Isogeometric collocation with smooth mixed degree splines over planar multi-patch domains

Abstract

We present a novel isogeometric collocation method for solving the Poisson's and the biharmonic equation over planar bilinearly parameterized multi-patch geometries. The proposed approach relies on the use of a modified construction of the C^s-smooth mixed degree isogeometric spline space [20] for s=2 and s=4 in case of the Poisson's and the biharmonic equation, respectively. The adapted spline space possesses the minimal possible degree p=s+1 everywhere on the multi-patch domain except in a small neighborhood of the inner edges and of the vertices of patch valency greater than one where a degree p=2s+1 is required. This allows to solve the PDEs with a much lower number of degrees of freedom compared to employing the C^s-smooth spline space [29] with the same high degree p=2s+1 everywhere. To perform isogeometric collocation with the smooth mixed degree spline functions, we introduce and study two different sets of collocation points, namely first a generalization of the standard Greville points to the set of mixed degree Greville points and second the so-called mixed degree superconvergent points. The collocation method is further extended to the class of bilinear-like G^s multi-patch parameterizations [26], which enables the modeling of multi-patch domains with curved boundaries, and is finally tested on the basis of several numerical examples.

Figures (13)

• Figure 1: Examples of a two-patch domain $\overline{\Omega}$ (left) and of a three-patch domain $\overline{\Omega}$ (right) with the patches $\Omega^{(i)}$ and their associated geometry mappings $\boldsymbol{F}^{(i)}$, with the edges $\Gamma^{(i)}$ (violet) and with the vertices $\boldsymbol{\Xi}^{(i)}$ (blue), where $x_1^{(i)} = \boldsymbol{F}^{(i)}(\xi_1,0)$ and $x_2^{(i)} = \boldsymbol{F}^{(i)}(0,\xi_2)$.
• Figure 2: The positions of extrema of all basis functions of the space $\mathcal{S}_h^{(\boldsymbol{s}+\boldsymbol{1},2\boldsymbol{s}+\boldsymbol{1}),\boldsymbol{s}}([0,1]^2)$ for $s=2$ and $k=8$. The five possible different variants of the space $\mathcal{S}_h^{(\boldsymbol{p}_1, \boldsymbol{p}_2),\boldsymbol{s}}([0,1]^2)$ depend on the number and position of the edges of $[0,1]^2$ which correspond to the inner edges of the multi-patch domain $\overline{\Omega}$. The blue, green and red dots correspond to the functions belonging 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 dashed gray lines denote the boundary of the subdomain $[h,1-h]^2$.
• Figure 3: The mixed degree Greville points for the spaces $\mathcal{S}_{1/9}^{(\boldsymbol{3},\boldsymbol{5}),\boldsymbol{2}}$ (left) and $\mathcal{S}_{1/9}^{(\boldsymbol{5},\boldsymbol{9}),\boldsymbol{4}}$ (right). The blue, green and red collocation points correspond to the basis functions from the spaces $\mathcal{S}_1([0,1]^2)$, $\overline{\mathcal{S}}_1([0,1]^2)$ and $\mathcal{S}_2([0,1]^2)$, respectively. The dashed gray lines denote the boundary of the subdomain $[h,1-h]^2$.
• Figure 4: All mixed degree superconvergent points for the spaces $\mathcal{S}_{1/9}^{(\boldsymbol{3},\boldsymbol{5}),\boldsymbol{2}}$ (left) and $\mathcal{S}_{1/9}^{(\boldsymbol{5},\boldsymbol{9}),\boldsymbol{4}}$ (right). The blue, green and red points are the clustered mixed degree superconvergent points that correspond to the basis functions from the spaces $\mathcal{S}_1([0,1]^2)$, $\overline{\mathcal{S}}_1([0,1]^2)$ and $\mathcal{S}_2([0,1]^2)$, respectively. The dashed gray lines denote the boundary of the subdomain $[h,1-h]^2$.
• Figure 5: Top row: Plots of the bilinear one-patch Domain A, given in \ref{['eq:DomainA']}, left, bilinear three-patch Domain B, presented in \ref{['eq:three_patch_domain_param']}, middle, and bilinear five-patch Domain C, given in \ref{['eq:five_patch_domain_param']}, right. Bottom row: The three multi-patch domains together with the exact solution \ref{['eq:exactSolution']}.

Theorems & Definitions (5)

• Proposition 1
• proof
• Example 1
• Example 2
• Example 3