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Mesh displacement based on Radial Basis Functions (RBF) interpolation is known for its ability to preserve the validity and quality of the mesh, even for large displacements, without being affected by mesh connectivity. However, in the case of large meshes, such as those used in real-world Computational Fluid Dynamics (CFD) applications, RBF interpolation, in its standard formulation, becomes excessively expensive. This paper proposes a cost reduction technique for mesh displacement based on RBF, by splitting the process into two steps. In the first step, named predictor, a data reduction algorithm that adaptively agglomerates mesh boundary nodes by reducing the RBF interpolation problem size is used. Upon completion of the first step, due to the agglomeration and the fact that the RBF interpolation is applied to the boundary nodes too, the so-displaced boundaries do not match the given displacements; thus, the position of the boundary nodes must be corrected during the second step, named corrector. The latter performs a local deformation based on RBF kernels with local support, to make the boundary conform to the known displacements of its nodes. The proposed method is accelerated by employing the Sparse Approximate Inverse preconditioner based on geometrical considerations and the Fast Multipole Method. The method and the programmed software are validated on three test cases related to the deformation of CFD meshes inside a duct and a turbine stator row as well as around a car model.

This paper presents a shape parameterization method based on morphing that acts directly on CAD-compatible Boundary-Representations (B-Rep), effectively integrated into aerodynamic shape optimization. The proposed technique requires the definition of a small number of "handles", which are strategically placed around or on the B-Rep shapes to be optimized. Displacement vectors associated with these handles are used as design variables in the optimization method. Using Radial Basis Functions (RBF) as an interpolation method, these displacements are transferred from the handles to the Non-Uniform Rational Basis-Splines (NURBS) control points of the B-Rep model; this approach offers the advantage that the updated surface remains in CAD format and is thus exportable to a STEP file. The proposed method combines two successive deformation steps. Each deformation is controlled by an independent set of handles, increasing the flexibility of the morphing action. A strategy for updating the CFD surface grid to the already changed B-Rep models enables the seamless inclusion of the proposed method into any optimization loop, avoiding any into-the-loop dependence on grid generation packages. The surface grid is updated by computing new nodal coordinates based on the updated NURBS parametric coordinates; these are computed according to changes in the parametric domain of trimmed surfaces by an RBF-based interpolation. The displacement of the surface nodes is then used to displace the CFD volume grid. The tool is differentiated and integrated into gradient-based optimization loops using the adjoint method to compute the gradient of the objective and constraint functions with respect to the surface nodal positions. The performance of the proposed method is assessed in four aerodynamic shape optimization problems, concerning a 2D airfoil, a duct, an aircraft model and a compressor stationary blading. An assessment of the proposed method based on parametric effectiveness, in the 2D case, is also presented.