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Extension of a Mesh Quality Metric for Elements With a Curved Boundary Edge or Surface

[+] Author and Article Information
Larisa Branets

Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 201 E. 24th St., ACES 6.430, 1 University Station C0200, Austin, TX 78712-0227larisa@ices.utexas.edu

Graham F. Carey

Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 201 E. 24th St., ACES 6.430, 1 University Station C0200, Austin, TX 78712-0227carey@cfdlab.ae.utexas.edu

J. Comput. Inf. Sci. Eng 5(4), 302-308 (Jun 15, 2004) (7 pages) doi:10.1115/1.2052827 History: Revised June 15, 2004; Received October 01, 2004

The formulation of a local cell quality metric [Branets, L., and Carey, G. F., 2003, Proceedings of the 12th International Meshing Roundtable, Santa Fe, NM, pp. 371–378;Engineering with Computers (in press)] for standard elements defined by affine maps is extended here to the case of elements with quadratically curved boundaries. We show for two-dimensional and three-dimensional simplex elements with quadratically curved boundaries that all cases of map degeneracy can be identified by the metric. Moreover, we establish a “maximum principle” which allows estimating the bounds on the quality metric. The nondegeneracy conditions for biquadratic quadrilaterals with one curved edge are also determined. The metric is implemented in an untangling/smoothing algorithm for improving unstructured meshes including simplex elements that have curved boundary segments. The behavior and efficiency of this algorithm is illustrated for numerical test problems in two and three dimensions.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

The quadratic map r in 5 of the regular reference triangle onto an arbitrary triangle

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Figure 2

Degenerate quadratic triangles with (a) one, (b) two, and (c) three curved edges (degeneracy at a vertex)

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Figure 3

Degenerate quadratic triangles

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Figure 4

Level sets as functions of position for lower midedge node in quadratic triangle with all other nodes fixed on the regular shape

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Figure 5

The quadratic map of the regular reference tetrahedron onto an arbitrary tetrahedron

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Figure 6

Quadratic map for quadrilaterals

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Figure 7

Quadrilateral with one curved edge

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Figure 8

Admissible positions for node 2 are inside the dashed contour

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Figure 9

“Restricted” biquadratic map

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Figure 10

Admissible positions for node 2 are inside the dashed contour

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Figure 11

Initial grids of curved triangles

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Figure 12

Smoothed/untangled grid of curved triangles

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Figure 13

From left to right: (a) curvilinear tetrahedral mesh inside a cylinder, (b) smoothing result with fixed cylindrical boundary, and (c) with free boundary

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