An improved finite point method for tridimensional potential flows

Abstract

At the local level, successful meshless techniques such as the Finite Point Method must have two main characteristics: a suitable geometrical support and a robust numerical approximation built on the former. In this article we develop the second condition and present an alternative procedure to obtain shape functions and their derivatives from a given cloud of points regardless of its geometrical features. This procedure, based on a QR factorization and an iterative adjust of local approximation parameters, allows obtaining a satisfactory minimization problem solution, even in the most difficult cases where usual approaches fail. It is known that high-order meshless constructions need to include a large number of points in the local support zone and this fact turns the approximation more dependent on the size, shape and spatial distribution of the local cloud of points. The proposed procedure also facilitates the construction of high-order approximations on generic geometries reducing their dependence on the geometrical support where they are based. Apart from the alternative solution to the minimization problem, the behaviour of high-order Finite Point approximations and the overall performance of the proposed methodology are shown by means of several numerical tests.

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

“…x x is a compact support exponential weighting function centred on the star point of the cloud [15] (Fixed Least-Squares (FLS)). The minimization of Eq.…”

A finite point method for adaptive three‐dimensional compressible flow calculations

“…[21] with the aim of reducing instabilities in the minimization procedure, especially those arising from non-appropriate local clouds of points. In addition to that, but from another perspective, we have recently presented an alternative approach towards robustness [22] intended to reduce the local approximation dependence on both, the spatial distribution of the cloud of points and the weighting function parameters. This ad hoc procedure, which is based on a QR factorization in conjunction with an iterative adjustment of the local approximation parameters, allows obtaining a satisfactory minimization problem solution for cases where usual approaches fail and avoids modifying the geometrical support where the local approximation is based on.…”

“…Also, in a previous work [22] we have dealt with high-order FP discretizations in a preliminary manner, exploring the FPM capabilities regarding p-adaptivity. This time, with the same objective in mind, i.e.…”

“…In this work, the procedure adopted to calculate the shape function and its derivatives is the following (cf. [22]). Given a certain cloud of points, first the direct inversion of matrix A is attempted.…”

“…The support of this function is isotropic, circular and spherical in two and three-spatial dimensions respectively. A detailed description of the effects of the free parameters w, k and  on the numerical approximation and some guidelines for their setting was presented in [22]. However, an important remark about the parameter  should be mentioned.…”