Approximation and interpolation splines on triangulations : doctoral thesis

english

doktorska dela

2013

Bernstein polynomial   Bézier surface   collocation matrix   cubic spline   Hermite interpolation   Lagrange interpolation   parametric surface   principal minor   spline surface   strain energy   triangular patch   triangulation

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

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.

12.05.2023