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Deformable Physics

Deformable Objects

A difficult problem Lots of specific solutions but no accepted general best method Applies to:

Soft bodies – incl. character animation Destruction Fluid simulation

Demo:

Pixelux Digital Molecular Matter

Used in LucasArts’ "Star Wars: The Force Unleashed".

Soft Body Dynamics

Visually Realistic physical simulations of motion and properties of deformable objects Shape and topology of objects changes

Points within the object move relative to one another However shape has tendency to retain it’s shape to some degree: elastic vs plastic E.g. muscle, fat, hair, plants, clothing and fabric

Soft-body Physics Engines

Havok Cloth

PhysX

Bullet

Deformation in Graphics Two classes of deformation

Geometric deformation Physically based deformation

Geometric Deformation

Solid Deformations

Hierarchical solid modeling transformation of surfaces Locally specified deformation by local/tangent transformation matrices Can be used for e.g. twisting, bending, tapering, … now mainstream

Alan H. Barr, Global and Local Deformations of Solid Primitives, Computer Graphics (Proceedings of SIGGRAPH 84). 18(3), pp. 21-30, 1984.

Free Form Deformations (FFD)

Global FFD Local FFD

Thomas W. Sederberg, Scott R. Parry, Free-Form Deformation of Solid Geometric Models, Computer Graphics (Proceedings of SIGGRAPH 86). 20(4), pp. 151-160, 1986.

FFD

Sabine Coquillart, Extended Free-Form Deformation: A Sculpturing Tool for 3D Geometric Modeling,Computer Graphics (Proceedings of SIGGRAPH 90). 24(4), pp. 187-196, 1990.

FFD

Working with control points can be awkward Apply (displacement) constraints directly to surface William M. Hsu, John F. Hughes, Henry Kaufman, Direct manipulation of free-form deformations, Computer Graphics (Proceedings of SIGGRAPH 92). 26(2), pp. 177-184, 1992.

Wires

Karan Singh, Eugene L. Fiume, Wires: A Geometric Deformation Technique, Proceedings

• f SIGGRAPH 98. pp. 405-414, 1998.

Physically Based Deformation

Elastically Deformable Models

Major contribution to physically based deformable models in graphics Lagrangian derivation of eqns of motion

) , ( ) ( t r t E t t t f r r r                   

Net externally applied forces Net instantaneous potential energy Damping density Mass density of body at a r(a, t): position of particle a at time time Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. Elastically Deformable Models, ACM SIGGRAPH 87, 205-214, 1987.

De Defor

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Solutions from Engineering

E.g. Finite element simulation Strive for accuracy Generally very slow Sometimes do not converge to a solution Don’t discern if the visual response is appealing – often provide

• ther kinds of info:

Visualization is somewhat different from computer graphics

Non-realtime Graphical Solutions

E.g. Special effects in movies and commercials Faster solutions, but not fast enough.

Animators don’t use techniques if the previews are too expensive.

Not as accurate as engineering Focus on Appeal Recent interest in “controllable” simulations

Real-time Solutions

Resolution Reduction

Blobby and coarse look Details disappear

Use specialized real-time techniques

Physics low-res, appearance hi-res (shader effects) Reduction of dimension from 3d to 2d or 2.5d (height field fluids, BEM) Level of detail (LOD) No equation solving, procedural animation for specific effects

Interactive Physics requirements

Solutions in Engineering and Off-line animation generally don’t fulfill the requirements of real-time physics Reality methods require:

Efficiency Most of the time is going to be used in the rendering process (especially in games). Interactive responses Low latency Visual and haptic minimum frequency constrains Stability Must guarantee stability If the model blows up we lose immersion Realism Interactivity vs. accuracy Plausibility vs. accuracy

Taxonomy of Deformable Simulation

We can categorise techniques based on:

Phenomena they are meant to simulate Formulation of the solution Space Discretization method used Time Discretization methods used

Phenomena Being Simulated

Elastic objects Plastic objects Fluids

Liquids Smoke, clouds fire

Cloth Hair

Formulation of solution

Eulerian

reference system attached to the space

Lagrangian

reference system attached to the

• bject

Semi-Lagrangian

Space Discretization

Mesh based techniques

A mesh joins the object nodes Large deformations are hard to simulate (fluids) Object boundary is explicitly calculated Mesh elements: segments, triangles, tetrahedrons, hexahedrons… Example: Mass-spring systems

