PDM research project

Splines on triangulations are also useful as finite elements for the numerical solution of partial differential equations. Many classical finite elements are defined on triangulations. Because of the inherent higher order of continuity, a finite element discretization (e.g., Galerkin projection) based on splines has a much smaller linear system than with classical Lagrange or Hermite finite elements. It results in a more efficient approach to find an accurate approximation of the solution of the differential equation. Moreover, spline finite elements can further simplify the design process of a product. The integration of CAD data in the finite element model is a major concern for an efficient design process. When splines are used for the design as well as for the finite element analysis, a considerable amount of conversion work can be avoided.

The construction of smooth bivariate spline functions on arbitrary triangulations, however, is no trivial task. A major difficulty is to determine the dimension of such spline spaces. In general, it is not (yet) possible to express the dimension in terms of geometric interesting characteristics of the triangulation (like the number of vertices and triangles). There are some exact results for particular choices of polynomial degree and smoothness. An alternative approach is using so-called macro-elements. Each triangle in the triangulation is then split in a particular way, e.g., the Powell-Sabin refinement.

Powell-Sabin (PS-) splines are piecewise quadratic polynomials with a global C¹-continuity. They are defined on arbitrary conforming triangulations, where all triangles are split into six smaller triangles (the so-called PS-refinement). PS-splines can be compactly represented in a stable normalized spline basis. This representation has an intuitively geometric interpretation involving tangent control triangles. Using these triangles one can interactively change the shape of the spline in a predictable way. These splines are appropriate in many application domains, e.g., for smoothing noisy measurement data, in surface modelling, and in finite element applications.

Quasi-hierarchical Powell-Sabin (QHPS-) splines are a hierarchical variant of the classical quadratic Powell-Sabin splines. They are C¹-continuous quadratic macro-elements defined on particular non-conforming triangulations. Such a triangulation is constructed in a hierarchical way using triadic splits, and is allowed to have dangling vertices (i.e., the triangles can contain vertices different from their own three vertices). Local refinement can be straightforwardly performed, such that an adaptive strategy is possible and the usual excessive computational and storage costs of a standard global refinement are avoided. That is an advantage over classical PS-splines, where the necessary conformity of the triangulation requires specialized shape-improvement and refinement propagation techniques to adopt local subdivision. For this QHPS-spline space, a normalized quasi-hierarchical basis can be constructed. The basis retains all the advantages of the classical PS-basis: the basis functions have a local support, they are positive, they form a partition of unity, they are stable, and the spline is locally adaptable by means of control triangles. In addition, the hierarchical triangulation admits local subdivision in a very natural way.

Higher order splines

Many practical problems (for instance, in data fitting and finite element applications) benefit from using higher order splines, especially when a highly accurate solution is required. Such splines converge more quickly to the exact solution. Therefore, it often satisfies to use coarse triangulations, resulting mostly in better conditioned discretizations. A spline with a higher order of continuity is also useful for the modelling of smooth surfaces. Surfaces with a continuous curvature are visually more pleasing.

There already exist some higher degree spline extensions on triangulations with a PS-refinement. Unfortunately, the construction of the corresponding basis is not as trivial as for the quadratic PS-splines. They need the composition of a so-called minimal determining set of points. The number of points in this set equals the dimension of the spline space. Using the Bernstein-Bézier polynomial representation and imposing the necessary continuity constraints, one can obtain a compact set of basis functions. The application of these higher order splines leads to some interesting research questions. Is it possible to associate with the proposed set of basis functions (or maybe another set) a geometric interpretation that enables an intuitive control of the spline surface? This is very useful in computer aided geometric design. Can such splines be implemented and manipulated in an efficient and easy way? How difficult is it to (locally) modify the order of the spline? This is important for the application of an adaptive p-refinement strategy (dynamical increase of the order). It is interesting to investigate the trade-off between p-refinement and h-refinement (refining the mesh).

We will extend the quasi-hierarchical approach, developed for PS-splines, to other spline spaces. The construction is essentially based on subdivision. It can be applied, for instance, to the above-mentioned higher-order Powell-Sabin splines. Some points of interest are the determination of the spline dimension and the stability of the constructed basis. It will also be useful to examine the quasi-hierarchical approach for tensor-product B-splines. This should enable local subdivision, without the loss of other interesting properties.

Multivariate splines

The extension to three-dimensional splines (or higher-dimensional) with similar properties as the two-dimensional PS-splines is a real challenge. At the moment, only practically usable splines are constructed on tessellations that consist of (semi-)regular simplices. Many practical problems (in the context of data fitting or numerical simulation) are defined on three-dimensional domains. Yet, the range of applications will be much larger when splines on tessellations with arbitrary simplices could be used.

We also want to develop subdivision rules for three-dimensional splines. In addition to global refinement, we will examine if local refinement is realizable. For this, we need to investigate how splines on locally refined (more irregular) tessellations can be constructed. Alternatively, we can keep regular tessellations, and create local flexibility using a quasi-hierarchical approach.