Fixture layout design based on two-stage method for sheet metal components

Abstract

The optimal fixture scheme can effectively reduce the deformation of sheet metal parts in the normal direction of their main plane. The fixture layout design requires a great deal of finite element analysis, which limits efficiency and quality of fixture design. This article proposes a two-stage method to decrease the number of finite element analyses for fixture layout design. First, three locating points are generated by using particle swarm optimization based on the rigid model. Second, the other points are optimized by establishing a response surface modeling based on radial basis function. Finally, through the case of rear bracket plate, this work elaborates two-stage fixture locating scheme design process. The results show that the two-stage method has good simulation effect and higher prediction accuracy.

Keywords

Fixture design two-stage method particle swarm optimization radial basis function

Introduction

Problem description

Sheet metal parts are widely used in many products such as automobiles, airplanes and appliances. Dimensional quality, which measured by its own variation, fixture configuration and assembly process variations, is an important factor influencing the final geometry of sheet metal assemblies. During manufacturing of a new product, the fixture-related dimensional errors constitute up to 40% of all dimensional variations during preproduction period, up to 70% during launching and 70%–100% during one and two shifts of production.¹ Therefore, fixture layout design is important in manufacturing to figure out the best configuration of a product to guarantee that the quality of a product or an assembly is insensitive to dimensional variations from components and fixtures.^2,3

Auto-bodies are assembled in a multistation sequential process, where at each station, fixtures are applied to locate and clamp the parts. Therefore, the fixtures play a critical role in controlling the position of the components and subassemblies on each station and on the final product quality.

Related work

Many scholars have studied fixture design problems since 1990s. The related work for fixture design is shown in Table 1. Wang⁴ built an arbitrary three-dimensional (3D) geometry model but restricted in discrete domain of locations for placing fixture elements of nonfrictional contacts. Wang and Pelinescu⁵ optimized the fixture layout through maximizing the determinant of information matrix. Cai et al.⁶ presented a robust fixture layout by using Euclidean norm of fixture sensitivity index. Carlson and Söderberg⁷ proposed a method of optimizing locating scheme through using the quadratic sensitivity equation. Qin et al.⁸ proposed to optimize fixture layout based on the degrees of freedom constrained. Lööf et al.⁹ presented an approach for optimizing positions of locators in a locating scheme to maximize robustness in defined critical dimensions. Vishnupriyan et al.¹⁰ proposed a genetic algorithm (GA) based on optimization method to get a layout of error containing locators for minimum machining variation satisfying the tolerance requirements and providing deterministic location. Above methods were used in deterministic workpiece location at single-station fixture design.

Current researches on fixture design focus on multistation assembly processes. Camelio et al.¹¹ suggested using influence coefficient to analyze the impact of fixture design to sheet metal assembly variation and showed that fixture deviation had greater influence on assembly variation than part deviation. According to Camelio et al.,¹¹ assembly variation depends on fixture positions in the presence of fixture variation. Therefore, the fixture layout should be optimized for reducing the final product tolerance. Kim and Ding³ proposed a methodology to design multifixture layouts in multistation assembly based on a station-indexed state-space model. Izquierdo et al.¹² proposed a method to achieve the robustness of fixture layout design by an optimal distribution of locators in a multistation assembly system. Phoomboplab and Ceglarek¹³ presented a two-step optimization method to improve process yield through determining an optimum set of fixture layouts for a given multistation assembly system. Huang et al.¹⁴ proposed an alternative sequential space filling strategy, which adopted sampling approaches to search optimal designs. All the above methods are based on the assumption of six-point locating principle; however, the assumption does not satisfy sheet metal components due to deformation during welding process. Cai et al.⁶ proposed a new “N-2-1” locating principle and proved that the new locating scheme was more suitable for sheet metal parts than the “3-2-1” locating principle. With the application of the principle of “N-2-1,” Cai¹⁵ presented the applications of a robust design strategy in locating pin layout design for sheet panels. The objective function of fixture scheme optimization applied the robust fixture layout model.⁶ Fan and Senthil Kumar¹⁶ studied that fixture locating layout was carried out with the robust design approach by combining the Taguchi method and the Monte Carlo statistical method in order to increase the quality of the final workpieces. Xing et al.¹⁷ used the heuristic algorithm to optimize locating schemes and applied the radial basis function (RBF) to reduce the number of the finite element analyses (FEAs). Xing¹⁸ also proposed a hybrid heuristic and orthogonal algorithm to optimize locating positions, which could only generate the satisfactory solution.

