Abstract
Science and engineering applications often have anisotropic physics and therefore require anisotropic mesh adaptation. In common with previous researchers on this topic, we use metrics to specify the desired mesh. Where previous approaches are typically heuristic and sometimes require expensive optimization steps, our approach is an extension of isotropic Delaunay meshing methods and requires only occasional, relatively inexpensive optimization operations. We use a discrete metric formulation, with the metric defined at vertices. To map a local sub-mesh to the metric space, we compute metric lengths for edges, and use those lengths to construct a triangulation in the metric space. Based on the metric edge lengths, we define a quality measure in the metric space similar to the well-known shortest-edge to circumradius ratio for isotropic meshes. We extend the common mesh swapping, Delaunay insertion, and vertex removal primitives for use in the metric space. We give examples demonstrating our scheme’s ability to produce a mesh consistent with a discontinuous, anisotropic mesh metric and the use of our scheme in solution adaptive refinement.
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Notes
The term error is used in the cited paper, however, the metric length can be defined as equivalent to the error and so we will use the term length to be consistent with usage later in the paper.
Note that this is not necessarily the triangle it was inserted to split!
This procedure fails only in rare near-boundary cases in two dimensions, although its three-dimensional analog is less robust; see Ollivier-Gooch [23] for more details.
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This work has been funded by the Canadian Natural Sciences and Engineering Research Council under Special Research Opportunities Grant SRO-299160.
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Pagnutti, D., Ollivier-Gooch, C. Two-dimensional Delaunay-based anisotropic mesh adaptation. Engineering with Computers 26, 407–418 (2010). https://doi.org/10.1007/s00366-009-0143-4
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DOI: https://doi.org/10.1007/s00366-009-0143-4