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Boundary parameter matching for isogeometric analysis using Schwarz-Christoffel mapping

Ye Ji, Matthias Möller, Yingying Yu, Chungang Zhu

TL;DR

This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations, and employs a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences.

Abstract

Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. The existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz-Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

Boundary parameter matching for isogeometric analysis using Schwarz-Christoffel mapping

TL;DR

This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations, and employs a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences.

Abstract

Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. The existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz-Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.
Paper Structure (20 sections, 3 theorems, 15 equations, 12 figures, 1 table)

This paper contains 20 sections, 3 theorems, 15 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Review for analysis-suitable parameterization in IGA
  4. Review for Schwarz-Christoffel mapping
  5. Problem statement and key ideas
  6. Problem statement
  7. Key ideas
  8. Methodology
  9. Schwarz–Christoffel mapping
  10. Schwarz-Christoffel parameter problem
  11. Geometry-preserving curve reparameterization technique
  12. Analysis-suitable paramterization
  13. Numerical experiments and comparisons
  14. Implementation details and parameterization quality metrics
  15. Role of boundary correspondence
  16. ...and 5 more sections

Key Result

Lemma 1

Let $N^{\bm{\Xi}}_{i, p}(\xi)$ be the $i$-th degree-$p$ B-Spline basis function defined over the knot vector $\bm{\Xi}$. Consider a scaled and translated knot vector $\hat{\bm{\Xi}} = s \bm{\Xi} + t$ with $s > 0$. Then, $N^{\hat{\bm{\Xi}}}_{i, p}(s \xi + t) = N^{\bm{\Xi}}_{i, p}(\xi)$ holds.

Figures (12)

  • Figure 1: Boundary parameter matching problem (Continental Shelf of the Gulf of Mexico): (a) Utilizing conventional chord-length parameterization for the boundary curves, (b) Applying the proposed Schwarz-Christoffel mapping approach for enhanced boundary correspondence.
  • Figure 2: Workflow diagram of the proposed boundary parameter matching approach via Schwarz-Christoffel mapping.
  • Figure 3: Notational conventions for the Schwarz–Christoffel mapping.
  • Figure 4: Schematic diagram for curve reparameterization and the resulting parameterization after curve parameterization.
  • Figure 5: Comparative visualization of scaled Jacobian $\vert \bm{\mathcal{J}} \vert_s$: (a) Parameterization via linear interpolation; (b) Improved parameterization employing PDE-Based approach.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Proposition 3