A vast variety of population-based optimization techniques have been formulated in recent years for use in different engineering applications, most of which are inspired by natural processes taking place in our environment. However, the mathematical and statistical analysis of these algorithms is still lacking. This paper addresses a comparative performance analysis on some of the most important nature-inspired optimization algorithms with a different basis for the complex high-dimensional curve/surface fitting problems. As a case study, the point cloud of an in-hand gas turbine compressor blade measured by touch trigger probes is optimally fitted using B-spline curves. In order to determine the optimum number/location of a set of Bezier/NURBS control points for all segments of the airfoil profiles, five dissimilar population-based evolutionary and swarm optimization techniques are employed. To comprehensively peruse and to fairly compare the obtained results, parametric and nonparametric statistical evaluations as the mathematical study are presented before designing an experiment. Results illuminate a number of advantages/disadvantages of each optimization method for such complex geometries’ parameterization from several different points of view. In terms of application, the final appropriate parametric representation of geometries is an essential, significant component of aerodynamic profile optimization processes as well as reverse engineering purposes.
The turbomachinery blades’ aerodynamic profile and structural strength play an important role in improvement of energy conversion efficiency as well as the reliability and availability of the machine in both land-based and aero applications. From aerodynamic point of view, geometry-based blade design optimization is of utmost importance for further enhancement of turbomachinery performance. On the other hand, a robust and precise reverse engineering of existing blades could also maintain their performance as near as possible to the reference design. In this way, precise and proper geometry modeling and parameterization of current blades are the first and sometimes the most important step in both optimization and reverse engineering missions which also, in turn, need an optimization effort [
At the same time, recent advances in both mathematical tools and optimization techniques enable further improvement in geometric modeling procedure. Irrespective of the goal of blade profile parameterization, that is, its use in inverse aerodynamic design or in its application to a direct numerical optimization, there are several parameterization methods with different impacts on the reversing/optimization process. For free-form shapes, however, parametric curves/surfaces including Bezier [
As mentioned above, a proper and precise curve fitting requires an unavoidable optimization mission. The nonlinear nature of sophisticated geometries, like turbomachinery blades’ shape, and availability of high-speed parallel computers for massive computation have resulted in the use of non-gradient-based and guided random search methodologies in curve/surface fitting problems [
Among these optimization techniques, genetic algorithm and differential evolution (DE) [
PSO and invasive weed optimization (IWO) [
In this paper, a study on a gas turbine compressor airfoil shape parameterization and optimum curve fitting as a main step to blade geometry optimization has been reported. For this purpose, two most important curve parameterization methods, that is, Bezier and NURBS curves, have been applied within the five above-mentioned population-based optimization loops (GA, RCGA, DE, PSO, and IWO) so that the best alternative solutions from different points of view can be discovered. To achieve this goal, the parametric procedures involving independence, normality and homoscedasticity are firstly discussed. Then, the nonparametric statistical analysis including Quade and Friedman aligned comparison tests, by applying several different post hoc procedures, are brought to perform a rigorous comparison among the performance of the optimization algorithms. After that, designing an experiment is done for each algorithm to finally illustrate the effectiveness of the proposed algorithms.
In the rest of the article, the theory behind the Bezier and nonuniform rational B-spline curves and different airfoil geometric representation techniques are first described in Section
The turbomachinery blade shape parameterization addresses the two-dimensional airfoil profile construction from either the direct handling of curves of airfoil shapes or the superposition of the camber line and thickness distribution around it. Between these two different approaches of geometry definition, however, many designers have considered distinct parameterized curves for the suction surface (SS), pressure surface (PS), leading edge (LE), and trailing edge (TE) of a blade section. In this article, direct handling of the curves of the airfoil shape has been used by dividing the profile from the hub to the shroud into five sections as shown in Figure
Measured points cloud for a section along the blade span, and four distinct segments of the airfoil for the curve fitting problem (a). Gas turbine compressor blade sections-view from the top (b).
