Issue 
Mechanics & Industry
Volume 24, 2023



Article Number  38  
Number of page(s)  13  
DOI  https://doi.org/10.1051/meca/2023031  
Published online  07 November 2023 
Regular Article
Multiobjective shape optimization of developable Bézierlike surfaces using nondominated sorting genetic algorithm
^{1}
College of Mathematics and Computer Application, Shangluo University, Shangluo 726000, PR China
^{2}
Department of Applied Mathematics, Xi'an University of Technology, Xi'an 710054, PR China
^{3}
Electronic Information and Electrical Engineering College, Shangluo University, Shangluo 726000, PR China
^{*} email: hugang@xaut.edu.cn
Received:
3
July
2022
Accepted:
18
August
2023
The shape optimization design of the developable surface is an important research topic in CAD/CAM, and it is widely used in engineering manufacturing. In this paper, NSGAII (the elitist nondominated sorting genetic algorithm) is used to study the multiobjective shape optimization problem of generalized cubic developable Bézierlike surfaces (GCDBLS, for short) to promote the application of GCDBLS in industrial software and engineering design. Firstly, the shape optimization of developable surfaces is transformed into the shape optimization of dyadic curves based on the pointtoplane duality theory. Secondly, a multiobjective shape parameter optimization model is developed based on three surface optimality criteria (the shortest arc length, the smallest energy, and the smallest curvature change rate of the dual curve). Finally, the results of shape parameter optimization of GCDBLS obtained by NSGAII are compared with MSSA and MOGOA to verify the feasibility and superiority of NSGAII in solving multiobjective shape optimization problems for developable surfaces and the flexibility of GCDBLS in the construction of developable surfaces.
Key words: Developable Bézierlike surfaces / multiobjective shape optimization / the elitist nondominated sorting genetic algorithm / shape parameter
© J. Lu et al., Published by EDP Sciences 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Developable surface is a special type of ruled surface in ComputerAided Design (CAD) and Computer Aided Manufacturing (CAM). Because it can be expanded into a complete plane without stretching and tearing, and it has high application value in industrial manufacturing [1]. Compared with other curved surfaces, this is also its biggest advantage, so scholars have done a lot of research on developable surfaces. Specifically, there are the structural design of developable surfaces [2–16], interpolation fitting [17–19], mesh approximation [20,21], smooth stitching [22–24] and shape optimization [25].
There are two main methods for constructing developable surfaces: one is pointing geometric representation [2–7], and the other is line or surface geometric representation [8–16], which is also called dual representation. The point geometric representation method regards the developable surface as a ruled surface. It adds certain constraints to express it as a tensor product surface in Euclidean space. However, there are two defects in the developable surface constructed by the point geometric representation method. On the one hand, the calculation of the nonlinear characteristic equation is more complicated in the construction process, and on the other hand, the constructed developable surface cannot give an explicit expression. Therefore, in order to solve the above mentioned two drawbacks, the surface geometry representation is proposed for constructing developable surfaces. The surface geometry representation method is to regard the developable surface as a curve in the 3D projective space for the construction. The developable surface constructed by this method can avoid the complicated calculation of the first method and has an explicit expression.
The surface geometric representation approach to surface construction is widely used. In 1993, Bodduluri et al. [8] used cubic Bernstein and Bspline basis functions to construct developable surfaces for the first time and gave explicit expressions for the constructed developable surfaces. However, the shape of the developable surface constructed by [8–10] is determined by its control plane and cannot be easily modified. Therefore, in order to control the shape flexibly, Zhou et al. [11] used CBézier basis functions to construct a type of CBézier developable surface with adjustable shape, but the constructed developable surface only contains one shape parameter, which can only be adjusted the global shape of the curved surface, but the shape of the developable surface cannot be adjusted locally, and the shape adjustability is limited. In order to overcome this defect, Hu et al. [14–16] used Bézierlike basis functions and HBézier basis functions to construct two types of developable surfaces with three shape parameters, generalized Bézierlike developable surface and generalized HBézier developable surface. They construct developable surfaces that can be flexibly modified by changing the size of the shape parameters of the corresponding developable surface given the control surface coordinates. Based on this property, the shape optimization problem of developable surfaces has become an important research topic in CAD and CAM field. However, there is still little research work on the shape optimization of developable surfaces worldwide. The related work in Section 2 introduces the current status of some techniques for optimizing the shape of developable surfaces. However, the related research works only construct singleobjective optimization problems based on a certain surface feature (shortest arc length, minimum energy, and minimum curvature rate of change) [25]. Therefore, this study extends the first shape parameter optimization problem for developable surfaces to multiobjective optimization.
