A reduced-order inverse distance weighting technique for the efficient mesh-motion of deformable interfaces and moving shapes in computational problems

Nowadays, despite the constant technological growth of the last twenty years, Computational Fluid Dynamics (CFD) problems represent a hard challenge for scientific computing because of their large demand on computational resources. In fact, in the context of aerodynamics optimization and design, CFD applications require to simulate many different possible realizations of a system which can become prohibitive in terms of computational and memory effort. These considerations have produced an intensive development of reduced order methods to provide high-fidelity simulations via efficient and low-dimensional models. In this thesis we focus on an efficient and flexible technique, namely the Inverse Distance Weighting (IDW). This strategy can be used to solve mesh motion problems in a Fluid-Structure Interaction (FSI) framework. IDW is an interpolation strategy which computes the displacement of the grid nodes starting from the movement of some data points, called control points. The original formulation of IDW uses as control points the nodes of the grid belonging to the interface and requires assembling a matrix [N_s × N_c], where N_s is the number of values to be interpolated and N_c indicates the number of control points. Since the number of interface nodes for a typical mesh motion problem can be very large (of the order of hundreds of thousands), the system to solve assumes significantly high dimensions. In order to reduce the computational and the memory load due to these huge dimensions, we implement an ad hoc algorithm which performs a geometrical and a model order reduction of the system: the former reduction is based on an iterative procedure which selects some relevant control points among the initial ones, while the latter reduction is inspired by the Proper Orthogonal Decomposition (POD) strategy to exploit the advantages of an Offline-Online splitting between ROM construction and evaluation of the solution. The implemented algorithm is validated on some examples, from benchmark cases to more complex three dimensional configurations such as a wing or a hull. All the reduction strategies developed are implemented using an efficient C++ object oriented code and an open-source Finite Element library (libMesh), while the processing of the grids is performed by the open-source visualization software Paraview.