Keywords: shape sensitivity analysis, optimal design, metal forming processes.

In the present work a sensitivity analysis for die extrusion problem with varying domain shape is considered: i.e. to find the shape of the back wall of the die that meets some design objective. This paper is focused on the design of shaped dies by using sensitivities and optimization techniques. To reach this objective, the internal surface of the die is optimally designed in order to obtain a desired velocity at the exit of the extrusion die. This is done using a high quality geometric model of the die-face shapes. Changes in shape will be performed by variations in the contact boundary between the work-piece and the die through a finite number of control parameters (design variables). The optimum values of process parameters will be determined by various techniques of mathematical programming. An objective function subject to constraints imposed by the metal forming problem must be constructed and minimised. The optimization technique requires derivatives of the cost function with respect to the design variables. They can be carried out following two alternative approaches known as the direct differentiation and adjoint variable methods. Here, the adjoint approach is chosen to evaluate shape sensitivities and the B-spline functions are used to represent the shapes of the dies.

In the extrusion process, the quality of the final products depends on being able to efficiently control the flow metal in the extrusion die. The contact condition between the work-piece and the die also induces non-uniform metal flow which causes formation of local defects. In order to reduce the possibility of failure, but also to improve the quality of the extruded product, the prediction of the extrusion process in the given design parameters is required. The finite element method allows us to provide the response of the process with fixed parameters.

The paper is organized as follows. In Section 2, we present a mathematical model based on a rigid-plastic material and the shape optimization problem. The discretization procedures are described in Section 3. In Section 4, the sensitivity analysis is described, and non-linear programming methods are used to minimize the objective function. In Section 5, various numerical tests are presented, and concluding remarks are discussed in Section 6.

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