Approximation and interpolation splines on triangulations : doctoral thesis

Kanduč, Tadej (author), Jaklič, Gašper (mentor)

In the thesis, some new results on correctness of polynomial Lagrange interpolation problem on triangles are presented. The results are based on positivity of principal minors of Bézier collocation matrices for non-parametric patches. L. L. Schumaker stated the conjecture, that for uniformly distributed domain points on triangle the corresponding collocation matrix has positive principal minors. The conjecture on the minors for polynomial total degree ▫$\le 17$▫ and for some particular configurations of domain points is confirmed. By stating the exact lower bound for the principal minors, the main conjecture is extended. A generalisation of domain points' positions imposing correctness of the interpolation problem is analysed for polynomial degree ▫$\le 4$▫. In the parametric case, two novel constructions solving Hermite interpolation problem (interpolation of points and tangent planes) are proposed. In the first one, a construction of good boundary curves of cubic triangular patches is analysed. The curves minimise an approximate strain energy functional. It is shown that the curves are regular and without shape defects. The shape of the curves is analysed with respect to a given shape parameter. The remaining free parameters of the spline surface are set in such a way that the patches have small Willmore energy. It is shown that a unique interpolant exists at mild presumptions. Next, a generalisation of macro-elements to the parametric case is considered. Hermite interpolation by two types of parametric ▫$C^1$▫ macro-elements on triangulations is presented in detail. Cubic triangular splines interpolate points and the corresponding tangent planes at domain vertices and approximate tangent planes at midpoints of domain edges. Quintic splines additionally interpolate normal curvature forms at the vertices. Control points of the interpolants are constructed in two steps. In the first one, uniformly distributed control points of a linear spline interpolant are projected to the interpolation planes. To ensure the smoothness conditions between patches, a correction of control points is obtained as the solution of a least square minimisation. The interpolation schemes inherit many desired properties from the functional case such as local and simple geometric construction and linear complexity. At the end, the interpolation schemes are tested in numerical examples and practical applications.

• Language: English
• Keywords: Bernstein polynomial, Bézier surface, spline surface, parametric surface, cubic spline, triangular patch, triangulation, Lagrange interpolation, Hermite interpolation, collocation matrix, principal minor, strain energy, collocation matrix, principal minor, strain energy
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [T. Kanduč]
• Year: 2013
• Number of pages: 103 str.
• PID: 20.500.12556/RUL-97345
• UDC: 519.65(043.3)
• COBISS.SI-ID: 16648281
• Publication date in RUL: 24.10.2017

Secondary language

• Language: Slovenian

Abstract

V disertaciji predstavimo nekaj novih rezultatov s področja korektnosti polinomske Lagrangeeve interpolacije nad trikotniki. Rezultati slonijo na pozitivnosti glavnih minorjev Bézierovih kolokacijskih matrik za neparametrične krpe. L. L. Schumaker je postavil naslednjo domnevo. Če izberemo enakomerno razporejene interpolacijske točke na trikotniku, potem so glavni minorji pripadajoče kolokacijske matrike pozitivni. V disertaciji pokažemo, da trditev velja za vse glavne minorje, če je totalna stopnja polinomov ▫$\le 17$▫, in za nekatere posebne razporeditve interpolacijskih točk. Omenjeno domnevo razširimo s postavitvijo natančne spodnje meje za vrednosti glavnih minorjev. Na koncu analiziramo korektnost interpolacijskega problema za splošnejšo lego točk in totalno stopnjo ▫$\le 4$▫. V parametričnem okolju predstavimo dve novi shemi, ki rešita Hermiteov interpolacijski problem (interpolacija točk in tangentnih ravnin). V prvi podrobno analiziramo konstrukcijo primernih robnih krivulj kubične trikotne krpe. Optimalne krivulje minimizirajo funkcional približne napetostne energije. Krivulje so regularne in brez zank ter osti. Kakovost krivulje študiramo v odvisnosti od danega parametra oblike. Preostale parametre kubičnega zlepka določimo tako, da imajo krpe majhno Willmorejevo energijo. Enolična rešitev interpolacijskega problema obstaja pri šibkih predpostavkah. Drugo shemo dobimo s posplošitvijo makro-elementov na parametričen primer. Podrobneje predstavimo dva tipa parametričnih ▫$C^1$▫ makro-elementov na triangulacijah, ki rešita Hermiteov interpolacijski problem. Kubični trikotni zlepki interpolirajo točke in pripadajoče tangentne ravnine v vozliščih triangulacije ter aproksimirajo tangentne ravnine na sredini povezav triangulacije. Zlepki stopnje pet v vozliščih dodatno interpolirajo forme normalnih ukrivljenosti. Kontrolne točke zlepkov konstruiramo v dveh korakih. V prvem, enakomerno razporejene kontrolne točke linearnega interpolanta projiciramo na interpolacijske ravnine. Da zadostimo pogojem gladkosti med trikotnimi krpami, popravke kontrolnih točk izračunamo kot rešitev po metodi najmanjših kvadratov. Interpolacijski shemi posedujeta veliko zaželenih lastnosti iz funkcijskega primera kot so lokalna in geometrijska konstrukcija ter linearna časovna zahtevnost. Na koncu interpolacijski shemi testiramo na različnih numeričnih primerih in v praktičnih aplikacijah.

• Keywords: Bernsteinov polinom, Bézierova ploskev, dvorazsežen zlepek, parametrična ploskev, kubičen zlepek, trikotna krpa, triangulacija, Lagrangeeva interpolacija, Hermiteova interpolacija, kolokacijska matrika, glavni minor, napetostna energija, Willmorejeva energija, minimizacija energije, makro-element