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

Motivation Simulation of fracturing materials in many different applications.

Motivation Requirements on fracturing algorithm:

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

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

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

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

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

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

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

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

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

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 Simulation loop: Time integrationGradient of displacement fieldStrainStressBody forceAdd external forcesStrain energy

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)

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 r  i (r)  by construction they build a first order partition of unity (PU)

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

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

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

Discontinuities Weight functionShape function Visibility Criterion

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,

Discontinuities Weight function Shape function Visibility Criterion Transparency Method

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

Re-sampling: Example

Part 2 Surface Handling

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

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

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

Crack Propagation: Example

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

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

Crack Events: Example

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

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

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

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

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

Thank you! Contact information Mark Richard Bart Phil Markus Leonidas J.