1. Meshless Animation of Fracturing Solids Mark Pauly Leonidas J. Guibas Richard Keiser Markus Gross Bart Adams Philip Dutré

2. Motivation Simulation of fracturing materials in many different applications.

3. Motivation Requirements on fracturing algorithm:

4. Motivation Simulation of fracturing materials in many different applications. Requirements on fracturing algorithm: brittle or ductile fracture

5. Motivation Simulation of fracturing materials in many different applications. Requirements on fracturing algorithm: brittle or ductile fracture arbitrary cracks

6. Motivation Simulation of fracturing materials in many different applications. Requirements on fracturing algorithm: brittle or ductile fracture arbitrary cracks control of fracture paths

7. Motivation Simulation of fracturing materials in many different applications. Requirements on fracturing algorithm: brittle or ductile fracture arbitrary cracks control of fracture paths highly detailed surfaces

8. Related Work O’Brien & Hodgins [99, 02] dynamic remeshing  element cutting  difficult to avoid ill- shaped elements

9. Related Work O’Brien & Hodgins [99, 02] dynamic remeshing  element cutting  difficult to avoid ill- shaped elements Molino, Bao & Fedkiw [04] virtual node algorithm  embedded surface in copied tetrahedra  restricted decomposition of tetrahedras

10. Meshless Methods Advantages sampling of the volume handling of large deformation (re-)sampling of the domain handling of discontinuities Drawbacks boundary conditions overhead for computing interpolation functions

11. Contributions A meshless animation framework for stiff- elastic and plasto-elastic materials that fracture handling of brittle and ductile fracture allows arbitrary crack initiation and propagation allows for easy control highly detailed surfaces due to decoupling of physics and surface representation

12. Overview Part 1: Physics Animation Meshless Continuum Mechanics Modeling Discontinuities Spatial Re-sampling Part 2: Surface Handling Surface Model Crack Initiation & Propagation Topological Events

13. Elasticity Model Meshless elasticity model derived from continuum mechanics. 1 x x+ux+u displacement field u Müller et al.: Point Based Animation of Elastic, Plastic and Melting Objects, SCA 2004 1 Simulation loop: Time integrationGradient of displacement fieldStrainStressBody forceAdd external forcesStrain energy

14. Discretization Discrete set of nodes {x i } Approximation of displacement field u: x uiui xixi u(x)   i  i (x) u i evaluation point summation over neighboring nodes i displacement vector of node i shape function of node i Derivation of shape functions using Moving Least Squares (MLS)

15. Discretization Shape functions  i :  i (x) =  i (x,x i ) p T (x) [M(x)] -1 p(x i ) weight function linear basis p(x) = [1 x] T moment matrix M(x) =  i  i (x,x i ) p(x i ) p T (x i ) Weight function  i (x,y):  i (x,y) =  i (r) = 1-6r 2 +8r 3 -3r 4 r  1 0r>1 r = ||x-y||/h i with h i the support radius of node i 0 1 1 0 r  i (r)  by construction they build a first order partition of unity (PU)

16. Discontinuities Only visible nodes should interact collect nearest neighbors perform visibility test crack

17. Discontinuities Only visible nodes should interact collect nearest neighbors perform visibility test crack

18. Discontinuities Problem: undesirable discontinuities of the shape functions not only along the crack but also within the domain crack

19. Discontinuities Weight functionShape function Visibility Criterion

20. Discontinuities Solution: transparency method 1 nodes in vicinity of crack partially interact by modifying the weight function:  i ’(x i,x j ) =  i (||x i -x j ||/h i + (2d s /κ) 2 ) crack dsds  crack becomes transparent near the crack tip Organ et al.: Continuous Meshless Approximations for Nonconvex Bodies by Diffraction and Transparency, Comp. Mechanics, 1996 1

21. Discontinuities Weight function Shape function Visibility Criterion Transparency Method

22. Re-sampling xixi crack Add simulation nodes when number of neighbors too small Shape functions adapt automatically! Local resampling of the domain of a node distribute mass adapt support radius interpolate attributes

23. Re-sampling: Example

24. Part 2 Surface Handling

25. Surface Animation All surfaces are represented using oriented point samples {s i } wrapped around the simulation nodes {p j } Deformation of surfels is computed from neighboring simulation nodes: surfels {s i } simulation nodes {p j } x i  x i +  j  i ’(x i,x j )(u j +  u j T (x j -x i )) same transparency weight

26. Crack Propagation Crack initiation where stress above threshold crack created by inserting 3 crack nodes  each carrying 2 opposing surfels  connection is crack front external force external force one fracture surface crack front

27. Crack Propagation Crack propagation propagate crack nodes along propagation direction re-project first and last node up-sample if necessary external force external force one fracture surface

28. Crack Propagation: Example

29. Crack Events Splitting when crack propagates through the material split front in two new fronts each one propagates independently block of material

30. Crack Events Merging when two fronts propagate close to each other merge fronts and associated fracture surfaces block of material

31. Crack Events: Example

32. Brittle Fracture Initial statistics: 4.3k nodes 249k surfels Final statistics: 6.5 k nodes 310k surfels Simulation time: 22 sec/frame

33. Controlled Fracture Initial statistics: 4.6k nodes 49k surfels Final statistics: 5.8 k nodes 72k surfels Simulation time: 6 sec/frame

34. Ductile Fracture Initial statistics: 2.2k nodes 134k surfels Final statistics: 3.3 k nodes 144k surfels Simulation time: 23 sec/frame

35. Conclusion Advantages decoupling of physics and surface representation dynamic adaptation of shape functions  during crack propagation  when re-sampling of spatial domain Drawbacks excessive fracturing  simulation nodes  visibility testing is still costly  each test = ray-surface intersection test

36. Future Work Real-time simulation simplification of algorithms efficient data structures efficient caching schemes Solve excessive up-sampling issue variant of the virtual node algorithm

37. Thank you! Contact information Mark Paulypauly@inf.ethz.ch Richard Keiserkeiser@inf.ethz.ch Bart Adamsbarta@cs.kuleuven.ac.be Phil Dutréphil@cs.kuleuven.ac.be Markus Grossgrossm@inf.ethz.ch Leonidas J. Guibasguibas@cs.stanford.edu