LR B-spline perspective for RM B-splines: construction and effortless refinements

TL;DR

The paper reframes RM B-splines as a refinement mechanism within the LR B-spline framework, enabling automatic refinements by leveraging established LR strategies and integrating RM B-splines into LR implementations. RM B-splines are generated from bilinear LR B-splines by multiplicity adjustments and can be evaluated on demand using existing de Boor-like routines, facilitating seamless adoption in IgA workflows. Numerical experiments reveal that RM B-splines expand the spline space compared to maximally smooth LR B-splines, yielding larger systems and sometimes no accuracy gain per DOF, though they can produce sparser meshes and enable efficient evaluation. Overall, the work provides a practical path to incorporate RM B-splines into LR-based isogeometric analysis with minimal changes to existing infrastructure.

Abstract

Reachable Minimally supported (RM) B-splines have been recently introduced as a novel B-spline--like basis. They feature local linear independence and admit a fast de Boor--like evaluation algorithm. These properties make them particularly attractive for applications in isogeometric analysis. In this note, we show that automatic mesh refinement procedures can be readily established by observing that RM B-splines are a special case of Locally Refined (LR) B-splines.

Figures (8)

• Figure 1: Example of tensor product mesh and some tensor product B-splines on it. In (a) the mesh $\mathcal{M}$. Given the degree $\mathbf{p} = (2, 2)$, the vertical meshline $\gamma$ has multiplicity $\mu(\gamma) = 3$, i.e., the maximal multiplicity allowed for $\mathbf{p}$. Such multiplicity is emphasized in the figures by drawing $\gamma$ with a triple line. In (b)--(d) we consider three different biquadratic B-splines $B_1, B_2$ and $B_3$ on $\mathcal{M}$. The highlighted regions correspond to their supports and their local tensor meshes are also coloured. In particular (part of) $\gamma$ is in all of them, as $\gamma_{B_1} = \gamma_{B_2} = \gamma_{B_3}$. However, it is here considered with different multiplicities: $\mu_{B_1}(\gamma_{B_1}) = 3, \mu_{B_2}(\gamma_{B_2}) = 2, \mu_{B_3}(\gamma_{B_3}) = 1$.
• Figure 2: Mesh refinement and secondary split of the generated B-splines to recover the minimal support property. Let $\mathbf{p} = (1, 1)$. In figure (a) we see a bilinear B-spline $B$ which is refined by the new meshline $\gamma_1$ (dashed). $B$ does not have minimal support on the refined mesh. We replace it with the two B-splines $B_1, B_2$ pictured in figure (b), involved in the knot insertion relation along the first direction. However, $B_1$ has not minimal support on the new mesh as well as its support is traversed by a meshline $\gamma_2$ on the mesh. Therefore, also $B_1$ is replaced using the knot insertion procedure. The final set of B-spline $\{B_2, B_3, B_4\}$ generated from the refinement of $B$ is illustrated in figure (c).
• Figure 3: Example of B-spline system in the interior of the mesh for $s = 1$. All the internal meshlines of the mesh have multiplicity $s + 1 = 2$. In the figures we show the local knot vectors and local tensor meshes of the B-splines belonging to the same system. Their local knot vectors, composed of $p + 2 = 5$ knots, in the first and second directions have been illustrated with full and empty dots respectively. Furthermore, double dots correspond to knots of multiplicity $2$ in the local knot vector. Such local knot vectors generate local tensor meshes, covering the support of the associated B-splines. Double dots correspond to double lines to emphasize the higher local meshline multiplicity considered. Although the 4 B-splines are all different, they have same support and hence belong to the same system.
• Figure 4: Examples of LR meshes which are admissible for RM B-splines. The meshes on the left column are built using the N$_2$S$_2$, the central using the HLR and the right using the EG refinements, respectively, for bilinear LR B-splines. All the meshes have been constructed using 7 iterations starting from the open boundary as initial tensor mesh for $\mathbf{p} = (1, 1)$. In the top row (a)--(c) we visually compare the refinements concentrated on three diagonal points. In the central row (d)--(f) we display the refinements along the diagonal. In the bottom row (g)--(i) we display the resulting meshes when refining along a circular arc. All the depicted LR meshes are admissible for RM B-splines of any degree by adjusting the meshline multiplicities as described in Proposition \ref{['prop']}.
• Figure 5: Comparisons of the cardinalities of sets of RM B-splines and LR B-splines of maximal smoothness, denoted by $\#\mathcal{R}$ and $\#\mathcal{L}$, of the same degree. The underlying LR meshes have meshlines (a) localized along the diagonal, (b) concentrated at three equidistant points on the diagonal, and (c) aligned along a circular arc, corresponding to the configurations shown in Figure \ref{['fig:ex']} for the RM B-splines. The meshes are generated using the $\mathrm{N}_2\mathrm{S}_2$, HLR, and EG refinement strategies. For RM B-splines, these strategies are applied to create LR meshes for LR B-splines of bidegree $(1,1)$, independently of the RM smoothness parameter $s$, as explained by Proposition \ref{['prop']}. For LR B-splines, the same refinement strategies are applied to build LR meshes for LR B-splines of bidegree $\mathbf{p} = (p,p)$ with $p = 2s+1$ and maximal smoothness. The horizontal axis corresponds to $s \in \{0,\ldots,10\}$, while the vertical axis displays the ratio $\#\mathcal{R}/\#\mathcal{L}$.

Theorems & Definitions (23)

• definition 1: Meshline multiplicity
• definition 2: Local tensor mesh & B-spline support
• definition 3: Local meshline multiplicity
• remark 1
• definition 4: Locally Refined (LR) B-splines
• definition 5: Locally Refined (LR) mesh
• remark 2
• definition 6: Local linear independence
• theorem 1: Characterization of Local Linear Independence with Non-Overloading
• remark 3