This paper proposes a damage detection method based on combined data of static and modal tests using particle swarm optimization (PSO). To improve the performance of PSO, some immune properties such as selection, receptor editing, and vaccination are introduced into the basic PSO and an improved PSO algorithm is formed. Simulations on three benchmark functions show that the new algorithm performs better than PSO. The efficiency of the proposed damage detection method is tested on a clamped beam, and the results demonstrate that it is more efficient than PSO, differential evolution, and an adaptive realparameter simulated annealing genetic algorithm.
A structural system or mechanical component continuously accumulates damage during their service life. The presence of damages may reduce the performance of a structure, such as decreasing the service life, or even progressing to catastrophic failure. In recent years, the damage assessment of structures has drawn wide attention from various engineering fields. Structural damage usually causes a decrease in structural stiffness, which produces changes in the vibration characteristics and static displacements of the structure. Major damage detection approaches can be clarified into three major categories, the static identification methods using static test data [
The usual damage detection methods minimize an objective function, which is defined in terms of the discrepancies between the vibration data or static data identified by testing and those which are computed from the analytical model. Traditional damage detection methods have some disadvantages such as the damage position and damage extent cannot be detected simultaneously and are not so efficient in detecting the damage extent and adopt local optimization methods which usually lead to a local minimum only.
In recent years, evolutionary algorithms have been extensively applied to damage detection and the related optimal sensor placement problems [
Addition to GA, some new techniques are proposed for engineering optimization problems, such as particle swarm optimization (PSO) [
This paper is organized as follows. Section
The analytical static model for an intact structure in the finiteelement formulation is
The characteristic evaluation of a dynamic undamaged structure can be expressed as
In (
In the context of discretized finite elements, structural damage can be represented by a decrease in the stiffness of the individual elements as
A uniform damage for the whole element has been assumed in (
In general, a small number of sensors will result in nonunique solutions. The sparsity of measurement can be overcome by increasing the number of loading conditions instead of increasing the number of sensors [
The first part is defined by the normalized difference between the measured and theoretical computed displacements. The second part is defined by the normalized difference between the measured and theoretically computed natural frequencies. Differences between displacements and natural frequencies are normalized to get a better representation of the relative change in response.
To include the uncertainty in the measured data and to study the sensitivity of IEPSO to noise, uniformly distributed random noise [
Inspired by a model of social interactions between independent animals seeking for food, PSO utilizes swarm intelligence to achieve the goal of optimization. Instead of using genetic operators to manipulate the individuals, each individual in PSO flies in the search space with a velocity which is dynamically adjusted according to its own flying experience and flying experience of its companions [
Shi and Eberhart [
They proposed that suitable selection of
An explicit maximum velocity or a constriction factor is usually utilized in PSO algorithms to control the exploration abilities of particles; however, it cannot prevent them from going outside the allowable solution space and hence produce invalid solutions. Four types of boundary conditions, namely, absorbing, reflecting, invisible, and damping, have been reported in the literature [
As for damage detection problem, generally only few elements are damaged and most elements are still intact. The parameter values of these intact elements are always in the upper bound [
AIS can be defined as abstract computational systems inspired by theoretical immunology and observed immune functions, principles, and models, applied to solve problems [
The antibodies present in a population set contain much information regarding the solution of the problem. Based on their affinity, the antibodies are selected to proliferate and produce clones. Traditionally, deterministic selection rule is adopted to select better antibodies for proliferation. However, deterministic selection rule selects only the best antibodies for proliferation, and that may lead to the premature convergence of the algorithm [
To overcome this difficulty, rankbased fitness assignment and roulette wheel selection rule is adopted. The fitness of an individual can be calculated as
In AIS, receptor editing allows an antibody to take large steps, landing in a locale where the affinity might be lower. However, occasionally the leap will lead to an antibody on the side where the region is more promising. From this locale, point mutations followed by selection can drive the antibody to reach the global optimum. Receptor editing offers the ability to escape from local optima [
For each individual
Given an individual
Two types of vaccines are adopted for function optimization and damage detection. For the function optimization problem, the vaccine can be abstracted from the best individuals of the population. If
In IEPSO, a population of particles is sampled randomly in the feasible space. Then the population executes PSO or its variants, including the update of position and velocity. After that, it executes receptor editing operator (nonuniform mutation) according to a certain probability
Algorithm IEPSO
Begin;
Initialize the population (generate a random population of
For
For each individual
Evolving individual
If (rand
If (rand
Evaluate individual
If (fit
If (fit
End;
Selection;
Calculate
Next
End;
In Algorithm
In the new algorithm, selection and vaccination can improve the convergence speed and receptor editing, helping the algorithm to avoid premature convergence.
Three nonlinear benchmark functions that are commonly used in literatures are adopted. Their formulas and variable ranges are shown in Table
Benchmark test functions.
Name  Formula 

