PhD Defence: THB-splines: Theory and Applications

THB-splines: Theory and Applications

For several decades the Computer–Aided Design (CAD) software libraries have been
using B–splines and NURBS as their basic spline representation for geometric objects. This
is due to the many desirable properties the B–splines possess. Unfortunately, for many years
the users of CAD software had to deal with the fact that the generalization of the B–spline
technology to higher dimensions, e.g. surfaces or volumes, is based on the tensor-product
construction, which precludes strictly localized refinement. The investigation of alternatives
to the tensor-product splines that support local refinement has gained significant momentum
in the past few years, mainly due to the advent of Isogeometric Analysis (IgA) where
adaptivity is needed for performing local refinement in numerical simulations.
In our work we focus on one of these adaptive spline technologies, namely on the recently
introduced truncated hierarchical B–splines (THB–splines). This technology does not only
support strictly localized refinement, but at the same time preserves the main properties of
the standard B–spline basis.
In the first part we present a short theoretical introduction to the topic of hierarchical
and truncated hierarchical splines. Subsequently we introduce an efficient implementation
of the fundamental algorithms needed for manipulation with THB–splines. To store the
subdomain structure of the THB–spline basis we employ a kD-tree data structure. This
is complemented by another data structure used for storing the information about the so
called active basis function. Using these two structures we obtain an efficient technique for
the construction and evaluation of THB–splines.
The main focus of our work is however on the application of the THB–spline technique
in research and real world applications. Firstly, we present an adaptive and automatic
surface reconstruction method used for the reconstruction of complex real world data. Using
both, synthetic and real world point data we demonstrate that surface fitting schemes
based on THB-spline representations lead to significant improvements of the reconstruction.
Furthermore, we show that the local THB-spline evaluation in terms of B-spline patches
can be properly combined with commercial geometric modeling kernels in order to convert
the multilevel spline representation into an equivalent standard B–spline CAD geometry.
Secondly, we explore the modelling capabilities of the THB–spline technology. Using a
simple interface between our THB–spline implementation and the Axel modeler we created
several simple THB–spline geometries by interactive control grid manipulation. We hope this
chapter also paves the way for introduction of THB–splines into commercial CAD software.
Lastly, we have investigated the performance of THB–splines in numerical simulations,
focusing on the influence of the truncation on the performance of multigrid solvers.