International Conference on Multivariate Approximation
September 24-27, 2011

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Locally refinable tensor product splines

Tom Lyche
The central idea of isogeometric analysis 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 as introduced by Sederberg et al is a promising alternative. T-splines are tensor product B-splines on a quadrilateral mesh with T-joins, called a T-mesh. A special case is quadrilateral hierarchical meshes. There are some unresolved problems with T-splines. For example, it has been observed that refinement along a diagonal in a T-mesh can lead to non-local refinement and rational basis functions can occur. To overcome some of these problems an alternative called LR-splines will be considered. Here local refinement of tensor product B-splines is specified as a sequence of inserted line segments parallel to the knot lines. Affected B-splines are split into two new B-splines using univariate knot insertion. We obtain a special case of a T-mesh, here named an LR-mesh. On the LR-mesh we obtain a collection of tensor product B-splines which form a partition of unity. The approach applies equally well in dimensions higher than two. In the bivariate case this collection of B-splines spans the full piecewise polynomial space on the LR-mesh and is independent of the order in which the segments are inserted.