Problems proposed by Airbus SAS and Powerflute companies
Optimization of fastener distribution during airframe assembly
One of the problems for ESGI 123 was provided by Airbus and related to the optimization of the wing assembly processing. As nowadays the demand for planes is very high, the company aims to reduce the time of wing production through minor changes in the current technological process.
One step of the assembling process is the connection of the outer surface of the wings with the underlying frame trough the installation of a template of temporal fasteners. These fasteners can be located only in pre-drilled holes i.e. a set of discrete positions spread all over the wing. Before this installation due to production tolerances and errors each individual wing and frame differ slightly that is why there is a random gap between these parts. The aim of fastener installation (which is the same for all the planes from one range) is to eliminate this gap for all wings. Now the number of temporary fasteners is around 30% of all holes, but the Airbus is interesting in decreasing this number, thence decreasing assembly time. So the question is: can this number be reducing without loss of quality?
The ASRP software allows getting the characteristic of the gap elimination for defined template of fasteners i.e. the probability of elimination, which describes the quality of contact. However, to calculate this probability the ASRP uses a set of solutions for non-linear contact problems for a big sample of the random gaps. Therefore, this characteristic is highly nonlinear and the only way to use it is to consider it as a “black box”. Hence, there is a so-called black box optimization problem with binary variables and the only way to find the global solution is the brute force (hard to implement since the large number of holes and the big sample of gaps).
The ESGI team suggested two ways of solving this optimization problem: one is decomposition of the initial domain and another is usage of the repeating patterns.
The first idea was to divide the initial domain (between wing surface and frame) into several subareas and work with each part sequentially. On every step we install one fastener somewhere inside this subarea and then move to the next subarea. This approach allows considering the impact of subareas on each other and reducing the gap all over the contact area. Inside one subarea we need to choose the most appropriate hole which somehow leads to optimal fastener arrangement. One way to do it is the brute force when we sequentially try to put a fastener in all the holes from the subarea and test the behavior of the gap. The team offered another approach: to predict the position of hole with the biggest gap. From uniformity of the gap we can conclude that if we have a triangle of the preinstalled fasteners the largest resulting gap will be somewhere near the barycenter of this triangle. So, the team suggested testing holes only near the barycenter of all triangles to reduce the number of calculations.
The second idea was to use the repeating patterns to cover all field of contact. The suggested pattern is a snake tracery. This pattern can be characterized by the length of the edges, the angels of the corners and the number of fasteners on each edge. The brute force can help to find the optimal values of this characteristic, through it means a lot of calculations.
Although these ideas do not guarantee the optimal solution but they are very promising as quite easy ways to get the approximate solutions. However, from mathematical point of view this optimization task is still waiting for new ideas.
Collapse of paperboard packages
Paperboard packages are widely used in our life to store many different products. Packages have to be designed to resist compressive loads, namely creep. However, mainly because of humidity conditions the creep rate increases which can lead to collapse of paperbox even at low levels of load. The main questions are: what causes the accelerated creep and how to design packages to resist better?
The problem was proposed by Powerflute within the ESGI123. The task was to investigate new material model that will include time and moisture dependency based on experimental data. First stage of this study was to define parameters of this material defining strain as some exponential series that depends on time and moisture with some constants that will be defined using experimental data. Here optimization procedure was used to fit theoretical curve into experimental data. For the second stage, theoretical equations were implemented in finite-element code as a user-defined material with moisture as variable parameter. These stages were done first for 1D model when paperbox loaded with constant pressure from the top is represented as a bar with the constant force applied in upper point of this bar. Then it was implemented with shell analogy when paperbox is represented as a shell with loaded edge. The results of this work provided good coincidence with the experimental data that allow us to conclude that this method is convenient for this problem.
The superviser of this problem was Joonas Sorvari from Lappeenranta University of Technology. Our team consisted of students from international master’s degree program “Continuum Mechanics” from SPbPU and two PhD students – Karunia Putra Wijaya from University of Koblenz and Landau and Anna Avranenko from Lappeenranta University of Technology.
It was a great opportunity to test our knowledge in practice with the real problem and to work in international team of students, especially with such supervisor as Joonas, who always was ready to help us and to explain repeatedly all pitfalls of this problem.
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