A comparative analysis of nature-inspired optimization approaches to 2d geometric modelling for turbomachinery applications

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

Multiple 2D self organising map network for surface reconstruction of 3D unstructured data

Surface reconstruction is a challenging task in reverse engineering because it must represent the surface which is similar to the original object based on the data obtained. The data obtained are mostly in unstructured type whereby there is not enough information and incorrect surface will be obtained. Therefore, the data should be reorganised by finding the correct topology with minimum surface error. Previous studies showed that Self Organising Map (SOM) model, the conventional surface approximation approach with Non Uniform Rational B-Splines (NURBS) surfaces, and optimisation methods such as Genetic Algorithm (GA), Differential Evolution (DE) and Particle Swarm Optimisation (PSO) methods are widely implemented in solving the surface reconstruction. However, the model, approach and optimisation methods are still suffer from the unstructured data and accuracy problems. Therefore, the aims of this research are to propose Cube SOM (CSOM) model with multiple 2D SOM network in organising the unstructured surface data, and to propose optimised surface approximation approach in generating the NURBS surfaces. GA, DE and PSO methods are implemented to minimise the surface error by adjusting the NURBS control points. In order to test and validate the proposed model and approach, four primitive objects data and one medical image data are used. As to evaluate the performance of the proposed model and approach, three performance measurements have been used: Average Quantisation Error (AQE) and Number Of Vertices (NOV) for the CSOM model while surface error for the proposed optimised surface approximation approach. The accuracy of AQE for CSOM model has been improved to 64% and 66% when compared to 2D and 3D SOM respectively. The NOV for CSOM model has been reduced from 8000 to 2168 as compared to 3D SOM. The accuracy of surface error for the optimised surface approximation approach has been improved to 7% compared to the conventional approach. The proposed CSOM model and optimised surface approximation approach have successfully reconstructed surface of all five data with better performance based on three performance measurements used in the evaluation

Gravitational Search Algorithm for NURBS Curve Fitting

By providing great flexibility non-uniform rational B-spline (NURBS) curves and surfaces are reason of preferability on areas like computer aided design, medical imaging and computer graphics. Knots, control points and weights provide this flexibility. Computation of these parameters makes the problem as a non-linear combinational optimization problem on a process of reverse engineering. The ability of solving these problems using meta-heuristics instead of conventional methods attracts researchers. In this paper, NURBS curve estimation is carried out by a novel optimization method namely gravitational search algorithm. Both knots and knots together weights simultaneous optimization process is implemented by using research agents. The high performance of the proposed method on NURBS curve fitting is showed by obtained results.Keywords: Non-uniform rational B-spline, gravitational search algorithm, meta-heuristi

Tabu search-based method for bézier curve parameterization

A very important issue in many applied fields is to construct the fitting curve that approximates a given set of data points optimally in the sense of least-squares. This problem arises in a number of areas, such as computer-aided design & manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. This is also a hard problem, because it is highly nonlinear, over-determined and typically involves a large number of unknown variables. A critical step in this process is to obtain a suitable parameterization of the data points. In this context, this paper introduces a new method to obtain an optimal solution for the parameterization problem of the least-squares fitting Bézier curve. Our method is based on a local search metaheuristic approach for optimization problems called tabu search. The method is applied to some simple yet illustrative examples for the cases of 2D and 3D curves. The proposed method is simple to understand, easy to implement and can be applied to any kind of smooth data points. Our experimental results show that the presented method performs very well, being able to fit the data points with a high degree of accuracy.This research has been financially supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. #TIN2012-30768, Toho University, the University of Cantabria, and the Instituto de Física de Cantabria, a mixed research center of the University of Cantabria and CSIC-Consejo Superior de Investigaciones Científicas.Peer Reviewe

Evalutionary algorithms for ship hull skinning approximation

Traditionally, the design process of a hull involves simulation using clay models. This must be done cautiously, accurately and efficiently in order to sustain the performance of ship. Presently, the current technology of Computer Aided Design, Manufacturing, Engineering and Computational Fluid Dynamic has enabled a 3D design and simulation of a hull be done at a lower cost and within a shorter period of time. Besides that, automated design tools allow the transformation of offset data in designing the hull be done automatically. One of the most common methods in constructing a hull from the offset data is the skinning method. Generally, the skinning method comprised of skinning interpolation and skinning approximation. Skinning interpolation constructs the surface perfectly but improper selection of parameterization methods may cause bumps, wiggles, or uneven surfaces on the generated surface. On the other hand, using the skinning surface approximation would mean that the surface can only be constructed closer to data points. Thus, the error between the generated surface and the data points must be minimized to increase the accuracy. Therefore, this study aims to solve the error minimization problem in order to produce a smoother and fairer surface by proposing Non Uniform Rational B-Spline surface using various evolutionary optimization algorithms, namely, Gravitational Search Algorithm, Particle Swarm Optimization and Genetic Algorithm. The proposed methods involve four procedures: extraction of offset data from line drawing plan; generation of control points; optimization of a surface; and validations of hull surfaces. Validation is done by analyzing the surface curvature and errors between the generated surface and the given data points. The experiments were implemented on both ship hull and free form models. The findings from the experiments are compared with interpolated skinning surface and conventional skinning surface approximation. The results show that the optimized skinning surfaces using the proposed methods yield a smaller error, less control points generation and feasible surfaces while maintaining the shape of the hull