As the first step, a standard coordinate measuring machine equipped with touch trigger probes has been employed to measure 2D point cloud of compressor blade in-hand (Figure
Here, the theory behind these mathematical methods and the characteristics of each will be briefly described.
The Bezier curve, proposed by Bezier in the early 1960s [
When this blending function is applied to the vertices of the polygon, the Bezier curve equation is found as
One of the major drawbacks of Bezier curves is that the curve degree is directly determined by the number of CPs. In addition, each CP has an impact on the shape of the whole curve. This becomes of specific concern when only some minor modifications on some specific places on the blade sections are required. Therefore, using B-spline curves is an alternative because of their local modification property.
In the early 1970s, Cox [
Based on this equation, the B-spline curve is defined as follows:
In these equations, numerical stability (due to the independence of polynomial degree from number of CPs), local shape control or local modification property (because of the influence range of each CP), coincidence of endpoints with first/last CPs, convex hull property (like Bezier curves), more versatile modification of the created curve, exact presentation of conic curves (circles, ellipses, etc.).
The above-mentioned characteristics are important, particularly when a researcher faces a real case study. For example, the last distinction can really be helpful for our specific problem in the case where the designer wants to construct two circles/ellipses for trailing/leading edges of the airfoil shape. In the present work, these two segments have also been modeled by free-form curves just like the other two segments.
Depending on whether the purpose of the designer is to create a completely new airfoil design or to make some minor aerodynamic improvements to the existing blade profile, a global geometry modeling approach or a localized parameterization method can be used, respectively. Considering a number of the important advantages for NURBS representation mentioned above and because of the prospective goal of the present continuing research to slightly improve the shape of the existing turbomachinery blades for some specific objectives in upcoming works, the application of NURBS curves seems to be more reasonable. Indeed, their affine invariance and local control property make them appropriate for such geometry handling. The comparison between Bezier and NURBS curves would be discussed further in the results section of the paper.
After skimming the background of Bezier/NURBS curves, the optimum data fitting problem using the defined tools and the planned approach is made clear in this section.
When the digitized points are available, model construction is continued by airfoil segmentation and CAD modeling. Segmentation subdivides the measured points into areas for various individual features [
Concerning the NURBS curve fitting specifically, various fitting schemes have been developed so far. Some researchers have identified both the CPs and the weights of a NURBS curve simultaneously by minimizing the sum of the squares of the distances from the measured points to the corresponding fitted curve points, while others have focused only on the optimization of the CPs’ location. Regarding the airfoil shape modeling, a third-order NURBS curve/surface has recently been employed to describe the suitable parametric geometry of 2D airfoil and 3D blades.
In this paper, the best CPs of both Bezier and NURBS curves among the candidates are searched for by using the least squares method. Although the proposed method can automatically determine the appropriate CPs, both number and location, at the same time, these optimal CPs will only be searched in a predefined range—based on the existing knowledge about the constraints of airfoil shape at hand—to save time and CPU efforts. Therefore, the CPs of these curves can be utilized as characteristic points. This procedure clearly needs an optimization effort due to the lack of knowledge about the exact position of the proper CPs at the first stage of curve fitting problem. Hence, the objective function which should be minimized is the sum of distances between the fitted curve and the original scanned data points. In such an optimization process, Bezier/NURBS CPs are the design variables, in fact, that have to be forced to the optimal locations.
The above-mentioned procedure is more clearly depicted in Figure
Errors between the original measured points and fitted curve.
The airfoil (as a profile section of the compressor blade) considered in this study consists of four Bezier/NURBS curves for four segments, and nine CPs identify each segment. Each CP has two coordinates
All optimization techniques in this paper have been used to locate the optimal position of the CPs for a set of open Bezier and/or NURBS curves according to the objective function represented in (
Bezier/NURBS optimum curve fitting flowchart.
This section briefly discusses the procedures, operators, and logic of all optimization algorithms used in this study. For this purpose, the optimization techniques are conceptually divided into two main categories based on their characteristics:
Key similarities and differences among DE, PSO, GA, and IWO procedures.
