## Abstract

High-dimensional design spaces cause simulation-driven shape optimisation to suffer from the curse of dimensionality, resulting in high computational costs. In this work, we employ dimensionality reduction [1] and a Bayesian optimisation [2] approach to reduce the design space's dimensionality and ease its exploration while reducing the number of required design evaluations, respectively. At the first step, statistical dependencies implicit in the design parameters encode essential latent features of the underlining shape, which form a lower-dimensional subspace while maintaining the maximum geometric variance of the original space. The feature extraction typically commences with the set of design parameters used to define the baseline design under consideration. However, if the original space created by these parameters is an orthotope, then the latent features are meaningless as they only define a new orientation of the original design space without capturing any geometric variance. Therefore, dimensionality reduction is performed on a discretised version of the shape modification vector, which modifies the shape for any realisation of the design parameter, resulting in a lower-dimensional subspace explicitly inducing the maximum geometric variances.

Afterwards, this space is explored and exploited for shape optimisation using the Bayesian approach. Design instances during this optimisation phase are evaluated using an Isogeometric Analysis (IGA) hydrodynamic solver [3], which guides the optimisation towards the global optima. To evaluate a design instance with the IGA solver, an accurate surface representation of the design is needed, which requires solving the inverse problem of extracting the surface representation from the lower-dimensional subspace. First, the higher-dimensional discretised version of the design is generated from the subspace to solve this problem. Then a NURBS representation of the design is approximated using this discretisation. The resulting NURBS approximation is commonly a surface with a large number of control points that is directly linked to the number of degrees of freedom (DoFs) employed in the IGA-enabled analysis tool. Consequently, evaluation of such designs, with mainly redundant control points (and DoFs), is time-consuming and defies the benefit of an enhanced accuracy with a low number of DoFs offered by IGA approaches. The benefit of dimensionality reduction performed may disappear in such a scenario. Therefore, an iterative step is introduced, before passing the design to the solver, to reduce the number of surface control points (and obviously analysis model DoFs) while maintaining a predefined geometric tolerance that guarantees the required accuracy for our application. The proposed approach is tested for a military and a commercial ship model parameterised with 27 and 26 parameters. The optimisation problem is solved to minimise the ship's wave-making resistance subject to pertinent constraints in each of these cases. The experimental studies presented in this work show the beneficial effects of the proposed approach in comparison to conventional optimisation.

REFERENCES

[1] W. Chen, M. Fuge, & J. Chazan. Design manifolds capture the intrinsic complexity and dimension of design spaces. Journal of Mechanical Design, 139(5), 2017.

[2] R. Martinez-Cantin. BayesOpt: a Bayesian optimization library for nonlinear optimization, experimental design and bandits. J. Mach. Learn. Res., 15(1), pp. 3735-3739, 2014.

[3] K. V. Kostas, A. I. Ginnis, C. G. Politis, and P. D. Kaklis. Ship-hull shape optimization with a T-spline based BEM–isogeometric solver. Computer Methods in Applied Mechanics and Engineering, 284, pp. 611-622, 2015.

Afterwards, this space is explored and exploited for shape optimisation using the Bayesian approach. Design instances during this optimisation phase are evaluated using an Isogeometric Analysis (IGA) hydrodynamic solver [3], which guides the optimisation towards the global optima. To evaluate a design instance with the IGA solver, an accurate surface representation of the design is needed, which requires solving the inverse problem of extracting the surface representation from the lower-dimensional subspace. First, the higher-dimensional discretised version of the design is generated from the subspace to solve this problem. Then a NURBS representation of the design is approximated using this discretisation. The resulting NURBS approximation is commonly a surface with a large number of control points that is directly linked to the number of degrees of freedom (DoFs) employed in the IGA-enabled analysis tool. Consequently, evaluation of such designs, with mainly redundant control points (and DoFs), is time-consuming and defies the benefit of an enhanced accuracy with a low number of DoFs offered by IGA approaches. The benefit of dimensionality reduction performed may disappear in such a scenario. Therefore, an iterative step is introduced, before passing the design to the solver, to reduce the number of surface control points (and obviously analysis model DoFs) while maintaining a predefined geometric tolerance that guarantees the required accuracy for our application. The proposed approach is tested for a military and a commercial ship model parameterised with 27 and 26 parameters. The optimisation problem is solved to minimise the ship's wave-making resistance subject to pertinent constraints in each of these cases. The experimental studies presented in this work show the beneficial effects of the proposed approach in comparison to conventional optimisation.

REFERENCES

[1] W. Chen, M. Fuge, & J. Chazan. Design manifolds capture the intrinsic complexity and dimension of design spaces. Journal of Mechanical Design, 139(5), 2017.

[2] R. Martinez-Cantin. BayesOpt: a Bayesian optimization library for nonlinear optimization, experimental design and bandits. J. Mach. Learn. Res., 15(1), pp. 3735-3739, 2014.

[3] K. V. Kostas, A. I. Ginnis, C. G. Politis, and P. D. Kaklis. Ship-hull shape optimization with a T-spline based BEM–isogeometric solver. Computer Methods in Applied Mechanics and Engineering, 284, pp. 611-622, 2015.

Original language | English |
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Number of pages | 1 |

Publication status | Published - 27 Sep 2021 |

Event | Virtual International Conference on Isogeometric Analysis - Lyon, France Duration: 27 Sep 2021 → 29 Sep 2021 https://iga2021.sciencesconf.org/program/graphic/date/2021-09-28 |

### Conference

Conference | Virtual International Conference on Isogeometric Analysis |
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Abbreviated title | VIGA 2021 |

Country/Territory | France |

City | Lyon |

Period | 27/09/21 → 29/09/21 |

Internet address |

## Keywords

- shape modification
- shape optimization
- wave resistance
- hull design
- dimensionality reduction
- marine engineering