1. Physically-Based Simulation of Objects Represented by Surface Meshes Matthias Muller, Matthias Teschner, Markus Gross CGI 2004

2. Outline Introduction Volume mesh construction Continuous model Finite element method Fracture rule Closing surface generation Future work

3. Introduction -1 Fracture simulation – Finite element method Very realistic Suitable for any kind of cracking simulation Complex and slow – Crack patterns Fast, simple Re-usable of crack pattern Visual effect are not acceptable – Spring model Less complex compare with FEM Create different threshold for each spring

4. Introduction -2

6. Volume Mesh Construction Volume mesh representation: Cube (efficiency) Preprocess: Guarantee no edge is longer than cube size Construct volume mesh

7. Continuous Model Hooke’s law In three dimensions, ε and σ can be expressed as 3 by 3 matrices

8. Finite Element Method -1 Let m 1, m 2, m 3, m 4 be the coordinates of tetrahedron Let x 1, x 2, x 3, x 4 be the deformed world coordinates A linear continuous deformation function p(u) Let b be barycentric coordinates in undeformed situation

9. Finite Element Method -2 To be more realistic: – Consider Shear elastic modalus

10. Fracture Plane Generation When internal stresses exceed the material threshold

11. Surface Fracturing When internal stresses exceed the material threshold Surface mesh needs to be fractured near face A new closing surface needs to be generated in order to keep the mesh watertight

12. Surface Mesh Do not cut any surface triangles during the fracture process Problem: Artifacts when big triangles are used to represent surface Solution: Subdivide large triangles as preprocessing step Preprocess: Guarantee no edge is longer than cube size

13. Closing Surface Generation -1

14. Closing Surface Generation -2

15. Future Work Apply rigid body system to particle system Cubes => particles Independent FEM structure: Coefficient changes when particles deforms – (x+ △ x, y+ △ y, z+ △ z) – Update strain and stress etc Design crack rule Render problem