20.500.12556/RUL-159278 Zlepki treh spremenljivk nad tetraedrsko particijo območja magistrsko delo Trivariate splines on tetrahedral partition 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 $\mathcal{N}_{\rm{WF}}$. 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 $\mathcal{N}_{\rm{WF}}$. 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 trivariate splines macroelement tetrahedral partition Bernstein basis polynomial De Casteljau algorithm blossom minimal determining set nodal minimal determining set Alfeld split Worsey-Farin split true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2024-07-05 08:15:18 2024-07-05 08:15:21 2024-07-14 03:36:11 0000-00-00 00:00:00 2024 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 139892 200587779 12085.pdf 12085.pdf 1 687519A0467D61BC479A615F1D8EB4F5 a31094215a93e35f0d3ee70d2b3fb792ce3700c5539fcb5b1f0455c4492b88d7 f4ff08a6-3a95-11ef-89ef-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=187244 Fakulteta za matematiko in fiziko 0 0 0