Shape optimization of automotive body frame using ...

Advances in Engineering Software 121 (2018) 235–249

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Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

Research paper

Shape optimization of automotive body frame using an improved genetic algorithm optimizer

T

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Qin Huan, Guo Yi, Liu Zijian , Liu Yu, Zhong Haolong State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: BIW frame Shape optimization Penalty-parameterless approach Improved genetic algorithm Meta-heuristic algorithms Integrated optimizer

At conceptual design stage, the cross-sectional shape design of automotive body-in-white (BIW) frame is a critical and intractable technique. This paper presents shape optimization using an improved genetic algorithm (GA) optimizer to promote the development of auto-body. The shape optimization problem is formulated as a mass minimization problem with static stiffness, dynamic eigenfrequency and manufacture constraints. Then the transfer stiffness matrix method (TSMM) proposed in our previous study is adopted for the exact static and dynamic analyses of BIW frame. Additionally, the scale vector method is introduced to remarkably reduce design variables. Especially, an integrated object-oriented GA optimizer, which employs penalty-parameterless approach to handle constraints, is developed to solve constrained single-objective and multi-objective optimization problems. The optimizer is benchmarked on 12 test functions and compared with a variety of current metaheuristic algorithms to demonstrate its validity and effectiveness. Lastly, the optimizer is applied to the solution of BIW shape optimization.

1. Introduction Automotive body-in-white (BIW) frame is the significant load-carrying component of automotive body, and it consists of semi-rigid connected thin-walled beams (TWBs) that are manufactured from multiple stamped metal sheets, which are assembled by spot welding and bolting [1], as shown in Figs. 1 and 2. The complex shapes and thicknesses of TWBs determine the cross-sectional properties, e.g., areas, moments of inertia and torsional constants, which profoundly affect the performances of automotive body, such as static stiffness, NVH (Noise, vibration and harshness), and crashworthiness. Thus, determining the optimal cross-sectional shapes and thicknesses of TWBs is one of the most important issues at conceptual design stage. However, to date, there is no available commercial software for cross-sectional shape design. Consequently, design engineers mostly rely on empirical and intuitive trial-and-error approach, which is laborious, time-consuming and unreliable, to design cross-sectional shape. At conceptual design phase, initial cross-sectional shapes are extracted from the benchmarking auto-body, selected from the cross-section database, or drawn by engineers. Engineers endeavor to design optimal cross-sectional shapes and thicknesses aiming at obtaining a lightweight BIW frame without violating the required performance targets and fabrication constraints. That is, this is typically a shape optimization problem.

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Vinot et al. [2] presented a shape optimization methodology for the design of thin-walled beam-like structures considering dynamic behavior. Apostol [3,4] proposed a general optimization method for arbitrary cross-section of a truss or beam. Nevertheless, studies in Refs. [2–4] paid little attention on handling manufacture and assembly constraints, which is one of the difficulties in BIW frame cross-sectional shape optimization. Yoshimura et al. [5] took two manufacture and assembly constraints into consideration. These constraints were introduced by Zuo [6–8]. The second difficulty arises from the large amount of design variables, especially for shape optimization problems of multiple cross-sections. Yim et al. [9] defined scale vectors as design variables rather than control point coordinates. The scale vector method notably reduced the amount of design variables, and enabled the cross-sectional optimization of multiple TWBs. The third difficulty is the solution of shape optimization. With fabrication constraints considered, BIW frame shape optimization is a constrained nonlinear optimization problem. On account of that these fabrication constraints cannot be explicitly expressed by formulas, thus gradient-based variables, e.g., the sensitivity of stamping constraint with respect to design variables cannot be calculated. Consequently, genetic algorithm (GA) is extensively used in the solution of this problem in conjunction with penalty method. However, how to set the penalty coefficients for penalty functions is a tough and inefficient technique. Deb et al. [10,11]

Corresponding author. E-mail address: [email protected] (Z. Liu).

https://doi.org/10.1016/j.advengsoft.2018.03.015 Received 15 November 2017; Received in revised form 16 March 2018; Accepted 25 March 2018 Available online 06 April 2018 0965-9978/ © 2018 Elsevier Ltd. All rights reserved.


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Fig. 1. BIW frame and its TWBs’ cross-sections. Fig. 3. A typical cross-sectional shape in BIW frame.

2. Formulation of cross-sectional properties and scale vector method 2.1. Formulation of cross-sectional properties A typical cross-section example of the TWB, e.g., the rocker, is illustrated in Fig. 3. Three stamped metal sheet parts are spot-welded together to form the cross-section. Every sheet comprises several segments, each of which can be regarded as a rectangle with certain length and thickness. Thus the cross-sectional area is namely n

A=

∑ l i ti

(1)

i=1

where n is the total number of segments of a specified cross-section; li is the length of the ith segment, and ti is thickness. The cross-sectional centroid coordinate is given by

Fig. 2. Fabrication of a typical cross-section for TWBs.

cy =

implemented different penalty-parameterless constraint-handling approaches for single-objective and multi-objective optimization problems. Moreover, studies in Refs. [2–9] are based on finite element method (FEM), which is an approximate method. This paper adopts the transfer stiffness matrix method (TSMM) proposed in our previous study [12], on account of that (a) firstly, at conceptual design stage, BIW frame can be simplified as a semi-rigid space frame structure, to which TSMM can be directly applied with less degrees of freedom; (b) furthermore, TSMM is an exact method both for static and dynamic analyses of framed structures, and it is more accurate than traditional FEM. Shape optimization of multiple cross-sections is implemented for lightweight design of BIW frame at conceptual design phase, with consideration of static bending stiffness, torsional stiffness, first-order free vibration eigenfrequency, and three manufacture constraints. Especially, an improved GA optimizer (IGA optimizer for short hereinafter), which employs penalty-parameterless approach to handle constraints, is developed to solve this constrained nonlinear optimization problem. The validity and effectiveness of the optimizer are demonstrated by benchmarking on multiple classical numerical examples, and then the optimizer is applied to BIW frame shape optimization. The remainder of this paper is organized as follows. In Section 2, the formulation of cross-sectional properties and scale vector method are reviewed and summarized. In Section 3, the shape optimization of multiple cross-sections is formulated. In Section 4, the development of the IGA optimizer is introduced. In Section 5, benchmark results and discussion of IGA optimizer are given. Afterwards, the shape optimization is solved using IGA optimizer in Section 6. Finally, conclusions are made in Section 7.

