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Adaptive RBF-FD method : PhD thesis
ID Slak, Jure (Author), ID Kosec, Gregor (Mentor) More about this mentor... This link opens in a new window

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Abstract
Radial-basis-function-generated finite differences (RBF-FD) is a method for solving partial differential equations (PDEs), which is developed into a fully automatic adaptive method during the course of this work. RBF-FD is a strong form meshless method, which means that it does not require a mesh of the problem domain, but uses only a set of nodes as the basis for the discretization. A large part of this PhD is dedicated to algorithms for meshless node generation. A new algorithm for construction of variable density meshless discretizations in arbitrary spatial dimensions is developed. It can generate points in the interior and on the boundary, has provable minimal spacing requirements, can generate $N$ points in $O(N\log N)$ time and the resulting node sets are compatible with RBF-FD. This algorithm is used as the basis of a newly proposed $h$-adaptive procedure for elliptic problems. The behavior of the procedure is analyzed on classical 2D and 3D adaptive Poisson problems. Furthermore, several contact problems from linear elasticity are solved, demonstrating successful adaptive derefinement and refinement, with the densest parts of the discretization being more than a million times denser than the coarsest. Finally, the software developed for this work and broader research is presented and published online as an open source library for solving PDEs with strong form methods.

Language:English
Keywords:meshfree methods, meshless methods, radial basis functions, partial differential equations, adaptivity, refinement, node generation, finite differences, scattered data
Work type:Doctoral dissertation (mb31)
Organization:FMF - Faculty of Mathematics and Physics
Year:2020
PID:20.500.12556/RUL-121511 This link opens in a new window
UDC:519.6
COBISS.SI-ID:31592707 This link opens in a new window
Publication date in RUL:13.10.2020
Views:741
Downloads:269
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Secondary language

Language:Slovenian
Title:Adaptivna RBF-FD metoda
Abstract:
Končne diference, generirane z radialnimi baznimi funkcijami (RBF-FD), so metoda za numerično reševanje parcialnih diferencialnih enačb (PDE), ki jo v delu razvijemo v avtomatsko adaptivno metodo. RBF-FD spada med brezmrežne metode, ki enačbe rešujejo v močni obliki. Brezmrežnost pomeni, da metoda ne potrebuje diskretizacije domene problema v obliki mreže, temveč lahko za diskreten izračun uporabi le množico ustrezno razporejenih točk. Velik del doktorata je posvečen algoritmom za generiranje diskretizacijskih točk. Razvit je tudi nov algoritem za konstrukcijo diskretizacij v poljubnih dimenzijah s prostorsko spremenljivo diskretizacijsko razdaljo. Algoritem je primeren za generiranje točk v notranjosti in na robu domene, dokazano ohranja predpisano minimalno razdaljo med točkami in potrebuje $O(N \log N)$ časa za generiranje $N$ točk. Konstruirane množice točk so kompatibilne z RBF-FD metodo in posledično algoritem uporabimo kot osnovo novega postopka za $h$-adaptivno reševanje eliptičnih problemov. Obnašanje postopka je analizirano na klasičnih dvo- in tro-dimenzionalnih Poissonovih problemih. Poleg tega je rešenih tudi več kontaktnih problemov iz linearne elastostatike, s katerimi pokažemo uspešno goščenje in redčenje diskretizacije, ki se avtomatsko prilagaja problemu, pri čemer razmerje med najgostejšimi in najredkejšimi deli diskretizacije naraste tudi do več milijonov. Na koncu je predstavljena programska oprema, razvita za to delo in širše raziskave, objavljena pa je tudi na spletu kot odprtokodna knjižnica, namenjena reševanju PDE v močni obliki z brezmrežnimi metodami.

Keywords:brezmrežne metode, radialne bazne funkcije, parcialne diferencialne enačbe, adaptivnost, zgoščevanje, generiranje točk, končne diference, razpršeni podatki

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