341–359

[41] Price K, Storn R M, Lampinen J A. Differential evolution: a practical approach to global optimization. Springer, 2006

[42] Civicioglu P, Besdok E. A conceptual comparison of the Cuckoo-search, particle swarm optimization, differential evolution and artificial bee colony algorithms. Artificial Intelligence Review, 2013, 39(4): 315–346

[43] Hegerty B, Hung C, Kasprak K. A Comparative Study on Differential Evolution and Genetic Algorithms for Some Combinatorial Problems. Mexican International Conference on Artificial Intelligence, 2009

[44] Gao Y, Spencer B F, Ruiz-Sandoval M. Distributed computing strategy for structural health monitoring. Structural Control and Health Monitoring, 2006, 13(1): 488–507

[45] Bernal D, Gunes B. Flexibility Based Approach for Damage Characterization: Benchmark Application. Journal of Engineering Mechanics, 2003, 130(1): 61–70

[46] Gao Y, Spencer B Jr, Bernal D. Experimental Verification of the Flexibility-Based Damage Locating Vector Method. Journal of Engineering Mechanics, 2007, 133(10): 1043–1049

[47] Qian J, Ji X, Xu Y. Two-stage damage diagnosis approach for steel braced space frame structures. Engineering Structures, 2007, 29(12): 3277–3292

[48] Vo-Duy T, Nguyen-Minh N, Dang-Trung H, Tran-Viet A, Nguyen-Thoi T. Damage assessment of laminated composite beam structures using damage locating vector (DLV) method. Frontiers of Structural and Civil Engineering, 2015, 9(4): 457–465

[49] Storn R. On the Usage of Differential Evolution for Function Optimization. in NAFIPS’96, IEEE, 1996, 519–523

[50] Kaveh A, Zolghadr A. Truss optimization with natural frequency constraints using a hybridized CSS-BBBC algorithm with trap recognition capability. Computers & Structures, 2012, 102–10314–27

[51] Xiang J, Liang M, He Y. Experimental investigation of frequency-based multi-damage detection for beams using support vector regression. Engineering Fracture Mechanics, 2014, 131: 257–268

[52] Shi Z, Law S, Zhang L. Structural damage detection from modal strain energy change. Journal of Engineering Mechanics, 2000, 126(12): 1216–1223

[53] Fu Y, Liu J, Wei Z, Lu Z. A two-step approach for damage Identification in plates. Journal of Vibration and Control, 2014

[54] Wei Z T, Liu J K, Lu Z R. Damage identification in plates based on the ratio of modal strain energy change and sensitivity analysis. Inverse Problems in Science and Engineering, 2016, 24(2): 265– 283

[55] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

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