n

1 A

∑ li ti yci and cz = i=1

1 A

n

∑ li ti z ci i=1

(2)

in which (yci, zci) is the coordinate of the ith segment centroid. The inertia moments Iy and Iz, and second area moment Iyz with regard to the centroid are calculated by n

Iy =

l t3

i i ∑ ⎛ 12 ⎜

i=1

⎝

n

Iz =

l t3

i i ∑ ⎛ 12 ⎜

i=1 n

Iyz =

cos2 θi +

sin2 θi +

⎝

li3 ti sin2 θi + li ti yci2 ⎟⎞ 12 ⎠

(3)

li3 ti cos2 θi + li ti z ci2 ⎞⎟ 12 ⎠

(4)

li3 ti − li ti3 sin 2θi + li ti yci z ci ⎟⎞ 24 ⎠ ⎝

∑⎛ ⎜

i=1

(5)

in which θi is the angle between the ith segment and the positive y axis. From Iy, Iz and Iyz, the principal inertia moments are derived as

Imax =

1 (Iy + Iz ) + 2

1 2 (Iy − Iz )2 + I yz 2

(6)

Imin =

1 (Iy + Iz ) − 2

1 2 (Iy − Iz )2 + I yz 2

(7)

The counterclockwise angle φ of principle inertia direction about the reference y axes is expressed as

φ=

−2Iyz ⎞ 1 tan−1 ⎜⎛ ⎟ 2 I ⎝ z − Iy ⎠

The general formula for the torsional constant is written as 236

(8)


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J=

system is

∑ Jopenj + pJsingle + qJdouble (9)

j=1

′ ⎤ ⎡ cos θ sin θ ⎤ ⎡ y12 ⎤ ⎡ y12 = ⎢ z ′ ⎥ ⎢− sin θ cos θ ⎥ ⎢ z12 ⎥ 12 ⎦ ⎣ ⎦ini ⎣ ⎦ ⎣

where Jopenj, Jsingle and Jdouble are torsional constants for the jth open section, single-cell closed section and double-cell closed section, respectively; no is the total number of open sections; p and q can be 0 or 1 for a specified cross-section of BIW frame. Fig. 3 is an example of double-cell closed section with 3 × 2 = 6 open sections, in which the upper cell and lower cell are filled with different colors. Jopenj is calculated by oj

Jopenj =

∑ i=1

′⎤ cos θ − sin θ ⎤ ⎡ SV 0 ⎤ ⎡ y12 ⎡ y12 ⎤ =⎡ ⎥ ⎢ z12 ⎥ ⎢ sin θ cos θ ⎥ ⎢ 0 1 ⎥ ⎢ z12 ′ ⎣ ⎦new ⎣ ⎦⎣ ⎦⎣ ⎦

li ti3 3

In this TWBs (all thickness) amount of

(10)

where oj is the total number of segments of the jth open section. Jsingle is given by

Jsingle =

4Ac2 c ∑i = 1 li / ti

4[Au2 (Ll / tl + Lr / tr ) + 2Au Al Lr / tr + Al2 (Lu / tu + Lr / tr )] (Lu Ll )/(tu tl ) + (Ll Lr )/(tl tr ) + (Lr Lu )/(tr tu )

(14)

paper, scale vectors, rotational angles and thicknesses of the stamped sheets of a specified TWB share an identical are defined as design variables to remarkably reduce the design variables.

3. Formulation of shape optimization (11)

Lightweight design of auto-body is of great significance for pursuing better energy efficiency when performance targets satisfy design requirements. Structural optimization is one of the main approaches for auto-body lightweight design; it contains three branches, i.e., topology optimization, shape optimization and sizing optimization. In this shape optimization problem, exact bending stiffness, torsional stiffness, firstorder eigenfrequency, and also three manufacture and assembly constraints are taken into account. In this section, shape optimization of multiple cross-sections is formulated. Define the design variable vector x as,

where c is the total number of segments, and Ac is the area enclosed by the perimeter. Jdouble is written as

Jdouble =

(13)

Then the new coordinate of no. 12 control point in the yoz coordinate system in terms of coordinate transformation about the scale coefficient SV is expressed as

(12)

where Au is the area of the upper closed polygon filled with gray, as shown in Fig. 3; Al is the area of the lower closed polygon filled with white, as shown in Fig. 3; Lu, Lr and Ll are the lengths of the upper sheet, reinforcement and lower sheet, respectively; tu, tr and tl are the thicknesses of the relevant sheets, respectively.

x = [θ , SV , t]

(15)

in which,

2.2. Scale vector method

⎧ θ = [θ1, θ2, ...,θnb] SV = [SV1, SV2, ...,SVnb] ⎨ ⎩ t = [t1, t2, ...,tnb]

As shown in Fig. 3, control points are classified as two types, i.e., the fixed point and movable point. Fixed points should be remained unchanged during the optimization phase to fulfill geometric layout or manufacture requirements. While movable points can be successively changed to obtain a better cross-sectional shape. For the multiple crosssectional shape optimization problem, the number of design variables may be so large that the optimization problem would be difficult to solve. For example, suppose that there are 10 TWBs to be optimized, and each TWB cross-section has 20 movable control points on average. There will be 20 × 2 × 10 = 400 design variables even without consideration of thicknesses. Additionally, each coordinate should be designated minimum and maximum bounds. The scale vector method [9] is introduced to facilitate the multiple cross-sectional shape design. As shown in Fig. 4, suppose that the initial coordinate of a movable control point, e.g., no. 12 point, is (y12,z12)ini. Firstly, rotate the yoz coordinate system with counter-clockwise angle θ to get the y'oz' coordinate system. The coordinate of no. 12 control point in the y'oz' coordinate

