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An adaptive arbitrary Lagrangian–Eulerian formulation is developed to compute the material flow and the temperature evolution during the three phases of the friction stir welding (FSW) process. It follows a splitting approach: after the calculations of the velocity/pressure and temperature fields, the mesh velocity is derived from the domain boundary evolution and from an adaptive refinement criterion provided by error estimation, and finally state variables are remapped. In this way, the unilateral contact conditions between the plate and the tool are accurately taken into account, so allowing one to model various instabilities that may occur during the process, such as the role played by the plunge depth of the tool on the formations of flashes, the possible appearance of non-steady voids or tunnel holes and the influence of the threads on the material flow, the temperature field and the welding efforts. This formulation is implemented in the 3D Forge3 FE software with automatic remeshing. The non-steady phases of FSW can so be simulated, as well as the steady welding phase. The study of different process conditions shows that the main phenomena taking place during FSW can be simulated with the right sensitivities.

SUMMARYWe suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.

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International audienceAn Arbitrary Lagrangian Eulerian (ALE) formulation was developed to simulate the different stages of the Friction Stir Welding (FSW) process with the FORGE3® F.E. software. A splitting method was utilized: a) the material velocity/pressure and temperature fields are calculated, b) the mesh velocity is derived from the domain boundary evolution and an adaptive refinement criterion provided by error estimation, c) P1 and P0 variables are remapped. The proposed ALE formulation is applied to FSW simulation. Steady state welding, but also transient phases are simulated, showing good robustness and accuracy of the developed formulation. Friction parameters are identified for an Eulerian steady state simulation by comparison with experimental results. Simulations of the transient plunge and welding phases help to better understand the deposition process that occurs at the trailing edge of the probe, and in particular possible void formation. Flexibility and robustness of the model allows investigating the influence of threads and tooling designs

This paper is the second part of a two‐part article about shape optimization of metal forming processes. This part is focused on numerical applications of the optimization method which has been described in the first paper. The main feature of this work is the analytical calculations of the derivatives of the objective function for a non‐linear, non‐steady‐state problem with large deformations. The calculations are based on the differentiation of the discrete objective function and on the differentiation of the discrete equations of the forging problem. Our aim here is to show the feasibility and the efficiency of such a method with numerical examples. We recall the formulation and the resolution of the direct problem of hot axisymmetrical forging. Then, a first type of shape optimization problem is considered: the optimization of the shape of the initial part for a one‐step forging operation. Two academic problems allow for checking the accuracy of the analytical derivatives, and for studying the convergence rate of the optimization procedure. Both constrained and unconstrained problems are considered. Afterwards, a second type of inverse problem of design is considered: the shape optimization of the preforming tool, for a two‐step forging process. A satisfactory shape is obtained after few iterations of the optimization procedure.

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