Kubični Clough-Tocherjevi zlepki : magistrsko delo

Stupica, Katarina (author), Knez, Marjetka (mentor)

V nalogi opišemo prostor kubičnih Clough-Tocherjevih zlepkov nad triangulacijo domene, pri čemer vsak trikotnik iz triangulacije z izbiro delilne točke razdelimo na tri manjše. Nad vsakim trikotnikom želimo dobiti kubični polinom, nad celotno triangulacijo pa ${\cal C}^1$ zlepek. Zanimal nas bo poseben podprostor tega prostora, prostor zreduciranih Clough-Tocherjevih zlepkov, za katerega lahko konstruiramo normalizirano bazo. Bazni zlepki imajo lokalni nosilec, so nenegativni in tvorijo particijo enote. Iskanje baze lahko prevedemo na geometrijski problem iskanja množice trikotnikov, ki morajo vsebovati določen nabor točk. To nas pripelje do strukture kontrolnih trikotnikov in stabilnega računanja zreduciranih Clough-Tocherjevih zlepkov.

• Language: Slovenian
• Keywords: Clough-Tocherjevi zlepki, B-zlepki, makro-elementi, določitvene množice
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-105133
• UDC: 519.6
• COBISS.SI-ID: 18474329
• Publication date in RUL: 28.10.2018

STUPICA, Katarina, 2018, Kubični Clough-Tocherjevi zlepki : magistrsko delo [online]. Master’s thesis. [Accessed 14 March 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=105133

Secondary language

• Language: English
• Title: Cubic Clough-Tocher splines

Abstract

In this thesis we present the Clough-Tocher space of cubic splines on a triangulation, where we make a refinement of the triangulation by splitting each triangle on three smaller ones. The spline is composed of cubic polynomials over each triangle and it is ${\cal C}^1$ over the whole triangulation. We focus on a special subspace - the reduced Clough-Tocher spline space and construct a normalized basis for it. The basis splines have a local support, they are nonnegative and they form a partition of unity. Geometrically, the basis construction problem is converted to a problem of finding a set of triangles that contain specific points. This leads us to control triangles and a stable way of computing with Clough-Tocher splines.

• Keywords: Clough-Tocher splines, B-splines, macro-elements, determining sets