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Numerical simulations have long been used in many engineering process industries for the virtual evaluation of responses from an accurate solution of the physics/mathematical models describing it. However, in most industrial cases, these simulations tend to be time-consuming, especially in studies involving parameter optimization studies/inverse analysis where a multitude of direct computations are necessary. Non-intrusive Model Order Reduction techniques (SSL-PGD, s-PGD) enable the construction of parametric solution spaces to ensure almost instantaneous responses to changes in input parameters. These parametrized solutions are built from snapshots generated from simulations using commercially available software, but the process parameters and re-meshing rules add a condition that the different snapshots be suitably parametrized for being able to compute and represent the parametric solutions using the MOR techniques. The snapshots generated in the case of sheet-metal forming simulations are 3D surfaces. Even though surface parametrization is a fairly easy process in the case of planar surfaces, the problem ceases to be trivial in the case of 3D surfaces. In this work, we present a conformal mapping procedure based on the Riemannian surface parametrization using the Yamabe-flow for the parametrization of 3D surfaces into planar 2D surfaces. These parametrized surfaces can then be interpolated to create parametric solution configurations for the full range of input parameters and hence allows the application of non-intrusive model order reduction techniques to simulations which include adaptive mesh refinement.

This work retraces the main recent advances in the so-called non-intrusive model order reduction, and more concretely, the construction of parametric solutions related to parametric models, with special emphasis on the technologies enabling allying accuracy, frugality and robustness. Thus, different technologies will be revisited beyond the usual metamodeling techniques making use of polynomial basis or kriging, for addressing multi-parametric models, with sometimes several tens of parameters, while keeping the complexity (DoE size) scaling with the number of parameters. Moreover, sparsity can be profitable for increasing accuracy while avoiding overfitting, and when combined with ANOVA-based decompositions the benefits are potentially huge.

The present work focuses on the reduced modelling of multi-component systems, in particular on a two stages stamping chain process. Starting from snapshots collected by using the commercial software PAM-STAMP, the non-intrusive sparse-PGD constructor is used in order to build a parametric transfer function of each operation in a separated representation, circumventing the problem of the curse of dimensionality. Moreover, in order to fill the gap between this deterministic solution and the measured fields and safely applied control strategies, data driven-modeling can be employed according to the new “hybrid twin” methodology.