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M. Technical Notes

Aspect Ratio, Triangulation Step, and Area Limitation

The aspect ratio (AR) is defined as:

A R = L 1 + L 2 h = 1 tan ( θ ) + 1 tan ( α ) 2 tan ( θ ) = 2 cos ( θ ) sin ( θ ) 1 sin ( θ )

Of the two angles shown the diagram, a3q and monotonic decrease of function 1/tan are used here. In addition, since angle q is the minimum, therefore 2cos(q)31

By analogy:

A R = L 1 + L 2 h = 1 tan ( θ ) + 1 tan ( α ) = sin ( θ + α ) sin ( θ ) sin ( α ) sin ( θ ) cos ( α ) + cos ( θ ) sin ( α ) sin ( θ ) sin ( α )

It means that to prove the above estimation it is necessary to demonstrate that

Since 90o3a3q, then

It remains to note, that .

Since area of a triangle is equal to (L1+L2)h/2, then it leads to, that is the above estimation for area, when angle q is constrained from below, results in to the above estimation of triangulation step.

Moreover, instead of estimation using the more accurate estimation we obtain


Matrix A (referred to as the Jacobian matrix) transfers points of the coordinate triangle to the real triangle. Matrix W transforms coordinate triangle into equilateral (ideal) triangle. Therefore matrix S=AW-1, which transforms ideal element into real one, defines deviation of real element from ideal one.

For any triangle, the free vertex which is located at the point x, and two other vertices – xi, xj, the Jacobian is equal to

det (xi – x, xj – x)

Algorithms Used

A - triangulation - is a classical Delaunay triangulation (see, for example, [9]). For other cases a version of J.Rupert [2] algorithm is used.


Basic smoothening algorithm is described in reference [10].


Node is moved to the point, which is averaging of locations M of the nearest nodes

x n = 1 M m = 0 M 1 x m


  1. M.Bern, D.Eppstein Mesh generation and optimal triangulation, In D.Z.Du and F.K.Hwang, editors, Computing in Euclidean Geometry, World Scientific, 1992, 23-90
  2. J.Rupert A Delaunay refinement algorithm for quality 2-dimensional mesh generation, Journal of Algorithms, 1995, 18(3), 548-585.
  3. J.R.Shewchuk Triangle: engineering a 2D quality mesh generator and Delaunay triangulator First Workshop on Applied Computational Geometry (Philadelphia, Pennsylvania), 124-133, Association for Computing Machinery, May 1996 (
  4. P.L. George, H. Borouchaki Automatic Mesh Generation : Applications to Finite Element Method. Paris, Hermés, 1998
  5. V.N.Parthasarathy, C.M. Graichen, A.F. Hathaway A comparison of tetrahedron quality measures, Finite Elements in Analysis and Design, Elsevier, Num 15, pp.255-261, 1993
  6. P. Knupp, Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities, Part I - A Framework for Surface Mesh Optimization, Int. J. Numer. Meth. Engr., Vol 48, Issue 3, pp 401-420, May 2000.
  7. J. Robinson, CRE Method of element testing and the Jacobian shape parameters, Eng. Comput., Vol 4, 1987.
  8. A. Oddy, J. Goldak, M. McDill, M. Bibby, A distortion metric for isoparametric finite elements, Trans. CSME, No. 38-CSME-32, Accession No. 2161, 1988.