Properties of Spline Spaces Over Structured Hierarchical Box Partitions

TL;DR

This work addresses what happens to the properties of the LRB-space when it is modified by local one-directional refinement at convex corners of, and along edges between dyadic refinement regions, and shows that such local modifications can reduce the number of B-splines over each element to the minimum prescribed by the polynomial bi-degree.

Abstract

Given a spline space spanned by Truncated Hierarchical B-splines (THB), it is always possible to construct a spline space spanned by Locally Refined B-splines (LRB) that contains the THB-space. Starting from configurations where the two spline spaces are equal, we adress what happens to the properties of the LRB-space when it is modified by local one-directional refinement at convex corners of, and along edges between dyadic refinement regions. We show that such local modifications can reduce the number of B-splines over each element to the minimum prescribed by the polynomial bi-degree, and that such local refinements can be used for improving the condition numbers of mass and stiffness matrices.

Figures (17)

• Figure 1: Spline spaces over the domain $\Omega = [0, 5]$. In \ref{['sub:open']}, the partition of unity is satisfied at the boundary by setting the knot multiplicity to $m = d + 1 = 4$. In \ref{['sub:ghost']}, the partition of unity is satisfied at the boundary by extending the domain to allow the full polynomial space to be spanned at the boundary elements. The shaded regions indicate the domain $\Omega$, and the spline space spanned by the B-splines over $\Omega$ are the same in both cases.
• Figure 2: The condition number of the mass matrix. We see that under repeated refinement, the condition numbers corresponding to spline spaces with open knot vectors (THB, LRB) tends towards the condition numbers corresponding to spline spaces with single knots (S-THB, S-LRB). We also see that a small local modification to reduce overloading in the LRB-space reduces the condition number of the mass matrix (S-LRB1, LRB1).
• Figure 3: The condition number of the stiffness matrix. Here the separation between S-LRB, S-THB, LRB and THB are seen in even greater effect.
• Figure 4: The meshes used for the preliminary comparison. In \ref{['sub:boundary_multiplicity_mesh_nomod']}, the unmodified mesh used for S-THB, S-LRB, THB and LRB. In \ref{['sub:boundary_multiplicity_mesh_mod']} the modified mesh used for S-LRB1 and LRB1. This mesh generates a few extra degrees of freedom.
• Figure 5: The eigenvectors corresponding to the smallest eigenvalue of the mass matrix for LRB (top row), and for S-LRB (bottom row) visualized over the hierarchical mesh after one, three and six refinements (left to right). Darker color indicates higher influence. As we see, the smallest eigenvalues for LRB is localized in the refined region after only one refinement. On the other hand, S-LRB is localized in the corners of the domain up until but not including six refinements, as shown for $n = 1$ and $n = 3$. The effect of the locally refined region dominates only after $n = 6$ refinements as in \ref{['sub:slrb_smallest_6']}.

Theorems & Definitions (22)

• remark 1
• definition 1
• remark 2
• definition 2
• definition 3: Informal
• remark 3
• definition 4
• definition 5
• definition 6
• definition 7