## Abstract

Parametric Sensitivity Analysis (PSA) investigates the sensitivity of parameters, defining the design space of a shape-optimisation problem, for tackling the challenges of the curse of dimensionality or decreasing the uncertainty in design's performance. However, the analytical implementation of PSA can often be tricky, especially if the chosen method requires the evaluation of high-dimensional integrals or if the baseline simulation codes do not provide an analytical solution to design performance. Therefore, PSA needs to be implemented with sampling methods, such as Monte Carlo sampling, which is highly susceptible to slow convergence and necessitates a sufficiently large number of samples for stable results, especially for high-dimensional problems.

In this work, we aim to address above the mentioned challenges associated with PSA by offloading the evaluation of parametric sensitivities from physical quantities to quantities, which are relatively inexpensive but, like physical metrics, provide important clues about the form, distribution and validity of the design. It is well known that shape's integral properties, such as geometric moments and their invariants serve as a geometric foundation for different designs' physical analyses. In this connection, we propose a geometric moment-dependent PSA approach, that harnesses the geometric variation of designs in the design space using geometric moments as a quantity of interest (QoI) to identify parametric sensitivities. These results can serve as prior estimates of parametric sensitivities with respect to physics. The selection of geometric moments in our work is motivated by the following baseline insights:

- It is very likely that physics analysis requires the computation of such integral properties of the geometry such as the stiffness and mass matrix, and moments of a domain are sufficient to ensure accurate integration of a large class of integrands.

- Like physics, geometric moments can also act as a compact shape signature or descriptor to a specific design falling in a specific category, which facilitates various shape processing tasks.

To validate our approach and experimentally demonstrate the effectiveness of geometric moments, we used two ship hulls parameterised with 27 and 26 parameters using two different techniques based on \textit{procedural deformation} (PD) and global modification function (GMF), respectively. In this setting, we use the {\it wave-resistance coefficient (C_w)} as the physical QoI, as it plays a crucial role in ship hull design. The longitudinal distribution of the hulls' geometry has a similar impact on geometric moments as C_w.

To commence, we construct the so-called shape-signature vector (MI^s), that will be used as shape descriptor and contains all the geometric moments up to s order. To align better with C_w, all moments in this vector are taken invariant with respect to translation and scaling. A Global Variance-Based Sensitivity Analyses (GVBSA) is performed for learning parametric sensitivities with respect to MI^s and C_w. Here MI^s is purely a vector quantity containing the moments of various orders while C_w is a scalar and computationally expensive one. Therefore, learning sensitivities to MI^s requires implementing a multivariate extension of GVBS, such as covariance decomposition, which provides generalised sensitivity indices of design parameters to all moments in MI^s.

The results from the experiments conducted in this study show a good correlation between the sensitive parameters obtained from C_w and MI^s, specifically with the fourth-order shape-signature vector MI^4. In the case of the PD-based hull, 7 parameters sensitive to MI^4 are also among the 8 parameters sensitive to C_w. Interestingly, similar results are obtained for the GMF-based hull, where 6 out of 7 sensitive parameters to C_w are also sensitive to MI^4. Afterwards, two different design spaces are constructed for both hull models, one with sensitive parameters obtained with C_w and the other with mathcalMI^4. Shape optimisation is performed in both spaces performed with a meta-heuristic optimisation approach. Final optimisation results showed that the design generated from design space constructed with sensitive parameters of C_w and MI^4 for both types of hulls offer similar performance. These results indicate that PSA performed with moments can reasonably estimate parameters' sensitivity to the design's physics with considerably reduced computational cost.

In this work, we aim to address above the mentioned challenges associated with PSA by offloading the evaluation of parametric sensitivities from physical quantities to quantities, which are relatively inexpensive but, like physical metrics, provide important clues about the form, distribution and validity of the design. It is well known that shape's integral properties, such as geometric moments and their invariants serve as a geometric foundation for different designs' physical analyses. In this connection, we propose a geometric moment-dependent PSA approach, that harnesses the geometric variation of designs in the design space using geometric moments as a quantity of interest (QoI) to identify parametric sensitivities. These results can serve as prior estimates of parametric sensitivities with respect to physics. The selection of geometric moments in our work is motivated by the following baseline insights:

- It is very likely that physics analysis requires the computation of such integral properties of the geometry such as the stiffness and mass matrix, and moments of a domain are sufficient to ensure accurate integration of a large class of integrands.

- Like physics, geometric moments can also act as a compact shape signature or descriptor to a specific design falling in a specific category, which facilitates various shape processing tasks.

To validate our approach and experimentally demonstrate the effectiveness of geometric moments, we used two ship hulls parameterised with 27 and 26 parameters using two different techniques based on \textit{procedural deformation} (PD) and global modification function (GMF), respectively. In this setting, we use the {\it wave-resistance coefficient (C_w)} as the physical QoI, as it plays a crucial role in ship hull design. The longitudinal distribution of the hulls' geometry has a similar impact on geometric moments as C_w.

To commence, we construct the so-called shape-signature vector (MI^s), that will be used as shape descriptor and contains all the geometric moments up to s order. To align better with C_w, all moments in this vector are taken invariant with respect to translation and scaling. A Global Variance-Based Sensitivity Analyses (GVBSA) is performed for learning parametric sensitivities with respect to MI^s and C_w. Here MI^s is purely a vector quantity containing the moments of various orders while C_w is a scalar and computationally expensive one. Therefore, learning sensitivities to MI^s requires implementing a multivariate extension of GVBS, such as covariance decomposition, which provides generalised sensitivity indices of design parameters to all moments in MI^s.

The results from the experiments conducted in this study show a good correlation between the sensitive parameters obtained from C_w and MI^s, specifically with the fourth-order shape-signature vector MI^4. In the case of the PD-based hull, 7 parameters sensitive to MI^4 are also among the 8 parameters sensitive to C_w. Interestingly, similar results are obtained for the GMF-based hull, where 6 out of 7 sensitive parameters to C_w are also sensitive to MI^4. Afterwards, two different design spaces are constructed for both hull models, one with sensitive parameters obtained with C_w and the other with mathcalMI^4. Shape optimisation is performed in both spaces performed with a meta-heuristic optimisation approach. Final optimisation results showed that the design generated from design space constructed with sensitive parameters of C_w and MI^4 for both types of hulls offer similar performance. These results indicate that PSA performed with moments can reasonably estimate parameters' sensitivity to the design's physics with considerably reduced computational cost.

Original language | English |
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Number of pages | 15 |

Publication status | Published - 24 Nov 2021 |

Event | Dagstuhl Seminar 21471: Geometric Modeling: Interoperability and New Challenges - Schloss Dagstuhl, Wardern, Germany Duration: 21 Nov 2021 → 26 Nov 2021 https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=21471 |

### Conference

Conference | Dagstuhl Seminar 21471 |
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Abbreviated title | Dagstuhl Seminar 21471 |

Country/Territory | Germany |

City | Wardern |

Period | 21/11/21 → 26/11/21 |

Internet address |

## Keywords

- expensive physical models
- geometric moment invariants
- sensitivity analysis
- shape optimisation