Range 

Sphere 

30 

Rosenbrock 

30 

Rastrigin 

30 

To evaluate the performance of the proposed IEPSO, the basic PSO and DE are used for comparisons. The version of DE used in this paper is known as DE/rand/1/bin, or “classic DE” [
The parameters used for IEPSO and PSO are recommended in [
Parameter values used in IEPSO.
Parameter  Function optimization  Damage detection 

Generation number  2000  100 
Population size  80  40 

0.1  0.05 

0.1  0.25 
SP  1.8  1.8 

5  5 
The three algorithms are executed in 50 independent runs. The mean fitness values of the best individual found during the 50 runs for the three functions are listed in Table
Mean best fitness values of three functions.
Function  DE  PSO  IEPSO 













Convergence processes of mean best fitness for three benchmark functions.
A clamped beam adopted from Wang et al. [
A clamped beam and its crosssection.
Statistical analysis is a good way to compare different stochastic algorithms. But the damage detection of a structure needs a long time. To compare different stochastic algorithms on these timeconsuming problems, usually adopt the best results obtained from several runs, and this can avoid chanciness in some extent [
The first case (case 1) has two damages in which element 1 has 50% reduction in Young’s modulus and element 9 has a severe 87.5% reduction in Young’s modulus. That means
The comparison of logarithmic best fitness values of three algorithms is shown in Figure
Comparison of damage detection results in case 1: (a) between theoretical value and IEPSO detected value; (b) between different algorithms.
Convergence processes in case 1: (a) best fitness; (b) SDFs of damaged elements.
A triple damage occurrence with different damage extents (case 2) is considered. In this example, element 2 has 10% reduction in Young’s modulus, element 9 has 10% reduction in Young’s modulus, and element 16 has 15% reduction in Young’s modulus. This means that
The comparison of logarithmic best fitness values of three algorithms is shown in Figure
Convergence processes in case 3: (a) best fitness; (b) SDFs of damaged elements.
Comparison of damage detection results in case 3: (a) between theoretical value and IEPSO detected value; (b) between different algorithms.
Damage detection performance of IEPSO is also compared with ARSAGA [
Comparison of damage detection results of case one by ARSAGA and IEPSO.
Element no.  Stiffness damage factors  

Exact  ARSAGA  IEPSO  
1  1.0 

1.0 
2 



3  1.0  1.0  1.0 
4  1.0  0.984  0.999 
5  1.0  0.992  1.0 
6  1.0  1.0  1.0 
7  1.0  1.0  1.0 
8  1.0  0.978  0.999 
9 



10  1.0  1.0  1.0 
11  1.0  1.0  1.0 
12  1.0  0.987  1.0 
13  1.0  1.0  1.0 
14  1.0  0.991  1.0 
15  1.0  1.0  1.0 
16 



17  1.0  1.0  1.0 
18  1.0  1.0  0.999 
19  1.0  1.0  1.0 
20  1.0  1.0  1.0 
To include the uncertainty in the measured data and to study the sensitivity of IEPSO to noise, different uniformly distributed random noise is added to measured data in the simulation as described (
The comparison of SDF of each element between theoretical values and IEPSO detected values under the condition 1% noise is shown in Figure
Comparison of damage detection results in case 1 with noise: (a) 1% noise is added to both static displacements and natural frequencies; (b) 5% noise is added to static displacements and 2% noise is added to natural frequencies.
It can be seen from these figures that noise decreases the accuracy of the algorithm. When noise increases, the misidentifications also increase. This may be because that large noise increases the uncertainty in theoretical response data and noise may be taken as damage by the procedure. From this point, the results detected by the procedure are still reasonable.
In this paper, a new IEPSO algorithm for structural damage identification is presented. It bases on a particle swarm optimization algorithm and introduces immune properties selection, receptor editing, and vaccination into it. Selection and vaccination can improve the convergence speed and receptor editing helps the algorithm to avoid premature convergence.
Boundary conditions for PSO are also discussed in the paper, and a combined boundary is adopted for damage detection problems. This boundary condition can avoid oscillating around the upper boundary and can quickly return to the feasible region around the lower boundary.
The feasibility of the methodology is first demonstrated through several numerical examples. Three benchmark functions are used to test the performance of IEPSO in complex function optimization problems. Numerical results show that IEPSO performs better than PSO and DE in most cases because its convergence speed is the fastest and converges to the best fitness value.
The proposed algorithm is then tested on damage detection problems of a clamped beam. Two damage cases can be detected quickly and accurately by the proposed algorithm when noise is free. Comparing with PSO and DE, IEPSO is more efficient in damage detection problems. The accuracy of IEPSO is also higher than ARSAGA in case 2. When different levels of noise are added, the accuracy of the algorithm is decreased. However, this is still reasonable because noise may be taken as damages in the beam during the damage detection process.
This work was supported by the National Natural Science Foundation of China (51109028, 90815024) and the Fundamental Research Funds for the Central Universities (DUT11RC(3)38).