Elsevier

Knowledge-Based Systems

Volume 240, 15 March 2022, 108071
Knowledge-Based Systems

An enhanced manta ray foraging optimization algorithm for shape optimization of complex CCG-Ball curves

https://doi.org/10.1016/j.knosys.2021.108071Get rights and content

Abstract

The shape optimization of complex curves is a crucial and intractable technique in computer aided geometric design and widely used in many product design and manufacturing fields involving complex curve modeling. In this paper, an enhanced manta ray foraging optimization (MRFO) algorithm is used to optimize the shape of complex composite cubic generalized Ball (CCG-Ball, for short) curves. Firstly, to solve the problems of shape optimization for Ball curves, we construct a class of new cubic generalized Ball basis, and then present the CCG-Ball curves with multiple shape parameters based on the constructed basis functions. The shapes of the curves can be modified and optimized easily by using the shape parameters. Secondly, the shape optimization of CCG-Ball curves is mathematically an optimization problem that can be efficiently dealt with by swarm intelligence algorithm. In this regard, an enhanced MRFO called WMQIMRFO algorithm, combined with control parameter adjustment, wavelet mutation and quadratic interpolation strategy, is developed to enhance its capability of jumping out of the local minima and improve the calculation accuracy of the native algorithm. Furthermore, the superiority of the WMQIMRFO algorithm is verified by comparing with standard MRFO, other improved MRFO and popular nature-inspired optimization algorithms on the well-known CEC’14 and CEC’17 test suite as well as four engineering optimization problems, respectively. Finally, by minimizing the bending energy of the CCG-Ball curves as the evaluation standard, the shape optimization models of the curves with 1th-order and 2th-order geometric continuity are established, respectively. The WMQIMRFO algorithm is utilized to solve the established models, and the CCG-Ball curves with minimum energy are obtained. Some representative numerical examples illustrate the ability of the proposed WMQIMRFO algorithm in effectively solving the shape optimization problems of complex CCG-Ball curves in terms of precision, robustness, and convergence characteristics.

Introduction

Computer aided geometric design (CAGD) originated from shape design of automobiles and airplanes, and its research object is geometric shape of various products [1]. CAGD studies issues such as representation, generation, storage, design, blending and fitting of curves and surfaces, which need to be realized by computer. Among them, Ball curves and surfaces constructed by Ball basis functions has been widely used and deeply studied because of their simple structure and excellent properties.

Ball [2], [3], [4], a British mathematician, defined rational cubic Ball curves for the first time in 1974 and took the Ball curves as mathematical basis for CONSURF fuselage surface modeling system of Warton former British Airways. Because basis functions of rational cubic Ball curves are limited to cubic, many scholars further study generalized Ball curves of high degree. Wang [5] proposed Wang-Ball curves in 1987. The Wang-Ball basis functions of high degree at head and end are quadratic, then degree of basis functions gradually increases from both ends to middle, and there is quadratic difference between adjacent basis functions. Said [6] proposed Said-Ball curves in 1989, but the constructed Said-Ball curves are still incomplete. Because they are derived from Hermite interpolation, the degree of basis functions can only be odd. Hu et al. [7] further perfected Said-Ball curves in 1996 by extending the degree of basis functions to even number, at this time, the Said-Ball curves can be of arbitrary degree. After that, Othman et al. [8] discussed dual basis functions of Said-Ball curves and their application.

Hu et al. [9] made comparative study of Wang-Ball curves, Said-Ball curves and Bézier curves in recursive evaluation, degree elevation and degree reduction algorithm and other aspects. The results showed that compared with Bézier curves, Said-Ball curves have better calculation speed in curves evaluation, degree elevation and degree reduction, and Wang-Ball curves have better performance in these aspects than Said-Ball curves. In general, generalized Ball curves not only have basic properties of Bézier curves, but also have faster calculation speed and higher calculated efficiency.

However, for given control points and corresponding Ball basis functions, the shape of generalized Ball curve is uniquely determined. The shape of generalized Ball curves cannot be modified unless adjusting the control points. In order to solve this problem, generalized Ball curves and surfaces with a shape control parameter are further proposed. According to properties of Wang-Ball curves and Said-Ball curves, Wu [10] proposed Said–Bézier generalized Ball curves and Wang–Said​ generalized Ball curves (abbreviated as SBGB curves and WSGB curves) in 2000. SBGB curves contain Said-Ball curves, Bézier curves, and several curves between them. WSGB curves contain Wang-Ball curves, Said-Ball curves, and several curves between them. By introducing position parameter into SBGB curves and WSGB curves, position of curves can be adjusted appropriately by selecting different value of position parameter. Based on Said-Ball curves, Hu et al. [11] proposed Said-Ball curves of degree 2M+2 with shape control parameter λ in 2009. Compared with traditional Said-Ball curves, they can control shape of curves by adjusting shape control parameter λ. In particular, when λ=0, Said-Ball curves of degree 2M+2 with shape parameter λ degenerate into Said-Ball curves of degree 2M+1 with shape parameter λ. Later, on the basis of Wang-Ball curves, Xiong et al. [12] proposed Wang-ball curves of degree N with shape control parameter λ in 2013. In particular, when λ=2, Wang-Ball curves of degree N with shape control parameter λ degenerate into traditional Wang-Ball curves of degree N.

