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3D Printing of Preclinical X-ray Computed Tomographic Data Sets
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Discrete differential geometry: the nonplanar quadrilateral mesh.

Carole J Twining1, Stephen Marsland

  • 1Imaging Science and Biomedical Engineering, University of Manchester, Manchester, United Kingdom. carole.twining@manchester.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
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Summary

This study develops discrete differential geometry for quadrilateral meshes, enabling exact conservation laws in physical models like electromagnetism. The new methods work on non-flat surfaces, unlike prior work focused on triangles.

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Area of Science:

  • Computational geometry
  • Discrete differential geometry
  • Mathematical physics

Background:

  • Existing discrete differential geometry primarily uses triangulated meshes.
  • Nonplanar quadrilateral meshes are crucial for accurate physical modeling in electromagnetism, fluid mechanics, and image analysis.
  • Current methods often approximate continuum theories on discrete spaces.

Purpose of the Study:

  • To construct a discrete differential geometry framework for nonplanar quadrilateral meshes.
  • To develop discrete analogs of fundamental geometric and physical concepts.
  • To ensure exact conservation laws on discrete meshes.

Main Methods:

  • Discretizing tensor calculus on quadrilateral meshes.
  • Developing discrete differential forms for mesh-based analysis.
  • Constructing discrete analogs of the Levi-Civita connection, torsion, and curvature.
  • Formulating discrete vector integral theorems.

Main Results:

  • Demonstrated that discretizing tensor calculus and differential forms yield different results on discrete meshes.
  • Successfully constructed mesh versions of the Levi-Civita connection, torsion, and curvature.
  • Developed discrete integral theorems that enforce exact conservation laws.
  • Created a discrete formulation of classical electromagnetism.

Conclusions:

  • The proposed discrete differential geometry on quadrilateral meshes provides an accurate and robust framework for physical modeling.
  • This formalism allows for exact conservation laws, surpassing approximations of continuum theories.
  • The method is applicable to various physical models, including electromagnetism and fluid dynamics.