As mentioned earlier, two evolutionary optimization methods customized and specialized by the authors are used to optimally fit the parametric curves to the airfoil point cloud. As the algorithms are based on Darwin’s theory of evolution, the optimization uses different mechanisms and operators to search for individuals that best adapt to the environment. In other words, individuals with a higher level of fitness have a greater chance to survive and continue in the optimization process.
GA was introduced by Holland [
The structure of a GA is composed of an iterative procedure through the following key steps [ creating an initial population of airfoil shapes with consideration of the existing experience, evaluation of the fitting performance of each profile in the population, selection of the best-fitted airfoils’ segments and reproduction of a new population, application of genetic operators, that is, crossover and mutation, iteration of steps II to IV until the maximum number of generations is reached.
The fitness value is associated with each individual, expressing the performance of any airfoil shapes created with respect to a fitting error function that should be minimized (
In addition, a real coding approach, instead of the standard GA coding strategy that is a binary pattern, is also used in this study. Using this approach, more adequate mutation operators can be defined while the destructive effects of crossover are reduced, specifically for a set of data points in the
DE is one of the newly emerging, but well-known, stochastic parallel direct search methods as well as a kind of population-based optimization algorithm. DE was presented by Price and Storn in 1997 [
As illustrated in Figure every individual of the population (each set of CPs for the problem in hand) undergoes mutation operation with scale factor (SF) a positive real number typically less than 1, mutation vector and target vector undergo crossover operation with crossover rate (CR), and trial vector is evaluated and compared with the fitness value of the target vector. The vector with the greater fitness value (the fitter CPs’ set) enters the next generation.
Recombination incorporates successful solutions from the previous generation, and mutation expands the search space, as in GA. Mutation, recombination, and selection continue until some stopping criterion is reached.
The present paper proposes an alternative method by implementing a DE algorithm for the curve fitting problem. This proposed method is compared with the result obtained from the GA method in the last section of the paper.
The theoretical bases for two swarm intelligence methods employed in the comparative study are described in this section.
A very popular swarm intelligence algorithm introduced by Kennedy and Eberhart is called particle swarm optimization [
In general, the PSO algorithm consists of three main steps as follows (Figure generating positions and velocities of particles, updating the velocities, updating the positions.
In this way, each particle refers to a point in a multidimensional space where its dimensions are related to the numbers of design variables. The positions of the points as well as the particles’ velocity change as the iterations, which are calculated in each step, proceed. In the first step, position
The first population is distributed uniformly in the space by this process. In the second step, PSO calculates new velocities to move the particles from positions in time,
Based on the above explanations, the velocity update formula takes the following form:
It is worth pointing out that when the effect of neighbor acceleration is considered in the velocity update (
Lastly, using the updated velocity vectors calculated by (
This algorithm is repeated until a stop criterion is reached. The stop criteria may be an iteration number or a specified error for the best global value.
Invasive weed optimization is an ecologically inspired optimization algorithm, newly developed by Mehrabian and Lucas [
IWO is implemented in this study with a problem dimension of ten and using the following procedure recursively: initializing a population (randomly spreadover the fitness evaluation for each member of the colony of weeds to produce seeds depending on its own and the colony’s lowest and highest fitness, reproduction (based on plant fitness and number of seeds), spatial dispersal (i.e., the generated seeds are randomly distributed over the search space by normally distributed random numbers with mean equal to zero, but varying standard deviation, SD), competitive exclusion (poor plants elimination).
The flowchart of the IWO method is also illustrated in Figure
The variation of the SD value with generation is defined as follows:
After reaching the maximum number of plants, the competitor exclusion mechanism activates in order to eliminate the plants with poor fitness in the generation. This mechanism is formulated so that it gives a chance to plants with lower fitness to reproduce, and if their offspring has a good fitness in the colony, then they will survive. The above-mentioned steps are repeated until the maximum number of iterations is exceeded.
This section presents several illustrations of compressor airfoil data fitting from different points of view. The goal is to show the performance of the proposed optimization methodologies to be used in such geometric modelling problems and to make a comparison among them. In order to have a fair comparison of the results, both the initial population and the stopping criteria are set to be identical for the relevant algorithms.
