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Submitted on 26 Nov 2018
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Dimension Reduction for Shape Optimization
David Gaudrie, Rodolphe Le Riche, Victor Picheny, Benoît Enaux, Vincent Herbert
To cite this version:
David Gaudrie, Rodolphe Le Riche, Victor Picheny, Benoît Enaux, Vincent Herbert. Dimension Reduction for Shape Optimization. Journées de la Chaire Oquaido 2018, Nov 2018, Cadarache, France. �hal-01934569�
Dimension Reduction for Shape Optimization
David Gaudrie
1,2, Rodolphe le Riche
2,3, Victor Picheny
4, Benoît Enaux
1, Vincent Herbert
11
Groupe PSA,
2Mines Saint-Étienne,
3CNRS LIMOS,
4Prowler.io
Context and past work
Multi-objective optimization of high dimen-sional systems
min
x∈X⊂Rd
(f1(x), . . . , fm(x))
Very tiny budget (≈ 100-200 evaluations), many objectives (m ≈ 6-8) ⇒ impossible for classical MO-EGO approaches to uncover the Pareto
Front (growing size of PY with m)
⇒ Target well-chosen parts of PY [1]
But what when d is large (& 50)?
CAD parameters and shapes
• x ∈ Rd design parameters
• Associated shapes Ωx approximately live
in a δ < d dimensional space
• Computation time of Ωx and φ(x)
negligi-ble compared to evaluation of f (x)
High-dimensional mapping
• Analyze many possible shapes ΩΩΩ :=
{Ωx, x ∈ X} in a high-dimensional space
Φ ⊂ RD, d << D
• Retrieve the δ dimensional manifold
em-bedded in RD
• Build the kriging surrogates and perform the optimization in this manifold
High dimensional representation of Ωx
φ : X → Φ
x 7→ φ(x)
• Characteristic function χΩx
• Signed distance to ∂Ωx
• Discretization of ∂Ωx
Choice of mapping φ is critical for identifying Ξ
Reduction by PCA in shape space Following the work of [2, 3, 4]
• Draw N designs x(i) ∈ X and compute
ΦΦΦ := (φ(x(i))>)i=1,...,N
• Analyze the variety of shapes by applying
a PCA on ΦΦΦ: the eigenvectors of ΦΦΦ>ΦΦΦ, vj
form a shape basis, ΦΦΦ(i) = ΦΦΦ+PDj=1 α(i)j vj
• Instead of x(i), work with its eigenbasis
coordinates: ααα(i) := (α(i)1 , . . . , α(i)D )>
• ααα(i), i = 1, . . . , N form a δ-dimensional
manifold Tests cases ● x1 ● x1 x2 ● x1 x2 x3 Characteristic function 10 5 Dim.2 0 -5 -10 -15 -20 -10 0 10 20 -10 -5 0 5 Dim.3 Dim.1 -30 10 5 0 Dim.2-5 -10 -15 -20 -10 0 10 20 -10 0 10 Dim.3 Dim.1 -30 15 10 5 Dim.2 0 -5 -10 -15 -10 -5 0 5 10 -10 -5 0 Dim.3 Dim.1 5 -15 Signed distance 4 2 Dim.2 0 -2 -4 -20 0 20 40 -4 -2 0 2 Dim.3 Dim.1 4 -40 15 10 5 Dim.2 0 -5 -10 -15 -20 0 20 40 -6 -4 -2 0 Dim.3 Dim.1 2 -40 15 10 5 Dim.2 0 -5 -10 -15 -10 0 10 20 -10 -5 0 5 Dim.3 Dim.1 10 -20 Discretization 4 2 Dim.2 0 -2 -4 -5 0 5 -4 -2 0 2 Dim.3 Dim.1 4 3 2 1 Dim.2 0 -1 -2 -3 -4 -2 0 2 4 6 -4 -2 0 2 4 Dim.3 Dim.1 -6 4 2 Dim.2 0 -2 -4 -2 0 2 4 -2 -1 0 1 Dim.3 Dim.1 2 -4
• Dimension reduction: ΦΦΦ(i) approximated
by retaining δ principal components from their eigenvalue contribution
ΦΦΦ(i) ≈ ΦΦΦ + Pδj=1 α(i)j vj
References
[1] D. Gaudrie, R. Le Riche, V. Picheny, B. Enaux and V. Herbert, Budgeted Multi-Objective Optimization with a Focus on the Central Part of the Pareto Front -Extended Version, arXiv preprint 1809.10482 (2018). [2] C. Goodall, Procrustes methods in the statistical analysis of shape, Journal of the Royal Statistical Society, Series B (Methodological), 285-339 (1991). [3] M. Stegmann and D. Gomez, A brief introduction
to statistical shape analysis, Informatics and mathe-matical modelling, Technical University of Denmark, DTU, 15(11), 2002.
[4] B. Raghavan, G. Le Quilliec, P. Breitkopf, A. Rassineux, J. M. Roelandt and P. Villon, Numerical assessment of springback for the deep drawing process by level set interpolation using shape manifolds, International journal of material forming, 7(4), 487-501 (2014).
Kriging in reduced basis
• φ = shape discretization: most efficient mapping to retrieve δ
• Principal axis: eigenshapes
Kriging on principal components: test cases
(a) Over-parameterized circle, x ∈ R39
→ δ = 3
(b) Meta NACA 3 dimensions, x ∈ R3
→ δ = 3
(c) Meta NACA 22 dimensions (bumpy
air-foil), x ∈ R22 → δ = 3, 6, 20
Conclusions
• Lower/true dimension retrieved
• Enhanced predictability in the reduced eigenbasis
• Shape discretization: better than charac-teristic function and signed distance
Further work
• Invariance of shape discretization under some permutations (points indexing, mul-tiple shapes)
• Use this framework to perform (multi-objective) optimization