Table 1.

Related work on fixture layout optimization.

	Single station	Multistation
Rigid model	Wang,⁴ Wang and Pelinescu,⁵ Cai et al.,⁶ Carlson and Söderberg,⁷ Qin et al.,⁸ Lööf et al.,⁹ Vishnupriyan et al.,¹⁰ Fan and Senthil Kumar¹⁶	Kim and Ding³, Izquierdo et al.,¹² Phoomboplab and Ceglarek,¹³ Huang et al.¹⁴
Sheet metal parts	Cai et al.,⁶ Cai¹⁵, Xing et al.,¹⁷ Xing¹⁸	Camelio et al.¹¹

Objective and organization

Methods of Xing et al.¹⁷ and Xing¹⁸ also conducted lots of FEA for the “N-2-1” locating principle but did not provide global optimal solutions for fixture layouts. In order to improve optimization precision and efficiency, this work presents a two-stage method to optimize the fixture layout of sheet metal parts. First is to generate the three optimal points based on a six-point locating principle using particle swarm optimization (PSO) to increase the optimization efficiency. Then, the other locating points are optimized through establishing a response surface modeling (RSM) based on RBF to decrease the numbers of FEA.

The remainder of this article is organized as follows. Section “Fixture scheme and evaluation function” presents fixture schemes and evaluation function of fixture layout optimization. Section “Fixture scheme optimization by PSO” uses the PSO algorithm to optimize three points based on rigid model. In section “Fixture scheme optimization based on RSM,” the other points are optimized by establishing an RSM based on RBF. The proposed methodology is illustrated by an example in section “Case of fixture design.” Finally, section “Discussions of the two-stage method” draws the conclusions.

Fixture scheme and evaluation function

According to Cai et al.,⁶ the “N-2-1” principle can be used to reduce the deformation in the normal direction of the main plane. This study uses the “4-2-1” locating scheme to optimize the fixture layout for simple sheet metal part, as shown in Figure 1. In order to compare the study results of the two-stage method and the hybrid heuristic and orthogonal algorithm, all the characteristics are exactly the same with those in Xing.¹⁸

Figure 1.

Sheet metal part (meshed).

The dimension of the sample is 300 mm × 300 mm with thickness of 1 mm. Young’s modulus is 20,700 N/mm² and Poisson’s ratio is 0.3. To analyze deformation in the normal direction of the main plane, five units of force is applied at the center points of four edges and whole plane, respectively (Figure 1). The plane is meshed by 12 × 12 elements, with a total of 169 nodes. Each node can be selected as one locating point, thus the optimization complexity is of $(169)$ ⁷, which cannot be analyzed through the enumerate method to get the global optimal fixture layout. Thus, the two-stage method was utilized to decrease the complexity of locating scheme optimization.

In order to evaluate different fixture schemes, the displacement of all the nodes in the surface is used as the objective function as shown in equation (1)

F (X) = \sum_{i = 1}^{N} w_{i} {(X)}^{2}

(1)

where F( X ) stands for the objective function and $w_{i}$ represents the displacement of the $i$ node. In this case, N is set to 169.

According to Cia et al.,⁶ the longer the distance between location points, the smaller the variation of the workpiece; thus, the principle of “2-1” can be applied first without influences of deformation of main plane. The positions and directions of the three nodes are (−150, −150, 0, 0, 1, 0), (−150, −150, 0, 1, 0, 0) and (−150, 150, 0, 1, 0, 0). The node numbers are 1, 1 and 157, respectively (as shown in Figure 1). The other four points are optimized according to the flowchart of the two-stage method.

The two-stage method is described in detail in the following sections. Since the “2-1” points are fixed, the three locating points are optimized by using the PSO algorithm based on rigid model, and then the other points are acquired through RSM based on RBF.

