20.500.12556/RUL-174321 Reševanje parcialnih diferencialnih enačb s predoločeno RBF-FD metodo magistrsko delo Solving partial differential equations with the oversampled RBF-FD method V delu je predstavljena in eksperimentalno analizirana brezmrežna metoda predoločenih končnih diferenc, generiranih z radialnimi baznimi funkcijami (predoločenih RBF-FD). Ta za aproksimacijo operatorja uporablja radialne bazne funkcije. Cilj metode je numerično reševanje parcialnih diferencialnih enačb, ki opisujejo skalarna in vektorska polja. V nasprotju s klasičnim RBF-FD pristopom metoda temelji na predoločenem sistemu enačb, ki upošteva obnašanje operatorja na razširjeni množici točk. Eden od ključnih izzivov, ki se pojavi zaradi predoločenega sistema, je zagotavljanje natančnosti pri izpolnjevanju robnih pogojev. V ta namen se uporabi skaliranje, ki uteži vpliv robnih pogojev. V delu sta izpeljani dve različici metode, ki temeljita na povezavi z zveznim primerom. Učinkovitost obeh različic je preverjena na testnem primeru Poissonove enačbe in z analizo deformacije vpetega nosilca. Izvedena je primerjava s klasično RBF-FD metodo in analiza vpliva skaliranja na natančnost ocene. Na koncu se posvetimo še nekaterim podrobnostim implementacije metode, kot so uporabljene strukture in reševanje dobljenih linearnih sistemov enačb. The work presents and experimentally analyzes a meshfree method of the oversampled Radial Basis Function-generated Finite Difference (oversampled RBF-FD) method. This method uses radial basis functions to approximate operators in order to numerically solve partial differential equations (PDE), which describe scalar and vector fields. In contrast to the classic RBF-FD approach, the method constructs an overdetermined system of linear equations that accounts for the behaviour of the operator on an extended set of points. One of the key challenges arising due to the overdetermined system is ensuring accuracy in the enforcement of boundary conditions. For this purpose, a scaling is used to weight the influence of the boundary conditions. Two variants of the method are derived, both of which are based on a connection with a continuous problem. Both versions are tested on the examples of the Poisson problem and the deformation of the cantilever beam. A comparison with the classic RBF-FD method is carried out, as well as an analysis of the influence of scaling on the accuracy of the approximation. Finally, we address some details of the implementation of the method, such as the data structures used and the approach used to solve the linear systems of equations. RBF-FD končne diference generirane z radialnimi baznimi funkcijami parcialne diferencialne enačbe predoločen RBF-FD brezmrežne metode RBF-FD radial basis function generated finite difference partial differential equations oversampled RBF-FD meshless true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2025-10-01 08:15:46 2025-10-01 08:15:48 2025-11-20 03:51:16 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 155100 250862595 19991.pdf 19991.pdf 1 F4CDA80126A2C2D33466AEDDF694B49E 0afcf6b7a91d6a224d83ee5980f475255de0d76d6fea89bc0e19fa658d7c0c89 e1169fd4-9e8d-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219110 Fakulteta za matematiko in fiziko 0 0 0