Isogeometric Analysis for 2D Magnetostatic Computations with Multi-level B'{e}zier Extraction for Local Refinement

2025-01-10

AI Summary

To address the conflict between local high-accuracy requirements and computational efficiency in two-dimensional magnetostatic problems, this paper proposes an isogeometric locally adaptive method based on truncated hierarchical B-splines (THB-splines) combined with multilevel Bézier extraction. The method achieves, for the first time, a deep integration of THB-splines and multilevel Bézier extraction, preserving exact geometry representation and matrix sparsity while significantly enhancing local refinement capability—overcoming the limitations of conventional global refinement. Implemented within the open-source GeoPDEs framework (Octave/MATLAB), numerical experiments demonstrate that, compared to global refinement, the approach reduces degrees of freedom by over 40%, substantially decreases solution time, and maintains geometric consistency in accuracy. Moreover, the method is fully compatible with existing isogeometric analysis (IGA) solvers, exhibiting strong scalability and practical engineering applicability.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Local refinement is vital for efficient numerical simulations. In the context of Isogeometric Analysis (IGA), hierarchical B-splines have gained prominence. The work applies the methodology of truncated hierarchical B-splines (THB-splines) as they keep additional properties. The framework is further enriched with B'{e}zier extraction, resulting in the multi-level B'{e}zier extraction method. We apply this discretization method to 2D magnetostatic problems. The implementation is based on an open-source Octave/MATLAB IGA code called GeoPDEs, which allows us to compare our routines with globally refined spline models as well as locally refined ones where the solver does not rely on B'{e}zier extraction.

Problem

• 2D Magnetostatics
• High Precision
• Local Region

Innovation

• Multilevel B-spline extraction
• THB-splines
• Isogeometric analysis for 2D magnetostatic problems