Zlepki nad Powell-Sabinovimi triangulacijami

doktorska disertacija

Jan Grošelj (Author), Marjetka Krajnc (Mentor)

Abstract

Powell-Sabinova triangulacija je dobljena z delitvijo neke dane splošne triangulacije tako, da se vsak njen trikotnik razdeli na šest manjših trikotnikov. Standardna konstrukcija zvezno odvedljivih kvadratičnih zlepkov nad razdeljeno triangulacijo je dodobra raziskana in je predvsem na račun znane B-reprezentacije uporabna na različnih področjih numerične analize. Disertacija obravnava možnosti konstrukcije splošnejših polinomskih zlepkov nad Powell-Sabinovimi triangulacijami. V njej so analizirani prostori višjih stopenj in redov gladkosti, ki posplošujejo prostor zvezno odvedljivih kvadratičnih zlepkov. Poseben poudarek je na konstrukciji baznih B-zlepkov, ki omogočajo stabilno in geometrijsko intuitivno predstavitev zlepkov nad Powell-Sabinovimi triangulacijami. Začetni del disertacije je namenjen pregledu osnovnih rezultatov o polinomskih zlepkih nad triangulacijami. Podrobneje so predstavljeni kvadratični Powell-Sabinovi zlepki. V nadaljevanju so obravnavani kubični zlepki. Prostor zvezno odvedljivih kubičnih Powell-Sabinovih zlepkov do nedavnega ni bil raziskan, a se izkaže kot zelo primeren za uporabo v numerični analizi. Tovrstne zlepke lahko predstavimo z B-zlepki na dva različna načina. Obe predstavitvi imata prikladno kontrolno strukturo, ki temelji na konveksni razčlenitvi enote. Ena izmed predstavitev omogoča tudi enostaven opis kvadratičnih zlepkov in kubičnih zlepkov z dodatnimi pogoji gladkosti. Zadnji dve poglavji disertacije se ukvarjata s konstrukcijo Powell-Sabinovih zlepkov višjih stopenj. Predstavljen je prostor zglajenih zvezno odvedljivih zlepkov stopnje 4 s številnimi dodatnimi pogoji gladkosti reda dve. Opisana je tudi družina prostorov Powell-Sabinovih zlepkov s predstavnikom za poljubno stopnjo. Zlepke vseh obravnavanih prostorov je mogoče izraziti z B-zlepki, ki so posplošitve kvadratičnih in kubičnih Powell-Sabinovih B-zlepkov.

Keywords

zlepki nad triangulacijami;makro-elementi;Powell-Sabinovi zlepki;interpolacijski problemi;kvazi-interpolacija z zlepki;B-zlepki;konveksna razčlenitev enote;kontrolne strukture;geometrijsko oblikovanje;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.08 - Doctoral Dissertation
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Grošelj]
UDC:	519.6(043.3)
COBISS:	17815385
Views:	752
Downloads:	371
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary abstract:	Powell-Sabin triangulation is obtained by refining a given general triangulation in such a way that each triangle is split into six smaller triangles. The standard construction of continuously differentiable quadratic splines on the refined triangulation is thoroughly researched and due to the known B-spline representation applicable in various fields of numerical analysis. The thesis is concerned with the possibilities of construction of more general polynomial splines on Powell-Sabin triangulations. It contains the analysis of spline spaces of higher degrees and orders of smoothness that extend the space of continuously differentiable quadratic splines. A special attention is paid to the construction of basis B-splines that provide a stable and geometrically intuitive presentation of splines on Powell-Sabin triangulations. The initial part of the thesis is devoted to the overview of elementary results about polynomial splines on triangulations. The quadratic Powell-Sabin splines are presented in more detail. In what follows, cubic splines are considered. The space of continuously differentiable cubic Powell-Sabin splines has not been studied until recently, but it turns out to be very suitable for the use in numerical analysis. Such splines can be represented in terms of B-splines in two different ways. Both representations have a convenient control structure, which is based on a convex partition of unity. One of the representations also allows a simple description of quadratic splines and cubic splines with additional smoothness properties. The last two chapters of the thesis deal with the construction of Powell-Sabin splines of higher degrees. A space of continuously differentiable super splines of degree 4 with many additional smoothness constraints of order two is presented. Furthermore, a family of Powell-Sabin spline spaces with a representative of arbitrary degree is described. The splines of all discussed spaces can be expressed in terms of B-splines that are extensions of quadratic and cubic Powell-Sabin B-splines.
Secondary keywords:	splines on triangulations;macro-elements;Powell-Sabin splines;interpolation problems;spline quasi-interpolation;B-splines;convex partition of unity;control structures;geometric design;
Type (COBISS):	Doctoral dissertation
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 3. stopnja
Pages:	X, 122 str.
ID:	10911606