Meshless techniques

No explicit mesh connecting particles Difficult to compute the object’s boundary (hard to draw) Suitable for large deformation: like fluids Example: SPH

Time Discretization

Explicit

Current state depends only on past states Faster Not unconditionally stable

Implicit

current state depends on past states and on the current state Requires solving a system of equations Generally slower Adds artificial damping (depending on the object properties) Unconditionally stable

t m t

1 i 1 i 1 i 1 i i 1 i

     

    

f v v v x x t m t

i 1 i 1 i i i 1 i

     

  

f v v v x x

Mass-spring Systems

Mass-spring Model

Body modelled as point masses: masses connected by (weightless) springs Hooke’s Law fof Damped Spring

Restoring force

Damping force (for now, lets just use viscous drag - effectively damps oscillation and other motions)

) ( r x f    

spring spring

k v f

drag drag

k  

Basic deformable objects

From erleben et al ch 8

Start with newtonian particle system For a particle system we know the accelerations acting on a particle at any time Assume Explicit Eurler Integration and that time is discretised as ti+1 = h + ti Second order Taylor approximation of position of particle at time ti+1 :

m m / f x x f       

                   

 

m ti

i i i 1 i 1 i

/ ) ( f x x x x x   

Leads to instability if step-size is too high particle will diverge Not unconditionally stable

Verlet Integration

Originated in molecular dynamics (central difference approximation: find x based on previous frame and next frame)

A velocity-free formulation (velocity implicitly represented by current and previous positions) in code (store current x and previous x): x = 2*x – x_prev + a *h^2; x_prev = x;

not always accurate but fast and stable

) ( ) ( ) ( ) ( t h h t t 2 h t

2a

x x x     

See; Jakobsen GDC 2001 talk: Advanced Character Physics http://www.floatingorigin.com/mirror/jacobson_01.shtml

• Provot. X, “Deformation constraints in a

mass-spring model to describe rigid cloth behaviour” Graphics Interface 2001, pp147-154

Cloth

Springs chosen in a certain structure to model deformable dynamic properties verlet integration An adaptive technique; Dynamic inverse procedure applied to cap deformations for excessive deformation rates More detail on Verlet: http://wapedia.mobi/en/Verlet_integration

Mass-spring system for cloth

Structural Springs between adjacent particles – resist stretching and compression. Usually relatively high k Shearing Springs connecting diagonal adjacent particles in a regular grid – relatively low k Bending Springs a.k.a flexion springs, prevent sharp ridges when folded: links every second or third particles. Usually very low k for cloth

Unstructured Meshes

Regular grid is not always applicable For an unstructured mesh:

Structural springs correspond to 1-neighbourhood Shearing and bending to a 2-neighbourhood

1 - neighbourhood 2 - neighbourhood

Implicit Integration

David Baraff, Andrew Witkin Large Steps in Cloth Simulation Siggraph 1998 http://ai.stanford.edu/~latombe/cs99k/ 2000/cloth.pdf

The difference in the two methods is that the forward method’s step is based solely on conditions at time t0 while the backward method’s step is written in terms of conditions at the terminus of the step itself.4

Integration Scheme Comparisons

http://femto.cs.uiuc.edu/~sbond/code/SpringMass/

Deformable Solids

Discretise object into 3d rectilinear grid (voxels)

In addition to shearing diagonals we need spatial diagonals that counter volume loss

Unstructured solid meshes can be voxelised

Mesh points in between grid nodes are updated by trilinear interpolation

Mesh coupling

Muller, M., Teschner, M., and Gross, M. Physically- Based Simulation of Objects Represented by Surface Meshes. In Proceedings of the Computer Graphics international 2004.

Tetrahedral decomposition

Split mesh up into tetrahedra Note that interior details are important (not just surface) Delauney tetrahedralization Springs between vertices and 2+ neighbourhoods depending on rigidity

http://tetgen.berlios.de/