Data-driven parameter and model order reduction for industrial optimisation problems with applications in naval engineering

In this work we study data-driven reduced order models with a specific focus on reduction in parameter space to fight the curse of dimensionality, especially for functions with low-intrinsic structure, in the context of digital twins. To this end we proposed two different methods to improve the accuracy of responce surfaces built using the Active Subspaces (AS): a kernel-based approach which maps the inputs onto an higher dimensional space before applying AS, and a local approach in which a clustering induced by the presence of a global active subspace is exploited to construct localized regressors. We also used AS within a multi-fidelity nonlinear autoregressive scheme to reduced the approximation error of high-dimensional scalar function using only high-fidelity data. This multi-fidelity approach has also been integrated within a non-intrusive Proper Oorthogonal Decomposition (POD) based framework in which every modal coefficient is reconstructed with a greater precision. Moving to optimization algorithms we devised an extension of the classical genetic algorithm exploiting AS to accelerate the convergence, especially for highdimensional optimization problems. We applied different combinations of such methods in a diverse range of engineering problems such as structural optimization of cruise ships, shape optimization of a combatant hull and a NACA airfoil profile, and the prediction of hydroacoustic noises. A specific attention has been devoted to the naval engineering applications and many of the methodological advances in this work have been inspired by them. This work has been conducted within the framework of the IRONTH project, an industrial Ph.D. grant financed by Fincantieri S.p.A

ATHENA: Advanced Techniques for High Dimensional Parameter Spaces to Enhance Numerical Analysis

ATHENA is an open source Python package for reduction in parameter space. It implements several advanced numerical analysis techniques such as Active Subspaces (AS), Kernel-based Active Subspaces (KAS), and Nonlinear Level-set Learning (NLL) method. It is intended as a tool for regression, sensitivity analysis, and in general to enhance existing numerical simulations' pipelines tackling the curse of dimensionality. Source code, documentation, and several tutorials are available on GitHub at https://github.com/mathLab/ATHENA under the MIT license

Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces

We propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape

A DeepONet multi-fidelity approach for residual learning in reduced order modeling

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented

An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment

In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is which is a common benchmark in ship hydrodynamics simulations

Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method

A new method to perform a nonlinear reduction in parameter spaces is proposed. By using a kernel approach it is possible to find active subspaces in high-dimensional feature spaces. A mathematical foundation of the method is presented, with several applications to benchmark model functions, both scalar and vector-valued. We also apply the kernel-based active subspaces extension to a CFD parametric problem using the Discontinuous Galerkin method. A full comparison with respect to the linear active subspaces technique is provided for all the applications, proving the better performances of the proposed method. Moreover we show how the new kernel method overcomes the drawbacks of the active subspaces application for radial symmetric model functions

Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition

Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise shi

PyGeM: Python Geometrical Morphing

PyGeM is an open source Python package which allows to easily parametrize and deform 3D object described by CAD files or 3D meshes. It implements several morphing techniques such as free form deformation, radial basis function interpolation, and inverse distance weighting. Due to its versatility in dealing with different file formats it is particularly suited for researchers and practitioners both in academia and in industry interested in computational engineering simulations and optimization studies