13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- an interval [xmin, xmax] normally determined in this type of modeling according to mechanical bases. The main reason to establish these limits is to make the search process more efficient by reducing its space. The initial population is produced by: - Initial solutions (individuals) proposed based on the expert opinion and on experimental observations; - Solutions chosen randomly in the search space. This allows, in fact, to start the search from various solutions of the search space incorporating expert opinion. Different numerical tests should be conducted and led to choice of stochastic operators as follows: - Selection type elitism, which allows to highlight the best individuals in the population. These are the most developed individuals which will participate in the improvement. Such a technique has the advantage of faster convergence to the best individuals to the detriment of individuals which seem less appropriate and could provide elements for the creation of new individuals. - Crossover scattered, which is cut individuals into several portions (2 or 3 portions) to obtain new individuals - Adapt feasible mutation which randomly generates directions that are adaptive compared to the last generation successful or not. The feasible region is limited by the constraints. A pitch length is selected along each direction in such a manner that the bounds constraints are satisfied. 3.2. Pattern Search Algorithm Pattern search is a direct search method. This method is employed for solving optimization problems that does not require any information about the gradient of the objective function. The pattern search begins at the initial point xo. At the first iteration the mesh size is 1 and the GPS (Generalized Pattern Search) algorithm adds the pattern vectors to the initial point xo to compute the following mesh points. The algorithm computes the objective function at the mesh points using the following approach: n i n i xm x v = + Δ (12) / ( ) min(( )) 1 i i j n f xm f xmj x xm = = + (13) Where i xm : the mesh points, nx : the current point, iv : the pattern vector and nΔ : the current mesh size. A pattern is a set of vectors {vi} that the PSA utilizes to define which points to search at each iteration. The set {vi} is determined by the number of independent variables in the objective function. For example, if there are three independent variables in the optimization problem, the default for a 2N positive basis consists of the following pattern vectors:
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