Before representing the experimental illustrations, however, it is expected to compare and analyse the results in a systematic approach recently designed and proposed [
Furthermore, the experimental results are presented by applying all optimization algorithms to the given airfoil shape parameterization process. The following figures and tables present the historical convergence of optimization algorithms during the generations, differences between Bezier and NURBS curves from different points of view, and their advantages and disadvantages, static and dynamic convergences of the used algorithms, fitting error in each case, and computational efficiency.
All coding and simulation carried out in this research have been implemented in MATLAB R2010b on a PC with Intel(R) Xeon(R) CPU X5650 @ 2.66 GHz with 2 processors and 32.0 GB RAM.
This section outlines the comparisons between the results obtained from the various algorithms. Studies conducted to prove the relative merits of the algorithms used for the curve fitting problem in hand are presented.
With respect to the no free lunch theorem in search and optimization, it is not possible to find one specific algorithm being better in behaviour for any given problems. Therefore, some criteria to select an appropriate optimization algorithm in each problem are sought. In this regard, statistical procedures including the parametric tests and nonparametric tests could be utilized for a fair comparison between the results. For this purpose, the required condition for the parametric tests is firstly described based on the published studies. These conditions are then checked for the obtained results in this work. Subsequently, the results of a nonparametric analysis using several methods are represented.
In order to use the parametric test, it is necessary to check independence, normality, and homoscedasticity procedures.
For the problem in hand, the independence of the results is obvious since they are independent runs of any algorithm with randomly generated initial population. At the same time, an observation is said to be normal when its behaviour follows a normal or Gauss distribution with a certain average value
The fitness values obtained from 30 independent runs.
RCGA | BGA | DE | PSO | IWO | |
---|---|---|---|---|---|
1 | 0.057 | 0.04081 | 0.037 | 0.8259 | 0.2795 |
2 | 0.0593 | 0.0401 | 0.037 | 0.0828 | 0.198 |
3 | 0.0739 | 0.04049 | 0.037 | 0.8142 | 0.3549 |
4 | 0.0537 | 0.042 | 0.037 | 1.8263 | 0.1485 |
5 | 0.0479 | 0.0467 | 0.037 | 1.1282 | 0.8118 |
6 | 0.0487 | 0.04077 | 0.037 | 0.9122 | 0.1485 |
7 | 0.072 | 0.04054 | 0.037 | 0.7141 | 0.1485 |
8 | 0.0514 | 0.03946 | 0.037 | 0.6595 | 0.866 |
9 | 0.064 | 0.04125 | 0.037 | 0.5901 | 1.4464 |
10 | 0.071 | 0.04348 | 0.037 | 0.8109 | 0.2169 |
11 | 0.0511 | 0.04127 | 0.037 | 0.7419 | 0.1485 |
12 | 0.0502 | 0.03894 | 0.037 | 0.1108 | 0.8994 |
13 | 0.0621 | 0.04075 | 0.037 | 1.7911 | 0.2255 |
14 | 0.0501 | 0.04204 | 0.037 | 0.7631 | 0.1889 |
15 | 0.0661 | 0.03943 | 0.037 | 0.245 | 0.1889 |
16 | 0.0493 | 0.03891 | 0.037 | 0.8892 | 0.1889 |
17 | 0.0534 | 0.03956 | 0.037 | 0.2679 | 0.8807 |
18 | 0.0511 | 0.03851 | 0.037 | 0.5432 | 0.7837 |
19 | 0.0522 | 0.03963 | 0.037 | 1.9922 | 0.2335 |
20 | 0.1003 | 0.04568 | 0.037 | 0.2032 | 0.1889 |
21 | 0.0592 | 0.04093 | 0.037 | 0.8074 | 0.1842 |
22 | 0.053 | 0.03999 | 0.037 | 0.7006 | 0.2008 |
23 | 0.0708 | 0.04271 | 0.037 | 0.7102 | 0.828 |
24 | 0.0624 | 0.03794 | 0.037 | 0.7437 | 0.2368 |
25 | 0.0627 | 0.04044 | 0.037 | 0.2942 | 1.5037 |
26 | 0.0645 | 0.04884 | 0.037 | 0.1837 | 1.1268 |
27 | 0.0579 | 0.04599 | 0.037 | 0.2674 | 1.1532 |
28 | 0.0531 | 0.04262 | 0.037 | 1.0189 | 0.1485 |
29 | 0.055 | 0.04305 | 0.037 | 1.4319 | 0.1775 |
30 | 0.1024 | 0.04206 | 0.037 | 1.717 | 0.2158 |
In this study, two different tests are used for numerical assessment of the normality. The first one is Kolmogorov-Smirnov (K-S test) which is a distribution-free and nonparametric test but could be modified to be employed as a goodness of fit test. In the special case of testing for normality of the distribution, data are standardized and compared with a standard normal distribution. The K-S test has this benefit to make no assumption about the distribution of data. The second test is Shapiro-Wilk (S-W test) that compares the ordered sample values with the corresponding order statistics from the specified distribution. The S-W test is most commonly used to assess a normal distribution.
