Four lectures on Locally Refined B-splines

Local refinement of spline represented models is essential in isogeometric analysis to facilitate compact geometry and analysis models and avoid unnecessary growth in the dimension of the spline space used. The three dominant approaches to locally refinable splines within isogeometric analysis are: • Hierarchical B-splines based on a hierarchy of nested splines spaces • T-splines where the refinement is specified in the T-mesh relating the vertices (coefficients) of the B-splines to the knot line segments in the parameter domain of the B-splines • Locally Refined B-splines (LR B-splines) where the refinement is specified directly in the parameter domain of the B-splines. While refinement of Hierarchical B-splines and LR B-splines is addressed from the structure of the spline spaces generated, refinement of T-splines is addressed from the T-mesh. The basis of all approaches is the refinement of a spline surface with a rectangular domain.' For all approaches a number of such rectangular parametric spline surfaces can be glued to form more complex patchworks of surfaces, where the number of surfaces meeting at a common vertex can be different from 4. The challenges of establishing proper continuity over the extraordinary points (vertices where 3, 5 or more surfaces meet) are the same for all approaches. For T-splines a T-mesh combining the T-meshes of all the rectangular T-splines surfaces is easy to construct, consequently providing one single composite T-mesh for the T-spline surface patch work. The vertices of the LR B-splines have the same geometric interpretation as the vertices of the T-splines. However, establishing adjacency relations between vertices in a T-spline fashion drastically reduces the modelling flexibility of the LR B-spline space, and this is not a feasible approach. Published theory of Hierarchical B-splines and T-splines is dominantly addressing the 2-variate case. The theory of LR B-splines is for the d-variate case, both with respect to splines spaces, their spanning properties, and when the B-splines form a basis. The extended T-grid of T-splines corresponds to the LR-mesh of LR B-splines. Consequently LR B-splines also form an extended theory for T-splines over rectangular domains, and provide more general conditions for situations when T-splines form a basis than the current fairly restrictive notion of analysis suitable T-splines. References [1] T. Dokken, T. Lyche y, K. F. Pettersen, Locally Refinable Splines over Box-Partitions. Preprint. http://www.sintef.no/Projectweb/Computational-Geometry/ [2] A. V. Vuong, C. Giannelli, B. Juttler, B. Simeon, A Hierarchical Approach to Adaptive Local Refinement in Isogeometric Analysis, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3554{3567. [3] T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng, and T. Lyche. T-spline simplication and local refinement. ACM Trans.Graph. (TOG), 23 (2004), pp. 276-283.