A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.Comment: 14 pages, 26 figure

Quantification of airfoil geometry-induced aerodynamic uncertainties - comparison of approaches

Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods since it is computationally expensive, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of Kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost

Differential quadrature method for space-fractional diffusion equations on 2D irregular domains

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L\'{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table

Sequential optimization of strip bending process using multiquadric radial basis function surrogate models

Surrogate models are used within the sequential optimization strategy for forming processes. A sequential improvement (SI) scheme is used to refine the surrogate model in the optimal region. One of the popular surrogate modeling methods for SI is Kriging. However, the global response of Kriging models deteriorates in some cases due to local model refinement within SI. This may be problematic for multimodal optimization problems and for other applications where correct prediction of the global response is needed. In this paper the deteriorating global behavior of the Kriging surrogate modeling technique is shown for a model of a strip bending process. It is shown that a Radial Basis Function (RBF) surrogate model with Multiquadric (MQ) basis functions performs equally well in terms of optimization efficiency and better in terms of global predictive accuracy. The local point density is taken into account in the model formulatio