Macro-element refinement schemes for THB-splines

Abstract

This paper introduces a novel adaptive refinement strategy for Isogeometric Analysis (IGA) using Truncated Hierarchical B-splines (THB-splines). The strategy is motivated by the fact that certain applications may benefit from adaptive refinement schemes, which lead to a higher degree of structure in the locally-refined mesh than usual, and building this structure a priori can simplify the implementation in those contexts. Specifically, we look at two applications: formulation of an L²-stable local projector for THB-splines a la Bézier projection [Dijkstra and Toshniwal (2023)], and adaptive structure-preserving discretizations using THB-splines [Evans et al. (2020), Shepherd and Toshniwal (2024)]. Previously proposed approaches for these applications require mesh modifications to preserve critical properties of the spline spaces, such as local linear independence or the exactness of the discrete de Rham complexes. Instead, we propose a macro-element-based refinement approach based on refining q=q₁×⋯×q_n blocks of elements, termed q-boxes, where the block size q is chosen based on the spline degree p and the specific application. • For the Bézier projection for THB-splines, we refine p-boxes (i.e., q=p). We show that THB-splines are locally linearly independent on p-boxes, which allows for a simple extension of the Bézier projection algorithm to THB-splines. This new formulation significantly improves upon the approach previously proposed by Dijkstra and Toshniwal (2023). • For structure-preserving discretizations, we refine (p+1)-boxes (i.e., q=p+1). We prove that this choice of q ensures that the mesh satisfies the sufficient conditions presented in Shepherd and Toshniwal (2024) for guaranteeing the exactness of the THB-spline de Rham complex a priori and in an arbitrary number of dimensions. This is crucial for structure-preserving discretizations, as it eliminates the need for additional mesh modifications to maintain the exactness of the complex during adaptive simulations. The effectiveness of the proposed framework is demonstrated through theoretical proofs and numerical experiments, including optimal convergence for adaptive approximation and the simulation of the incompressible Navier-Stokes equations.