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Several applications of computational fluid dynamics require to simulate many different possible realizations of a system, thus yielding relevant computational challenges and, very often, large demand on computational resources. This is the case, for instance, of optimization, control and design problems in aerodynamics. A possible way to alleviate this computational burden is provided by reduced order models (ROMs), that is, low- dimensional, efficient models which are fast to solve, but also able to approximate well the underlying high-fidelity simulations.
In this work we analyse and implement a Reduced Basis (RB) method for the rapid and reliable solution of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape. This method allows to capture the essential flow features by means of a handful of degrees of freedom, and to keep under control the error with respect to a high-fidelity solution, all over the parameter space.
For the construction of our RB method we rely on a high-fidelity approximation technique given by an Isogeometric Boundary Element Method (IGA-BEM), thus lead- ing to a very efficient Isogeometric Reduced Basis (IGA-RB) Method for the reduction of shape-dependent problems. We have decided to rely on a Galerkin-Boundary Element Method because it enables a preliminary reduction of the problem dimension, through a suitable boundary integral formulation, and the chance to treat external flows in (possibly) infinite domains. On the other hand, Isogeometric Analysis allows a direct interface with CAD tools, in view of possible extensions to complex applications of in- dustrial interest. Moreover, in order to ensure a suitable Offline/Online decomposition between ROM construction and evaluation, a suitable Empirical Interpolation Method has been applied.
We have adopted two different strategies for the construction of the reduced spaces, namely the Proper Orthogonal Decomposition (POD) and a Greedy algorithm, by showing the main analogies and differences for the case at hand, and their computational performances. Finally, we validate the results – obtained both with the high-fidelity IGA-BEM method and the reduced order models – with respect to experimental data and numerical codes (Xfoil), showing in both case a great agreement.