Compared with traditional Ball curves, the shape adjustability of generalized Ball curves with a shape control parameter is improved, but this is not enough to meet people’s demand for flexibility of curves. It can only make curves swing simply on one or both sides of Ball curves, and the shape adjustability is very limited. Therefore, Liu [13] proposed quadric Ball curves with two shape control parameters λ,μ in 2011, and the shape adjustability of curves is improved. One of purposes in this paper is to propose a class of new cubic generalized Ball (CG-Ball, for short) curves with multiple shape control parameters. By introducing shape parameters to reconstruct basis functions, the shape of curves can be controlled more flexibly by changing multiple shape parameters. At the same time, it is a challenging task to design complex graphs by single Ball curve. Therefore, this paper also studies 1th-order and 2th-order (i.e. G1 and G2) continuity conditions between adjacent Ball curves, so as to study the shape optimization problem of composite CG-Ball (CCG-Ball, for short) curves with whole G1 and G2 smooth blending.

When studying the shape optimization problem of combined Ball curves with whole G1 and G2 smooth blending, a problem arises: how to select the values of shape control parameters to make generated curves have optimal shape? In order to generate CCG-Ball curves with optimal shape, it is necessary to establish evaluation criteria for optimal shape of curves. Energy method is a commonly used method to optimize shape of curves and surfaces, that is, aiming at minimum stretching energy, bending energy or twisting energy of curves and surfaces, using range of shape control parameters and other constraints to control shape of curves, and the calculation formula of shape control parameters is derived [14]. When optimizing shape of curves and surfaces, different objective functions can be selected according to different design requirements. There are many research literature on shape optimization of curves and surfaces, such as Refs. [15], [16], [17], [18], [19]. This paper intends to establish the shape optimization model of CCG-Ball curves with whole G1 and G2 smooth blending respectively according to the requirements of the smoothest curves. But due to CCG-Ball curves have multiple local shape control parameters, objective function in established optimization model is highly nonlinear, which is difficult to find the minimum value of objective function by traditional optimization methods. Therefore, this paper tentatively adopts popular swarm intelligence optimization algorithm in recent years to solve shape optimization model of CCG-Ball curves.

Swarm intelligence optimization algorithms simulate social behavior of creatures. The most classical algorithm is particle swarm optimization (PSO) [20]. In recent years, other swarm intelligence optimization algorithms are proposed including firefly algorithm (FA) [21], gray wolf optimizer (GWO) [22], whale optimization algorithm (WOA) [23], grasshopper optimization algorithm (GOA) [24], Harris hawks optimization (HHO) [25], barnacles mating optimizer (BMO) [26], slime mold algorithm (SMA) [27], manta ray foraging optimization (MRFO) [28], etc. Among them, MRFO algorithm is a new meta-heuristic algorithm proposed by Zhao et al. in 2020, which is inspired by intelligent behavior of manta rays. MRFO algorithm has three kinds of foraging operators, namely chain foraging, spiral foraging and somersault foraging, which can effectively improve optimization ability of MRFO algorithm from different aspects. MRFO algorithm has the advantages of strong global search ability, few parameters, strong robustness and so on. Therefore, MRFO algorithm has been applied to solve some practical engineering optimization problems [29], [30], [31], [32], [33], [34], [35], [36].

