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.

Issue Section:
Special Issue Papers
Keywords:
mesh generation, computational geometry, maximum principle
Topics:
Algorithms, Jacobian matrices, Shapes, Computational geometry, Computers, Dimensions, Mesh generation
1.
Yuan
,
K. Y.
,
Huang
,
Y. S.
, and
Pian
,
T. H. H.
, 1994, “
Inverse Mapping and Distortion Measures for Quadrilaterals with Curved Boundaries
,”
Int. J. Numer. Methods Eng.
0029-5981,
37
, pp.
861
–
875
.
Crossref
Search ADS
 
2.
Chen
,
Z.
,
Tristano
,
J. R.
, and
Kwok
,
W.
, 2003, “
Combined Laplacian and Optimization-based Smoothing for Quadratic Mixed Surface Meshes
,”
Proceedings of the 12th International Meshing Roundtable
, Santa Fe, NM, pp.
361
–
370
.
3.
Baart
,
M. L.
, and
McLeod
,
R. J. Y.
, 1986, “
Quadratic Transformations of Triangular Finite Element in Two Dimensions
,”
IMA J. Numer. Anal.
0272-4979,
6
, pp.
475
–
487
.
Crossref
Search ADS
 
4.
Mitchell
,
A. R.
,
Phillips
,
G.
, and
Wachspress
,
E.
, 1971, “
Forbidden Shapes in the Finite Element Method
,”
J. Inst. Math. Appl.
0020-2932,
8
, pp.
260
–
269
.
Crossref
Search ADS
 
5.
Field
,
D. A.
, 1983, “
Algorithms for Determining Invertible Two- and Three-Dimensional Quadratic Isoparametric Finite Element Transformations
,”
Int. J. Numer. Methods Eng.
0029-5981,
19
, pp.
789
–
802
.
Crossref
Search ADS
 
6.
Baart
,
M. L.
, and
Mulder
,
E. J.
, 1987, “
A Note on Invertible Two-Dimensional Quadratic Finite Element Transformations
,”
Commun. Appl. Numer. Methods
0748-8025,
3
, pp.
535
–
539
.
Crossref
Search ADS
 
7.
Frey
,
A. E.
,
Hall
,
C. A
, and
Porsching
,
T. A.
, 1978, “
Some Results on the Global Inversion of Bilinear and Quadratic Isoparametric Finite Element Transformations
,”
Math. Comput.
0025-5718,
32
, pp.
725
–
749
.
Crossref
Search ADS
 
8.
Branets
,
L.
, and
Carey
,
G. F.
, 2003, “
A Local Cell Quality Metric and Variational Grid Smoothing Algorithm
,”
Proceedings of the 12th International Meshing Roundtable
, Santa Fe, NM, pp.
371
–
378
.
9.
Branets
,
L.
, and
Carey
,
G. F.
, “
A Local Cell Quality Metric and Variational Grid Smoothing Algorithm
,”
Engineering with Computers
(in press).
10.
Branets
,
L. V.
, and
Garanzha
,
V. A.
, 2001, “
Global Condition Number of Trilinear Mapping. Application to 3D Grid Generation
,”
Proceedings of the Minisymposium in the International Conference, “Optimization of finite-element approximations, splines and wavelets,”
St.-Petersburg, Russia, pp.
45
–
60
.
11.
Garanzha
,
V. A.
, 2000, “
Barrier Variational Generation of Quasi-Isometric Grids
,”
Computational Mathematics and Mathematical Physics
,
40
, pp.
1617
–
1637
.
12.
Strang
,
G.
, and
Fix
,
J.
, 1973,
Analysis of Finite Element Method
,
Prentice-Hall
, Englewood Cliffs, NJ.
13.
Branets
,
L.
, and
Carey
,
G. F.
, 2004, “
Smoothing and Adaptive Redistribution for Grids with Irregular Valence and Hanging Nodes
,”
Proceedings of the 13th International Meshing Roundtable
, Williamsburg, VA, pp.
333
–
344
.
Copyright © 2005
 by American Society of Mechanical Engineers
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