Generally speaking, most engineering design problems are multiobjective optimization problems. In the specific optimization process, the subobjectives are mostly restricted to each other, so there is no way to get a single solution so that multiple subobjectives can reach the optimum at the same time. The optimal solution set obtained by multiobjective optimization can guarantee that each solution in the solution set can make all subobjectives as optimal as possible. According to the pointtoplanes duality theory in 3D projective space, the shape optimization of a developable surface is transformed into the shape optimization of the corresponding dual curve in the dual space. Based on this conversion process, three optimization criteria (shortest arc length, minimum energy, and minimum curvature rate of change) related to the surface construction characteristics are introduced and subjected to multiobjective optimization [25]. Furthermore, in order to verify the effect of picking shape parameters optimized by different objective functions on the surface shape, three surfacerelated optimization objective functions are freely combined, resulting in three twoobjective optimization models and one threeobjective optimization model. However, it is complicated for traditional mathematical methods to solve multiobjective optimization problems. Therefore, this paper will use a multiobjective evolutionary algorithm to solve this problem and verify the effects of different objective functions on shape parameter optimization.
The earliest multiobjective evolutionary algorithm is the Vector Evaluation Genetic Algorithm (VEGA), which was proposed by David Schaffer et al. in [26]. Compared with the traditional weighting method to solve the multiobjective optimization problem, the results obtained by VEGA have higher population diversity, but because the algorithm has not adopted the Pareto optimal mechanism, it is easy to converge to the local optimal and has certain defects. By 1995, Debet al. [27] proposed a multiobjective genetic algorithm based on nondominated sorting (NSGA), which improves the diversity and convergence of the understanding set, but the algorithm is more troublesome because of its high computational complexity, a large amount of calculation, lack of elite strategies, and need to specify shared parameters manually. On the basis of NSGA, Deb et al. [28] introduced the elite strategy, the crowding degree comparison operator and the fast nondominated sorting method based on grading to overcome those shortcomings and obtained the fast nondominated sorting genetic algorithm with the elite strategy (NSGAII). NSGAII greatly improves the calculation speed and optimization ability of NSGA, and there is no need to specify shared parameters manually. After that, scholars successively proposed Multiobjective Particle Swarm Optimization Algorithm [29] (MOPSO), Evolutionary Multiobjective Optimization Algorithm Based on an Artificial Immune System [30] (MISA), Multiobjective Evolutionary Algorithm Based on Decomposition [31] (MOEA/D), Multiobjective Salp Swarm Algorithm [32] (MSSA), Multiobjective Grasshopper Optimization Algorithm [33] (MOGOA), etc. However, with its powerful optimization capabilities, NSGAII has been one of the most popular multiobjective evolutionary algorithms so far. This article is going to use NSGAII to solve the optimization problem of GCBLDS. In addition, the optimization results of GCBLDS obtained by NSGAII are experimentally compared with other multiobjective optimization algorithms in this study, which is used to verify the superiority of the NSGAII algorithm in solving the optimization of GCBLDS shape parameters.
The main scientific contributions of this study are as follows:
Based on the pointtoplanes duality theory, the shape optimization of developable surfaces is transformed into the shape optimization of dual curves in the corresponding dual space.
A multiobjective optimization model based on the shortest arc length, the smallest energy, and the smallest curvature change rate of the dual curve is established.
Using NSGAII to solve the multiobjective generalized cubic developable Bézierlike surfaces established with different combinations of objective functions, the optimal shape parameters satisfying the optimization criterion are obtained.
The structure of the article is as follows: In the first section, we give the display expressions of the generalized Bézierlike curve family and surface family and the optimization criteria of the generalized Bézierlike developable surface. And establish four multiobjective optimization models according to the optimization criteria. Section 2 introduces the basic knowledge related to the multiobjective optimization problem. Section 3 explains the ideology and algorithm steps of the NSGAII algorithm. Section 4 gives a set of control plane coordinates and then uses NSGAII to solve the optimization model established in the first section and record the experimental results. At last, the conclusion is outlined in Section 5.