Table
Results of normality tests.
GA | DE | PSO | IWO | |
---|---|---|---|---|
|
0.1066 | 0.1027 | 0.8102 | 0.5304 |
|
0.0802 | 0.2507 | 0.8012 | 0.4304 |
Finally, passing these normality tests allows us to state with 90% confidence that the data fit the normal distribution with no significant departure from the normality.
Since the research work deals with the real values in this study and based on the promising results from the previous section, the results of parametric tests seem enough for a reliable statistical analysis. For more evaluation of the all given algorithms’ behaviour, the use of nonparametric statistics is also considered in addition to the parametric test. The performance of any algorithm with respect to the remaining ones is studied, and based on that, it has been determined if these results offer better performance for each one. Here, all results are reported with respect to the level of significance
Tables
Post hoc results by Quade comparison test (reference algorithm: DE).
|
Algorithms |
|
|
Holm Hochberg Hommel | Holland | Rom | Finner | Li |
---|---|---|---|---|---|---|---|---|
4 | PSO | 5.522134 | 0 | 0.0125 | 0.012741 | 0.013109 | 0.012741 | 0.045765 |
3 | IWO | 5.063582 | 0 | 0.016667 | 0.016952 | 0.016667 | 0.025321 | 0.045765 |
2 | RCGA | 3.02449 | 0.002491 | 0.025 | 0.025321 | 0.025 | 0.037739 | 0.045765 |
1 | BGA | 1.512245 | 0.130472 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
Post hoc results by Friedman aligned comparison test (reference algorithm: DE).
|
Algorithms |
|
|
Holm |
Holland | Rom | Finner | Li |
---|---|---|---|---|---|---|---|---|
4 | PSO | 7.321863 | 0 | 0.0125 | 0.012741 | 0.013109 | 0.012741 | 0.007325 |
3 | IWO | 5.384422 | 0 | 0.016667 | 0.016952 | 0.016667 | 0.025321 | 0.007325 |
2 | RCGA | 0.63888 | 0.522901 | 0.025 | 0.025321 | 0.025 | 0.037739 | 0.007325 |
1 | BGA | 0.175321 | 0.860828 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
To compare an optional reference algorithm with others, Holm procedure driven
|
Algorithms |
|
Unadjusted |
Holm |
---|---|---|---|---|
10 | DE versus PSO | 8.981462 | 0 | 0.005 |
9 | DE versus IWO | 8.164966 | 0 | 0.005556 |
8 | BGA versus PSO | 6.531973 | 0 | 0.00625 |
7 | BGA versus IWO | 5.715476 | 0 | 0.007143 |
6 | RCGA versus DE | 4.898979 | 0.000001 | 0.008333 |
5 | RCGA versus PSO | 4.082483 | 0.000045 | 0.01 |
4 | RCGA versus IWO | 3.265986 | 0.001091 | 0.0125 |
3 | RCGA versus BGA | 2.44949 | 0.014306 | 0.016667 |
2 | BGA versus DE | 2.44949 | 0.014306 | 0.025 |
1 | PSO versus IWO | 0.816497 | 0.414216 | 0.05 |
Lastly, contrast estimation is obtained to calculate approximately the performance difference of the algorithms, two by two (Table
Contrast estimation between averages of results considering all pairwise comparisons.
