Réduction de modèles pour des problèmes d optimisation et d identification en calcul de structures Piotr Breitkopf Rajan Filomeno Coelho, Huanhuan Gao, Anna Madra, Liang Meng, Guénhaël Le Quilliec, Balaji Raghavan, Liang Xia, Manyu Xiao Sorbonne Universités, Université de Technologie de Compiègne Laboratoire Roberval, UMR 7337 UTC-CNRS Séminaire Aristote, 8 novembre 2016, École Polytechnique, Palaiseau.
An example of a complex shape to be parameterized How to find the lowest number of parameters that allow to obtain all the «admissible» shapes? 2
Two kinds of problems t=1 t=2 t=3? t=4 t=5 IdenMficaMon interpolamon t=? t=t new 3
Parameterization issues - high number of design variables - implicit technological constraints - intrinsic dimensionality much lower then the number of variables 4
Problems of interest Highly dimensional input spaces : reduce the number of parameters (highly constrained CAD) [Raghavan et al., 2013] Find the smallest number of parameters for «natural» shapes (biomedical, post-springback) [Gonzalez et al., 2015] [Le Quilliec et al., 2015] Characterize measured shapes (Image correlation) [Meng et al., 2015] [Madra 5 et al., 2015]
Shape space Problem: dimension of the shape space is generally very high 6
Discrete shape manifold Hypothesis: all admissible shapes belong to a lower-dimensional manifold embedded in shape space 7
Smooth shape manifold All intermediate shapes may be obtained by interpolamon on the manifold 8
Interpolation in shape space Cartesian distance has to be replaced by geodesic distance 9
Projection in shape space Experimentally measured shapes generally do not belong to the manifold due to the noise
Taxonomy of NLDR methods Proper orthogonal decomposition (POD) covariance matrix of data points, and computes a position for each point. Multidimensional Scaling (MDS) matrix of pair-wise distances between all points, and computes a position for each point. Isomap (Tenenbaum, de Silva, and Langford) generalization of MDS assumes that the pair-wise distances are only known between neighboring points > pair-wise geodesic distances Locally-Linear Embedding (LLE, Roweis and Saul) find the low-dimensional embedding of points, such that each point is still described with the same linear combination of its neighbors Other methods Laplacian eigenmaps kernel PCA Hessian LLE Manifold unfolding... Proposed approach explicit nonlinear manifold construction based on local information intrinsic dimensionality estimation custom-built manifold walking algorithms 11
Agenda Explicit construction of the manifold in reduced shape space admissible shape Tools hypothesis of existence of a smooth manifold intrinsic dimensionality and intrinsic variables Predictor-corrector manifold walking techniques Level set representation of structural shapes POD Fukunaga Olsen (also LLE under development) Diffuse Approximation Examples shape optimization springback minimization (talk by Guénhaël Le Quilliec) inverse problem (talk by Meng Liang) optimization with categorical variables (talk by Manyu Xiao) 12
Level set surface representation 13
Proper Orthogonal Decomposition 14
Estimation/detection of dimensionality p 1 α 6 0 1 2 1 0 1 2 α 5 2 1 0 1 2 3 α 4 15
Diffuse Approximation of the manifold 1 α 6 0 1 2 1 0 1 2 α 5 2 1 0 1 2 3 α 4 16
Manifold walking algorithms Target shape: α T Local manifold (I =1) α 3 Projection giving: P 1 α 1 α 2 Global manifold (unknown) 17
Manifold walking algorithms Target shape: α T Local manifold (I =2) Projection giving: P 2 α 3 α 1 α 2 Global manifold (unknown) 18
Manifold walking algorithms Target shape: α T Local manifold (I =3) Projection giving: P 3 α 3 α 1 α 2 Global manifold (unknown) 19
Manifold walking algorithms Here P 4 P 3 convergence. Target shape: α T Local manifold (I =4) Projection giving: P 4 α 3 Preserved points from I =3 α 1 α 2 Global manifold (unknown) 20
test case 1 : 3D shape optimization CAD design 92 parameters unknown number of implicit design constraints admissible shape criterion CAD/mesh generator success 8% success rate in DOE 21
test case 2 : Quantification of springback 22
Example on a 3D complex shape: automotive strut tower punch. 23
Optimization with 2 tool geometry design variables Target Optimum 2D α-manifold 2D α-manifold Target α 3 0 α 3 Optimum 10 α 2 0 10 20 α 1 30 20 10 0 10 20 0 α 4 0 20 10 0 10 α 2 Target shape Limitation plane z 100 50 0 0 50 100 150 200 250 x y 100 50 0 50 100 Optimal zero level set 24
Material characterization by image correlation and inverse analysis Instrumented indentation test F F(N) 200 150 100 50 35CD4 a g b c d e f h i h R=0,5 mm Experimental setup 0 0 10 20 30 40 P-h curve h(µm) Indenter & Specimen scanning device Imprint shape 25
experimental imprint shape (C100 steel) 0.04 0.02 0-0.02 undeformed shape piling up -0.04 max depth -0.06-0.08-0.1 Scanned experimental imprint -0.12-0.14 0 0.1 0.2 0.3 0.4 0.5 u u Imprint shape voxelized with resolumon 200X200 Step length of scanning is 10µm; 26
Finite element simulation SimulaMon Parameters : Coulomb fricmon coefficient: Maximum force: F max = 500 N. Geometric parameters: µ = 0.1; Dimensions of specimen: 59mm 59mm Radius of Indenter: r = 0.5mm SimulaMon informamon: u u Four-node axisymmetric elements(cax4) Elements: 4394 (specimen) and 6070 (indenter) Postulated consmtumve law An isotropic Hollomon s law with two parameters E σ = σ y work-hardening exponent σ y yield stress σ E = 210GPa; y [50,400] n ε n n [0.1,0.5], υ = 0.3; 27
Manifold of admissible shapes 0.04 0.02 undeformed shape FE h 0-0.05-0.1 x imprint -0.15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 α 3 0-0.02-0.04-0.06-0.08-0.1-0.12-0.14 exp Scanned experimental imprint 0 0.1 0.2 0.3 0.4 0.5 c 2 c 1 * c 2 * c * O 3 2 1 c 2 * * α 2 c 0 c * 1 Design space c 1 α- α 1 Manifold in shape space 28
Panning & zooming CombinaMon of zooming and panning (Table.3). 29
Zooming convergence of snapshots Experimental imprint and numerical snapshots (zooming algorithm). 30
Smoothing aspects 31
Panning & zooming numerical results 32
Conclusion and prospects A machine learning approach for representation of complex shapes concepts shape space shape manifold intrinsic dimensionality a family of manifold walking algorithms applications Extensions metal forming inverse identification problems shape optimization discrete optimization with categorical variables [Gao et al, 2016] Towards simultaneous reduction of both input and output spaces for interactive simulationbased structural design [Raghavan et al. 2013] 33
Thank you for your attention 34