2 Related work
Shape optimization of extensible surfaces constructs shapeoptimal extensible surfaces based on certain requirements or objectives. The implementation of this technique relies on the shape parameters that are available at the time of construction of the spreading surface. However, the conventional method of constructing an expandable surface lacks additional shape parameter. In addition, considering the complexity of the optimization process of spreading surfaces constructed based on shape parameters, many studies have introduced metaheuristic algorithms into the optimization of spreading surfaces and achieved effective optimization results. Some studies of metaheuristic algorithms for spreading surface optimization are presented below to describe the current state of the art.
Hu and Wu et al. invoked an adaptive cuckoo search algorithm with the shape optimization of generalized developable HBézier surfaces. They developed three shape optimization models for generalized developable HBézier surfaces using the duality between points and planes. Then, the adaptive cuckoo search algorithm is applied to solve the solution of the shape optimization model for generalized developable HBézier surfaces [25].
An improved marine predator algorithm is proposed and used for the shape optimization problem of the spreading spherical surface. Hu et al. establish a class of novel shapeadjustable generalized cubic Ball basis functions and construct shape optimization models for the envelope and spine curve spreading surfaces with the energy of the minimized Ball surface as the evaluation criterion. Then, they optimize the shape parameters of Ball surfaces by an improved marine predator algorithm based on a quasiopposition strategy and a differential evolutionary algorithm established [35].
Lu and Su et al. developed an enhanced Salp Swarm algorithm and used it to optimize the shape of the developable Bézierlike surfaces. First, the shape optimization problem of the generalized developable class of Bézierlike surfaces is transformed into the problem of minimizing the arc length, energy, and rate of change of curvature of the dyadic curves based on the duality principle of point and plane. Then the shape optimization model based on three developable surfaces with arc length, energy, and curvature rate of change is developed. In addition, the Morlet wavelet mutation strategy is introduced into the Salp Swarm algorithm to improve its optimization capability and is used to solve the shape optimization of developable surfaces [36].
Hu and Zhu et al. apply the multistrategy marine predator algorithm to the optimization of approximately developable surfaces. First, Lévy, Brownian motions, and optimal encounter rate strategies are introduced into the marine predator algorithm to improve the diversity of the algorithm's population. Then, a model for designing an approximately developable surface by interpolating two given spatial curves is developed with the evaluation criterion of the maximum developability degree of the spreading surface. Finally, the proposed algorithm is used to solve the optimized model to obtain the approximate developable surface with the maximum developability degree [37].
An enhanced JS algorithm for shape optimization of combinatorial cubic generalized ball surfaces is proposed. First, a new combinatorial cubic generalized ball surface is constructed, and the shape optimization of the Ball surface is investigated by an enhanced jellyfish search algorithm. In this study, the surface shape optimization models with G^{1} and G^{2} geometric continuity are developed separately by minimizing the Ball surface energy as the evaluation criterion [38].
A particle swarm optimization algorithm based on GHTBézier developable surface shape optimization was proposed by BiBi et al. First, the GHTBézier developable surface shape optimization model is established with the developability degree of the straight grain surface as the objective function and the shape parameters as the optimization variables. Then, the PSO technique is used to search for the optimal shape control parameters within the value range of shape parameters and then visualize the developable surface with high accuracy exploitability [39].
3 Preliminaries
3.1 Generalized cubic developable Bézierlike surfaces
Generalized cubic developable Bézierlike surfaces are constructed based on control planes with Bernstein basis functions and contain three shape parameters. By controlling the three shape parameters, the generalized cubic developable Bézierlike surface allows the position and shape of the developable surface to be freely adjusted, thus increasing the flexibility of developable surface design and improving the ability to solve problems in engineering appearance design. However, the introduction of multiple shape parameters also implies whether the selection of parameters is reasonable for different surface design cases. Therefore, based on the abovementioned advantages of the surfaces and the problems that arise, this study considers a multiobjective optimization of the control parameters of the generalized cubic developable Bézierlike surface to obtain the optimal parameters.
Given control points P ∈ R^{u}(u = 2, 3), P = [P_{0}, P_{1}, P_{2}, P_{3}], the matrix expression of the generalized cubic Bézierlike curves can be written as [14]
where
t and λ_{i} are the family parameter and shape control parameters of respectively M is called coefficient matrix.
