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A characterization of linear independence of THB-splines in $\mathbb{R}^n$ and application to Bézier projection

Kevin Dijkstra, Deepesh Toshniwal

TL;DR

This work extends the Bézier projection framework to truncated hierarchical B-splines (THB-splines) by introducing projection elements that avoid local linear dependence (overloading). It establishes a two-step projection process on non-overloaded macro-elements and provides a priori error estimates, demonstrating optimal convergence in 2D and detailing an adaptive refinement scheme. The approach preserves THB-spline structure while enabling local projections and stable global recovery, with numerical results that corroborate theoretical rates and compare favorably to existing THB projections. The methodology opens avenues for more aggressive local refinement and scalable projection-based techniques in isogeometric analysis.

Abstract

In this paper we propose a local projector for truncated hierarchical B-splines (THB-splines). The local THB-spline projector is an adaptation of the Bézier projector proposed by Thomas et al. (Comput Methods Appl Mech Eng 284, 2015) for B-splines and analysis-suitable T-splines (AS T-splines). For THB-splines, there are elements on which the restrictions of THB-splines are linearly dependent, contrary to B-splines and AS T-splines. Therefore, we cluster certain local mesh elements together such that the THB-splines with support over these clusters are linearly independent, and the Bézier projector is adapted to use these clusters. We introduce general extensions for which optimal convergence is shown theoretically and numerically. In addition, a simple adaptive refinement scheme is introduced and compared to Giust et al. (Comput. Aided Geom. Des. 80, 2020), where we find that our simple approach shows promise.

A characterization of linear independence of THB-splines in $\mathbb{R}^n$ and application to Bézier projection

TL;DR

This work extends the Bézier projection framework to truncated hierarchical B-splines (THB-splines) by introducing projection elements that avoid local linear dependence (overloading). It establishes a two-step projection process on non-overloaded macro-elements and provides a priori error estimates, demonstrating optimal convergence in 2D and detailing an adaptive refinement scheme. The approach preserves THB-spline structure while enabling local projections and stable global recovery, with numerical results that corroborate theoretical rates and compare favorably to existing THB projections. The methodology opens avenues for more aggressive local refinement and scalable projection-based techniques in isogeometric analysis.

Abstract

In this paper we propose a local projector for truncated hierarchical B-splines (THB-splines). The local THB-spline projector is an adaptation of the Bézier projector proposed by Thomas et al. (Comput Methods Appl Mech Eng 284, 2015) for B-splines and analysis-suitable T-splines (AS T-splines). For THB-splines, there are elements on which the restrictions of THB-splines are linearly dependent, contrary to B-splines and AS T-splines. Therefore, we cluster certain local mesh elements together such that the THB-splines with support over these clusters are linearly independent, and the Bézier projector is adapted to use these clusters. We introduce general extensions for which optimal convergence is shown theoretically and numerically. In addition, a simple adaptive refinement scheme is introduced and compared to Giust et al. (Comput. Aided Geom. Des. 80, 2020), where we find that our simple approach shows promise.
Paper Structure (27 sections, 7 theorems, 63 equations, 8 figures, 2 algorithms)

This paper contains 27 sections, 7 theorems, 63 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. B-splines and Bézier projection
  3. Univariate B-splines
  4. Multivariate B-splines
  5. Bézier projection for B-splines
  6. Pseudo-local linear independence of THB-splines
  7. Construction of THB-splines
  8. Some assumption on the hierarchical meshes
  9. An example of overloaded mesh elements in ${\mathbb R}^1$
  10. Characterizing non-overloaded mesh elements in ${\mathbb R}^n$
  11. Bézier projection for THB-splines
  12. General formulation of the projector
  13. Step 1: Local projections on $\widehat{\Omega}^e$
  14. Step 2: Weighted-averaging to form the global projection on $\Omega$
  15. Possible choices for $\mathbb{V}(\widehat{\Omega}^e)$
  16. ...and 12 more sections

Key Result

lemma 3.1

Let $\Omega^e$ be a well-behaved border element, and let $\tau_{\vec{t}_1}(\Omega), \tau_{\vec{t}}(\Omega) \subset \widehat{\Omega}^e$, $|t_1^i|\geq |t^i|$ for all $i$, such that they are both contained in the same level $\ell-1$ element. If $T_j$ is the truncation of some level $\ell-1$ B-spline an

Figures (8)

  • Figure 1: An initial projection on to local collections of mesh elements, and a global smoothing step to produce a global projection of the target onto a THB-spline space.
  • Figure 2: The two main steps of Univariate Bézier B-spline projection. The Target function is initially projected on to a local polynomial basis of degree $p$, on every element. These discontinuous polynomial functions are smoothed to obtain a global projection in the B-spline space $\mathbb{B}_{\boldsymbol{\xi}}$.
  • Figure 3: An example of an HB-spline basis (top) and a corresponding THB-spline basis (bottom). The blue splines are from the first level and the orange splines are from the second one. Notice that the total sum of all splines (red line) is 1 for the THB-spline basis.
  • Figure 4: The basis functions of a quadratic THB-spline space consisting of two levels are shown. While the blue coloured element is overloaded, the combination of the blue and orange is not.
  • Figure 5: Numerical Convergence rates for projection of $f(x,y) = \sin(\pi x)\sin(\pi y)$ onto the THB-spline space built for the domain hierarchy from \ref{['eq:local-estimates-numerical-problem']}. The numerical rates (solid lines) are compared to Theorem \ref{['thm:LocalTHBsplineProjEst']} (dashed lines).
  • ...and 3 more figures

Theorems & Definitions (25)

  • definition 1.1
  • definition 3.1
  • definition 3.2
  • definition 3.3
  • definition 3.4
  • definition 3.5
  • definition 3.6
  • definition 3.7
  • definition 3.8
  • lemma 3.1
  • ...and 15 more