The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, twodimensional (2D) incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-Cartesian node arrangements and large sizes, i.e., in the order of 10 5 nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favorably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains.
One of the goals of new CAE (Computer Aided Engineering) software is to reduce both time and costs of the design process without compromising accuracy. This result can be achieved, for instance, by promoting a “plug and play” philosophy, based on the adoption of automatic mesh generation algorithms. This in turn brings about some drawbacks, among others an unavoidable loss of accuracy due to the lack of specificity of the produced discretization. Alternatively it is possible to rely on the so called “meshless” methods, which skip the mesh generation process altogether. The purpose of this paper is to present a fully meshless approach, based on Radial Basis Function generated Finite Differences (RBF-FD), for the numerical solution of generic elliptic PDEs, with particular reference to time-dependent and steady 3D heat conduction problems. The absence of connectivity information, which is a peculiar feature of this meshless approach, is leveraged in order to develop an efficient procedure that accepts as input any given geometry defined by a stereolithography surface (.stl file format). In order to assess its performance, the aforementioned strategy is tested over multiple geometries, selected for their complexity and engineering relevance, highlighting excellent results both in terms of accuracy and computational efficiency. In order to account for future extensibility and performance, both node generation and domain discretization routines are entirely developed using Julia, an emerging programming language that is rapidly establishing itself as the new standard for scientific computing.
The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Functiongenerated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations.Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains , showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches.These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method.
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