Multiple damage detection and localization in beam-like and complex structures using co-ordinate modal assurance criterion combined with firefly and genetic algorithms

2.1. Genetic algorithm for damage detection

Genetic Algorithm, originally developed by Holland [24], combines algorithms to solve optimization problems using the principles of evolution. It demonstrated excellent operations in combinatorial optimization that have a finite solution. Potential solutions are sought through a population of individuals, i.e. candidate solutions, which evolve towards an optimum solution. Each individual has a set of properties, known as chromosomes, which can be mutated and altered during the optimization process. The initial population usually generates random individuals and the evolution progressed in an iterative way. After each iteration, a new population, called a generation, is created. During the process of GA, a fitness function, which has the value of objective function in an optimization problem, is evaluated in each iteration. The individuals that best fit the objective function are selected and used in the next generation after recombination and possibly random mutation and reproduction. The iteration process continues until either a satisfactory minimization of the objective function is obtained or the maximum defined number of iterations is reached.

In our study, the individuals are the sets of damage parameters and the chromosomes are the damage parameters themselves, i.e. the stiffness of the structure at different locations, and the fitness or objective function to be minimized is based on COMAC as it will be presented later in section 2.3. All numerical studies presented in this paper are implementing in MATLAB. In applying GA, the following steps are programmed:

Step 1: Creating a random initial population of individuals and elements’ stiffness as chromosomes.

Step 2: Creating a sequence of new populations at each iteration using the current population. To achieve this step, the algorithm does the following:

• Compute the score of each individual by calculating its fitness.

• Convert the fitness scores into a more usable range of values.

• Select individuals or members, called parents, based on their scores.

• Select individuals or members, called elite, which have lower score, and pass them to the next generation.

• Produce children from the parents by making random changes to a single parent (mutation) or combining a pair of parents (crossover).

• Form the next generation using the current population and its children.

Step 3: Stop the algorithm when the stopping criteria is met. In our implementation, the maximum number of iteration is set equal to 100.

2.2. Firefly Algorithm for damage detection

The Firefly Algorithms focus on the variation of light intensity and formulation of the attractiveness. To simplify this concept, we can always consider that the attractiveness of a firefly is determined by its brightness, which in turn is associated with the encoded objective function based on frequencies, e.g. comparing the measured and calculated frequencies as in our case, where the brightness $I$ of a firefly at a particular location x can be chosen as $I\left(x\right)\infty f\left(x\right)$. However, as the attractiveness $b$ is relative, it should be seen in the eyes of the beholder or judged by the other fireflies [25]. Thus, it will vary with the distance $r_{ij}$ between firefly $i$ and firefly $j$.

Some of the flashing characteristics of fireflies can be realized in order to develop firefly-inspired algorithms. For the description of Firefly Algorithm (FA) that is used herein, three rules will be considered:

1) All fireflies are unisex so that one firefly will be attracted to other fireflies regardless of their sex;

2) Attractiveness is proportional to their brightness, thus for any two flashing fireflies, the less bright one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If there is no brighter one than a particular firefly, it will move randomly;

3) The brightness of a firefly is affected or determined by the landscape of the objective function. For a maximization problem, the brightness can simply be proportional to the value of the objective function.

In addition, light intensity decreases with the distance from its source, and light is also absorbed in the media, so the attractiveness variation with the degree of absorption must be allowed. In the simplest form, the light intensity $I\left(r\right)$ varies according to the inverse square law:

where $I_s$ is the intensity at the source. For a given medium with a fixed light absorption coefficient $\gamma$, the light intensity $I$ vary with the distance r, i.e.:

where $I_0$ is the original light intensity.

As a firefly’s attractiveness is proportional to the light intensity seen by adjacent fireflies, the attractiveness $\beta$ of a firefly can be defined by:

where ${\beta }_0$ is the attractiveness at $r=$ 0.

In our study the parameters used are ${\beta }_0=$ 1, $\alpha \in$[0, 1], $\gamma =$ 1 and $\lambda =$ 1.5. In addition, if the scales vary significantly in different dimensions such as −10⁵ to 10⁵ in one dimension while, say, −0.001 to 0.01 along the other, it is a good idea to replace $\alpha$ by $\alpha S_k$,where the scaling parameters $S_k$ ($k=$ 1,..., $d$) in the $d$ dimension should be determined by the actual scales of the problem of interest.

2.3. Objective function

The objective function used in this study is based on the Co-ordinate Modal Assurance Criterion (COMAC) [26]. The COMAC is calculated over a set of mode pairs, which can be numerical/numerical, numerical/experimental or experimental/experimental set of data. It is an extension of the classical well-known Modal Assurance Criterion (MAC), which is widely used by researchers for modal correlation. The main intension of the COMAC is to identify which measured degrees of freedom contribute negatively to a low value in the MAC matrix. In other words, it localizes the mismatch in the structure between two mode pairs of modal data. In damage detection application, these mode pairs are for those of damaged and intact structures. The two modal vector COMAC uses the eigenvectors for each degree of freedom ($i$) in the structure and is given by [26]:

where ${\left\{{\phi }_A\right\}}_{ir}$ and ${\left\{{\phi }_D\right\}}_{ir}$ are the healthy and damaged mode shapes, respectively, $i$ is the indices for (measured) degree of freedom number, $r$ is the indices for mode shape number and $n$ is the total number of modes considered in the calculation. The COMAC values vary between 0 and 1 and for a good correlation it should be close to 1. Therefore, in damage detection application, COMAC should be far below 1 at a damaged location in the structure. Based on COMAC, the objective function to be minimized is defined as follows:

