Zlepki treh spremenljivk nad tetraedrsko particijo območja : magistrsko delo

Šenica, Ana (author), Knez, Marjetka (mentor)

V magistrski nalogi si ogledamo lokalno konstrukcijo polinomskih zlepkov treh spremenljivk nad poljubno tetraedrsko particijo $\triangle$. Pri tem uporabimo reprezentacijo polinomov treh spremenljivk stopnje $n$ nad posameznim tetraedrom $T \in \triangle$ v Bernsteinovi bazi in jo povežemo z množico domenskih točk $\mathcal{D}_{n,T}$. Ogledamo si učinkovit in stabilen izračun odvodov polinomov v ogliščih, na robovih, na ploskvah in v notranjosti tetraedrov s pomočjo De Casteljaujevega algoritma in razcveta. Na koncu vpeljemo tri konkretne prostore $C^1$ superzlepkov nad tetraedrsko particijo $\triangle$, njeno Alfeldovo drobitvijo $\triangle_{\rm{A}}$ in Worsey-Farinovo drobitvijo $\triangle_{\rm{WF}}$, poiščemo njihove minimalne nodalne določitvene množice $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ in $\mathcal{N}_{\rm{WF}}$ ter s tem pokažemo, da gre za prostore $C^1$ polinomskih makroelementov. Z njihovo pomočjo nato poiščemo rešitve Hermitovega interpolacijskega problema, določenega z $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ oziroma $\mathcal{N}_{\rm{WF}}$.

• Language: Slovenian
• Keywords: zlepki treh spremenljivk, makroelement, tetraedrska particija, Bernsteinov bazni polinom, De Casteljaujev algoritem, razcvet, minimalna določitvena množica, minimalna nodalna določitvena množica, Alfeldov razcep, Worsey-Farinov razcep
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-159278
• UDC: 519.6
• COBISS.SI-ID: 200587779
• Publication date in RUL: 05.07.2024

ŠENICA, Ana, 2024, Zlepki treh spremenljivk nad tetraedrsko particijo območja : magistrsko delo [online]. Master’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=159278

Secondary language

• Language: English
• Title: Trivariate splines on tetrahedral partition

Abstract

In the master's thesis we consider local construction of polynomial trivariate splines over a tetrahedral partition $\triangle$. For the representation of trivariate polynomials of degree $n$ over a tetrahedron $T$ we use the Bernstein basis and connect it to its set of domain points $\mathcal{D}_{n,T}$. We take a look at efficient and stable computation of derivatives at the vertices, on the edges, on the faces and in the interior of tetrahedra using De Casteljau algorithm and polynomial blossoms. We further introduce three $C^1$ superspline spaces over tetrahedral partition $\triangle$, its Alfeld refinement $\triangle_{\rm{A}}$ and Worsey-Farin refinement $\triangle_{\rm{WF}}$ and find their nodal minimal determining sets $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ and $\mathcal{N}_{\rm{WF}}$. Consequently, these spaces are indeed $C^1$ macroelement spaces. They are used for finding the solutions of Hermite interpolation problems defined by $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ and $\mathcal{N}_{\rm{WF}}$.

• Keywords: trivariate splines, macroelement, tetrahedral partition, Bernstein basis polynomial, De Casteljau algorithm, blossom, minimal determining set, nodal minimal determining set, Alfeld split, Worsey-Farin split