According to the pointtoplanes dual theorem, combining equation (1), the generalized cubic Bézierlike plane family equation can be obtained as [14]
in equation (2), T and M have the same values as in equation (1), and Q_{i} = (a_{i}, b_{i}, c_{i}, d_{i}), i = 0, 1, 2, 3 represents the coordinates of the control plane. Equation (2) can also be expressed as:
F_{i,4}(t)(i = 0, 1, 2, 3) are the generalized cubic Bézierlike basis functions, its computational formula are as follows:
See equation (4) below
Let
Equation (2) can be written as:
(a) Generalized Bézierlike envelope developable surfaces
According to the definition of developable surface, the envelope surface of the singleparameter family of planeis a developable surface, so the generalized Bézierlike envelope developable surface can be expressed as
where
(b) Generalized Bézierlike spine curve developable surfaces
Generalized Bézierlike spine curve developable surfaces can be defined as:
where
See equation (10) below
v ∈ (− ∞ , + ∞), t ∈ [0, 1],the value of u(t) is the same as equation (8).
3.2 Optimization criteria
Based on the shape adjustability of the GCBLDS, under the premise of keeping the control plane of the developable surface unchanged, the shape parameters of the developable surface can be adjusted to make the developable surface have any shape. However, in many practical engineering problems, it is often necessary to make the developable surface meet certain requirements based on specific constraints. At this time, it is necessary to determine the corresponding criteria to determine the value of the shape parameter. For the GCBLDS designed by the duality theory in 3D projective space, if its control plane is regarded as the control vertex of the curve, the optimization of the developable surface can be converted to the corresponding dual curve optimization. The following three optimization criteria [25] for curves are given:
(a) The shortest arc length of dual curves
The arc length of the surface has an important effect on the construction of the surface, so the arc length is minimized to obtain a betterconstructed surface. The arc length of a generalized cubic Bézierlike curve L(t Ω), Ω = {λ : λ_{1}, λ_{2}, λ_{3}} can be calculated based on the principle of simplicity by the following formula:
(b) Minimum energy of dual curves
The energy value of the parametric surface is used as an approximate measure of the smoothness of the surface. The smaller the energy value, the smoother the curve. The constructed surface should be as smooth as possible for the practical application process. The energy function of the curve is:
(c) Dual curves hold smallest curvature variation
Since curvature variation has an important effect on the shape of the surface, a better surface is obtained by combining curvature variation with energy minimization. The curvature change rate of the curve can be measured by the following equation (13):
Equations (11)–(13) are three criteria for curve optimization.
3.3 Optimization models
All three criteria (shortest arc length, minimum energy, and minimum curvature variation) significantly affect the shape of generalized cubic developable Bézierlike surfaces but focus on different surface configurations. In order to verify the effect of selecting shape parameters optimized by different objective functions on the surface shape and to select a suitable optimization model, three surfacerelated optimization objective functions are freely combined to form three twoobjective optimization models and one threeobjective optimization model. In the following, we present these models.
Establish a twoobjective optimization model K1 based on the shortest arc length and minimum energy of the dual curve:
A twoobjective optimization model K2 is established by combining the minimum arc length of the dual curve and the minimum rate of curvature change criterion:
Establish a twoobjective optimization model K3 based on the minimum energy of the dual curve and the minimum rate of curvature change:
A threeobjective optimization model K4 is established based on the shortest arc length, the smallest energy, and the smallest curvature change rate of the dual curve:
The models K1–K4 established by combining different optimization criteria, which will be solved later.
4 Elitist nondominated sorting genetic algorithm (NSGAII)
4.1 Multiobjective optimization problems
For common multiobjective optimization problems, its objective function can be expressed as:
where M(λ) is called the objective function, and λ = (λ_{1}, λ_{2}, ..., λ_{D}) are the decision variable of M(λ), Ω = [L_{i}, U_{i}]⊂R_{D} is search space, L_{i} and U_{i} are the lower bound and upper bound of the ith dimension decision variable, D is the dimension of the decision variables, n is the number of subobjective functions.
Definition 1. Pareto domination
For any λ_{1}, λ_{2}∊Ω, λ_{1} is said to dominates λ_{2} if and only if ∀i ∈ {⥃ 1, 2, ⋯ n}makes m_{i}(λ_{1})≤m_{i}(λ_{2}), and∃ ⥃ j ∈ {⥃ 1, 2 ⋯ n, j ≠ i ⥃}makes m_{i}(λ_{1})<m_{i}(λ_{2}), which marked as λ_{1} ≺ λ_{2}, λ_{1} is called nondominated solution, and λ_{2} is called dominated solution.
