Locally Refined Splines

The central idea of isogeometric analysis is to replace traditional Finite Elements by NonUniform Rational B-splines (NURBS) to provide accurate shape description and higher order elements in analysis. Since the introduction of the idea by T. Hughes in 2005 excellent results have been obtained documenting the potential of isogeometric analysis. However, it has also been demonstrated that NURBS do not support the local refinement needed in efficient isogeometric analysis.  T-splines offer some local refinement but have been shown to be linearly dependent in specific configurations, a situation not acceptable in analysis. Rather than adapting technology such as T-splines developed for Computer Aided Geometric Design (CAGD), Locally Refined Splines (LR-splines) are developed from the dual viewpoints of CAGD and analysis. T-splines represent the refinement in the grid of control vertices, a natural approach in CAGD. However, this hides the dimensionality of the spline space, gives refinement a fairly large footprint, and can in certain situations trigger an unwanted growth in the number of coefficients. LR-splines code the refinement directly in the parameterization of the spline space by insertion of knot line segments. This approach gives refinement a small footprint, provides control of the dimensionality of the spline space, and supplies a basis composed of tensor product B-spline basis functions with different levels of refinement. The LR-spline basis also satisfies the important partition of unity property without resorting to rational scaling as in T-splines. The talk will report on the current status of the development of LR-splines, and show the first examples of the flexible refinement of LR-splines.  