Issue 
Int. J. Simul. Multisci. Des. Optim.
Volume 5, 2014



Article Number  A18  
Number of page(s)  4  
DOI  https://doi.org/10.1051/smdo/2013014  
Published online  26 February 2014 
Article
A location optimization method for aircraft weaklyrigid structures
The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Northwestern Polytechnical University, 710072
Xi’an, Shaanxi, P.R. China
^{*} email: yangyuan0824@mail.nwpu.edu.cn
Received:
23
June
2013
Accepted:
23
October
2013
Since aircraft weaklyrigid structure has large size and weak stiffness, there has serious deformation during assembly process. The current deformation analysis theory of rigid assembly is not applicable. Based on the N21 (N > 3) locating principle, this paper presents a methodology for weaklyrigid parts. An optimization algorithm combines finite element analysis and nonlinear programming methods to find the optimal number and position of the locating points in order to minimize the assembly deformation. An example application study is presented to demonstrate the optimization procedure and its effectiveness by using the software of ABAQUS.
Key words: Weaklyrigid structures / N21 Locating principle / Location optimization / ABAQUS
© Z.Q. Wang et al., Published by EDP Sciences, 2014
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://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
By virtue of high strength, large deflection, and light weight, weaklyrigid structures are widely used in the aerospace industry, as important parts of fuselage, wings, horizontal stabilizer, and vertical stabilizer. Different from rigid structures, they always tend to distort and deform during assembly, which causes the assembly deformation [1, 2]. The problems may adversely affect the final product functionality and the process performance. For this reason, many domestic and foreign scholars and technicians have carried out the research [3–7]. More research concentrates on the auto body assembly positioning, while little on weaklyrigid for aircraft assembly. This paper focuses on the impact of locators (number and position) on the dimensional quality. The locator layout is based on an N21 locating principle, which has more than three locators in the primary plane. The optimization approach combines finite element analysis and nonlinear programming methods in finding the optimal number of locators as well as their positions that minimizes assembly deformation.
2 Optimization algorithm of locating points
As described above, it is better to use the N21 (N > 3) locating principle than the 321 scheme for weaklyrigid assemblies to avoid the assembly deformation and assure the assembly quality. Figure 1 shows a typical N21 locating principle (N = 6).
Figure 1
N21 Locating principle of weaklyrigid structures. 
For weaklyrigid structures assembly, there is a set of points on components, called key measurement points (KMPs), which are utilized to evaluate the assembly dimensional quality [8, 9]. In this study, we choose the boundary points and midpoints as KMPs. Optimization objective function is set to the deformation caused by gravity at KMPs, and it is convenient and rapid to calculate the deformation using ABAQUS. According to required precision, a proper mesh generation should be performed on the component. Then calculate the deformation with finite element method. Residual strain and work hardening of weaklyrigid component caused by metal stamping is not considered in this study.
2.1 Objective function and constraints
Suppose that there are L number of KMPs. The deformation of KMPs in X and Yaxes of part coordinate under the action of dead load is so small that we assume its location in X and Yaxes is in an ideal position. So the paper focuses on the deformation in Zaxis:(1) (2) (3)where is a vector which stands for N number of design variables or the position of locating points. Therefore, equation (1) can be rewritten as:(4)where j = 1, 2, …, N, V _{ j } stands for the position of the jth locating point.
is the optimization objective function. L is the number of KMPs, and the deformation of the KMPs is considered in . stands for normal displacement of the ith KMP. and denote respectively equality constraints and inequality constraints of design variables, that is, the region in which the locators can exist. is a nonlinear function of , while and are linear functions. Considering local deformation, the deformation displacement of KMPs should subject to constraint condition as follows:(5)where Ω is the allowed maximum displacement for each locating point.
2.2 Optimization method
From the mathematical model, it is seen that the nonlinear objective function is like a “blackbox” and its properties are unknown [10]. Aiming at this optimization problem, the procedure can be described with the following steps:

Perform a proper mesh generation on the component according to the principle of mesh generation, and set a couple of supporting points at certain intervals in the feasible area of the component. Then number the points 1, 2, …, N.

