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Isogeometric Analysis for 2D Magnetostatic Computations with Multi-level Bézier Extraction for Local Refinement

Andreas Grendas, Michael Wiesheu, Sebastian Schöps, Benjamin Marussig

TL;DR

This work addresses the need for local, geometry-preserving refinement in isogeometric analysis of magnetostatic problems by combining truncated hierarchical B-splines (THB-splines) with multi-level Bézier extraction (MBE). The proposed approach enables adaptive refinement at the element level while preserving CAD geometry, facilitated by a local extraction operator that maps Bernstein elements to the hierarchical basis in a multipatch setting. An a posteriori least-squares estimator with Dorfler marking drives refinement, and the method is demonstrated on a 2D multipatch magnetostatics problem, showing faster convergence with fewer degrees of freedom compared to uniform refinement. The results indicate that THB-splines with MBE integrate well with existing Bézier-based IGA codes and are effective for complex geometries, providing practical benefits for magnetostatic simulations in engineering applications.

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 Bézier extraction, resulting in the multi-level Bézier 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 Bézier extraction.

Isogeometric Analysis for 2D Magnetostatic Computations with Multi-level Bézier Extraction for Local Refinement

TL;DR

This work addresses the need for local, geometry-preserving refinement in isogeometric analysis of magnetostatic problems by combining truncated hierarchical B-splines (THB-splines) with multi-level Bézier extraction (MBE). The proposed approach enables adaptive refinement at the element level while preserving CAD geometry, facilitated by a local extraction operator that maps Bernstein elements to the hierarchical basis in a multipatch setting. An a posteriori least-squares estimator with Dorfler marking drives refinement, and the method is demonstrated on a 2D multipatch magnetostatics problem, showing faster convergence with fewer degrees of freedom compared to uniform refinement. The results indicate that THB-splines with MBE integrate well with existing Bézier-based IGA codes and are effective for complex geometries, providing practical benefits for magnetostatic simulations in engineering applications.

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 Bézier extraction, resulting in the multi-level Bézier 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 Bézier extraction.
Paper Structure (11 sections, 17 equations, 10 figures)

This paper contains 11 sections, 17 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. B-splines
  4. Multi-level structure with THB-splines
  5. Standard Bézier extraction
  6. Multi-level Bézier extraction
  7. Problem assembly for numerical simulation
  8. Posteriori error estimator
  9. Numerical example
  10. Conclusions
  11. Ackowledgments

Figures (10)

  • Figure 1: B-spline function with knot vector, $\Xi = [0,0,0,0.25,0.5,0.75,0.75,1,1,1]$ and degree $p=2$. Left: basis functions. Right: curve with control points.
  • Figure 2: Reference domain with the basis function in 2D, as tensor product structure.
  • Figure 3: Left: a univariate quadratic THB-spline with two additional levels of refinement. Right: a 2D representation of a tensor product quadratic THB-splines.
  • Figure 4: Left: B-spline basis and curve. Right: Bézier basis and curve.
  • Figure 5: Translation from hierarchical spline space to local Bernstein polynomials.
  • ...and 5 more figures