(16)

where nb denotes the number of TWBs to be optimized. The exact bending stiffness, torsional stiffness and first-order eigenfrequency of free vibration, which are the crucial properties of BIW frame, can be determined with the use of TSMM. The relevant test conventions for static analysis are shown in Fig. 5, where constraints 1, 2 and 3 denote the displacements in the global X, Y and Z axes are constrained, respectively; the initial and deformed wireframes of the BIW are drawn by black and red curves, respectively. Meanwhile, three manufacture and assembly constraints are considered to assure the fabrication demands of metal sheets. The first two fabricating constraints were first proposed by Yoshimura [5]. Because the metal sheet is formed using stamping, the draft angle should be equal to or greater than 90○, as shown in Fig. 6(a). Also, the intersection of metal sheets is impossible for assembly process, as shown in Fig. 6(b). In our practice, we encounter with a special case that the flanging may be enclosed by a cell, and point ‘A’ becomes an interior point of the cell, as shown in Fig. 6(c). This case also should be avoided for that it has no practicability. We name point ‘A’ an invalid interior point. Therefore, the shape optimization problem can be formulated as

m = f (x) ⎧ min x ⎪ ⎪ ⎧ c1 (x) = δ − δallowable ≤ 0 ⎪ ⎪ c2 (x) = ϕ − ϕallowable ≤ 0 ⎪ ⎪ ⎪ c (x) = freqallowable − freq ≤ 0 ⎪ ⎪ 3 ⎨ s. t . c (x) = n = 0 aa ⎪ ⎨ 4 ⎪ ⎪ c5 (x) = nip = 0 ⎪ ⎪ c (x) = n = 0 ii ⎪ ⎪ 6 ⎪ LB ≤ x ≤ UB ⎪ ⎩ ⎩ Fig. 4. Coordinate transformation about the scale vector.

(17)

where m is the BIW mass; ci (i = 1–6) are the constraints; δ is the 237


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Fig. 5. Static stiffness test conventions.

maximum vertical deflection in Fig. 5(a), and δallowable is its allowable limit value; ϕ is the twist angle in Fig. 5(b), and ϕallowable is its allowable limit value; freq is the first-order natural frequency, and freqallowable is its allowable limit value; naa, nip, nii are the numbers of acute draft angles, intersection points and invalid interior points, respectively, as shown in Fig. 6; vectors LB and UB are the lower and upper bounds of x, respectively.

(3) It can solve constrained optimization problems using penaltyparameterless approach. (4) It adopts elitist strategy to achieve better convergence and prevent the loss of good solutions. (5) The computational complexity should not be too high. (6) Diversity-preservation mechanism should be used to reduce the possibility of getting trapped in local convergence in single-objective optimization and achieve better spread of solutions in multiobjective optimization.

4. The development of IGA optimizer

Motivated by these requirements, we develop the binary IGA optimizer. The multi-objective optimization function is based on NSGA-II and not involved in shape optimization in this paper, so only the singleobjective function of this optimizer is introduced in detail. Different from Ref. [10], we propose a strategy enlightened by NSGA-II [11] for handling constraints in single-objective optimization, and conveniently integrate single-objective and multi-objective optimization algorithms into the IGA optimizer. The basic procedure for this strategy is identical to that in NSGA-II. However, several concepts are redefined to suit single-objective optimization. Firstly, some primary concepts are given to make this paper self-contained. Thereafter, we address the main process of constrained single-objective optimization in IGA optimizer. Eventually, object-oriented development of this optimizer is introduced.

GA is a typical evolutionary algorithm, which is a population-based approach and capable of solving linear and nonlinear optimization problems without gradient information [13], thus it is extensively used to solve the constrained nonlinear optimization problem in Eq. (17) in conjunction with penalty method. To be readily embedded in our “Automotive Body Conceptual Design” toolbox (“ABCD” toolbox for short hereinafter) [12], a GA optimizer coded in MATLAB is preferable. A binary GA toolbox developed by the University of Sheffield (“USGA” toolbox for short hereinafter) [14] is widely used in engineering optimization. However, for constrained optimization problems, it should be used in conjunction with penalty method whose penalty coefficients usually are intractable to set and vary from problem to problem. The initiator of NSGA-II (nondominated sorting genetic algorithm II), i.e., Deb, released the relevant GA codes in C/C++ [15] but not in MATLAB language. In addition, optimizers for single-objective and multiobjective are separated because different strategies are taken for constraint handling and fitness function definition [10,11]. Seshadri developed a NSGA-II optimizer coded in real for multi-objective optimization and provided open sources [16]. Nevertheless, constrained optimization problems, which are common in engineering design practices, are not addressed. Thus, a GA optimizer having these following merits is desired.

4.1. Some primary concepts 4.1.1. Mathematical form The mathematical form of the optimization problem should be formulated as a minimum optimization problem, namely

⎧ x = [x1, x2, …, x n] ⎪ minf (x) ⎪ x ⎨ c (x) ≤ 0i=1, 2, …, m ⎪ s. t . ⎧ i ⎪ ⎨ ⎩ LB ≤ x ≤ UB ⎩

(1) It can solve optimization problems with continuous and/or discrete design variables. (2) It can solve optimization problems with single-objective and multiple objectives.

(18)

where vector x is the design variable; n and m are the numbers of

Fig. 6. Manufacture and assembly constraints. 238


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dimensions of design variables and constraints, respectively; vectors LB and UB are the lower and upper bounds of x, respectively. 4.1.2. Domination definition for feasible solutions Suppose that xi (xi = [x1i , x 2i , …, x ni]) is the ith solution (i.e., the ith individual) of a specified population. For arbitrary two feasible solutions xp and xq, xp is defined to dominate xq and symbolically represented as xp≺xq, if the objective function value of the former solution is smaller, i.e., f(xp) < f(xq). 4.1.3. Constraint violation for infeasible solutions An infeasible solution xp is defined to have a smaller constraint violation than another infeasible solution xq, if any following condition is true. (1) The number of violated constraints of xp is smaller than that of xq. (2) The number of violated constraints of xp is equal to that of xq, and the summation of violated constraints of xp is smaller than that of x q.