RCGA | BGA | DE | PSO | IWO | |
---|---|---|---|---|---|
RCGA | 0 | 0.016 | 0.021 | −0.619 | −0.226 |
BGA | −0.016 | 0 | 0.005 | −0.635 | −0.243 |
DE | −0.021 | −0.005 | 0 | −0.64 | −0.248 |
PSO | 0.619 | 0.635 | 0.64 | 0 | 0.392 |
IWO | 0.226 | 0.243 | 0.248 | −0.392 | 0 |
All the algorithms were run several times with the set of parameters based on the existing experience in order to find the best possible solution.
To begin with, because GA-based optimization is broadly used and more well-known than other methods employed in the paper and exhibited good performance in statistical analysis, this algorithm has been selected to make a reliable comparison between NURBS and Bezier curve fitting for airfoil shape parameterization. The best regulations for this optimization algorithm involve 120 chromosomes as the initial population selected from the whole solution space, and maximum number of generations is set to 300. Selection method is Roulette wheel, and probabilities of two-point crossover and mutation are set to 1.0 and 0.20, respectively. In order to make a fair comparison of the results, the termination criteria as well as all other parameters are set to be the same for both curve fitting tools. Finally, the real-number genes for design variables’ definition with the method of least squares for evaluation of curve fitting fitness were successfully used as shown in the following illustrations.
Figure
Optimum curve fitted using Bezier and NURBS methods (a) and global (b) and local (c) modification properties of Bezier and NURBS, respectively.
Another important characteristic that should be investigated is the local/global modification property. As shown in Figures
Table
Comparison of NURBS and Bezier properties’ effect on some implementation criteria.
5 control points | 10 control points | |||||
---|---|---|---|---|---|---|
Degree of curve | Fitting error [m] | Comp. burden [s] | Degree of curve | Fitting error [m] | Comp. burden [s] | |
Bezier | 4 | 0.1376 | 60 | 9 | 0.0617 | 203 |
NURBS | 3 | 0.0518 | 287 | 3 | 0.0393 | 475 |
Although the computational time of NURBS curve fitting is generally higher than that of Bezier curve fitting, the results in Table
As a result, the remaining implementation is done by using NURBS representation as the tool of geometry modeling in airfoil curve fitting.
After a GA-based comparison of Bezier and NURBS properties and an investigation of their effects on airfoil geometry parameterization, the results obtained from all the developed optimization methods using NURBS representation are compared here in terms of static and dynamic convergences, final fitness values, and computing time. This is to see whether these methods can be successfully used in optimally solving these specific curve fitting problems. To the best knowledge of the authors, this is the first time that the DE and IWO algorithms have been profitably applied to a NURBS curve fitting optimization problem.
Table
Evolutionary and swarm algorithms’ regulations for comparison task.
RCGA | BGA | DE | PSO | IWO | |
---|---|---|---|---|---|
Number of Iterations | 250 | 250 | 250 | 250 | 250 |
Population/swarm/plant size | 120 | 120 | 120 | 180 | 10 |
Selection method | Roulette wheel | Stochastic uniform | — | — | — |
Crossover | Two-point, 0.90 | Single-point, 0.95 | 0.5 | — | — |
Mutation | Uniform, 0.35 | Uniform, 0.30 | — | — | — |
Scale factor | — | — | 0.5 | — | |
Self-confidence factor | — | — | — | 1.5 | — |
Swarm confidence factor | — | — | — | 1.5 | — |
Neighbor accel. factor | — | — | — | 1.0 | — |
Velocity weight at the beginning | — | — | — | 0.75 | — |
Velocity weight at the end | — | — | — | 0.25 | — |
Number of plants | — | — | — | — | 10 |
Min. Number of seeds | — | — | — | — | 1 |
Max. Number of seeds | — | — | — | — | 10 |
Modulation index | — | — | — | — | 3 |
Standard deviation, initial | — | — | — | — | 10 |
Standard deviation, final | — | — | — | — | 0.001 |
Taking the metaheuristic nature of the algorithms used in this part into account, RCGA, BGA, DE, PSO, and IWO are run several times. Because of rationally convergence of the algorithms, the final fitness values have been too close to each other for the unique objective function and parameters in several runs. However, all results presented in this paper are the mean values of those runs. Moreover, as addressed before, all results have only been illustrated for one of the segments (segment 2 according to Figure
Figure
Static convergence of RCGA, BGA, DE, PSO, and IWO.
