On a class of polynomial triangular macro-elements

AbstractIn this paper we present a new class of polynomial triangular macro-elements of arbitrary degree which are an extension of the classical Clough-Tocher cubic scheme. Their most important property is that the degree plays the role of a tension parameter, since these macro elements tend to the plane interpolating the vertices data. Graphical examples showing their use in scattered data interpolation are reported

A reduced Hsieh–Clough–Tocher element with splitting based on an arbitrary interior point

AbstractWe present formulas for a reduced Hsieh–Clough–Tocher (rHCT) element with splitting based on an arbitrary interior point. These formulas use local barycentric coordinates in each of the subtriangles and are not significantly more complicated than formulas for an rHCT element with splitting based on the centroid

Using Least Squares to Construct Improved Clough-Tocher Interpolant

In this thesis, a quartic Clough-Tocher interpolation scheme is introduced, and additional modifications, to adjust the macro-boundary and the order of continuity across domain triangles, are provided to improve both the mathematical and the visual quality of the resulting surface. Furthermore, a proof is given to show the convergence of the interpolation scheme under some specific constraints

Scattered data fitting on surfaces using projected Powell-Sabin splines

We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ω embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens

Generalized Finite Element Systems for smooth differential forms and Stokes problem

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8 figures and several comments adde

B-spline-like bases for C2C^2 cubics on the Powell-Sabin 12-split

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps