arXiv subjects

Gregor Kosec

Publications and source records attributed to Gregor Kosec.

40 records · Page 3Linked to original sources

arXiv2019

On generation of node distributions for meshless PDE discretizations

In this paper we present an algorithm that is able to generate locally regular node layouts with spatially variable nodal density for interiors of arbitrary domains in two, three and higher dimensions. It is demonstrated that the generated node distributions are suitable to use in the RBF-FD method, which is demonstrated by solving thermo-fluid problem in 2D and 3D. Additionally, local minimal spacing guarantees are proven for both uniform and variable nodal densities. The presented algorithm has time complexity $O(N)$ to generate $N$ nodes with constant nodal spacing and $O(N \log N)$ to generate variably spaced nodes. Comparison with existing algorithms is performed in terms of node quality, time complexity, execution time and PDE solution accuracy.

math.NA↗

arXiv2019

Adaptive Radial Basis Function-generated Finite Differences method for contact problems

This paper proposes an original adaptive refinement framework using Radial Basis Functions-generated Finite Differences method. Node distributions are generated with a Poisson Disk Sampling-based algorithm from a given continuous density function, which is altered during the refinement process based on the error indicator. All elements of the proposed adaptive strategy rely only on meshless concepts, which leads to great flexibility and generality of the solution procedure. The proposed framework is tested on four gradually more complex contact problems, governed by the Cauchy-Navier equations. First, a disk under pressure is considered and the computed stress field is compared to the closed form solution of the problem to assess the basic behaviour of the algorithm and the influence of free parameters. Second, a Hertzian contact problem, also with known closed form solution, is studied to analyse the proposed algorithm with an ad-hoc error indicator and to test both refinement and derefinement. A contact problem, typical for fretting fatigue, with no known closed form solution is considered and solved next. It is demonstrated that the proposed methodology can be used in practical application and produces results comparable with FEM without the need for manual refinement or any human intervention. In the last case, generality of the proposed approach is demonstrated by solving a 3-D Boussinesq's problem of the concentrated normal traction acting on an isotropic half-space.

math.NA↗

arXiv2019

Refined Meshless Local Strong Form solution of Cauchy-Navier equation on an irregular domain

This paper considers a numerical solution of a linear elasticity problem, namely the Cauchy-Navier equation, using a strong form method based on a local Weighted Least Squares (WLS) approximation. The main advantage of the employed numerical approach, also referred to as a Meshless Local Strong Form method, is its generality in terms of approximation setup and positions of computational nodes. In this paper, flexibility regarding the nodal position is demonstrated through two numerical examples, i.e. a drilled cantilever beam, where an irregular domain is treated with a relatively simple nodal positioning algorithm, and a Hertzian contact problem, where again, a relatively simple h-refinement algorithm is used to extensively refine discretization under the contact area. The results are presented in terms of accuracy and convergence rates, using different approximations and refinement setups, namely Gaussian and monomial based approximations, and a comparison of execution time for each block of the solution procedure.

math.NA↗

arXiv2017

An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.

math.NA↗