A C^s-smooth mixed degree and regularity isogeometric spline space over planar multi-patch domains

TL;DR

This work addresses the need for high-order smooth isogeometric bases on planar multi-patch domains by constructing a C^s-smooth mixed degree and regularity spline space. The authors design an underlying mixed space on the unit square with p2=2s+1, r2=s near edges/vertices and interior components with degree p1 and r1=p1-1, enabling a reduction of degrees of freedom while preserving C^s continuity across patch interfaces. They provide explicit constructions for Case A ($p1=s+1$) and Case B ($p1=2s+1$), prove key properties, and extend the framework to bilinear-like G^s parameterizations. The isogeometric Galerkin method is applied to the biharmonic and triharmonic equations, with numerical results showing optimal convergence rates in relevant norms and substantial DOF savings compared to uniform high-degree spaces. The approach broadens the practicality of high-order IGA on complex multi-patch geometries and points to future extensions to other PDEs and geometric settings.

Abstract

We construct over a given bilinear multi-patch domain a novel $C^s$-smooth mixed degree and regularity isogeometric spline space, which possesses the degree $p=2s+1$ and regularity $r=s$ in a small neighborhood around the edges and vertices, and the degree $\widetilde{p} \leq p$ with regularity $\widetilde{r} = \widetilde{p}-1 \geq r$ in all other parts of the domain. Our proposed approach relies on the technique [35], which requires for the $C^s$-smooth isogeometric spline space a degree at least $p=2s+1$ on the entire multi-patch domain. Similar to [35], the $C^s$-smooth mixed degree and regularity spline space is generated as the span of basis functions that correspond to the individual patches, edges and vertices of the domain. The reduction of degrees of freedom for the functions in the interior of the patches is achieved by introducing an appropriate mixed degree and regularity underlying spline space over $[0,1]^2$ to define the functions on the single patches. We further extend our construction with a few examples to the class of bilinear-like $G^s$ multi-patch parameterizations [33,35], which enables the design of multi-patch domains having curved boundaries and interfaces. Finally, the great potential of the $C^s$-smooth mixed degree and regularity isogeometric spline space for performing isogeometric analysis is demonstrated by several numerical examples of solving two particular high order partial differential equations, namely the biharmonic and triharmonic equation, via the isogeometric Galerkin method.

Figures (9)

• Figure 1: The basis functions from the space $\mathcal{S}_h^{(p_1, p_2),{(r_1,r_2)}}([0,1])$, where $p_1=s+1$, $r_1=s$ (left), and $p_1=2s+1$, $r_1=2s$ (right), with $s=2$ and $k=5$. The blue and red functions are B-splines from the spaces $\mathcal{S}_h^{{p}_1,{r_1}}([0,1])$ and $\mathcal{S}_h^{{p}_2,{r_2}}([0,1])$, respectively, while the green functions are the truncated ones. The dashed blue functions are the ones from $\mathcal{S}_h^{{p}_1,{r_1}}([0,1])$ before the truncation. The dashed gray vertical lines split the interval $[0,1]$ into $[0,h]$, $[h,1-h]$ and $[1-h,1]$.
• Figure 2: Instances of some basis functions (above) and the positions of extrema of all basis functions (below) of the spaces $\mathcal{S}_h^{(\boldsymbol{p}_1,\boldsymbol{p}_2),(\boldsymbol{r}_1,\boldsymbol{r}_2)}([0,1]^2)$ for $p_1=s+1, r_1=s$ (left) and $p_1=2s+1, r_1=2s$ (right) with $s=2$ and $k=7$. The blue, green and red functions belong to the spaces $\mathcal{S}_1([0,1]^2)$, $\mathcal{\overline{S}}_1([0,1]^2)$ and $\mathcal{S}_2([0,1]^2)$, respectively. The gray region denotes the subdomain $[h,1-h]^2$.
• Figure 3: A particular example of domain $\overline{\Omega}$ with the open patches $\Omega^{(i)}$, defined by respective geometry mappings $\boldsymbol{F}^{(i)}$ (blue color), open edges $\Gamma^{(i)}$ (orange color) and the vertices $\boldsymbol{\Xi}^{(i)}$ (red color).
• Figure 4: The parameterization of the two-patch domain $\overline{\Omega^{(i_0)} \cup \Omega^{(i_1)}}$ with the common edge $\Gamma^{(i)}$ and the associated geometry mappings $\boldsymbol{F}^{(i_0)}$ and $\boldsymbol{F}^{(i_1)}$.
• Figure 5: Plots of the bilinear three-patch domain \ref{['eq:verticesThreePatch']}, left, of the bilinear five-patch domain \ref{['eq:verticesFivePatch']}, middle, and of the bilinear-like $G^2$ three-patch domain \ref{['eq:three_patch_domain_bilinearLike']}, right.

Theorems & Definitions (11)

• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Theorem 3
• proof
• Example 1
• Example 2
• Example 3