Fig. 7. Dominated front sets for feasible solutions projected in xj-f(x) plane.

4.1.4. Constrained-domination and constrained-dominated sorting An arbitrary solution xp is defined to constrained-dominate another arbitrary solution xq and symbolically represented as xp≺cxq, if any of the following conditions is true. (1) xp is feasible and xq is not. (2) xp and xq are both feasible and xp dominates xq. (3) xp and xq are both infeasible and xp has a smaller constraint violation. The corresponding pseudocode and MATLAB code of constraineddominated sorting are shown in Appendix A. Firstly, the first dominated front set F1 is solved. For each solution, Sp and np are calculated, where Sp is a set of numbers of the solutions that the solution xp constraineddominates, and the domination count np denotes the total number of solutions which constrained-dominate the solution xp. If np is equal to 0, then p belongs to the first dominated front set F1, and the rank (fitness) of the pth solution xp is designated as 1. Then, other dominated front sets Fi (i ≥ 2) are solved. For each solution xp whose number p in the first dominated front set F1, each solution xq whose number q in the relevant set Sp is visited. If for any solution xq, its domination count nq vanishes after subtracting 1, then this solution number q will be placed in a separate set Q, q belongs to the second dominated front set F2, and the rank of qth solution xq is designated as 2. This process continues until all dominated front sets are determined.

Fig. 8. The composition of dominated front set Fi.

crowding distances for infeasible solutions can be calculated in the identical way. The corresponding pseudocode and MATLAB code of constrained-dominated sorting are given in Appendix B.

4.1.5. Crowding distance and crowding distance sorting In NSGA-II, crowding distance is defined in objective function space to obtain better spread of solutions in multi-objective optimization; while in the single-objective optimization of IGA, we define crowding distance in design parameter space to decrease the possibility of getting trapped in local convergence in single-objective optimization. Fig. 7 depicts the solutions projected in the xj-f(x) plane, where xj is the jth design parameter and f(x) is the objective function. Suppose that there are NF dominated front sets, and a specified dominated front set Fi comprises v solution numbers, as shown in Fig. 8, where xk is the kth solution of set Fi, and Xj is the set which consists of the jth design parameters of the solutions. Firstly, the solutions in Fi are sorted in the jth design parameter dimension in ascending order of magnitude according to design parameter value. The crowding distances of the solutions whose design parameter values are smallest and largest are assigned as infinite. All other intermediate solutions are assigned a distance value equal to the absolute normalized difference in the design parameter values of two adjacent solutions. This calculation is continued with other design parameters to obtain the overall crowding distance value. Note that Fig. 7 only shows the feasible solutions,

4.2. Main process The flowchart for the main process of IGA optimizer is illustrated in Fig. 9. Initially, a binary parent population P0 including NInd individuals is created randomly. Each individual is concatenated by a continuous chromosome and a discrete chromosome to take account of continuous and discrete optimization problems. Next, the binary-decimal conversion of P0 is evaluated to obtain its objective function values, constraints, ranks, dominated front sets, crowding distances as introduced in Section 4.1. Under this evaluation, genetic manipulations, i.e., the tournament selection, single-point crossover, and binary mutation operators, are applied to P0 to generate an offspring population Q0 of size NInd. Populations P0 and Q0 are combined to get a population R0 (R0 = P0∪Q0) of size 2NInd, which is evaluated to introduce elitism. Then, R0 is used to generate the new population P1 of size NInd, and P1 is assigned as the new iteration population P0. Lastly, P0 is evaluated and continued to be iterated until the generation number NGen exceeds the maximum generation number MaxGen. The procedure for generating the new iteration population P0 is 239


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Fig. 9. The flowchart for the main process of IGA optimizer.

accommodated. Then, crowding distance sorting is carried out for the set Fi in descending order to choose the best solutions needed to fill all population slots. It should be noted that if the size of F1 is equal to or greater than NInd, crowding distance sorting will not be implemented. Now, it can be observed that IGA optimizer can be viewed as a generalized edition of NSGA-II. Therefore the overall computational complexity of IGA optimizer is O(M × NInd2), where M is the number of objective function(s).

4.3. Object-oriented development of IGA optimizer Object-oriented programming (OOP) [17] has advantages of encapsulation, polymorphism, inheritance, and code reusability, so it is introduced to develop IGA optimizer. The Unified Modeling Language (UML) diagram for IGA optimizer is shown in Fig. 11, where the Function class can be rewritten to solve different optimization problems. Then IGA optimizer is integrated into the “ABCD” toolbox to promote the development of automotive body at conceptual design stage.

Fig. 10. The procedure for generating the new iteration population P0.

shown in Fig. 10. Firstly, constrained-dominated sorting is implemented for the combined population R0. Suppose that there are u dominated front sets. Solutions whose numbers belonging to the top several front sets are putted into the new population P1. Suppose that the set Fi is the last dominated front set beyond which no other set can be 240


