XIII ADM - XV INGEGRAF
Cassino, June 3th, 2003
Napoli, June 4th and June 6th, 2003
Salerno, June 5th, 2003
Propagation Study in the Longest-edge
Refinement of Unstructured 3D
Triangular Meshes
José Pablo Suárez
University of Las Palmas de Gran Canaria.
Spain
Motivation:Finite Element Method
“Virtual Prototyping solutions use finiteelement analysis and advanced calculus
to accurately predict the product’s
operating perfomance1”
1Product
data quality and collaborative engineering
Contero, M.; Company, P.; Vila, C.; Aleixos, N. IEEE Computer Graphics and Applications,
22-3, pp. 32-42, (2002)
Overview of simulation tools for computer-aided production engineering
P. Klingstam, P. Gullander
Proc. Advanced Summer Inst. (ASI 97), Am. Soc. Mechanical Eng., New York 1997
Motivation:Finite Element Method
Structural Mechanics
Plane stress analysis of a mechanical component.
Navier’s equation in 2D
SWF
Unknown
s
Loads
AVI
Motivation:Finite Element Method
Heat Flow in a room
Heat equation in 2D
SWF
AVI
Motivation:Finite Element Method
Dynamic meshes to solve a nonlinear fire propagation problem:
3D modelling for FEM

Mechanical components
Refinement procedure
Propagation inherent in any refinement
procedure based on the longest edge
t
Longest-Edge bisection is good –> minimum angle does not vanish
“Domino” effect in propagation
Ominous problem for mesh generation:
Large propagation increase elements  increase complexity
difficult parallelization ...
2D Refinement propagation
Refinement inside R induces propagation outside R due to
conformity process by longest edge.
2D Refinement propagation
Definition (2D-Longest-Edge Propagation Path, 2D-LEPP) The 2D-LongestEdge Propagation Path of any triangle t is the set of all neighbor triangle (by the
longest edge) having respective longest edge greater than or equal to the longest
edge of the preceding tetrahedra in the path.
Definition (3D-Longest-Edge Propagation Path, 3D-LEPP) The 3D-LongestEdge Propagation Path of any tetrahedron t is the set of all neighbor tetrahedra (by
the longest edge) having respective longest edge greater than or equal to the
longest edge of the preceding tetrahedra in the path.
In 2D: problem solved
Theorem.- The successive application of the 4 Triangles Longest-Edge
partition to an initial triangular mesh produces a sequence o meshes
such that:
LEPP  2 when n tends to infinity
Suárez J.P., Plaza, A. and Carey G.F. The propagation problem in longest-edge based refinement algorithms,
Submitted to International Journal for Numerical Method in Enginnering, 2003
LEPP statistics report for Experiment 1
A Canonical Liu-Joe tetrahedron. We repeatedly apply uniform refinement following the 8T-LE partition
(5 steps of uniform refinement). We get the finest mesh with 32768 tetrahedra.
# Tet.
Mean
Median
Std
Max
Min
8
0.7500
1
0.4629
1
0
64
1.8125
1
1.0820
3
0
512
2.3906
3
0.9466
3
0
4096
2.6914
3
0.7306
3
0
32768
2.8447
3
0.5379
3
0
LEPP statistics report for Experiment 2
Canonical Liu-Joe tetrahedron. We repeatedly apply uniform refinement following the Standard partition (6
steps of uniform refinement). We get the finest mesh with 262144 tetrahedra.
# Tet.
Mean
Median
Std
Max
Min
8
0.7500
1
0.4629
1
0
64
3.3594
1
4.4521
22
0
512
5.5723
3
6.1966
32
0
4096
7.1755
3
8.2435
88
0
32768
8.0102
3
9.3316
88
0
262144
8.2431
3
10.1415
88
0
LEPP statistics report for Experiment 3
Delaunay type mesh with 1927 tetrahedra and then we apply two uniform
refinement steps to get a final fine mesh with 123328 elements.
# Tetrahedra
LEPP Mean
1927
97.7000
15416
24.1951
123328
13.2069
Conclusions
The 3D-Longest-Edge Propagation Path:

1. Has a statistical mean approaching to a fix constant that is
dependent on the partition type used in the refinement and on the
initial mesh.

2. Has maximum and minimum values also dependent on the
partition type used in the refinement.

3. As the uniform refinement steps increase, the statistical mean get
stable around a fixed constant.

4. We gave numerical evidence showing that propagation in 3D is not
an ominous problem affecting efficiency or degeneracy of the meshes,
as long as regular meshes and regular partitions are used.

This is an useful basis for engineers who often uses meshing/
refinement algorithms for a variety of application problems.
XIII ADM - XV INGEGRAF
Cassino, June 3th, 2003
Napoli, June 4th and June 6th, 2003
Salerno, June 5th, 2003
Propagation Study in the Longest-edge
Refinement of Unstructured 3D
Triangular Meshes
José Pablo Suárez
University of Las Palmas de Gran Canaria.
Spain