Identification by genetic algorithms

The identification of the parameters of the sensing scheme (types, numbers and location) could be done by an effective algorithm, the genetic algorithm (GA). It is a search procedure that uses the mechanics of natural selection and natural genetics wherechromosomes can be coded in two different ways: either as binary vectors or as real vectors. The sum of all bits, which represent one search variables, is called “Gene”, and the sum of all genes collected in a binary vector is called “chromosome”. For the initialization, a starting population P(t = 0) of n individuals is stochastically generated based on uniform probability within the given bounds. Then, the evaluation and interpretation of the objective function value provides a measure for the "fitness value". So to evolve towards the next generation of generally better solutions, the GA selects the highest performing candidates from the current generation using "survival- of-the-fittest" learning and the selection probability for the recombination is calculated.

The best solutions are then recombined with each other through an operation called “crossover” to form some new solutions which are used to replace the worst solutions of the original population. This type of recombination is defined by two steps:

 at first individuals chosen for the recombination are mixed and then two by two individuals are chosen as parents;

 in the second step, the parents’ chromosomes are recombined according to different crossover schemes.

Another type of recombination is the mutation which consists on finding a new region of the search space and avoiding the convergence to a suboptimum by exchanging values in the chromosome.

In general, the population size is kept constant so it is necessary to decide which individuals should survive or be substituted for the next generation, this step is called “substitution”. There are different kinds of substitutions like the elitism or cancellation of n worst elements or cancellation of n stochastically chosen individuals, etc. The process is then repeated until the desired fitness value is reached.

4. Damage identification procedure

The occurrence of various crack patterns in a structure takes place during construction and/or after completion. A structure component develops cracks whenever the stress in the components exceeds its strength (Figure 1). Some types of cracking indicate a structural issue, when others do not indicate any type of issue other than normal weathering. Whatever the cause, it still remains important to detect a crack at its early age in order to avoid serious failures.

In the case of bridges, the sources of primary loading include the own weight of materials from which the structure was built, traffic, weather conditions, natural catastrophes and loading conditions experienced during construction. Some of these loads act permanently so they are considered as dead loads while the others are not permanent, so they are called live loads. However, the secondary loads are those due to temperature change, construction eccentricities, shrinkage of structural materials, settlement of foundation, or other such loads.

Figure 1: Cracks on a concrete beam after loading

Because of the applied loads, bridge structures accumulate damage during their service life and the actual structure response to loading is degraded from the predicted design performance. Some of the most frequent defects on the elements of a bridge structure are: lateral movements or rotation of the substructure, excessive vertical displacement of the superstructure, cracks and open joints between the segments of the concrete, concrete cover depth defects, corrosion of the reinforcement, etc.

For instance, if we want to study the case where cracks occur in a concrete bridge due to traffic loads:

A defect dj in the structure will cause a degradation in the structure which will

affect its mechanical properties. To detect the damage, one or more sensors like the linear variable differential transformer (LVDT) can be implemented. The deflection v given by these sensors will depend on a number of parameters such as the load Pi and its location bl, the position of the sensor kl,

the span l, the Young’s modulus E and the moment of inertia I, in addition to the parameters characterizing the defect like its position cl and the induced reduction of the moment of inertia αI.

In order to identify the defect parameters, the optimization problem should be set such as its solution leads to the best fitting of the defect identification. The fitness function is defined by the sum of quadratic difference between the calculated and observed deflections; this function takes the form:

f = ∑𝑁𝑖=1(𝑣𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑− 𝑣𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑)2 (1)

The best fitting is the one which leads to the minimum value of f, which should be ideally zero. However, due to uncertainties in the defect evolution and measurement techniques, the zero is never reached and the best solution will appear when f takes the closest value to zero.

As mentioned above, the genetic algorithms are effective to solve this problem. Figure 2 shows the flow chart of the genetic algorithm procedure. At first, a population of chromosomes is randomly created. Each chromosome containing several 8-bit variables, representing the defect parameters to calculate. Then, some individuals of the population are recombined by crossover with a given probability; this latter is chosen as 0.6. The others are subjected to mutation where one bit of the chromosome is switched to another bit (i.e. 0 is switch to 1, and 1 is switched to 0, at a single gene of the chromosome) with a given probability; a value of 0.3 is chosen herein for mutation probability.

The crossover and mutation will result in new offspring being created. Subsequently, a truncation selection is applied on the new population in order to select the individuals with the best fitness.

This procedure (i.e. population evolution) is repeated many times until the best fitness is reached. The computation time depends on the size of each population and the number of generations to reach convergence.

Figure 2: Flow chart of the genetic algorithm.