magistrsko delo
Viktor Cvrtila (Author), Bor Plestenjak (Mentor), Jure Slak (Co-mentor)

Abstract

Metodo RBF-FD (metodo končnih diferenc, generiranih z radialnimi baznimi funkcijami) lahko razumemo kot posplošitev metode končnih diferenc. Za razliko od FDM, ta metoda ni omejena na pravokotne mreže, temveč lahko rešuje PDE na razpršenih točkah. Metoda RBF-FD sodi med brezmrežne metode, ker ni potrebno, da te točke tvorijo mrežo. To je ugodno, saj je generiranje ustreznih mrež, kot na primer triangulacij pri metodi končnih elementov, pogosto računsko zahtevno. Bistvena prednost metode FDM je, da rešitev izračuna kot rešitev sistema linearnih enačb z značilno pasovno matriko, kar olajša analizo metode. V tem magistrskem delu bomo obravnavali lastnosti analogne matrike pri RBF-FD; najprej bomo preučili lastnosti teh matrik na pravokotni mreži. Ker ti rezultati niso koristni za praktične primere, bomo nato pogledali, ali se ugodne lastnosti ohranijo, če pravokotno mrežo perturbiramo. V nadaljevanju bomo vpeljali algoritem za diskretizacijo domen, da ustvarimo bolj realne nabore razpršenih točk. Nazadnje bomo pogledali, kakšne so lastnosti matrik, če metodo RBF-FD uporabimo na takih naborih točk.

Keywords

matematika;radialne bazne funkcije;metoda končnih diferenc;Laplaceova enačba;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [V. Cvrtila]
UDC: 519.6
COBISS: 81737731 Link will open in a new window
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Other data

Secondary language: English
Secondary title: Properties of matrices generated by RBF-FD
Secondary abstract: The RBF-FD (radial-basis-function generated finite differences) method for solving PDEs can be interpreted as a generalisation of the finite difference method. Unlike the latter it is not restricted to grids, but can solve problems on scattered sets of nodes. It belongs to the category of meshless methods, as it does not require that the nodes form a mesh. This is useful, because generating meshes, such as triangulations for FEM, is often computationally difficult. One major advantage of the finite difference method, however, is that the matrix involved in finding the numerical solution of a problem is of a characteristic banded shape. This makes analysing the method easier. The aim of this master’s paper is to explore the properties of analogous matrices, produced by RBF-FD. First, the properties of such matrices are considered when the discretisation forms a regular grid. As this is not particularly usefull in practice, a more general case is considered, when the discretisation is a perturbed grid. Then, a domain discretisation algorithm is presented, which produces more realistic sets of nodes. Finally we consider the properties of the matrices, when RBF-FD is used to solve problems on such sets.
Secondary keywords: mathematics;radial basis function;RBF-FD;finite difference method;Laplace equation;
Type (COBISS): Master's thesis/paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages: VII, 69 str.
ID: 13729671
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