Abstract
Parametric shape optimization aims at minimizing an objective function f(x) where x are CAD parameters. This task is difficult when f(⋅) is the output of an expensive-to-evaluate numerical simulator and the number of CAD parameters is large. Most often, the set of all considered CAD shapes resides in a manifold of lower effective dimension in which it is preferable to build the surrogate model and perform the optimization. In this work, we uncover the manifold through a high-dimensional shape mapping and build a new coordinate system made of eigenshapes. The surrogate model is learned in the space of eigenshapes: a regularized likelihood maximization provides the most relevant dimensions for the output. The final surrogate model is detailed (anisotropic) with respect to the most sensitive eigenshapes and rough (isotropic) in the remaining dimensions. Last, the optimization is carried out with a focus on the critical dimensions, the remaining ones being coarsely optimized through a random embedding and the manifold being accounted for through a replication strategy. At low budgets, the methodology leads to a more accurate model and a faster optimization than the classical approach of directly working with the CAD parameters.
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Notes
Even if pruning the αj components for \(j>d^{\prime }\) (see comments at the end of Section 2.2), \(n<d^{\prime }\) may hold.
As explained at the end of Section 2, we can restrict all calculations to α’s \(d^{\prime }\) first coordinates. Even though \(d^{\prime }\ll D\), it has approximately the same dimension as d, hence the optimization is still carried out in a high-dimensional space.
In this article, the mapping T(⋅) is the composition of ϕ(⋅) with the projection onto a subspace of \((\mathbf v^{1},\dotsc ,\mathbf v^{D})\).
Since we do not know the convexity of \(\mathcal A\), the projection might not be unique.
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Funding
This research was partly funded by a CIFRE grant (convention #2016/0690) established between the ANRT and the Groupe PSA for the doctoral work of David Gaudrie.
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Replication of results
Comparisons between variants of EGO algorithms working in the original X space or in a reduced eigencomponents space have been presented in Sections 3 and 4. Two examples out of three are analytical and easily reproducible. To facilitate a replication of results, the pseudo-code of the final approach we propose, AddGP(\(\boldsymbol \alpha ^{a} + \boldsymbol \alpha ^{\overline {a}}\))-EI embed with replication, is given in Section 5. The penalized maximum likelihood was implemented in the R language extending the kergp package. The additive GP was built using the kergp package. The maximization of the EI was carried out by the genoud package.
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Gaudrie, D., Le Riche, R., Picheny, V. et al. Modeling and optimization with Gaussian processes in reduced eigenbases. Struct Multidisc Optim 61, 2343–2361 (2020). https://doi.org/10.1007/s00158-019-02458-6
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DOI: https://doi.org/10.1007/s00158-019-02458-6