Propagation Study in the Longest-edge Refinement of Unstructured 3D

Triangular Meshes

University of Las Palmas de Gran Canaria.

Spain

José Pablo Suárez

XIII ADM - XV INGEGRAF 

Cassino, June 3th, 2003Napoli, June 4th and June 6th, 2003

Salerno, June 5th, 2003

Motivation:Finite Element Method

1Product data quality and collaborative engineeringContero, 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 engineeringP. Klingstam, P. Gullander Proc. Advanced Summer Inst. (ASI 97), Am. Soc. Mechanical Eng., New York 1997

“Virtual Prototyping solutions use finite-element analysis and advanced calculus to accurately predict the product’s operating perfomance1”

Motivation:Finite Element Method

Structural MechanicsPlane stress analysis of a mechanical component.

Navier’s equation in 2D

Loads

Unknowns

SWF

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

t

Refinement procedure

Propagation inherent in any refinement procedure based on the longest edge

Longest-Edge bisection is good –> minimum angle does not vanishminimum angle does not vanish

“Domino” effect in propagation

Ominous problem for mesh generation:

Large propagation increase elements increase complexity

difficult parallelization ...

2D Refinement propagationRefinement inside R induces propagation outside R due to conformity process by longest edge.

2D Refinement propagationDefinition (2D-Longest-Edge Propagation Path, 2D-LEPP) The 2D-Longest-Edge 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-Longest-Edge 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.

LEPP statistics report for Experiment 2Canonical 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.

LEPP statistics report for Experiment 3

# Tetrahedra 1927 15416 123328 LEPP Mean 97.7000 24.1951 13.2069

Delaunay type mesh with 1927 tetrahedra and then we apply two uniform refinement steps to get a final fine mesh with 123328 elements.

ConclusionsThe 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 This is an useful basis for engineers who often uses meshing/ refinement algorithms for a variety of application problems. algorithms for a variety of application problems.

Propagation Study in the Longest-edge Refinement of Unstructured 3D

Triangular Meshes

University of Las Palmas de Gran Canaria.

Spain

José Pablo Suárez

XIII ADM - XV INGEGRAF 

Cassino, June 3th, 2003Napoli, June 4th and June 6th, 2003

Salerno, June 5th, 2003