Guenhael Le Quilliec

Surrogate POD Models for Parametrized Sheet Metal Forming Applications

The aim of this work is to present a POD (Proper Orthogonal Decomposition) based surrogate approach for sheet metal forming parametrized applications. The final displacement field for the stamped work-piece computed using a finite element approach is approximated using the method of snapshots for POD mode determination and kriging for POD coefficients interpolation. An error analysis, performed using a validation set, shows that the accuracy of the surrogate POD model is excellent for the representation of finite element displacement fields. A possible use of the surrogate to assess the quality of the stamped sheet is considered. The Green-Lagrange strain tensor is derived and forming limit diagrams are computed on the fly for any point of the design space. Furthermore, the minimization of a cost function based on the surrogate POD model is performed showing its potential for solving optimization problems.

Springback assessment based on level set interpolation and shape manifolds in deep drawing

In this paper, we introduce an original shape representation approach for automatic springback characterization. It is based on the generation of parameterized Level Set functions. The central idea is the concept of the shape manifold representing the design domain in the reduced-order shape-space. Performing Proper Orthogonal Decomposition on the shapes followed by using the Diffuse Approximation allows us to efficiently reduce the problem dimensionality and to interpolate uniquely between admissible input shapes, while also determining the smallest number of parameters needed to characterize the final formed shape. We apply this methodology to the problem of springback assessment for the deep drawing operation of metal sheets.

Fat Latin Hypercube Sampling and Efficient Sparse Polynomial Chaos Expansion for Uncertainty Propagation on Finite Precision Models: Application to 2D Deep Drawing Process

In the context of uncertainty propagation, the variation range of random variables may be many oder of magnitude smaller than their nominal values. When evaluating the non-linear Finite Element Model (FEM), simulations involving contact/friction and material non linearity on such small perturbations of the input data, a numerical noise alters the output data and distorts the statistical quantities and potentially inhibit the training of Uncertainty Quantification (UQ) models. In this paper, a particular attention is given to the definition of adapted Design of Experiment (DoE) taking into account the model sensitivity with respect to infinitesimal numerical perturbations. The samples are chosen using an adaptation of the Latin Hypercube Sampling (Fat-LHS) and are required to be sufficiently spaced away to filter the discretization and other numerical errors limiting the number of possible numerical experiments. In order to build an acceptable Polynomial Chaos Expansion with such sparse data, we implement a hybrid LARS+Q-norm approach. We showcase the proposed approach with UQ of springback effect for deep drawing process of metal sheet, considering up to 8 random variables.

Surrogate POD models for building forming limit diagrams of parameterized sheet metal forming applications

The aim of this work is to present a surrogate POD (Proper Orthogonal Decomposition) approach for building forming limit diagrams at minimum cost for parameterized sheet metal formed work-pieces. First, a Latin Hypercube Sampling is performed on the design parameter space. Then, at each design site, displacement fields are computed using the popular open-source finite element software Code_Aster. Then, the method of snapshots is used for POD mode determination. POD coefficients are interpolated using kriging. Furthermore, an error analysis of the surrogate POD model is performed on a validation set. It is shown that on the considered use case the accuracy of the surrogate POD model is excellent for the representation of finite element displacement fields. The validated surrogate POD model is then used to build forming limit diagrams (FLD) for any design parameter to assess the quality of stamped metal sheets. Using the surrogate POD model, the Green-Lagrange strain tensor is derived, then major and minor principal deformations are determined at Gauss points for each mesh element. Furthermore, a signed distance between the forming limit curve in rupture and the obtained cloud of points in the plane (e2, e1) is computed to assess the quality of the formed workpiece. The minimization of this signed distance allows determining the safest design for the chosen use case.

A two-pronged approach for the assessment of springback variability for sheet metal forming applications

Springback assessment for sheet metal forming processes is a challenging issue which requires to take into account complex phenomena (physical non linearities and uncertainties). We highlight that the stochastic analysis of metal forming process requires both a high precision and low cost numerical models and propose a two-pronged methodology to address these challenges. The deep drawing simulation process is performed using an original low cost semi-analytical approach based on a bending under tension model with a good accuracy for small random perturbations of the physical and process parameters. The springback variability analysis is performed using an efficient stochastic metamodel, namely a sparse version of the polynomial chaos expansion.