Problem
Research questions and friction points this paper is trying to address.
Isogeometric Analysis
Truncated Hierarchical B-splines
Multi-dimensional B-splines and NURBS Optimization
Multi‐level Bézier extraction of truncated hierarchical B‐splines for isogeometric analysis
🤖 AI Summary
Truncated hierarchical B-splines (THB) face challenges in isogeometric analysis (IGA), including poor compatibility with legacy finite element (FE) codes and limited adaptivity due to global matrix assembly and loss of partition-of-unity.
Method: This paper proposes a decoupled multilevel Bézier extraction framework that constructs local, direct THB-to-Bézier element mapping operators—bypassing global stiffness matrix assembly while strictly preserving the partition-of-unity property. The approach enhances sparsity of basis functions and computational flexibility. A novel local construction operator enables seamless integration into standard FE workflows within the GeoPDEs platform.
Contribution/Results: Numerical experiments on Poisson problems demonstrate that, at comparable accuracy, the proposed method significantly reduces adaptive refinement overhead compared to classical THB schemes, thereby improving practical applicability in engineering simulations.
📝 Abstract
Multivariate basis splines (B‐splines) and Non‐uniform rational B‐splines (NURBS) lack adaptivity due to their tensor product structure. Truncated hierarchical B‐splines (THB‐splines) provide a solution for this. THB‐splines organize the parameter space into a hierarchical structure, which enables efficient approximation and representation of functions with different levels of detail. The truncation mechanism ensures the partition of unity property of B‐splines and defines a more scattered set of basis functions without overlapping on the multi‐level spline space. Transferring these multi‐level splines into Bézier elements representation facilitates straightforward incorporation into existing finite element (FE) codes. By separating the multi‐level extraction of the THB‐splines from the standard Bézier extraction, a more general independent framework applicable to any sequence of nested spaces is created. The operators for the multi‐level structure of THB‐splines and the operators of Bézier extraction are constructed in a local approach. Adjusting the operators for the multi‐level structure from an element point of view and multiplying with the Bézier extraction operators of those elements, a direct map between Bézier elements and a hierarchical structure is obtained. The presented implementation involves the use of an open‐source Octave/MATLAB isogeometric analysis (IGA) code called GeoPDEs. A basic Poisson problem is presented to investigate the performance of multi‐level Bézier extraction compared to a standard THB‐spline approach.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Multilevel Truncated Hierarchical B-splines
Hierarchical Bezier Method
Isogeometric Analysis
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