Fixture scheme optimization by PSO

This section focuses on the process of fixture layout optimization based on rigid model. The three optimized points are in same directions (0, 0, 1) but different positions. The results of fixture locating scheme can be calculated by the robust fixture design proposed by Cai et al.⁶ If the directions of “N” and “2-1” are not orthogonal, the algorithm for robust fixture design can be used to calculate different directions and positions of locating points.

Objective function for PSO

The translational and orientation variations at any key product/process characteristic point (KPC) can be expressed as a vector

δ q_{0} = - J^{- 1} Φ_{R} δ R_{f}

(2)

where the Jacobian is $J_{i} = [- n_{xi}, - n_{yi}, - n_{zi}, 2 (n_{zi} y_{i} - n_{yi} z_{i}), 2 (n_{xi} z_{i} - n_{zi} x_{i}), 2 (n_{yi} x_{i} - n_{xi} y_{i})]$ , $Φ_{R} = [\begin{matrix} 0 n_{i}^{T} 0 \end{matrix}], i = 1, 2, \dots, 6$ and $δ R_{f}$ means the deviations of fixture points.

The purpose of fixture design is to choose set of locating points to reduce the $δ q$ as much as possible. Therefore, the fitness function, which is to optimize fixture scheme, should satisfy the principle of minimizing the variation propagation, as shown in equation (3)

H (X) = ‖ J^{- 1} (X) ‖

(3)

where X is the set of locating points and $‖ J^{- 1} (X) ‖$ is the Euclid norm of the inverse Jacobian. Three points in any two or three are not allowed to overlap; otherwise the Jacobian matrix cannot be inversed.

Particle coding and decoding

The PSO suggested by Kennedy and Eberhart¹⁹ was based on observations of social behavior of animals, such as bird flocking where the PSO algorithm adopted a population of candidate solutions, called particles, with their positions initialized randomly from the search space. Each particle is assigned with a velocity, also initialized randomly, according to its own experiences and those of its companions. One of the advantages of the PSO method is that previously visited best positions were remembered.

The particle is generated from the node numbers, which are decomposed into x and y directions. As the plane part of Figure 1 is meshed by $12 \times 12$ elements, the range of each direction is from 0 to 12. The particle length is 12 for the three locating points, of which the front four values are taken as the first point, the middle four values as the second and last four values as the third. In this case, the coordinates of each node are shown in equation (4)

x_{nd} = - 150 + 25 n d_{x}, y_{nd} = - 150 + 25 n d_{y}, z_{nd} = 0

(4)

where $x_{nd}, y_{nd} and z_{nd}$ are node coordinates (mm), $n d_{x}$ is the value of the node in x direction and $n d_{y}$ is the value of the node in y direction.

For example, one particle is (050707081011). Then, the first node number is 97, the second node number is 112 and the third node number is 154. The coordinates and directions of three points are (−25, 25, 0, 0, 0, 1), (25, 50, 0, 0, 0, 1) and (100, 125, 0, 0, 0, 1), respectively. Thus, the Jacobian matrix is shown below and the objective function is 5.53

[\begin{matrix} 0 & - 1 & 0 & 0 & 0 & - 300 \\ - 1 & 0 & 0 & 0 & 0 & 300 \\ - 1 & 0 & 0 & 0 & 0 & - 300 \\ 0 & 0 & - 1 & 0 & 50 & 0 \\ 0 & 0 & - 1 & 100 & - 50 & 0 \\ 0 & 0 & - 1 & 250 & - 200 & 0 \end{matrix}]

In the initialization of particle swarm population, if there points in any two or three overlap, it is an unfeasible particle that can be assigned with a bigger value such as 1000.0 to reduce the evolution of these particles.

