Parameter identification and shape optimization

Simulation of metal forming processes using the Finite Element Method (FEM) is a well established procedure, being nowadays possible to develop alternative approaches, such as inverse methodologies, in solving complex problems. In the present paper, two types of inverse approaches will be discussed, namely the parameter identification and the shape optimization problems. The aim of the former is to evaluate the input parameters for material constitutive models that would lead to the most accurate set of results respecting physical experiments. The second category involves determining the initial geometry of a given specimen leading to a desired final geometry after the forming process. The purpose of the present work is then to formulate these inverse problems as optimization problems, introducing a straightforward methodology of process optimization in engineering applications such as metal forming and structural analysis. To reach this goal, an integrated optimization approach, using a finite element code together with a numerical optimization program, was employed. A gradient-based optimization method, as a combination of the steepest-descent method and the Levenberg-Marquardt techniques, was used. Numerical applications in the parameter optimization category include, namely, the characterization of a non-linear elasto-plastic hardening model and the determination of the parameters for a nonlinear hyperelastic model. It is also discussed the simultaneous identification of both constitutive material model parameters and the friction coefficient parameters. From the point of view of shape optimization problems, the determination of the initial geometry of a specimen in a upsetting billing problem as well as a methodology for defining the most suited blank shape to be formed in a square cup, are discussed. The final results for both categories show that this kind of algorithms have great potential for future developments in more demanding and realistic benchmarks. It is also worth noting that the presented integrated methodology can be easily applied to a first introduction of optimization techniques and numerical simulation to undergraduate courses in engineering.