Definition 2. Pareto optimal solution
λ^{*}∊Ω is called pareto optimal solution if and only if λ∊Ω makes λ ≺ λ^{*}.
Definition 3. Paretooptimality Set (PS)
The set consisting of all Pareto optimal solutions is called the Paretooptimality Set, denoted as PS:
Definition 4. Paretooptimality Front (PF)
Given a multiobjective optimization problem M(λ), PF is defined as:PF = {⥃ M(λ^{*}) ⥃  ⥃ λ^{*} ∈ PS},
PF is the image of PS in the target space.
4.2 Algorithmrelated concepts
NSGAII is obtained by adding some improvement strategies on the basis of NSGA. These strategies can not only improve the calculation progress of the algorithm but also speed up the convergence speed of the algorithm. Let's briefly introduce these improvement strategies.
(a) Tournament selection
In the traditional genetic algorithm (GA), the roulette selection method is usually used, while in NSGAII, the tournament selection method is used in order to select the outstanding individuals faster. The main ideas are: Randomly select m (m < N) individuals from a population P_{i} with a population size of N, select the individual with the best fitness value to enter the next generation population P_{i}_{+1} and repeat this step until the number of individuals in the new population P_{i}_{+1} reaches N to stop the selection.
(b) Fast nondominated sorting
For the ith individual in the population P, record the number of individuals dominated by the ith individual (n(i)) and the set of other individuals dominated by the ith individual (s(i)). The steps of the fast nondominant sorting algorithm are as follows:
Step 1. Let k = 1
Step 2. Store all individuals with n(i) = 0 in the population P into the set F_{k};
Step3. Find the set S(i) of individuals dominated by the ith individual in the set F_{k}, iterate through all j individuals in the set S(i), and perform the operation n(j) = n(j)1, if n(j) = 0, store the individual j into the set G;
Step4. Take F_{k} as the kth nondominated individual solution set, and assign the individual nondominated ranking rank in the current set F_{k}: i = r_{rank};
Step5. Take the set G as the current population P, k = k+ 1, and repeat Step2∼Step4 until all individuals in the population are graded.
(c) Crowdedness
Crowdedness implies the density of nondominated solutions in the neighborhood of the solution. All solutions in the set of nondominated solutions are sorted in ascending order according to the fitness value of each objective function. According to Figure 1, for candidate solution i, its crowding degree i_{d} is expressed as the average side length of the rectangle formed by neighboring candidate solutions i−1 and i+1. In addition, the crowdedness of two candidate solutions on the boundary is assumed to be infinite. A larger i_{d} means more sparsity and better diversity of solutions near candidate solution i. Crowdedness is calculated as follows:
Fig. 1 Crowdedness of solutions with two objectives. 
(d) Elite retention strategy
The elite retention strategy is mainly used in the algorithm to generate a new parent population. The main ideas are: first, merge the parent population P_{i} of the same current generation with the offspring population S_{i} to obtain a temporary population R_{i}, and the population size is 2N. Then, N is selected in population R_{i}. Two factors are considered for the selection of individuals: nondominant ranking level and crowding degree. The selection gives preference to the individuals with the top nondominant ranking level. However, when faced with two individuals with the same ranking and only one slot exists, the individual with a high degree of crowding is selected for the new parental population. Finally, the population is traversed, and N superior individuals are selected as the new parent population P_{i}_{+1} according to the pros and cons relationship between every two individuals.
4.3 Algorithm steps
The basic principles of NSGAII can be summarized as the following steps:
Step 1. Set the maximum number of iterations of the algorithm to maxGen and Gen = 1. Then randomly generate an initial population P_{i} with a population size of N, that is, the initial parent population;
Step 2. Perform nondominated sorting on the individuals in the population P_{i}, then select individuals through the tournament to perform crossover and mutation operations to obtain the offspring population S_{i}, and the population size is also N;
Step 3. Gen = Gen+1;
Step 4. For the current parent population P_{i} and offspring population S_{i}, use the elite retention strategy to select the new parent population P_{i}_{+1};
Step 5. Let P_{i} = P_{i}_{+1}, and perform a fast nondominant sorting of the individuals in P_{i}. Then select, cross, and mutate to obtain a new offspring population S_{i}_{+1};
Step 6. S_{i} = S_{i}_{+1} and repeat steps 3 through 5 until maxGen = Gen and the algorithm ends.