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composite functions are similar to the real-world search spaces in the aspects of that they have a mass of local optima, and the landscapes in different regions are various and complicated. An algorithm is required to have an appropriate balance ability between exploration and exploitation to approximate the global optimum. Consequently, composite functions can benchmark both the exploration and exploitation abilities of an algorithm. IGA and comparison algorithms are run 30 times with identical population size (or search agent number, particle number) and iteration number, i.e., 30 and 1000 respectively. Nine well-known current metaheuristic algorithms are chosen for comparison, i.e., the Salp Swarm Algorithm (SSA) [20], Grasshopper Optimization Algorithm (GOA) [21], Whale Optimization Algorithm (WOA) [22], Grey Wolf Optimizer (GWO) [23], Particle Swarm Optimization (PSO) [24], Gravitational Search Algorithm (GSA) [25], Bat-inspired Algorithm (BA) [26], MultiVerse Optimizer (MVO) [27], and Harmony Search (HS) [28]. Also, USGA [14] is compared as a conventional GA optimizer to benchmark the performance of elitist strategy in IGA. The statistical results (average, standard deviation and rank) are reported in Table 1, where the rank denotes the order of average sorted from the smallest to biggest. As can be seen in the tables, IGA performs well in composite benchmark functions. It may be concluded that IGA has good balance ability between exploitation and exploration. Moreover, since the difficulties of composite benchmark functions resemble that of the actual search spaces, this may indicate that IGA could perform well in the complicated engineering design problems. On the other side, IGA outperforms USGA in most of the benchmark functions. We may conclude that the GA optimizer adopting elitist strategy achieves better convergence and has better exploitation ability.

Fig. 11. The UML diagram for IGA optimizer.

5. Benchmark results and discussion of IGA optimizer In this section, 12 benchmarking tests (6 unconstrained and 6 constrained optimization problems, and all of them are minimization problems) are carried out to validate the efficiency of IGA optimizer and draw a reasonable conclusion of its performance. The tests are implemented on MATLAB R2015a, running under Windows 7 Ultimate. And the MATLAB version is R2015a. Different crossover probability and mutation probability are set for different problems, thus these parameters are not given in this paper. A variety of metaheuristic algorithms are utilized to evaluate and compare the experimental results; and the corresponding parameters of the algorithms are set according to the reported literature.

5.2. Benchmark of constrained functions Six typical constrained engineering optimum design problems are utilized to benchmark the constrained optimization ability of IGA. They are respectively the cantilever beam design problem with 1 constraint [29], three-bar truss design problem with 3 constraints [29], tension/ compression spring design problem with 4 constraints [23], pressure vessel design problem with 4 constraints [23], welded beam design problem with 7 constraints [23], and speed reducer design problem with 11 constraints [30]. The descriptions and mathematical

5.1. Benchmark of unconstrained functions Six composite functions (F1–6) chosen from the CEC 2005 technical report [18] are used to benchmark the unconstrained optimization ability of IGA. Composite functions are the state-of-the-art challenging benchmark functions, which are the shifted, rotated, expanded, and combined version of classical functions [19]. As shown in Fig. 12,

Fig. 12. 2-D version of composite benchmark functions. 241


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Table 1 Statistical results of composite benchmark functions. Algorithm

Statistic

F1

F2

F3

F4

F5

F6

IGA

Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank Ave Std. Rank Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank Ave. Std. Rank

4.6667E+01 7.7607E+01 4 3.6667E+01 5.5560E+01 3 1.2000E+02 1.2148E+02 9 1.2092E+02 1.2964E+02 10 8.2187E+01 1.1517E+02 5 1.1333E+02 1.0742E+02 8 3.3333E+00 1.8257E+01 1 1.3550E+02 1.0742E+02 11 8.6676E+01 8.1719E+01 6 9.3251E+01 4.4602E+01 7 2.0017E+01 6.1033E+01 2





5.4426E+02 6.8397E+01 1 6.0864E+02 1.8421E+01 2 8.4643E+02 1.3786E+01 7 6.7317E+02 1.9652E+02 3 8.6055E+02 1.2324E+02 9 7.5169E+02 1.9432E+02 4 8.1497E+02 1.1347E+02 5 8.7604E+02 1.9432E+02 10 8.1565E+02 1.6172E+01 6 8.4686E+02 1.2769E+01 8 9.0004E+02 1.5626E−01 11

SSA [20]

GOA [21]

WOA [22]

GWO [23]

PSO [24]

GSA [25]

BA [26]

MVO [27]

HS [28]

USGA [14]

6. Shape optimization of multiple cross-sections using IGA optimizer

formulations of the test functions are available in the relevant references. For all the problems, IGA is executed 30 times to find the best solution and report the statistical results. While the population size and iteration number are set according to the complexity of each problem, e.g., the number of design variables and number of constraints. Tables 2–12 list the best solutions found by a variety of algorithms and the relevant statistical results, where the abbreviations of the algorithms can be reviewed in the relevant references, and “SD”, “NFEs”, and “NA” stand for the standard deviation, number of function evaluations and not available, respectively. It can be concluded from Tables 2–12 that compared with other algorithms, the solutions found by IGA are quite close to the best solutions; but the function evaluations of IGA are larger than the state-of-the-art metaheuristic algorithms, such as ALO [29], WCA [34] and TLBO [43]; besides, IGA offers better solutions with fewer function evaluations compared with the considered GAs, such as GA2 [40] and GA3 [44]. Taken together, it could be summarized that the constraint-handling strategy proposed in IGA is efficient.

The 10 TWBs of the BIW side frame, as shown in Fig. 1, are taken for shape optimization. For the design variables, i.e., θ, SV, and t, the lower bounds are set as 0 radian, 0.5, and 0.8 mm, respectively; and the upper bounds are π/2 radian, 1.5, and 2.0 mm, respectively. According to the benchmarking experiment of the car body [12], the allowable limit values for constraints δ, ϕ and freq are defined to be 0.8250 mm, 0.1910° and 26.6000 Hz, respectively. The initial thicknesses for inner and outer sheets are 1.50 mm, and for reinforcement sheets are 1.00 mm. In the optimization process, control points on the outer sheet and flanging are defined as fixed points. The well-regarded GWO [23] optimizer collocating with the penalty method and WCA [34] are utilized to validate the optimization results of IGA optimizer. For all the optimizers, identical population size and iteration number are set, i.e., 100 and 50, respectively. All the optimizers are run 10 times to find the best solution. The best solutions, performance targets, and convergence curves of the mass function obtained by the three optimizers are presented in Tables 13 and 14, and Fig. 13, respectively; the initial and optimized cross-sectional shapes using IGA optimizer are depicted in Fig. 14. It can be concluded from Table 14 and Fig. 13 that (a) all the three optimizers acquire reasonable lightweight solutions; (b) in terms of the best solution, IGA outperforms GWO [23] and WCA [34] in this constrained engineering design problem with identical function evaluations. In addition, GWO optimizer should be run several times to tune the coefficients in the penalty method, while IGA and WCA handle constraints automatically in a single run. It should be noted that IGA, GWO and WCA take, respectively, about 65 min, 50 min, and 50 min on average to solve the optima. That is, GWO and WCA run faster than IGA. This should be attributed to the combination of the parent population and offspring population in Section 4.2 that increases the