Updating the particle position and velocity

At each iteration k, the particles are represented by a vector $X$ in multidimensional space to characterize its position, as shown in equation (5)

X = {x_{1}^{k}, x_{2}^{k}, \dots, x_{N}^{k}}

(5)

The vector $V$ is used to characterize its velocity, as shown in equation (6)

V = {v_{1}^{k}, v_{2}^{k}, \dots, v_{N}^{k}}

(6)

Each particle changes its position $x_{i}^{k}$ in each iteration. The new position $x_{i}^{k + 1}$ of the $ith$ particle is biased toward its best position $p_{i}^{k}$ with best function value, referred to as personal best or $pBest$ , and the global best position $p_{g}^{k}$ , referred to as the global best or $gBest$ . The $gBest$ is the best position in the set

P = {p_{1}^{k}, p_{2}^{k}, \dots, p_{N}^{k}}

(7)

where $p_{i}^{0} = x_{i}^{0}, i = 1, 2, \dots, N$ .

At each iteration k, the position $x_{i}^{k}$ is updated by a velocity $v_{i}^{k + 1}$ that depends on three factors: (1) current velocity $v_{i}^{k}$ , (2) the weight difference vector $(p_{i}^{k} - x_{i}^{k})$ and (3) the weight difference vector $(p_{g}^{k} - x_{i}^{k})$ . The set $X$ is updated for the next iteration using equation (8)

x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1}

(8)

The velocity $v_{i}^{k + 1}$ can be calculated as

v_{i}^{k + 1} = {wv}_{i}^{k} + r_{1} c_{1} (p_{i}^{k} - x_{i}^{k}) + r_{2} c_{2} (p_{g}^{k} - x_{i}^{k})

(9)

The parameter $w$ is commonly equal to 0.9. The parameters $r_{1}$ and $r_{2}$ are uniformly distributed random numbers in $[0, 1]$ and $c_{1}$ and $c_{2}$ are equal to 2.²⁰ The set $P$ is updated when the new positions are calculated through equation (10)

p_{i}^{k + 1} = {\begin{matrix} x_{i}^{k + 1} & if f (x_{i}^{k + 1}) < f (p_{i}^{k}) \\ p_{i}^{k} & if f (x_{i}^{k + 1}) \geq f (p_{i}^{k}) \end{matrix}

(10)

Flowchart of the PSO

The PSO algorithm maintains a population of particles. The PSO starts with a random population of particles, then generates new populations of particles through updating the positions and terminates when the stopping condition is met. The flowchart of the PSO is shown in Figure 2.

Figure 2.

Flowchart of PSO.

Step 1. Initializing parameters.

Step 1-a: Initializing iteration counter k = 0.

Step 1-b: Initializing N random positions of theparticles $x_{i}^{k} (i = 1, 2, \dots, N)$ .

Step 1-c: Initializing N random velocities $v_{i}^{k} (i = 1, 2, \dots, N)$ .

Step 1-d: Initializing N $pBest$ $p_{i}^{k} (i = 1, 2, \dots, N)$ .

Step 2. Evaluating particles.

Step 2-a: Checking the feasibility of particles.

Step 2-b: Calculating the fitness value of each particle $x_{i}^{k}$ by using equation (3).

Step 2-c: Setting $p_{g}^{k}$ equal to the best $pBest$ in $P$ .

Step 3. Judging whether the stop condition is met. If the maximum iteration is met, the algorithm terminates; otherwise the algorithm goes to step 4.

Step 4. Generating a new population of PSO.

Step 4-a: Updating $V$ : Calculating $v_{i}^{k + 1}$ by using equation (9).

Step 4-b: Updating $X$ : Calculating $x_{i}^{k + 1}$ by using equation (8).

Step 4-c: Updating $P$ : Updating $p_{i}^{k}$ by using equation (10).

Step 4-d: Updating $gBest$ : $p_{g}^{k + 1} = min (f (p_{i}^{k + 1})), i = 1, 2, \dots, N$

Step 5. Repeating from step 2 to step 4.

Three-point optimization of the sheet metal part

In order to optimize the three locating points, the PSO first sets fifty particles. The maximum iteration can be set to 50. According to the flowchart of the PSO, the optimization process of the minimum fitness value is shown in Figure 3.

Figure 3.

Curve of the fitness value.

