Abstract
Starting from a tensor product B-spline space and its collection of B-splines a collection of Locally Refined B-Splines (LR B-splines) is generated in a stepwise refinement process that successively inserts mesh-rectangles in the box-partition over which the spline space is defined. For each step the collection of LR B-splines is refined until the collection only consists of B-splines that have minimal support with respect to the box-partition.
As experienced with refinement of T-splines an LR B-spline refinement process does not necessarily result in a collection of linearly independent B-splines. However, algorithms that check if linear independence can be ensured after a refinement exist:
• None overloaded elements – AS the refinement starts from a tensor product B-spline of degrees
• None overloaded B-splines – Check that there are no B-splines with only overloaded elements.
• Peeling algorithm – Remove all B-splines from the collection of LR B-splines that cannot be part of a linear dependency relation by first removing all none overloaded B-splines, remaining elements that are just covered by one B-splines cannot be included in a linear dependency relation, successively remove B-splines covering such elements.
• Hand-in-hand – Check that the dimension increase of the spline space spanned by the LR B-splines match the dimension increase of the spline space over the box-partition.
Linear independence can also be addressed by looking at the relations between LR B-splines that take part in a linear dependency relation.
• At all convex corners of region containing only linearly dependent B-splines there is at least one B-spline nested within another B-spline.
• The linear dependency configurations observed so fare all fall into the same pattern provided a change of the sequence of refinements is done. First nested B-splines are introduced at the corners of the linearly dependency region. Then nested B-splines not touching the boundary are created by refinement of one or more nested corner B-splines.
The above results we be used for discussing the linear independence over box-partitions box-partitions conforming to the knot lines of hierarchical B-splines.
REFERENCES
[1] Dokken, Tor; Nørtoft, Peter. Isogeometric Analysis of Navier-Stokes Flow Using Locally Refinable B-Splines. I: SAGA – Advances in ShApes, Geometry, and Algebra. Results from the Marie Curie Initial Training Network. Springer 2014 ISBN 978-3-319-08634-7. s. 299-318
[2] Johannessen, Kjetil Andre; Kvamsdal, Trond; Dokken, Tor. Isogeometric analysis using LR B-splines. Computer Methods in Applied Mechanics and Engineering 2014 ;Volum 269. s. 471-514 Dokken, Tor; Lyche, Tom; Pettersen, Kjell Fredrik.
[3] Polynomial splines over locally refined box-partitions. Computer Aided Geometric Design 2013; Volum 30.(3) s. 331-356
As experienced with refinement of T-splines an LR B-spline refinement process does not necessarily result in a collection of linearly independent B-splines. However, algorithms that check if linear independence can be ensured after a refinement exist:
• None overloaded elements – AS the refinement starts from a tensor product B-spline of degrees
• None overloaded B-splines – Check that there are no B-splines with only overloaded elements.
• Peeling algorithm – Remove all B-splines from the collection of LR B-splines that cannot be part of a linear dependency relation by first removing all none overloaded B-splines, remaining elements that are just covered by one B-splines cannot be included in a linear dependency relation, successively remove B-splines covering such elements.
• Hand-in-hand – Check that the dimension increase of the spline space spanned by the LR B-splines match the dimension increase of the spline space over the box-partition.
Linear independence can also be addressed by looking at the relations between LR B-splines that take part in a linear dependency relation.
• At all convex corners of region containing only linearly dependent B-splines there is at least one B-spline nested within another B-spline.
• The linear dependency configurations observed so fare all fall into the same pattern provided a change of the sequence of refinements is done. First nested B-splines are introduced at the corners of the linearly dependency region. Then nested B-splines not touching the boundary are created by refinement of one or more nested corner B-splines.
The above results we be used for discussing the linear independence over box-partitions box-partitions conforming to the knot lines of hierarchical B-splines.
REFERENCES
[1] Dokken, Tor; Nørtoft, Peter. Isogeometric Analysis of Navier-Stokes Flow Using Locally Refinable B-Splines. I: SAGA – Advances in ShApes, Geometry, and Algebra. Results from the Marie Curie Initial Training Network. Springer 2014 ISBN 978-3-319-08634-7. s. 299-318
[2] Johannessen, Kjetil Andre; Kvamsdal, Trond; Dokken, Tor. Isogeometric analysis using LR B-splines. Computer Methods in Applied Mechanics and Engineering 2014 ;Volum 269. s. 471-514 Dokken, Tor; Lyche, Tom; Pettersen, Kjell Fredrik.
[3] Polynomial splines over locally refined box-partitions. Computer Aided Geometric Design 2013; Volum 30.(3) s. 331-356