In this work, we propose and test a method to expedite Global Sensitivity Analysis (GSA) in the context of shape optimisation of free-form shapes. To leverage the computational burden that is likely to occur in engineering problems, we construct a Shape-Signature-Vector (SSV) and propose to use it as a substitute for physics. SSV is composed of shapes’ integral properties, in our case geometric moments and their invariants of varying order, and is used as quantity-of-interest (QoI) for prior estimation of parametric sensitivities. Opting for geometric moments is motivated by the fact that they are intrinsic properties of shapes’ underlying geometry, and their evaluation is essential in many physical computations as they act as a medium for interoperability between geometry and physics. The proposed approach has been validated in the area of computer-aided ship design with regard to the capability of global- and composite-SSV to reveal parametric sensitivities of different ship hulls for the wave-making resistance coefficient (Cw), which is a critical QoI towards improving ship’s efficiency and thus decreasing emissions. More importantly, the longitudinal distribution of the volume below the ship’s floating waterline, which is measurable via geometric moments, has an impact on Cw. Through extensive experimentation, we show a strong correlation between the sensitive parameters obtained with respect to SSV and those based on Cw. Consequently, we can estimate parameters’ sensitivity with considerably reduced computational cost compared to when sensitivity analysis is performed with respect to Cw. Finally, two design spaces are constructed with sensitive parameters evaluated from SSV and Cw, and spaces’ quality and richness are analysed in terms of their capability to provide an optimised solution.

In shape optimisation problems, subspaces generated with conventional dimension reduction approaches often fail to extract the intrinsic geometric features of the shape that would allow the exploration of diverse but valid candidate solutions. More importantly, they also lack incorporation of any notion of physics against which shape is optimised. This work proposes a shape-supervised dimension reduction approach. To simultaneously tackle these deficiencies, it uses higher-level information about the shape in terms of its geometric integral properties, such as geometric moments and their invariants. Their usage is based on the fact that moments of a shape are intrinsic features of its geometry, and they provide a unifying medium between geometry and physics. To enrich the subspace with latent features associated with shape’s geometrical features and physics, we also evaluate a set of composite geometric moments, using the divergence theorem, for appropriate shape decomposition. These moments are combined with the shape modification function to form a Shape Signature Vector (SSV) uniquely representing a shape. Afterwards, the generalised Karhunen–Loève expansion is applied to SSV, embedded in a generalised (disjoint) Hilbert space, which results in a basis of the shape-supervised subspace retaining the highest geometric and physical variance. Validation experiments are performed for a three-dimensional wing and a ship hull model. Our results demonstrate a significant reduction of the original design space’s dimensionality for both test cases while maintaining a high representation capacity and a large percentage of valid geometries that facilitate fast convergence to the optimal solution. The code developed to implement this approach is available at https://github.com/shahrozkhan66/SSDR.git.

The paper presents a multi-fidelity global metamodel for expensive computer simulations, developed as an essential part of efficient simulation-based design optimization under uncertainty. High- and low-fidelity solvers are managed through a multi-fidelity adaptive sampling procedure. The multi-fidelity approximation is built as the sum of a low-fidelity-trained metamodel and the metamodel of the difference (error) between high- and low-fidelity simulations. The metamodels are based on dynamic stochastic radial basis functions, which provide the prediction along with the associated uncertainty. New training points are placed where the prediction uncertainty is maximum. The prediction uncertainty of both the low-fidelity and the error metamodel is considered for the adaptive refinement of the low- and high-fidelity training set, respectively. The method is demonstrated through three analytical test problems and one simple industrial application in ship hydrodynamics. The fitting error of the multi-fidelity metamodel is used as evaluation metric. The comparison with a high-fidelity-trained metamodel shows the effectiveness of the present method.