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.

[1]  W. Manning,et al.  Measuring overfitting and mispecification in nonlinear models , 2011 .

[2]  P. Villon,et al.  An Introduction to Moving Least Squares Meshfree Methods , 2002 .

[3]  Peter Z. G. Qian,et al.  CONSTRUCTION OF NESTED SPACE-FILLING DESIGNS , 2009, 0909.0598.

[4]  Piotr Breitkopf,et al.  Diffuse response surface model based on moving Latin hypercube patterns for reliability-based design optimization of ultrahigh strength steel NC milling parameters , 2011 .

[5]  Daniel M. Dunlavy,et al.  Formulations for Surrogate-Based Optimization with Data Fit, Multifidelity, and Reduced-Order Models , 2006 .

[6]  Michael S. Eldred,et al.  Sparse Pseudospectral Approximation Method , 2011, 1109.2936.

[7]  Peter Z. G. Qian.  Nested Latin hypercube designs , 2009 .

[8]  Roger Ghanem,et al.  Numerical solution of spectral stochastic finite element systems , 1996 .

[9]  Slawomir Koziel,et al.  Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction , 2010, J. Comput. Sci..

[10]  M. Lemaire,et al.  Stochastic finite element: a non intrusive approach by regression , 2006 .

[11]  P. Breitkopf,et al.  Numerical Methods in Contact Mechanics: Yastrebov/Numerical Methods in Contact Mechanics , 2013 .

[12]  Boxin Tang,et al.  NESTED SPACE-FILLING DESIGNS FOR COMPUTER EXPERIMENTS WITH TWO LEVELS OF ACCURACY , 2009 .

[13]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[14]  B. Sudret,et al.  Reliability-based design optimization using kriging surrogates and subset simulation , 2011, 1104.3667.

[15]  I. A. Burchitz,et al.  Numerical product design: Springback prediction, compensation and optimization , 2008 .

[16]  Tomas Jansson,et al.  Reliability analysis of a sheet metal forming process using Monte Carlo analysis and metamodels , 2008 .

[17]  Roger Ghanem,et al.  Ingredients for a general purpose stochastic finite elements implementation , 1999 .

[18]  D. Wei,et al.  Optimization and tolerance prediction of sheet metal forming process using response surface model , 2008 .

[19]  Hengan Ou,et al.  Finite element modelling and optimisation of net-shape metal forming processes with uncertainties , 2012 .

[20]  W. T. Martin,et al.  Transformations of Weiner Integrals Under Translations , 1944 .

[21]  Olivier P. Le Maître,et al.  Polynomial chaos expansion for sensitivity analysis , 2009, Reliab. Eng. Syst. Saf..

[22]  Z. J. Yang,et al.  Study on structure of AlGaN on AlN interlayer by synchrotron radiation XRD and RBS , 2007 .

[23]  Raphael T. Haftka,et al.  Multi-fidelity design of stiffened composite panel with a crack , 2002 .

[24]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[25]  P. Villon,et al.  Moving least squares response surface approximation: Formulation and metal forming applications , 2005 .

[26]  Alexander I. J. Forrester,et al.  Multi-fidelity optimization via surrogate modelling , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  P. A. Newman,et al.  Optimization with variable-fidelity models applied to wing design , 1999 .

[28]  Johann Sienz,et al.  Formulation of the Optimal Latin Hypercube Design of Experiments Using a Permutation Genetic Algorithm , 2004 .

[29]  R. Lewis,et al.  An Overview of First-Order Model Management for Engineering Optimization , 2001 .

[30]  Trevor Hastie,et al.  Linear Methods for Regression , 2001 .

[31]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[32]  Christopher J. Roy,et al.  A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing , 2011 .

[33]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[34]  Timothy W. Simpson,et al.  Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come - Or Not , 2008 .

[35]  George N. Karystinos,et al.  Fast generation of space-filling Latin hypercube sample designs. , 2010 .

[36]  R. Ghanem.  Probabilistic characterization of transport in heterogeneous media , 1998 .

[37]  C. Bucher,et al.  A fast and efficient response surface approach for structural reliability problems , 1990 .

[38]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[39]  George N. Karystinos,et al.  GENERATING A MAXIMALLY SPACED SET OF BINS TO FILL FOR HIGH DIMENSIONAL SPACE-FILLING LATIN HYPERCUBE SAMPLING , 2011 .

[40]  T. Hesterberg,et al.  Least angle and ℓ1 penalized regression: A review , 2008, 0802.0964.

[41]  Bernard Rolfe,et al.  Multivariate modelling of variability in sheet metal forming , 2008 .

[42]  P. A. Newman,et al.  Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models , 2001 .

[43]  Piotr Breitkopf,et al.  A two-pronged approach for springback variability assessment using sparse polynomial chaos expansion and multi-level simulations , 2014 .

[44]  G. Venter,et al.  An algorithm for fast optimal Latin hypercube design of experiments , 2010 .

[45]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[46]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

[47]  Michael S. Eldred,et al.  Second-Order Corrections for Surrogate-Based Optimization with Model Hierarchies , 2004 .

[48]  Bruno Sudret,et al.  Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach , 2008 .

[49]  J.W. Bandler,et al.  Space mapping optimization of microwave circuits exploiting surrogate models , 2000, 2000 IEEE MTT-S International Microwave Symposium Digest (Cat. No.00CH37017).

[50]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[51]  Guangyao Li,et al.  Multi-fidelity optimization for sheet metal forming process , 2011 .

[52]  H. Elman,et al.  DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS , 2008 .

[53]  G. Gary Wang,et al.  Review of Metamodeling Techniques in Support of Engineering Design Optimization , 2007 .

[54]  Soo-Chang Kang,et al.  An efficient response surface method using moving least squares approximation for structural reliability analysis , 2010 .