The crossover and mutation operations in the algorithm operation process are the same as the traditional genetic algorithm [34]. In order to express the specific process of the algorithm more clearly, a detailed explanation is given in the flowchart (Fig. 2).
Fig. 2 Flowchart of NSGAII algorithm. 
5 Multiobjective numeric experiments for the case of developable surfaces
For the four optimization models K1–K4 proposed in the Section 2, in this section, NSGAII, MSSA and MOGOA are used to solve the models, and the optimization results obtained are compared. Finally, this paper demonstrates the effectiveness of NSGAII in solving the GCBLDS problem by an example. In this experiment, the population size and the number of calculation iterations are both set to 200.
5.1 Evaluation index
Generation Distance (GD) is used to describe the degree of approximation between the obtained Pareto front and the real Pareto front. The calculation formula is
Inverted Generational Distance (IGD) is a comprehensive performance evaluation index. It is used to evaluate the convergence performance and distribution performance of the algorithm. The formula is expressed as
where P is the solution set on the obtained Pareto front, and λ is any individual in P; Similarly, P^{*} is the solution set on the real Pareto optimal front, λ^{*} is any individual in P^{*} ; P and P^{*} represent the size of solution set P and P^{*} respectively; d(λ, P^{*} ) is the Euclidean distance between individual λ and the closest individual in the solution set P^{*} , d(λ, P^{*} ) is the Euclidean distance between individual λ^{*} and the closest individual in the solution set P; The smaller the value of GD and IGD, the better the convergence and distribution of the algorithm. When the value is 0, it means that the obtained Pareto frontier is completely consistent with the real Pareto frontier, and the better the solution obtained.
Computing time T(s): Record the total time from the beginning to the end of the calculation, which can measure the calculation speed of the algorithm.
5.2 Numerical experimental analysis of generalized cubic Bezierlike surface
Given a generalized cubic Bezierlike curve, the control plane coordinates are as follows:
Table 1 records the optimized control parameters and objective function values of the three algorithms under four optimization models K1–K4. Considering that the final optimal solution set obtained from the multiobjective optimization is not unique, the optimization results provided by the three algorithms in the table are all optimal solution sets. The provided optimization control parameters and objective functions are used for the model building of generalized cubic Bessel curves to verify the advanced nature of the optimization results.
The results in the table show that for the optimization results of the K1 model, the optimal control parameters λ_{1} and λ_{2} for MSSA, NSGAII and MOGOA are 0.055877511 and −0.009776853. only the third control parameter is different. Also, for the results of the optimal objective function, MSSA, NSGAII and MOGOA are Pareto optimal solutions. For the K2 model, the optimal control parameters λ_{1} and λ_{2} are the same for MSSA and MOGOA, which are 0.055877511 and −0.009776853. the optimal control parameters of NSGAII are different from all other algorithms, and the differences are large. In addition, for the results of the optimized objective function, MSSA, NSGAII and MOGOA are all Pareto optimal solutions. For the K3 model, the optimal control parameters λ_{1} are the same for NSGAII and MOGOA, while the chemical control parameters λ_{2} are the same for all three algorithms. In addition, for the results of the optimized objective function, MSSA, NSGAII, and MOGOA are Pareto optimal solutions. For the K4 model, the optimal control parameters λ_{1} and λ_{2} of MSSA and MOGOA are the same, which are 0.19115202 and −0.00918123. In addition, the optimal control parameters λ_{3} of MSSA and NSGAII are the same. In addition, for the results of the optimized objective function, MSSA, NSGAII and MOGOA are Pareto optimal solutions.
The comparative results of the K1–K4 models MSSA, NSGAII, and MOGOA in GD, IGD, and T(s) are given in Table 2. Analysis of the data in Table 2 reveals that NSGAII obtains the minimum mean, minimum and maximum values of GD for all four optimization models. Also, NSGAII obtained the mean, minimum and maximum values of the smallest IGD under all four optimization models. Finally, NSGAII not only obtains the best optimization results but also its computational speed is much faster than the other two algorithms, which indicates the advantage of the method for solving optimization problems of this type of exploitable surfaces.
Figures 36 show the real Pareto front and the obtained Pareto front when the optimization models are K1–K4 respectively, the blue curve is real Pareto front and the red is the obtained Pareto front. The higher the agreement between the two curves, the better the calculation result. Combining the data in Table 2, it can be found that when the same number of iterations and population size are set, the optimization results of using NSGAII to solve the four optimization models are better than MSSA and MOGOA, and NSGAII is significantly faster in calculation.
