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Subdivision-based isogeometric analysis for second order partial differential equations on surfaces

Abstract

We investigate the isogeometric analysis approach based on the extended Catmull–Clark subdivision for solving the PDEs on surfaces. As a compatible technique of NURBS, subdivision surfaces are capable of the refinability of B-spline techniques, and overcome the major difficulties of the interior parameterization encountered by the isogeometric analysis. The surface is accurately represented as the limit form of the extended Catmull–Clark subdivision, and remains unchanged throughout the h-refinement process. The solving of the PDEs on surfaces is processed on the space spanned by the Catmull–Clark subdivision basis functions. In this work, we establish the interpolation error estimates for the limit form of the extended Catmull–Clark subdivision function space on surfaces. We apply the results to develop the approximation estimates for solving multiple second-order PDEs on surfaces, which are the Laplace–Beltrami equation, the Laplace–Beltrami eigenvalue equation and the time-dependent Cahn–Allen equation. Numerical experiments confirm the theoretical results and are compared with the classical linear finite element method to demonstrate the performance of the proposed method.

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Acknowledgements

Qing Pan is supported by National Natural Science Foundation of China (No.11671130), and Construct Program of the Key Discipline in Hunan Province (No.19K056).

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Correspondence to Qing Pan.

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Pan, Q., Rabczuk, T. & Yang, X. Subdivision-based isogeometric analysis for second order partial differential equations on surfaces. Comput Mech 68, 1205–1221 (2021). https://doi.org/10.1007/s00466-021-02065-7

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Keywords

  • Isogeometric analysis
  • Extended Catmull–Clark subdivision
  • A priori error estimates
  • Laplace–Beltrami equation
  • Laplace–Beltrami eigenvalue equation
  • Cahn–Allen equation