Many scholars have made some improvements to this algorithm on specific problems. Elaziz et al. [37] proposed an improved MRFO algorithm based on fractional-calculus for solving global optimization and image segmentation. Hassan et al. [38] proposed an improved MRFO algorithm for solving cost-effective emission dispatch problems. Xu et al. [39] proposed an developed MRFO algorithm for exergy analysis and optimization of high-temperature proton exchange membrane fuel cell HT-PEMFC. Izci et al. [40] proposed an improved MRFO algorithm using opposition-based learning for solving optimization problems. Feng et al. [41] proposed an improved MRFO algorithm for minimizing energy consumption in building shape optimization problem. Micev et al. [42] proposed an improved MRFO algorithm based on hybrid simulated annealing for optimal design of automatic voltage regulation controller. Razak et al. [43] proposed a spiral-based MRFO algorithm for optimizing PID control of flexible manipulator. Sheng et al. [44] proposed a balanced MRFO algorithm for PEMFCs system identification. Compared with other optimization algorithms, MRFO algorithm also has a good performance in solving shape optimization model of CCG-Ball curves. However, there still are opportunities to improve MRFO algorithm to overcome its disadvantages, such as premature convergence and easy to fall into local optimum. Another purpose of this paper is to propose an enhanced MRFO called WMQIMRFO algorithm combining the control parameter adjustment strategy, wavelet mutation strategy and quadratic interpolation strategy, which can be used to solve the established shape optimization model of CGG-Ball curves. The contributions of the works can be summarized as follows:

(a) Construct a class of new cubic generalized Ball (CG-Ball, for short) curves with multiple shape parameters, deduce the G1 and G2 geometric continuity conditions of the curves, and define the composite CG-Ball (CCG-Ball, for short) curves by using the deduced conditions. The CCG-Ball curves have more preferable shape adjustability than the traditional Ball curves, and their optimal shapes can be obtained by optimizing the shape control parameters.

(b) Propose an enhanced MRFO called WMQIMRFO algorithm with three improvement strategies, and verify the superiority of this algorithm by comparing other improved MRFO algorithms and other intelligent optimization algorithms on CEC2014 and CEC2017 test sets. The three improvement strategies work as follows:

Control parameter adjustment strategy, changing control parameter from linear to nonlinear increasing, can better balance global exploration ability and local exploitation ability of algorithm;

Morlet wavelet mutation strategy, by adding mutation operation, can enhance the ability of algorithm to jump out of local optimum;

Quadratic interpolation strategy, using the extreme point of quadratic curve to approximate the extreme point of objective function, can quickly strengthen the convergence of algorithm.

(c) Establish the shape optimization models of CCG-Ball curves, use the proposed WMQIMRFO algorithm to solve the models, and obtain the CCG-Ball curves with optimal shape.

The remainder of the paper is arranged as follows. In Section 2, a class of new cubic generalized Ball curves with multiple shape parameters are constructed, and the G1 and G2 continuity conditions of CCG-Ball curves are studied. In Section 3, an enhanced MRFO algorithm is proposed. To discuss the performance of proposed WMQIMRFO algorithm, it is compared with other improved MRFO algorithms and other optimization algorithms on CEC2014 test set, CEC2017 test set and four practical engineering problems to verify that this algorithm is highly competitive. In Section 4, according to requirement of the smoothest curves, the shape optimization models of CCG-Ball curves with whole G1 and G2 smooth blending are established respectively. The proposed enhanced MRFO algorithm is used to solve the models, and three numerical examples of shape optimization of CCG-Ball curves are given. The Section 5 is simple conclusion and future prospects of this paper.

Section snippets

Construction of CCG-Ball curves

Compared with traditional Ball curves, the shape adjustability of generalized Ball curves with single shape control parameter is improved, but it is still very limited. In this section, we define the CCG-Ball curves, which are composed of N CG-Ball curves with three shape parameters to be blended. At the same time, the proposed CG-Ball curves can degenerate into traditional cubic Ball curves, or cubic Ball curves with a shape parameter or two shape parameters. The one of advantages for proposed

Overview of proposed manta ray foraging optimization

Shape optimization models of the CCG-ball curves

The smoothness of CCG-Ball curves can be approximately measured by their value of bending energy. And smaller value of bending energy means the smoothness of curve is better. Assuming that the bending energy of the jth curve of CCG-Ball curves is expressed by Ej(Ωj), then there is Ej(Ωj)=01Pj(uuj1hj;Ωj)2dt,u[uj1,uj]where hj=ujuj1 and Ωj=(αj,βj,γj),j=1,2,,n represent shape parameters of the jth CG-Ball curve.

After further calculation, we have Pj(uuj1hj;Ωj)2=B0,4(uuj1hj)P0,j2+

Conclusion and future works

In this paper, a new class of CG-Ball curves with multiple parameters is constructed, CCG-Ball curves are defined, which are composed of N CG-Ball curves to be blended, and the G1 and G2 continuity conditions of CCG-Ball curves are studied. Compared with traditional Ball curves and Ball curves with single parameter or double parameters, CCG-Ball curves have the following two advantages: one is that they can flexibly adjust the shape of curves as a whole or locally by changing shape control

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are very grateful to the Reviewers for their insightful suggestions and comments, which helped us to improve the presentation and content of the paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 51875454 and 61772416).

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