Efficient Contouring on Unstructured Meshes for Partial Differential Equations

ABSTRACT

We introduce three fast contouring algorithms for visualizing the solution of partial differential equations based on the PCI (pure cubic interpolant). The PCI is a particular piecewise bicubic polynomial interpolant defined over an unstructured mesh. Unlike standard contouring approaches, our contouring algorithms do not need a fine-structured approximation and work efficiently with the original scattered data. The basic idea is to first identify the intersection points between contour curves and the sides of each triangle and then draw smooth contour curves connecting these points. We compare these contouring algorithms with the built-in Matlab contour procedure and other contouring algorithms. We demonstrate that our algorithms are both more accurate and faster than the others.

REFERENCES

• 1: Emma L. Bradbury , Wayne H. Enright, Fast contouring of solutions to partial differential equations, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.418-439, December 2003 [doi>10.1145/962437.962441]
• 2: W. H. Enright, Accurate approximate solution of partial differential equations at off-mesh points, ACM Transactions on Mathematical Software (TOMS), v.26 n.2, p.274-292, June 2000 [doi>10.1145/353474.353482]
• 3: Goldani-Moghaddam, H. 2004. Efficient contouring on unstructured meshes. M.S. thesis, University of Toronto.
• 4: Goldani-Moghaddam, H. and Enright, W. H. 2005. The PCI: A scattered data interpolant for the solution of partial differential equations. In Proceedings of the International Conference on Adaptive Modeling and Simulation (ADMOS). Barcelona, Spain.
• 5: MathWorks. 2008. MATLAB online documentation, 12th ed. http.www.mathworks.com/access/ helpdesk/help/helpdesk.html.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Approximation of surfaces and contours
      G.1.8 Partial Differential Equations
          Subjects: Elliptic equations

General Terms:
Algorithms, Performance

Keywords:
PDE, Visualization, contour, scattered data, unstructured mesh