Surface fitting for quasi scattered data from coordinate measuring systems

Non-uniform rational B-spline (NURBS) surface fitting from data points is wildly used in the fields of computer aided design (CAD), medical imaging, cultural relic representation and object-shape detection. Usually, the measured data acquired from coordinate measuring systems is neither gridded nor completely scattered. The distribution of this kind of data is scattered in physical space, but the data points are stored in a way consistent with the order of measurement, so it is named quasi scattered data in this paper. Therefore they can be organized into rows easily but the number of points in each row is random. In order to overcome the difficulty of surface fitting from this kind of data, a new method based on resampling is proposed. It consists of three major steps: (1) NURBS curve fitting for each row, (2) resampling on the fitted curve and (3) surface fitting from the resampled data. Iterative projection optimization scheme is applied in the first and third step to yield advisable parameterization and reduce the time cost of projection. A resampling approach based on parameters, local peaks and contour curvature is proposed to overcome the problems of nodes redundancy and high time consumption in the fitting of this kind of scattered data. Numerical experiments are conducted with both simulation and practical data, and the results show that the proposed method is fast, effective and robust. What’s more, by analyzing the fitting results acquired form data with different degrees of scatterness it can be demonstrated that the error introduced by resampling is negligible and therefore it is feasible

Immunological-based approach for accurate fitting of 3D noisy data points with Bézier surfaces

Free-form parametric surfaces are common tools nowadays in many applied fields, such as Computer-Aided Design & Manufacturing (CAD/CAM), virtual reality, medical imaging, and many others. A typical problem in this setting is to fit surfaces to 3D noisy data points obtained through either laser scanning or other digitizing methods, so that the real data from a physical object are transformed back into a fully usable digital model. In this context, the present paper describes an immunologicalbased approach to perform this process accurately by using the classical free-form Bézier surfaces. Our method applies a powerful bio-inspired paradigm called Artificial Immune Systems (AIS), which is receiving increasing attention from the scientific community during the last few years because of its appealing computational features. The AIS can be understood as a computational methodology based upon metaphors of the biological immune system of humans and other mammals. As such, there is not one but several AIS algorithms. In this chapter we focus on the clonal selection algorithm (CSA), which explicitly takes into account the affinity maturation of the immune response. The paper describes how the CSA algorithm can be effectively applied to the accurate fitting of 3D noisy data points with Bézier surfaces. To this aim, the problem to be solved as well as the main steps of our solving method are described in detail. Some simple yet illustrative examples show the good performance of our approach. Our method is conceptually simple to understand, easy to implement, and very general, since no assumption is made on the set of data points or on the underlying function beyond its continuity. As a consequence, it can be successfully applied even under challenging situations, such as the absence of any kind of information regarding the underlying function of data

Parametric design and optimization of high speed train nose

Aiming at shortening the design period and improve the design efficiency of the nose shape of high speed trains, a parametric shape optimization method is developed for the design of the nose shape has been proposed in the present paper based on the VMF parametric approach, NURBS curves and discrete control point method. 33 design variables have been utilized to control the nose shape, and totally different shapes could be obtained by varying the values of design variables. Based on the above parametric method, multi-objective particle swarm algorithm, CFD numerical simulation and supported vector machine regression model, multi-objective aerodynamic shape optimization has been performed. Results reveal that the parametric shape design method proposed here could precisely describe the three-dimensional nose shape of high speed trains and could be applied to the concept design and optimization of the nose shape. Besides, the SVM regression model based the multi-points criterion could accurately describe the non-linear relationship between the design variables and objectives, and could be generally utilized in other fields. No matter the simplified model or the real model, the aerodynamic performance of the model after optimization has been greatly improved. Based on the SVR model, the nonlinear relation between the aerodynamic drag and the design variables is obtained, which could provide guidance for the engineering design and optimization

Parametric design and optimization of high speed train nose

Aiming at shortening the design period and improve the design efficiency of the nose shape of high speed trains, a parametric shape optimization method is developed for the design of the nose shape has been proposed in the present paper based on the VMF parametric approach, NURBS curves and discrete control point method. 33 design variables have been utilized to control the nose shape, and totally different shapes could be obtained by varying the values of design variables. Based on the above parametric method, multi-objective particle swarm algorithm, CFD numerical simulation and supported vector machine regression model, multi-objective aerodynamic shape optimization has been performed. Results reveal that the parametric shape design method proposed here could precisely describe the three-dimensional nose shape of high speed trains and could be applied to the concept design and optimization of the nose shape. Besides, the SVM regression model based the multi-points criterion could accurately describe the non-linear relationship between the design variables and objectives, and could be generally utilized in other fields. No matter the simplified model or the real model, the aerodynamic performance of the model after optimization has been greatly improved. Based on the SVR model, the nonlinear relation between the aerodynamic drag and the design variables is obtained, which could provide guidance for the engineering design and optimization