Iteration number in which the best solution is achieved | Final objective value (sum of errors in millimeters) | |
---|---|---|
RCGA | 180 | 50.2 |
BGA | 144 | 40.8 |
DE | 135 | 37.7 |
PSO | 181 | 203.2 |
IWO | 94 | 148.8 |
Convergence history of all given algorithms.
Additionally, a comparison between the dynamic convergence characteristics of each of the algorithms employed has been made in Figure
Dynamic convergence characteristics of evolutionary and swarm intelligence algorithms.
Finally, in order to confirm the effectiveness of the planned methods to optimize the curve fitted to the airfoil shape of GT blades, 20 optimized CPs, knot vectors, and NURBS curves found by both evolutionary and swarm intelligence algorithms have been shown in Figure
Airfoil shapes represented with optimal control points derived from evolutionary (a) and swarm intelligence (b) methodologies.
Last but not least, in order to compare the computational efficiency of all four optimization algorithms used in this study, the CPU time is averaged over 30 independent runs for each method. The results are shown in Table
Computational efficiency (average of 30 runs).
Real-coded GA | Binary GA | DE | PSO | IWO | |
---|---|---|---|---|---|
CPU time (Sec) | 240 | 218 | 216 | 14 | 23 |
This paper presents the results of a study where a binary and real-coded genetic algorithm, a differential evolution, a particle swarm, and a newly developed invasive weed optimization methods were implemented using Bezier and NURBS tools to optimally create the well-parameterized airfoil shapes on existing turbomachinery blades. The approach could be used for both precise reverse engineering and aerodynamic shape optimization purposes [
First, the touch trigger probes are employed to measure the point cloud of an existing gas turbine compressor blade sections. After that, the model construction is continued by segmentation of the measured points to four areas. Then, CAD modeling searches for the best CPs of Bezier and NURBS curves among the candidates by using the least-squares fitting technique in the objective function of the aforementioned optimization algorithms.
One important conclusion is that the NURBS curves, because of their special properties such as local modification and numerical stability, are certainly more suitable tools for such complex curve/surface fitting problem in turbomachinery than the Bezier curves. Regarding the planned optimization algorithms, the behavior of all the given evolutionary and swarm intelligence techniques is inclusively evaluated from both statistical and experimental points of view. As a result, GA and DE—as two of the most well-known evolutionary algorithms—as well as PSO and IWO—as the swarm intelligence methods—generally showed their ability in optimization of such complex curve fitting processes. Nevertheless, several significant behaviour differences are detected from a comprehensive comparison of their results. As a result of nonparametric test, it can be concluded that the evolutionary optimization algorithms obviously outperform the swarm intelligence ones. At the same time, while B/RC GA and DE demonstrate their superior performance in terms of reaching minimum fitting error, IWO has the best static convergence. IWO also shows the finest behaviour in dynamic convergence, in areas where GA results exhibit the highest fluctuations, and this is because of its mutation operator. Also, swarm intelligence optimization methods are observed to have a significantly better computational efficiency than evolutionary algorithms.
Lastly, since all the results obtained from the well-organized proposed optimization strategies demonstrate that the final optimum fitted curves have good agreement with the original measured points, selecting one among them to be employed in a computer-aided geometric design optimization process strongly depends on the final goals and priorities.