In real optimization problems where the objective function is computed based on simulation packages, direct search optimization is the most effective technique for computing objective function derivatives. The direct search method can be performed for single and multi-
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objective functions. In single point optimization the variables are varied in positive or negative x-axis and for multi-objective optimization the design variable are modified in both positive and negative x-axis. The optimization algorithms which are based on this method is known as one- variable at a time approach. The common direct search based algorithm is genetic algorithm (GA). GA technique is a computerized global optimization algorithm which can perform robust effective optimization but with large cycle time. The computing time in the design optimization using GA can be reduced by combining response surface approximation (RSA) with direct optimization.
5.3.2.1 Genetic Algorithm:
Genetic algorithm is a computerized optimization approach which is based on search optimization method. In this algorithm the solution is obtained by natural selection evolution mechanism. The GA is a common optimization approach for aerodynamic design optimization and it is proven as effective optimization technique for finding the optimal global solution for the problems which have many local optimal points (Zhu and Chan 1998, Gen and Cheng 2000). GA scheme is used in the case of maximization design problems and for minimization goals the objective function is modified to be maximized by applying appropriate transformation to the fitness function. The fitness function F(x) is a non-negative function that is derived from the objective function and used in genetic operations. In the case of maximization problems, the fitness function is equal to the objective function (𝐹(𝑥) = 𝑓(𝑥)). However, in minimization problems the fitness function is transformed to become a maximization problem as described by (Deb 2012):
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The GA optimization starts normally by generating random population of design points. This population is evaluated by applying four basic operations to find the optimum solution. As shown in
Figure 5.2 the main GA operators include selection, crossover, mutation, and estimation which are described below (Deb 2012).
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Selection Operator:
This operator is also known as reproduction operator which is applied to generate new candidates (Children) using good candidates in current population (parents) in order to produce a mating pool.
Crossover Operator:
This operator is used to generate a new solution by creating a new chromosome which is compared to parent candidates. There are two types of crossover operators: Arithmetic and Heuristic. In Arithmetic crossover, the two parent candidates are combined to create new children candidates as:
Child1 = λ (parent1) + (1 − λ)parent2 Child2 = (1 − λ)(parent1) + λ(parent2) Where λ is crossover probability between 0 and 1.
The Heuristic operator works based on the fitness between the best and worst individuals to create a new candidate as:
Child1 = Best Parent + λ[Best Parent − Worst Parent] Child2 = Best Parent
Mutation Operator:
This operator is the key operator in GA scheme which is used to alter the genes in the chromosome to create a new good string which can be added to the population. This new gene helps the GA search around the obtained solution for local optimum in short time. The mutation is performed using mutation probability (𝑃𝑚) which is normally set at 10% of the population size.
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Elitism Operator:
In order to ensure that the best performed solution will be considered, elitism operator copies two best candidates to be included in the next generation.
5.3.2.2 Multi-objective Genetic Algorithm:
In turbine development, the designer needs to optimize the blade profile with more than one objective function which makes the optimization of turbine design a multi-objective problem. Multi-objective genetic algorithm (MOGA) is well known as one of the most powerful optimization techniques in turbomachinery design optimization which can be applied for non- linear continuous and discrete parameters in both response surface and direct optimization with more than one option (Obayashi, et al. 2000, Yang and Xiao 2014).
Multi- objective genetic algorithm is a normal GA with multi-objective function (min/max) subjected to inequality and equality constrains. According to (Coello, et al. 2002) MOGA can be mathematically formulated in a vector form as:
The objective function vector: 𝐹(𝑋) = [𝑓1(𝑋), 𝑓2(𝑋), … … 𝑓𝑘(𝑋)]𝑛 Subject to: 𝑔𝑖(𝑋) ≤ 0 𝑖 = {1, … . 𝑚}
ℎ𝑗(𝑋) = 0 𝑗 = {1, … . 𝑝}
Where k is the dimensional space of the objective functions 𝑔𝑖(𝑋) is the inequality constrains, and ℎ𝑗(𝑋) is the equality constrains.
In MOGA technique the solution trade-offs are divided into dominated and non-dominated solutions based on Pareto-Optimal population ranking (Fonseca and Fleming 1993). The dominated solution group is considered as the efficient population and the optimum candidate is chosen with respect to one function without worsening the rest of objective functions.
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The objective functions in this MOGA are in conflict together and as a result the objective functions are combined in one scalar fitness function. As shown in Figure 5.3 the solution in MOGA optimization is found by searching in the multi-dimensional space (Murata and Ishibuchi 1995).
Figure 5.3 MOGA optimization solution(Murata and Ishibuchi 1995).