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Two-dimensional metric tensor visualization using pseudo-meshes

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Abstract

Riemannian metric tensors are used to control the adaptation of meshes for finite element and finite volume computations. To study the numerous metric construction and manipulation techniques, a new method has been developed to visualize two-dimensional metrics without interference from an adaptation algorithm. This method traces a network of orthogonal tensor lines, tangent to the eigenvectors of the metric field, to form a pseudo-mesh visually close to a perfectly adapted mesh but without many of its constraints. Anisotropic metrics can be visualized directly using such pseudo-meshes but, for isotropic metrics, the eigensystem is degenerate and an anisotropic perturbation has to be used. This perturbation merely preserves directional information usually present during metric construction and is small enough, about 1% of the prescribed target element size, to be visually imperceptible. Both analytical and solution-based examples show the effectiveness and usefulness of the present method. As an example, pseudo-meshes are used to visualize the effect on metrics of Laplacian-like smoothing and gradation control techniques. Application to adaptive quadrilateral mesh generation is also discussed.

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Acknowledgements

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for its financial support. Furthermore, please note that the solutions, meshes and iconic tensor visualizations were plotted using medit, a mesh visualization program developed by Pascal J. Frey of the French National Institute for Research in Computer Science and Control (INRIA) as well as VU, a configurable visualization software tool for the display and analysis of numerical solutions developed by Benoît Ozell at the now defunct Centre for Research on Computation and its Applications (CERCA), Montréal, Québec, Canada.

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Authors and Affiliations

  1. Department of Mechanical Engineering, École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, Canada, H3C 3A7

    Ko-Foa Tchon & Ricardo Camarero

  2. Department of Computer Engineering, École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, Canada, H3C 3A7

    Julien Dompierre, Marie-Gabrielle Vallet & François Guibault

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  1. Ko-Foa Tchon
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  2. Julien Dompierre
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  3. Marie-Gabrielle Vallet
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  4. François Guibault
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  5. Ricardo Camarero
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Correspondence to Ko-Foa Tchon.

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Tchon, KF., Dompierre, J., Vallet, MG. et al. Two-dimensional metric tensor visualization using pseudo-meshes. Engineering with Computers 22, 121–131 (2006). https://doi.org/10.1007/s00366-006-0012-3

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