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 B\'ezier 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 B\'ezier 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 B\'ezier 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.

翻译：本文针对等几何分析提出了一种基于截断分层B样条的新型自适应细化策略。该策略通过增强局部细化网格在特定应用中的适应性，简化了算法实现过程。我们重点研究两个关键应用领域：基于Bézier投影的THB样条$L^2$稳定局部投影算子[Dijkstra and Toshniwal (2023)]，以及采用THB样条的结构保持离散化方法[Evans et al. (2020), Shepherd and Toshniwal (2024)]。传统方法需要修改网格以保持局部线性独立性和离散de Rham复形精确性等重要性质。本文提出的方法引入了基于宏单元的细化技术，通过细化$\vec{q} = q_1\times\cdots\times q_n$个单元块（称为$\vec{q}$-盒）来实现，其中块尺寸$\vec{q}$由样条次数和应用需求共同决定。对于Bézier投影，我们细化$\vec{p}$-盒（即$\vec{q} = \vec{p}$），确保THB样条在这些单元块内保持局部线性独立，从而实现了Bézier投影算法的直接扩展，较Dijkstra和Toshniwal (2023)的方法有显著改进。在结构保持离散化方面，我们细化$(\vec{p+1})$-盒（即$\vec{q} = \vec{p}+\vec{1}$），证明该选择满足Shepherd和Toshniwal (2024)提出的任意维度THB样条de Rham复形精确性充分条件。这一关键特性使得自适应模拟无需额外的网格修改。通过理论证明和数值实验（包括自适应逼近的最优收敛性及不可压缩Navier-Stokes方程模拟）验证了本框架的有效性。