Table 2 Comparison results for the cantilever design problem. Value

IGA

ALO [29]

SOS [31]

CS [32]

MMA [33]

GCA_I [33]

x1 x2 x3 x4 x5 f(x) NFEs

6.033373 5.314924 4.48952 3.492740 2.14335 1.33997 30,000

6.01812 5.31142 4.48836 3.49751 2.158329 1.33995 14,000

6.01878 5.30344 4.49587 3.49896 2.15564 1.33996 15,000

6.0089 5.3049 4.5023 3.5077 2.1504 1.33999 25,000

6.0100 5.3000 4.4900 3.4900 2.1500 1.3400 NA

6.0100 5.30400 4.4900 3.4980 2.1500 1.3400 NA

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Table 3 Comparison of the best solution obtained from different algorithms for the three-bar truss design problem. Value

IGA

WCA [34]

ALO [29]

DEDS [35]

MBA [36]

PSO-DE [37]

x1 x2 c1(x) c2(x) c3(x) f(x)

0.788729 0.408096 −2.32E−12 −1.464274 −0.535726 263.895845

0.788651 0.408316 0.000000 −1.464024 −0.535975 263.895843

0.788663 0.408283 NA NA NA 263.895843

0.78867513 0.40824828 0.000000 −1.464101 −0.535898 263.895843

0.7885650 0.4085597 −5.29E−11 −1.4637475 −0.5362524 263.895852

0.7886751 0.4082482 −5.29E−11 −1.463747 −0.536252 263.895843

Table 4 Comparison of statistical results obtained from various algorithms for the three-bar truss design problem. Algorithm

Worst

Mean

Best

SD

NFEs

IGA WCA [34] DEDS [35] MBA [36] PSO-DE [37] SC [38] HEAA [39]

263.95189 263.896201 263.895849 263.915983 263.895843 263.903356 263.896099

263.901000 263.895903 263.895843 263.897996 263.895843 263.969756 263.895865

263.895845 263.895843 263.895843 263.895852 263.895843 263.895846 263.895843

1.05E−02 8.71E−05 9.7E−07 3.93E−03 4.5E−10 1.3E−02 4.9E−05

30,000 5250 15,000 13,280 17,600 17,610 15,000

Table 5 Comparison of the best solution obtained from different algorithms for the tension/compression spring design problem. Value

IGA

WCA [34]

CPSO [49]

GA3 [44]

HSA [41]

DE [42]

x1 (d) x2 (D) x3 (N) c1(x) c2(x) c3(x) c4(x) f(x)

0.051760 0.358421 11.191034 −1.03E−04 −5.09E−06 −4.056588 −0.726546 0.012667

0.051680 0.356522 11.300410 −1.65E−13 −7.9E−14 −4.053399 −0.727864 0.012665

0.051728 0.357644 11.244543 −0.000845 −1.26E-05 −4.051300 −0.727090 0.012675

0.051989 0.363965 10.890522 −1.26E−03 −2.54E−05 −4.061337 −0.722697 0.012681

0.05115438 0.34987116 12.0764321 0.000000 −0.000007 −4.0278401 −0.7365723 0.012671

0.051609 0.354714 11.410831 −0.000039 −0.000183 −4.048627 −0.729118 0.012670

Table 6 Comparison of statistical results obtained from various algorithms for the tension/compression spring design problem. Algorithm

Worst

Mean

Best

SD

NFEs

IGA WCA [34] TLBO [43] GA2 [40] GA3 [44] CPSO [49] DEDS [35] HEAA [39]

0.012694 0.012952 NA 0.012822 0.012973 0.012924 0.012738 0.012665

0.012681 0.012746 0.012666 0.012769 0.012742 0.012730 0.012669 0.012665

0.012667 0.012665 0.012665 0.012704 0.012681 0.012674 0.012665 0.012665

1.12E−05 8.06E−05 NA 3.94E−05 5.90E−05 5.20E−04 1.3E−05 1.4E−09

50,000 11,750 10,000 900,000 80,000 240,000 24,000 24,000

Table 7 Comparison of the best solution obtained from different algorithms for the pressure vessel design problem. Value

IGA

WCA [34]

CPSO [49]

GA2 [40]

GA3 [44]

DE [42]

x1 (Ts) x2 (Th) x3 (R) x4 (L) c1(x) c2(x) c3(x) c4(x) f(x)

0.815752 0.403932 42.248583 174.814712 −3.54E−04 −8.80E−04 −166.627947 −65.185288 5957.9898

0.7781 0.3846 40.3196 200.0000 −2.95E−11 −7.15E−11 −1.35E−06 −40.0000 5885.3327

0.812500 0.437500 42.091266 176.746500 −0.000139 −0.035949 −116.382700 −63.253500 6061.0777

0.812500 0.434500 40.323900 200.000000 −0.034324 −0.052847 −27.105845 −40.00000 6288.7445

0.812500 0.437500 42.097398 176.654050 −2.01E−03 −3.58E−02 −24.7593 −63.3460 6059.9463

0.812500 0.437500 42.098411 176.637690 −6.68E−07 −0.035881 −3.683016 −63.36231 6059.7340

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Table 8 Comparison of statistical results obtained from various algorithms for the pressure vessel design problem. Algorithm

