# Design Optimization of the Aeronautical Sheet Hydroforming Process Using the Taguchi Method
## 1 Introduction  
In recent years, the development of new aluminium alloys adapted to the specific needs of every part of the aircraft, lighter, more resistant to corrosion, more tolerant to damage, weldable, and cheaper than carbon fibre, has allowed the aluminium alloy sheet to continue having an important presence in aeronautical structures, as demonstrated in the new generations of single-aisle aircraft. Aluminium alloy sheet forming processes forging in rubber pad presses, also known as hydroforming processes, were developed in the 20 th century with the arrival of metallic fuselage aircraft to be able to use a more economical and faster method than manual forming [1]. By means of hydroforming, the contour flaps are folded into sheet pieces from 0.5 to 3 mm, with a flat core, generally belonging to ribs or to aeronautical structural elements. The flaps are used to join other pieces by rivets and can be straight or curved (concave, convex, or mixed). Also, by means of hydroforming, smaller stampings are formed in the core of the pieces whose objective is to increase the rigidity without increasing the weight of the piece. In these stampings, the pressure acts on a free radius, forming the sheet by biaxial membrane tensions in the area of the sheet that is in contact withradiuses of the block, and bending stresses are generated accompanied by the traction produced by the dragging of the rubber. The variability in the results is an unsolved problem in hydroforming, considering that it can lead to the need for unproductive operations on the finished piece to return it to the desired shape, as in the case of distortions caused by elastic recovery, or to render it unusable by localised necking or ductile fracture [2]. For a given geometry, the appearance of defects depends on parameters directly related to the hydroforming process, such as: (i) technological parameters (like the pressure, the hardness of the rubber, the friction between the sheet and the rubber or between the sheet and the forming block), and (ii) parameters related to the formability of the sheet (like Young's module, the strength coefficient, the anisotropy indexes or the hardening exponent of the material). The technological parameters of the process are controllable, while the parameters related to the formability of the sheet are difficult to control or it is expensive to reduce their variability. The main defects that appear in the hydroforming are the breaks produced by high tensile stresses, the elastic recovery that takes place when diminishing the forces of the forming, changing the form of the piece, and the wrinkles that occur when shaping the sheet in areas with less perimeter (see Figure 1). Different studies have analyzed aspects for the optimization of hydroforming processes. For example, (i) [PERSON] and [PERSON] [3] studied the microhydroforming process of metallic sheets considering the flow stress size effect. The authors used the hydraulic pumping test to investigate the size effects of the material, both grain size and specimen size. The results determined that the material flow curves decrease when reducing both relations; (ii) a similar study was carried out by [PERSON] et al. [4]. In their work, the authors analyzed the relationship between the geometry and the grain size on copper sheet hydroforming processes. The results showed that as the grain size approaches the thickness of the sheet, the maximum formation height and the final pressure decrease; (iii) [PERSON] et al. [5] used finite element methods to reduce the production process of aeronautical materials from four to three steps. The results allowed them to reduce the time of production, although the thinning process constitutes the longest process, due to the combination of other subprocesses; and (iv) [PERSON] et al. [6] developed a methodology for determining the hydraulic pressure necessary to perform hydroforming processes in noncircular pieces. The results achieved reductions of 5.5% in the thickness of the pieces.