SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2011-2012
Jeudi 8 mars 2012 :
Tom LYCHE
(Université d'Oslo, Norvège)
Locally Refinable Splines on Box Partitions.
The motivation for this work comes from isogeometric analysis. The central idea is to replace traditional Finite Element spaces by NonUniform Rational B-Splines (NURBS) to provide accurate shape description and elements with higher degrees and smoothness. Since the introduction of the idea in 2005 by Tom Hughes and co-workers, promising results have been obtained documenting its potential. However, it has also been demonstrated that NURBS do not support the local refinement needed in efficient finite element analysis. To overcome this deficiency the use of T-splines is a promising alternative. T-splines uses tensor product B-splines on a quadrilateral mesh with T-joins, called a T-mesh. There are some unresolved problems with T-splines. For example, T-splines are not always linearly independent, it has been observed that refinement along a diagonal in a T-mesh can lead to non-local refinement, and it leads to rational basis functions.
To overcome some of these problems we present a general theory for an alternative called Locally Refinable splines or LR-splines. In two dimensions we obtain a special case of a T-mesh, here named an LR-mesh. However, the concept works in any space dimension, local refinement is guaranteed, the tensor product basis functions are piecewise polynomials, they span the full piecewise polynomial space on the underlying partition, and simple strategies guarantees linear independence.
This is joint work with Tor Dokken and Kjell Fredrik Pettersen at SINTEF, Oslo.