According to Figure 3, the minimum fitness value is not continuously decreasing because the PSO algorithm easily falls into local optimal solutions. Therefore, the coefficients of particle velocity should be adjusted when the minimum fitness value stays the same during several iterations. The optimal particle is (000306121203) according to the flowchart of the PSO algorithm, thus the three nodes are 40, 163 and 52 by decoding the particle, whose coordinates and directions are (−150, −75, 0, 0, 0, 1), (0, 150, 0, 0, 0, 1) and (150, −75, 0, 0, 0, 1), respectively. In this case, the objective value is $4.576 \times 10^{- 2}$ through the FEA, a little larger than the value in Xing,¹⁸ which is $3.96 \times 10^{- 2}$ . However, the sheet metal part in Xing¹⁸ is located based on the “4-2-1” locating principle scheme. The results show that the fixture layout design by PSO not only decreases the displacement of sheet metal part but also reduces the deformation in normal direction of main plane.

Fixture scheme optimization based on RSM

In the final section, the three points are generated based on the PSO. Then, the other points are optimized by using an RSM based on RBF in this section. Commonly, RBF includes Gaussians, thin-plate splines and multiquadrics, of which multiquadrics function can be utilized in handling aircraft shape design problem of surface fitting.²¹ Since the part deformation is very complex in different fixture schemes, the study uses the multiquadrics function to establish the RSM between the part deformation and fixture schemes. The multiquadrics function is shown in equation (11)

φ (r) = (r^{2} + c)^{β}, β > 0, β \notin N

(11)

The above function has two main variables: $c$ and $β$ , which can be initially assigned to 8 and 0.5, respectively, according to Xing.¹⁸ Therefore, this work only needs to calculate the coefficient matrix for establishing the RSM, for which the flowchart is shown in Figure 4.

Figure 4.

Flowchart of response surface model.

To construct the RSM, the first step is to design fixture layout samples, which are selected by using interpolation. In this work, the samples are equally spaced nodes in x and y directions. The next step is to set multiquadrics function as a basis function. Based on the above conditions, the coefficient matrix is established between fixture schemes and node displacements. When the RSM is created, the fitting results can be calculated by inputting the test samples. Then, the results can be compared to the value calculated by the FEA. If the error can be accepted, the operation will stop; otherwise, the parameters of RBF would be reselected.

For Figure 1, five points are set in x and y directions, then 25 samples are used to establish the RSM. The node numbers of all samples are shown in equation (12)

n d_{i, j} = 1 + 4 i + 4 j \times 13; i, j = 0, 1, \dots, 4

(12)

The objective function value can be calculated based on the samples. The RSM is established through the flowchart (Figure 4). The relationship of the node number and part deformation is shown in Figure 5.

Figure 5.

The response surface model of the sheet metal part.

According to Xing,¹⁸ the steps of the hybrid heuristic and orthogonal algorithm can be described as follows:

Step 1. Determining the initial fixture layout.

Step 2. Selecting one locating point to be optimized.

Step 3. Designing the samples of fixture layouts and calculating the results by the FEA.

Step 4. Establishing the RSM and optimizing the locating point.

Step 5. Repeating from step 2 to step 4 until four points have been optimized.

Step 6. Optimizing the final fixture scheme by using orthogonal design.

The objective value of evaluating deformation can be compared by the above steps for different methods, as shown in Figure 6. The objective values are 0.105 by the orthogonal design, 0.039 by the heuristic algorithm and 0.027 by the hybrid algorithm. Based on the results, the hybrid algorithm is proved to have the best optimization precision.

Figure 6.

Sheet metal deformation under optimal locating layout.

As shown in Figure 5, the optimal solution is $5.98 \times 10^{- 3}$ . The node number is 84. According to Figure 6, the objective value is $2.7 \times 10^{- 2}$ . The value for this case is only 22% of that in literature.¹⁸ The optimization accuracy is largely improved. Moreover, the two-stage method only needs 25 times of FEA. The hybrid heuristic and orthogonal algorithm needs 145 times of FEA. Therefore, the optimization efficiency of the two-stage method is greatly proved.

