Keywords: sheet metal forming, inverse approach, evolutionary optimization, V-bending, punch design, die design, spring-back.

Bending of sheet metal is one of the most widely used processes in the metal working industry. In sheet metal forming processes the part being formed conforms closely the tool shape. After the load is released and the tools removed, the part changes its geometrical shape. This phenomenon called spring-back is the main defect of U and V shaped parts, undergoing significant modification of the angles between bottom, sidewall and flange. Spring-back parameters are mainly influenced by the following factors: punch and die radii, punch and die angles, initial clearance, friction conditions, blank-holder force, draw beads geometry, sheet thickness, elastic modulus, Poisson's coefficient, blank material and constitutive behaviour of the material in plastic field [1].

The numerical simulation of sheet metal forming processes is extensively used for the analysis and the design of industrial parts to avoid the long and expensive experimental tryout procedures. In this paper we present a new developed numerical approach to optimize process parameters of sheet metal forming that iterates over the forming analysis performed by an available commercial finite element code.

In sheet metal forming an inverse problem searches the initial blank and tool parameters in order to form the prescribed product. The shape inverse problem can be formulated as an optimisation problem where the objective is to minimize the gap between the finite element simulation final geometry and the desired one. The main hypothesis and aspects of the formulation of the inverse approach, design variables, objective, quality and cost functions and functionality constraints are outlined and then the evolutionary optimization procedure, a developed numerical algorithm based on a genetic search supported by an elitist strategy is presented [2,3]. The new optimisation process is based on an iterative process coupling the forming simulation and an evolutionary algorithm. The algorithm performs iteratively aiming the improvement of the fitness value associated to the optimisation problem.

The iterative process runs as follows:

(i)

An initial population is generated by randomly selecting individuals. For each individual of the population an independent numerical simulation of punch and spring-back is performed. Each simulation will produce a product with an associated fitness function value.

(ii)

A new population of solutions is generated from the previous using the genetic operators: Selection, Crossover, Elimination/Substitution and Mutation.

(iii)

Again, for each individual of the population an independent numerical simulation of the process is performed. Each simulation will produce a final product with an associated fitness value.

(iv)

The optimisation program checks if the stopping criterion is satisfied. If the convergence condition is not satisfied, the iterative process continues from (ii); if the convergence condition is satisfied the design objectives are met and the iterative process stops.

A V-bending test application is presented to demonstrate the applicability of the proposed method considering several relevant parameters including punch and die radii, punch and die angles and blank-holder force. The optimised results approximate the desired V-shape.

The presented results also suggest that more process parameters should be added to the design optimisation vector in other to obtain optimal shapes closer to the desired ones.

1

R.H. Wagoner, W. Gan, K. Mao, S. Price, F. Rasouli, "Design of Sheet Forming Dies for Springback Compensation", The 6th International ESAFORM Conference on Material Forming, Salerno, Italy, April 28-30, 7-14, 2003.

2

C.C. António, C.F. Castro, L.C. Sousa, "Optimization of metal forming processes", Computers and Structures, accepted for publication, 2004. doi:10.1016/S0045-7949(04)00128-2

3

C.F. Castro, C.A.C. António, L.C. Sousa, "Optimisation of shape and processes parameters in metal forging using genetic algorithms", J. Materials Proc. Technology, 146, 356-364, 2004. doi:10.1016/j.jmatprotec.2003.11.027

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