1. 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, 2003 Napoli, June 4 th and June 6 th, 2003 Salerno, June 5 th, 2003

2. Motivation: Finite Element Method 1 Product 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 “Virtual Prototyping solutions use finite- element analysis and advanced calculus to accurately predict the product’s operating perfomance 1 ”

3. Motivation: Finite Element Method Structural Mechanics Plane stress analysis of a mechanical component. Navier’s equation in 2D Loads Unknown s SWF AVI

4. Motivation: Finite Element Method Heat Flow in a room Heat equation in 2D SWF AVI

5. Motivation: Finite Element Method Dynamic meshes to solve a nonlinear fire propagation problem:

6. 3D modelling for FEM Mechanical components

7. t Refinement procedure Propagation inherent in any refinement procedure based on the longest edge minimum angle does not vanish Longest-Edge bisection is good –> minimum angle does not vanish

8. “Domino” effect in propagation Ominous problem for mesh generation: Large propagation  increase elements  increase complexity  difficult parallelization...

9. 2D Refinement propagation Refinement inside R induces propagation outside R due to conformity process by longest edge.

10. 2D Refinement propagation Definition (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.

11. 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

12. 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.

13. 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.

14. 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.

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

16. 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, 2003 Napoli, June 4 th and June 6 th, 2003 Salerno, June 5 th, 2003