Case of fixture design

This work also uses the rear bracket plate part in Xing¹⁸ to illustrate the operation of the two-stage method. The rear bracket plate part influences the assembly variation of rear taillight. Thus, it is very important to design the fixture layout of this part. Figure 7(a) shows the rear bracket plate part, whereas Figure 7(b) shows the 18 units of force that will cause the part to deform.

Figure 7.

Rear bracket plate components: (a) the rear bracket plate and (b) the forces and the constraints.

The layout of the rear bracket plate uses the “4-2-1” principle, in which “2-1” utilizes the original hole (pin) and slot (pin). The hole (pin) can be decomposed into two points: (5281.6, −694, 1157.5, 0, 1, 0) and (5281.6, −694, 1157.5, 0, 0, 1). The slot (pin) can control one degree of freedom, and the point is (5318, −668, 867, 0, 1, 0). According to the two-stage method, the particle swarm algorithm optimizes the first three points. The final point is acquired through establishing the RSM.

According to the flowchart of PSO, the number of particle population is set to 50 and the maximum iteration is 50. Because the main plane of rear bracket plate is very complicated, it is very difficult to optimize the three points according to the node numbers. The approximate area should be confirmed based on the main plane of the part, as shown in Figure 8. The four vertexes can be calculated from the dimensions of the part.

Figure 8.

Approximate area of the rear bracket plate.

The approximate area is meshed by $L$ partitions in x direction and $K$ partitions in y directions. Then, coordinates of each node can be written as shown in equation (13)

{\begin{matrix} y_{j} = \frac{y_{T} - y_{D}}{K} • j + y_{D}; y_{T} = \frac{y_{2} - y_{1}}{L} • i + y_{1}, \\ y_{D} = \frac{y_{3} - y_{4}}{L} • i + y_{4}; j = 1, \dots, K \\ z_{i} = \frac{z_{R} - z_{L}}{L} • i + z_{L}; z_{R} = \frac{z_{2} - z_{3}}{K} • j + z_{3}, \\ z_{L} = \frac{z_{1} - z_{4}}{K} • j + z_{4}; i = 1, \dots, L \end{matrix}

(13)

In this work, K and L are set to 25 and 50, respectively. The coordinates of the four vertexes are (−796.7, 830), (−744, 1180), (−643, 1172) and (−643.8, 803). The optimal particle is (255000292500) and the minimum objective value is 42.7. According to equation (10), the three points are (P1: 5305.7, −744, 1180), (P2: 5311, −640.8, 1034.6) and (P3: 5390.1, −795.7, 830). The curve of fitness value is shown in Figure 9.

Figure 9.

Curve of the minimum fitness value.

The objective function value is not the optimal in the principle of optimal scheme based on PSO since the part is too complicated. To overcome the contradiction of part deformation and variation propagation, the work utilizes the following steps to select the three points:

Step 1. Analyzing the center point P0 of the existing three points (P1, P2 and P3).

Step 2. Setting partitions from the center point to each vertex is the same.

Step 3. Analyzing the part deformation by FEA.

Step 4. Calculating the objective function value according to equation (1).

Based on the above steps, the three best points are (OP1: 5137, −739, 867.8), (OP2: 5130, −638, 1028) and (OP3: 5283, −702, 1124), as shown in Figure 10.

Figure 10.

(a) Fixture scheme based on rigid model and (b) part deformation of new optimal three points.

In this study, the multiquadrics function is used to establish the RSM. The 40 nodes are designed as the samples of fixture schemes, which are shown in Figure 11(a). The RSM is created according to Figure 4, which is expressed as Figure 11(b) through inputting every node coordinates.

Figure 11.

Response surface model of the rear bracket plate: (a) fixture scheme based on rigid model and (b) deformation of new optimal three points.

The objective function value is 0.441 using the two-stage method in the “4-2-1” principle. The best value is 1.26 according to literature.¹⁸ The optimization accuracy is largely improved and the times of the FEA are decreased. Figure 12 compares the optimization fixture schemes of the two methods: (1) the part deformation of optimization fixture scheme by heuristic algorithm and (2) the part deformation of optimization fixture scheme by the two-stage method. It can be seen that the efficiency of the two-stage method is better than that of the heuristic algorithm.

