Macro-element Refinement schemes for THB-Splines: Applications to Bézier Projection and Structure-Preserving Discretizations

TL;DR

This work introduces a macro-element refinement strategy for THB-splines using $oldsymbol{q}$-boxes to enable application-driven adaptivity without mesh modification. By selecting $oldsymbol{q}=oldsymbol{p}$, it yields a constructive, locally linearly independent Bézier projector for THB-splines across dimensions. Selecting $oldsymbol{q}=oldsymbol{p}+oldsymbol{1}$ ensures the THB-spline de Rham complex remains exact on adaptive meshes, enabling stable structure-preserving simulations of problems like incompressible Navier–Stokes and Maxwell-type PDEs. Theoretical results on non-overload conditions are complemented by numerical experiments showing optimal convergence and effective adaptive structure-preserving discretizations.

Abstract

This paper introduces a novel adaptive refinement strategy for Isogeometric Analysis (IGA) using Truncated Hierarchical B-splines (THB-splines). The proposed strategy enhances locally-refined meshes for specific applications, simplifying implementation. We focus on two key applications: an $L^2$-stable local projector for THB-splines via Bézier projection [Dijkstra and Toshniwal (2023)], and structure-preserving discretizations using THB-splines [Evans et al. (2020), Shepherd and Toshniwal (2024)]. Previous methods required mesh modifications to retain crucial properties like local linear independence and the exactness of discrete de Rham complexes. Our approach introduces a macro-element-based refinement technique, refining $\vec{q} = q_1\times\cdots\times q_n$ blocks of elements, termed $\vec{q}$-boxes, where the block size $\vec{q}$ is determined by the spline degree and application. For the Bézier projection, we refine $\vec{p}$-boxes (i.e., $\vec{q} = \vec{p}$), ensuring THB-splines are locally linearly independent in these boxes, which enables a straightforward extension of the Bézier projection algorithm, greatly improving upon Dijkstra and Toshniwal (2023). For structure-preserving discretizations, we refine $(\vec{p+1})$-boxes (i.e., $\vec{q} = \vec{p}+\vec{1}$), demonstrating that this choice meets the sufficient conditions for ensuring the exactness of the THB-spline de Rham complex, as outlined by Shepherd and Toshniwal (2024), in any dimension. This critical aspect allows for adaptive simulations without additional mesh modifications. The effectiveness of our framework is supported by theoretical proofs and numerical experiments, including optimal convergence for adaptive approximation and simulations of the incompressible Navier-Stokes equations.

Figures (11)

• Figure 1: An HB-spline basis \ref{['fig:HB-THB-comparison-HB']} and a THB-spline basis \ref{['fig:HB-THB-comparison-THB']}. The green and orange highlighted basis functions represent the (T)HB-splines at levels 1 and 2, respectively. The sum of all basis functions is shown in purple, showing that the THB-splines form a partition of unity.
• Figure 2: An element mesh $\Omega^{e_{\ell}}$ (top) and a $\boldsymbol{q}$-box mesh $\Omega^{E_{r,\ell}^{}}$ (bottom, $q=2$) of levels $\ell=1,2$. Like mesh elements, $\boldsymbol{q}$-boxes are bisected when refined.
• Figure 3: In (a), a $\boldsymbol{q}$-box mesh is depicted for $\boldsymbol{q}=(2,2)$ and the associated element mesh with dashed lines. In (b) and (c), respectively, the active border and well-behaved $\boldsymbol{q}$-boxes are highlighted. Observe that well-behaved $\boldsymbol{q}$-boxes can either be active or non-active.
• Figure 4: On the left a well-behaved $\boldsymbol{p}$-box $\Omega^{E_{\ell}^{}}$ of level $\ell$ is depicted. The middle depicts the active and deactivated $\boldsymbol{p}$-boxes in $\Omega^{E_{\ell}^{}}$ per level. The border $\boldsymbol{p}$-boxes are highlighted in grey. On the right, the intermediate THB-spline spaces ${\mathcal{T}_{k}^{}}$ are given for the levels $\ell,\ell+1$ and $\ell+2$.
• Figure 5: Convergence rate for various spline degrees $p$ and the theoretical convergence rates according to Theorem \ref{['thm:LocalTHBsplineProjEst']}. The same refinement domain is used for all cases for a fair comparison.

Theorems & Definitions (51)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.1
• Proposition 3.2