Worst

Mean

Best

SD

NFEs

IGA WCA [34] TLBO [43] GA2 [40] GA3 [44] CPSO [49] HEAA [39]

6580.3433 6590.2129 NA 6308.4970 6469.3220 6363.8041 6059.714335

6184.1022 6198.6172 6059.71434 6293.8432 6177.2533 6147.1332 6059.714335

5957.9898 5885.3327 6059.714335 6288.7445 6059.9463 6061.0777 6059.714335

245.2762 213.0490 NA 7.4133 130.9297 86.4500 3.62E−10

50,000 27,500 10,000 900,000 80,000 240,000 30,000

Table 9 Comparison of the best solution obtained from different algorithms for the welded beam design problem. Value

IGA

WCA [34]

CPSO [49]

GA3 [44]

GWO [23]

MBA [36]

x1 (h) x2 (l) x3 (t) x4 (b) c1(x) c2(x) c3(x) c4(x) c5(x) c6(x) c7(x) f(x)

0.205218 3.481587 9.036823 0.205731 −0.372018 −1.481705 −0.054010 −5.13E−04 −0.182085 −0.080218 −3.431970 1.725597

0.205728 3.470522 9.036620 0.205729 −0.034128 −3.49E−05 −1.19E−06 −3.432980 −0.080728 −0.235540 −0.013503 1.724856

0.202369 3.544214 9.048210 0.205723 −13.655547 −78.814077 −3.35E−03 −3.424572 −0.077369 −0.235595 −4.472858 1.728024

0.205986 3.471328 9.020224 0.206480 −0.103049 −0.231747 −5E−04 −3.430044 −0.080986 −0.235514 −58.646888 1.728226

0.205676 3.478377 9.03681 0.205778 NA NA NA NA NA NA NA 1.72624

0.205729 3.470493 9.036626 0.205729 −0.001614 −0.016911 −2.40E−07 −3.432982 −0.080729 −0.235540 −0.001464 1.724853

Table 10 Comparison of statistical results obtained from various algorithms for the welded beam design problem. Algorithm

Worst

Mean

Best

SD

NFEs

IGA WCA [34] TLBO [43] GA2 [40] GA3 [44] CPSO [49] CAEP [45]

1.731117 1.744697 NA 1.785835 1.993408 1.782143 3.179709

1.729277 1.726427 1.728447 1.771973 1.792654 1.748831 1.971809

1.725597 1.724856 1.724852 1.748309 1.728226 1.728024 1.724852

3.18E−02 4.29E−03 NA 1.12E−02 7.47E−02 1.29E−02 4.43E−01

50,000 46,450 10,000 900,000 80,000 240,000 50,020

computation burden.

of BIW frame. Scale vector method is introduced to dramatically reduce the design variables. In the shape optimization problem, three fabrication constraints are considered to promote the cross-sectional shape more practice. Especially, an integrated, objected-oriented, and penalty-parameterless genetic algorithm optimizer, i.e., the IGA optimizer, is developed to facilitate the solution of this constrained nonlinear optimization problem. IGA optimizer is capable of solving single-

7. Conclusions In this paper, shape optimization of multiple cross-sections is implemented to promote the conceptual design of automotive body frame. TSMM is employed for the exact static and dynamic structural analyses

Table 11 Comparison of the best solution obtained from different algorithms for the speed reducer design problem. Algorithms

IGA WCA [34] PSO-DE [37] SHO [30] GWO [23] MFO [46] MVO [27] SCA [47] GSA [25] HS [28]

Optimum variables

f(x)

x1 (b)

x2 (m)

x3 (z)

x4 (l1)

x5 (l2)

x6 (d1)

x7 (d2)

3.500412 3.500000 3.500000 3.50159 3.506690 3.507524 3.508502 3.508755 3.600000 3.520124

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

17 17 17 17 17 17 17 17 17 17

7.303039 7.300000 7.300000 7.300000 7.380933 7.302397 7.392843 7.300000 8.300000 8.370000

7.721213 7.715319 7.800000 7.800000 7.815726 7.802364 7.816034 7.800000 7.800000 7.800000

3.350381 3.350214 3.350214 3.35127 3.357847 3.323541 3.358073 3.461020 3.369658 3.366970

5.286765 5.286654 5.2866832 5.28874 5.286768 5.287524 5.286777 5.289213 5.289224 5.288719

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2994.957952 2994.471066 2996.348167 2998.5507 3001.288 3009.571 3002.928 3030.563 3051.120 3029.002


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Table 12 Comparison of statistical results obtained from various algorithms for the speed reducer design problem. Algorithm

Worst

Mean

Best

SD

NFEs

IGA WCA [34] SC [38] PSO-DE [37] DLEC [48] DEDS [35] HEAA [39]

2996.051600 2994.505578 3009.964736 2996.348204 2994.471066 2994.471066 2994.752311

2995.400303 2994.474392 3001.758264 2996.348174 2994.471066 2994.471066 2994.613368

2994.957952 2994.471066 2994.744241 2996.348167 2994.471066 2994.471066 2994.499107

0.29451182 7.4E−03 4.0 6.4E−06 1.9E−12 3.6E−12 7.0E−02

50,000 15,150 54,456 54,350 30,000 30,000 40,000

Table 13 The optimum solutions of the shape optimization. Section number

Optimum solution θ(radian)

1 2 3 4 5 6 7 8 9 10

SV (dimensionless)

t (mm)

