Method for Optimizing the Kinematic Scheme of Stretch-Forming Parts from Sheet Blanks

The modeling of the stretch-forming process a sheet on a punch is considered. A numerical method has been developed for solving the optimal control problems of the stretch-forming process sheet blanks. The values of the residual displacements are considered as an optimization criterion. The algorithms implemented in CAE allow calculating the optimal parameters for the operation of the CNC stretch-forming press. The simulation results are compared with full-scale experiments on the shaping of sheet parts on a stretch-forming press T-600.


Introduction
One of the ways to obtain details of aircraft structures is the stretch-forming process. These are details such as skins and frame elements, which are obtained by bending with tension from sheet and profile blanks on numerically controlled stretch presses. The accuracy of manufacturing details by the stretch forming method is greatly influenced by such technological conditions of the shaping process as the geometry of the punch and the kinematics of the stretch forming press control bodies.
At the present time, the stretch forming punch is manufactured according to electronic models of details without taking into account the material springback. In this case, the elastic recovery of the material (springback) can have a significant effect on the accuracy of the detail manufacture. In practice, in order to minimize deviations of the formed workpiece from the given geometry, at the last stage of wrapping, calibrating tension with a high level of elongation is applied, up to rupture of the workpiece. This solution leads to a decrease in the plastic resource of the workpiece material and does not always provide the required shaping accuracy.
Existing software tools and methods for calculating control programs do not always provide accurate manufacturing of details, which increases their production cycle due to manual revision and the search for optimal modes of shaping and adjusting control programs. To reduce the production preparation time, it is necessary, when creating control programs, to take into account the stress-strain state of the workpiece material during its deformation according to the loading scheme.
Various methods of stretch forming on presses are proposed: taking into account the displacements of the workpiece relative to the punch [1,2], taking into account a certain sequence of the stretch forming kinematic schemes with stages alternating with loading and unloading [3,4]. CAE-systems makes it possible to simulate the stretch forming processes that implement various deformation schemes and select the best ones [5][6][7], but for a complete analysis of possible solutions, it is necessary to develop a method for finding the optimal shaping scheme.
The calculation of the tooling, which provides the necessary anticipatory sheet shape, can be performed by the iterative method for solving inverse problems [8,9]. As an optimization method, the method of dynamic programming is considered, implemented in finite element algorithms for solving [10].
The results of modeling and optimization are compared with the data of full-scale experiments carried out on a T-600 press for shaping the skin of an aircraft detail. It is assumed that some form of the punch is known. In this case, the problem of kinematic shaping will include the problem of deformation in plasticity and elastic unloading. The problems of mechanics are formulated by variational principles with functionals:

Formulation and method for solving problem
where c W -is the contact potential [11,12]; potential form are given by , , , , , components of the elastic constant tensor; p kl ε -are the plastic strain rates [12], ij ε , ij ε   -are the velocity components of the current and residual strains of Green-Lagrange, ij σ , ij ρare the current and residual components of the second stress tensor of Piola-Kirchhoff, The problem of optimal deformation is to find the movement trajectory of the clamps in the stretch forming process on the punch, which ensured the minimum deviations of the sheet after unloading from the surface of the punch. This problem includes the equations of the deformable solid mechanics, obtained from the stationarity conditions (1), and the optimization functional where α -is the component of residual displacements, which determines the panel deflection, * u α  -is the given components of the sheet displacement to the punch, ensuring the adherence of the sheet to the punch surface.
Taking into account the discretization of variation (1), finite element equations are formed to solve the contact problem [11,12,13] t t t t +∆ +∆ where ( ) t α ∆U  -α -components of the nodes residual displacements increment caused by the increment of current displacements. Due to the springback of the sheet, the value in brackets (4) will be positive. Equations (3) and optimization criteria (4) form discrete optimal control problem. In such a formulation, the Bellman function is constructed and the problem is solved by the dynamic programming method [14].

Numerical results of solving the optimization problem and experimental data
The solution of the modeling and optimization problem of the sheet stretch forming process is carried out by the finite element method in MSC.Marc (Fig.1.). Due to symmetry, only the right side of the sheet is considered. A workpiece with a thickness of 3 mm and a length of 1000 mm is specified by a set of 4-angle elements for 2D analysis (plane strain conditions). The contact interaction of bodies at the loading stage a workpiece with a punch is determined with a friction coefficient equal to 0.  The minimum is taken over those numbers m for which the nodes lie in the feasible region n G and belong to the hyperplane n Σ . The implementation of the dynamic programming method for this problem is similar to [10]. Fig. 2 shows variants of deformation trajectories from point A to B for the right edge of the sheet when stretch-formed with a mesh at N=4, M=1 and N=2, M=2 (the optimal path is highlighted in bold). The displacements are indicated in mm. Trajectory curves are approximated by elliptical arcs. In the case of the grid N=2, M=2, as can be seen, the optimal trajectory deviates from the ellipse curve (Fig. 2b). In criterion (4), instead of the workpiece surface, the workpiece point with maximum displacements (control point) is considered.
Full-scale experiments were carried out on a stretch-forming press T-600 using equipment for shaping the skin of an aircraft detail (Fig. 1). The workpieces are strips from a sheet of 1163T material 200 mm wide, 2000 mm long and 3 mm thick. The experimental results were carried out for three loading trajectories (bold, dashed and dash-dotted curves in Fig. 2a) and are presented in Fig. 3, and in comparison with modeling -in Fig. 4. These trajectories correspond to three loading schemes: bending -stretching (B-S), stretchingbending -stretching (S-B-S) and stretching -bending (S-B). Fig. 4 shows the deviations of the control point after unloading the sheet, depending on measurements and samples in comparison with the results of modeling for three loading schemes. Experimental studies carried out to assess the reliability of the results of mathematical modeling showed a practical coincidence for various schemes of stretchforming sheet details on a T-600 press.

Conclusions
In the considered optimization method, the original problem is reduced to a sequence of auxiliary simpler minimization problems. This method allows us to consider various loading paths and at the same time reduces the amount of calculations compared to a simple enumeration of all possible deformation paths, since non-optimal paths are excluded during the calculation. According to the results of modeling and full-scale experiments, the optimal kinematic shaping scheme is bending followed by stretching.
The developed numerical method, implemented in the CAE-system, makes it possible to take into account the complex geometry of the detail, material properties and makes it possible, even at the stage of production preparation, to optimize the shaping parameters to reduce the cycle of the technological process. This research was financially supported by the Russian Science Foundation (grant № 21-11-00165).