Macro-element Refinement schemes for THB-Splines: Applications to B\'ezier Projection and 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 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.