IGA

GWO

WCA

IGA

GWO

WCA

IGA

GWO

WCA

0.1612 0.1674 0.4683 0.6725 1.0948 0.4238 0.3470 0.5466 0.8629 1.3574

1.4298 1.3098 0.4569 1.5019 1.3258 1.2320 0.9587 0.0975 0.5299 0.1895

0.0853 1.5646 0.1774 1.5642 1.5708 0.0631 1.2913 1.2937 1.0977 0.0628

0.9710 0.9761 0.9362 1.1581 0.9370 1.2560 0.8572 0.7651 0.8883 0.7850

0.9844 0.9696 0.9547 1.1558 0.9455 0.8829 1.2105 0.7672 0.8451 0.8976

0.9814 0.9894 0.9812 0.7707 0.7817 0.8987 0.9425 1.1202 1.2249 0.7675

1.0545 0.8270 0.8434 0.9067 1.6669 1.2399 1.3044 1.3537 1.0463 0.9032

1.0300 0.8140 0.8088 0.9404 1.6081 1.5003 1.9003 1.2623 0.9218 0.8801

1.0261 0.8000 0.8000 0.8419 2.0000 1.7919 1.3620 1.2994 0.8000 0.8061

Table 14 Performance targets solved by the three optimizers. Performance target

Optimizer

Initial

Optimized

Variation (

BIW mass (kg)

IGA GWO WCA IGA GWO WCA IGA GWO WCA IGA GWO WCA

240.6519

227.0986 227.8088 227.3040 8.2592 × 103 8.2246 × 103 8.2531 × 103 9.4289 × 103 9.4638 × 103 9.4254 × 103 28.2185 28.1653 28.7190

−5.63% −5.34% −5.55% 2.24% 1.81% 2.17% 3.76% 4.14% 3.72% 6.31% 6.11% 8.19%

Bending stiffness (N/mm)

Torsional stiffness (N · m/°)

1st-order frequency (Hz)

8.0781 × 103

9.0875 × 103

26.5447

Optimized − Initial Initial

× 100% )

objective and multi-objective problems with continuous and/or discrete variables. Benchmark tests on 12 classical optimization problems and comparison with multiple well-regarded algorithms prove the validity of this optimizer. Afterwards, the shape optimization is carried out with use of IGA optimizer. Lastly, IGA optimizer is integrated into our “ABCD” toolbox to facilitate the automotive body conceptual development. It should be noted that in this paper some simplifications are carried out, e.g., (a) relevant design variables of different cross-sections are defined with identical bounds. This may be modified in the practice; (b) for a specified TWB, the thicknesses of inner, outer and reinforcement panels are defined to be identical, while in fact these values may be different. In addition, shape optimization in this paper is formulated as a single-objective optimization problem with only continuous variables. In the future, we will design the BIW using multiple materials, aiming at

Fig. 13. Convergence curves obtained via IGA, GWO and WCA optimizers.

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Fig. 14. Initial and optimized cross-sectional shapes obtained by IGA optimizer.

finding a solution that satisfies both objectives of minimum mass and cost. This is a constrained multi-objective optimization with both continuous and discrete variables, and will be researched in our further studies.

Acknowledgment Authors acknowledge financial support from the National Natural Science Foundation of China (No. 51475152).

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Fig. 14. (continued)

Appendix A. Pseudocode and MATLAB code of constrained-dominated sorting.

% Solve F1 for each xp ∈ P Sp = ∅ np = 0 F1 = ∅ prank = 0 for each xq ∈ P if (xp≺cxq) Sp = Sp∪{q} else if (xq≺cxp) np = np + 1 end if end for if (np = 0) prank = 1 F1 = F1∪{p} end for

% Solve Fi (i ≥ 2) i=1 do while Fi ≠ ∅ Q=∅ for each p ∈ Fi for each q ∈ Sp nq = nq − 1 if (nq = 0)

% Solve F1 % Initialization Rank = zeros(NInd,1); F{1} = []; for p = 1:NInd Sp{p} = []; np(p) = 0; for q = 1:NInd [Judge]= DominationJudge(p,q,ObjVal,Constraint); if Judge == −1% p constrained-dominates q Sp{p} = [Sp{p} q]; elseif Judge == 1% q constrained-dominates p np(p) = np(p) + 1; end end if np(p) == 0 Rank(p) = 1; F{1} = [F{1} p]; end end % Solve Fi (i ≥ 2) i = 1; while ∼isempty(F{i}) Q = []; NumP = length(F{i}); for j = 1:NumP p = F{i}(j); NumQ = length(Sp{p}); 247


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qrank = i + 1 Q = Q∪{q} end if i=i+1 Fi = Q end for end for end while

if NumQ >= 1 for k = 1:NumQ q = Sp{p}(k); np(q) = np(q) − 1; if np(q) == 0 Rank(q) = i + 1; Q = [Q q]; end end end End i = i + 1; [temp, index] = sort(Q); F{i} = temp; end % delete the last empty dominated front set NF = length(F); F(NF) = [];

Note: DominationJudge is the function to judge the constrained-domination relationship between the pth and qth solutions.

Appendix B. Pseudocode and MATLAB code of crowding distance sorting

NF = length(F) for i = 1 to NF v = length(Fi) Distance(xp) = 0 (p = 1,2, …,v) for j = 1 to n Xj = sort(Xj) Distance(x1) = ∞ Distance(xv) = ∞ for k = 2 to (v − 1) Distance(x k)=

Distance(x k) + end for end for end for

x kj + 1 − x kj − 1

% Initialization NF = length(F); Distance = cell(NF,1); for i = 1:NF Fi = F{i}; v = length(Fi); Distance{i} = zeros(v,1); for j = 1:length(ChromReal(1,:)) Xj = ChromReal(Fi,j); % sort using each design parameter

max(Xj ) − min(Xj )

[AscendingXj, index ]= sort(Xj); % maximum and minimum XjMin = AscendingXj(1); XjMax = AscendingXj(v); % so that boundary points are always selected Distance{i}(index(1)) = inf; Distance{i}(index(v)) = inf; % for all other points for k = 2:(v − 1) Distance{i}(index(k)) = Distance{i}(index(k)) + (AscendingXj(k + 1) − AscendingXj(k − 1)) / (XjMax − XjMin); end end end

Note: n is the total number of design parameters in an individual; the definitions of Xj, xk and x jk can be referenced in Fig. 8; ChromReal is the binary-decimal conversion of chromosome of a specified population.

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