Calculate under gravity, if is too large, it should adjust the spacing of the supporting points, so that is controlled within a certain range;

Sequentially remove the supporting points 1, 2, 3, ..., N, calculate when each supporting point was removed, and find the point which has the minimal impact on the component and take it away. Now, there are (N – 1) number of supporting points left.

Among the (N – 1) supporting points, remove each one sequentially, then search the point having minimal impact on the component and take it away. Now the number of the supporting points is reduced to (N − 2).

Repeat step 4 until a certain stopping criterion is met.
Due to the lack of a close form function for , the objective function gradient vector g can be approximated using finite difference method:(6)where ΔX _{ i }, defined as ΔX _{ i } = [0, …, δ _{ i }, …, 0]^{ T }, is a perturbation vector of the design variable in the idirection. Now, the number and position of positioning points is preferred. To sum up, the whole optimization process can be illustrated in Figure 2.
Figure 2
Flow chart of the optimization method. 
3 Application example
3.1 Problem description
In this section, the optimization methodology is illustrated by a sheet metal part. It has dimensions 400 × 350 × 1.5 mm^{3}. The parameters of the material can be shown as Table 1. Generally speaking, the KMPs need to be determined according to the demand of practical production, and in this paper, we choose the boundary points and midpoints as the KMPs. In order to be more representative, the point having the maximum deformation is also chosen as one of the indicators to measure parts deformation, we call it KMP10. The configuration of 10 KMPs is shown as Figure 3. Now, we have (9 + 1) number of KMPs and (3 × 4) number of initial locating points.
Figure 3
Mesh generation of sheet metal part. 
The parameters of the material.
The constraint condition of locating point (x _{ j }, y _{ j }):(7) (8)
The constraint condition of the deformation in normal direction of each KMP:(9)
The finite element method result in this study only considers the gravity of the part, but in some real situations, other constraint conditions during assembly should be taken into consideration, such as the deformation caused by clamps, etc. And we must take them into consideration in the finite element analysis, if the deformation has a large effect on the sheet metal part.
3.2 Optimization result
After having removed the point having the minimal impact on the part in turn, the deformation cloud diagram in ABAQUS (N = 12, 11, 10, 9, 8, 7, 6, 5) is shown in Figure 4.
Figure 4
Result comparison in the software of ABAQUS. 
Table 2 shows the deformation displacement and its square sum when N = 5, 6 and 7 respectively. It can be seen, when it has five locators, there are three KMPs (KMP6, KMP9 and KMP10) whose deformation displacement exceeds the set maximum allowable displacement. When it has six locators, the deformation and its square sum of each KMP are decreased significantly, and the set accuracy is met. As the number of locators is increased to 7, with the increment of the number of locators, the deformation displacement of each KMP still has a downward trend. But considering the cost of clamping equipment and the increment of coupling error may caused by too many overpositioning, select the “621” locator layout scheme in the case study is reasonable. At the same time, the example also verified the “321” positioning principle is difficult to meet the positioning requirements for weaklyrigid structures.
Objective function values when N = 5, 6, 7 (×10^{−3} mm).
4 Conclusions
For the N21 (N > 3) locating principle of the weaklyrigid structures, the location performance not only depends on the number, but also the position of the locators. In this paper, combining finite element analysis and nonlinear programming methods, an optimization method is proposed. With the using of finite element software of ABAQUS, the optimal number of locators as well as their positions can be found. Furthermore, a case study also illustrated the effectiveness and practicability of the proposed method.
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant No. 51375396).
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Cite this article as: Wang ZQ, Yang Y, Kang YG & Chang ZP: A location optimization method for aircraft weaklyrigid structures. Int. J. Simul. Multisci. Des. Optim., 2014, 5, A18.
All Tables
All Figures
Figure 1
N21 Locating principle of weaklyrigid structures. 

In the text 
Figure 2
Flow chart of the optimization method. 

In the text 
Figure 3
Mesh generation of sheet metal part. 

In the text 
Figure 4
Result comparison in the software of ABAQUS. 

In the text 
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