Figure 12.

Comparison of the part deformation through different methods: (a) part deformation by the heuristic algorithm and (b) part deformation by the two-stage method.

Discussions of the two-stage method

The case study of the rear bracket plate in section “Case of fixture design” illustrates the effectiveness of the two-stage method. The proposed method can improve optimization precision and efficiency for fixture layout designing, especially in the “4-2-1” principle. For example, the number of the FEAs for the simple sheet metal part decreases from 145 to 25 and the objective value of the deformation in normal direction of main plane reduces from 0.027 to 0.00598. The two-stage method is divided into two steps: (1) three points are generated by using the PSO algorithm and (2) the other points are optimized through the RSM. This could not lead to global optimization solutions in theory. Instead, it leads to satisfactory solutions.

Since the FEA for sheet metal parts takes a long time, the global optimal results are almost impossible to be calculated by the enumeration method, the GA or other algorithms. Each FEA for the sheet metal part (Figure 1) takes about 50 s (Intel Pentium Dual CPU, 2.20-GHz, 2.19-GHz, 0.99-GB random-access memory (RAM)), including editing of input files of ABAQUS, analyzing the finite elements and reading displacements of all nodes, which even based on the operation of secondary development of ABAQUS. In this case, the complexity of the sheet metal part is $C_{169}^{4}$ , thus the optimal fixture layout needs about 455,488 h through the enumeration method. The optimal results of fixture design need about 70 h by using GA or PSO if the population size is 50 and the maximum generation is 100. However, the fixture layout design may be an impossible mission if the sheet metal components such as rear door or body side are very complex. For the rear bracket plate, the number of all nodes is 3481 and then the complexity of fixture design is $C_{3481}^{4}$ . Each time of the FEA is about 154 s (Intel Pentium Dual CPU, 2.20-GHz, 2.19-GHz, 0.99-GB RAM). The optimal fixture layout calculated needs about $7.0 \times 10^{8}$ years by the enumeration method. The optimal results of fixture design needs about 2140 h through GA if the population size is 100 and the maximum generation is 500. The rear bracket case came from an assembly testing of the Cadillac rear lamp bracket, which includes three parts. The case of the rear bracket plate in this article is the first part. The part is located in the principle of “3-2-1” of the assembly testing. For reducing assembly variation, the part is located according to “4-2-1” in this article. The deformation of main plane is smaller by the method of two-stage method than that by the other current methods.^17,18 Because the principle of “4-2-1” is not mature in practice, the new fixture layout cannot be used to the assembly test. However, the assembly variation is reduced based on the principle of “4-2-1” according to the method of influence coefficient.¹¹ For optimizing N points of the large sheet metal parts, the first three points can be gotten by the PSO algorithm based on rigid model and the other points are optimized by the RSM based on RBF. The PSO algorithm only needs several minutes. The RSM can largely reduce times of FEA, and displacements of all nodes are read by using secondary development of ABAQUS. Therefore, the two-stage method of this article is suitable to optimize fixture layout.

Conclusion

The fixture layout is critical for product quality. In order to reduce optimization time of fixture layout, the two-stage method is proposed to optimize fixture scheme. Three locating points are generated by using PSO algorithm based on rigid model, and the other points are optimized through RSM based on RBF. The two-stage method can generate satisfactory solutions. Finally, the rear bracket plate is utilized to illustrate the flowchart of the two-stage method. The results show that the two-stage method can improve the optimization efficiency and greatly reduce the number of FEAs. The method of this article is effective for designing fixture layout, which differs from the current methodologies in three aspects:

The two-stage method is proposed to optimize the fixture layout of sheet metal parts, which can improve the optimization precision and efficiency.

The linear variation analysis model is applied to optimize three points in the flowchart of the PSO algorithm, which can generate the global fixture layout based on the rigid model.

The RSM based on RBF is used to decrease the number of FEAs for improving the optimization efficiency.

Footnotes

Funding

This study was supported by National Natural Science Foundation of China (51105241) and Shanghai Natural Science Foundation (11ZR1414700).

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