Dynamic mode decomposition (DMD) is a model reduction algorithm developed by Schmid (Schmid 2010). Since then has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. It is used for a data-driven model simplification based on spatiotemporal coherent structures. DMD relies only on the high-fidelity measurements, like experimental data and numerical simulations, so it is an equation-free algorithm. Its popularity is also due to the fact that it does not make any assumptions about the underlying system. See (Kutz et al. 2016) for a comprehensive overview of the algorithm and its connections to the Koopman-operator analysis, initiated in (Koopman 1931), along with examples in computational fluid dynamics. In the last years many variants arose, such as multiresolution DMD, compressed DMD, forward backward DMD, and higher order DMD among others, in order to deal with noisy data, big dataset, or spurius data for example.

We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameter space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes and to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

Model Order Reduction, roughly speaking, can be summarized in two parts: an offline and an online part. In the first, one constructs the database of solutions, or snapshots, for proper selected parameters. In the latter the database is used for a fast evaluation of the quatity of interest for a new parameter (see for example Schilders, Van der Vorst, and Rommes 2008; Hesthaven et al. 2016). Choices can be made either for the selection of the parameters in the construction of the database or on how the database is used to approximate the manifold of the snapshots.