In order to determine the eigenvectors required to calculate COMAC for intact and damaged structures, finite element simulations were carried out. Classical Euler-Bernoulli beam element was programmed in MATALAB and eigenvalue problem is solving to determine the modal data. The equation of motion can be obtained for free vibrations, in the absence of damping, and is written as:

where [$M$] and [$K$] are the mass and stiffness matrices, respectively and $\left\{u\right\}$ is the displacement vector. By solving Eq. (6) for the eigenvalues and eigenvectors, the data required for COMAC are obtained. In applying GA to beam structures, hard constrains of the stiffness by specifying maximum and minimum values are considered. The maximum value is the stiffness of the intact beam, while the minimum value is set equal to zero.

3.1. Simply supported beam

For a simply supported beam shown in Fig. 1, the following geometrical and mechanical properties are used: length $l=$ 6 m, cross-section area $A=$ 0.09 m², density $\rho =$ 2500 kg/m³ and stiffness $EI=$ 20.25 ×10⁶ N.m² [27]. A finite element model is constructed using 10 beam elements as shown in Fig. 1. The intact beam is modelled using a constant stiffness, $EI$, for all elements, while the damaged beam is modeled by reducing $EI$ of some specific elements. The beam is macroscopically considered homogeneous. Three different damage scenarios, D1, D2 and D3, were considered using different locations of damage and a reduction in stiffness in the damaged elements:

Scenario D1: A single damage at element 2 – reduction in stiffness of 50 %.

Scenario D2: A single damage at element 8 – reduction in stiffness of 30 %.

Scenario D3: A multiple damage at element 2 and 8 – reduction in stiffness of 50 % in element 2 and 30 % in element 8.

Fig. 1Simply-supported beam structure discretized in 10 elements

Fig. 2The iteration process of objective function based on FA and GA - damage scenario D1

The predicted damage locations and severities, as well as the convergence rates, for both proposed algorithms, are shown in Figs. 2-7 for all considered damage scenarios D1 to D3, respectively. From Figs. 2-7, it can be seen that good results are obtained using FA and GA, which predict accurately the location of damage and its severity with a small amount of error. Furthermore, the damage detection algorithm based on FA provides mush faster convergence rate that that based on GA. It can be also observed that damage scenario D2 has a slightly faster convergence that scenario D1. This can be due to the fact that in D2 the damage is more towards the middle of the beam that in case of D1. For the multiple damage scenario, D3, this study shows that FA and GA together with COMAC can effectively identify multiple damages in beams. It can also be observed that the rate of convergence of damage scenarios D1 and D2 is faster than that of D3. This is because D1 and D2 have only a single damage, whereas D3 has two damages at two different locations.

Fig. 3Evolutionary process of damage index – damage scenario D1

Fig. 4The iteration process of objective function based on FA and GA – damage scenario D2

Fig. 5Evolutionary process of damage index – damage scenario D2

Fig. 6The iteration process of objective function based on FA and GA – damage scenario D3

Fig. 7Evolutionary process of damage index – damage scenario D3

3.2. Damage of a two-dimensional frame structure

We consider now a more complex structure, namely a two-dimensional frame structure as shown in Fig. 8. It consists of 17 beam elements, 10 nodes and 30 DOFs. This structural system is used to evaluate the ability of the proposed methodology to detect damage in a more complex structure. As it has been seen in the previous numerical example, FA has a better performance than GA and provides faster convergence, therefore only FA is used in this analysis. The material constants of the frame structure are: elastic-modulus $E=$ 20×10⁹ N/m² and density $\rho =$ 7780 kg/m³. In this study, a multiple damage scenario, D4, is considered using different locations and severities of damage:

Scenario D4: Multiple damages – reduction in stiffness of 20 %, 30 %, 40 % and 50 % in elements 5, 8, 14 and 17, respectively.

The predicted damage locations and severities, as well as the convergence rates, are shown in Figs. 9 and 10 for damage scenario D4 using FA. From Figs. 9 and 10, it can be seen that good identification of damage locations and severities are obtained. This demonstrates the accurate prediction of our approach in case of multiple damages in complex structures. Regarding convergence rate, it can be seen that the damage scenario D4 shows slower convergence rate when compared to scenarios D1 and D2. Again, this sounds as logic results since scenario D4 contains four locations of damages with different severities, whereas D1 and D2 contain one location. Thus, the more damage locations in the structure, the slower the convergence rate and the more CPU time required.

Fig. 8A two-dimensional frame structure discretized in 17 beam elements

Fig. 9The iteration process of objective function based on FA – damage scenario D4

Fig. 10Evolutionary process of damage index – damage scenario D4

It is worth mentioning that the objective function used in FA, Eq. (4), depends on the boundary conditions. This is because COMAC is a function of modal vectors, which depend on boundary conditions. If the boundary conditions are changed to clamped-clamped or free-free, a different modal vector will be produced for not only the intact girder, but for also the damaged one. By using these two modal vectors in Eq. (4) and applying FA, damage can be detected in a way similar to that for the simply supported case.