Figures 710 show the generalized cubic Bezierlike developable surfaces obtained by optimizing the optimization models K1–K4 with the optimal shape parameters of a nondominated solution set obtained by the three algorithms MSSA, NSGAII and MOGOA. First, it can be noticed from Figure 7 that the transitions of the generalized cubic Bezierlike developable surfaces constructed by the NSGAII optimized algorithm are smoother. In addition, according to the experimental results in Figure 8, the generalized cubic Bessellike developable surface constructed by the NSGAII optimization algorithm is steeper on both sides and has a more pronounced slope. This phenomenon is mainly caused by the large difference between the parameters optimized by NSGAII in the K2 model and those of MSSA and MOGOA. For the results of model K3 in Figure 9, the generalized cubic Bezierlike developable surfaces constructed by the NSGAII optimized algorithm are gentler around the top of the slope. In addition, the overall tilt of the generalized cubic Bezierlike developable surface constructed by the algorithm optimized by MSSA is more pronounced. The main reason for this is that the optimization results of λ_{1} and λ_{3} are significantly different from the NSGAII and MOGOA algorithms. Finally, for the results of model K4 in Figure 10, the differences existing in the generalized cubic Bezierlike developable surfaces constructed by the optimization of the three algorithms are insignificant. It is mainly due to the fact that the three objective functions make the optimization process more rational and comprehensive. The optimization results of the three algorithms do not differ significantly in shape.
Optimization results of control parameters and objective functions for models K1–K4.
Comparison results of MSSA, NSGAII, and MOGOA for models K1–K4 in GD, IGD, and T(s).
Fig. 3 Pareto optimal front of model K1. 
Fig. 4 Pareto optimal front of model K2. 
Fig. 5 Pareto optimal front of model K3. 
Fig. 6 Pareto optimal front of model K4. 
Fig. 7 GCBLDS optimized based on model K1. 
Fig. 8 GCBLDS optimized based on model K2. 
Fig. 9 GCBLDS optimized based on model K3. 
Fig. 10 GCBLDS optimized based on model K4. 
6 Conclusions
In this paper, NSGAII is used to optimize the multiobjective shape of GCDBLS. First, according to the dual principle of point and plane in 3D projective space, three optimization criteria for shape optimization of developable surfaces are proposed, and a multiobjective optimization model is established by freely combining two or three of the three optimization criteria. Secondly, it introduces the corresponding basic knowledge in the multiobjective optimization problem and the thought theory of the NSGAII algorithm. Finally, use NSGAII to solve the proposed model to obtain the optimal shape parameters. Compared with the other two multiobjective optimization algorithms, it is found that NSGAII has obvious advantages for solving this type of multiobjective shape optimization problem. However, the use of multiple optimization criteria is experimentally found to increase the time complexity.
For future research directions, attempts can be made to improve NSGAII by combining other algorithms and introducing effective search operators so as to improve the multiobjective optimization performance of NSGAII. Finally, many complex singleobjective geometric optimization problems are also extended to multiobjective problems, such as shape optimization of the CQGSBall surfaces in [40].
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 51875454).
Conflicts of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Data availability statement
All data generated or analyzed during this study are included in this published article (and its supplementary information files).
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Cite this article as: J. Lu, X. Su, J. Zhong, G. Hu, Multiobjective shape optimization of developable Bézierlike surfaces using nondominated sorting genetic algorithm, Mechanics & Industry 24, 38 (2023)
All Tables
Optimization results of control parameters and objective functions for models K1–K4.
Comparison results of MSSA, NSGAII, and MOGOA for models K1–K4 in GD, IGD, and T(s).
All Figures
Fig. 1 Crowdedness of solutions with two objectives. 

In the text 
Fig. 2 Flowchart of NSGAII algorithm. 

In the text 
Fig. 3 Pareto optimal front of model K1. 

In the text 
Fig. 4 Pareto optimal front of model K2. 

In the text 
Fig. 5 Pareto optimal front of model K3. 

In the text 
Fig. 6 Pareto optimal front of model K4. 

In the text 
Fig. 7 GCBLDS optimized based on model K1. 

In the text 
Fig. 8 GCBLDS optimized based on model K2. 

In the text 
Fig. 9 GCBLDS optimized based on model K3. 

In the text 
Fig. 10 GCBLDS optimized based on model K4. 

In the text 
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