towards a unified dynamics solver for computer graphicsreleased in maya 8.5 (ncloth) : 2007...
TRANSCRIPT
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Nucleus
Jos StamSenior Research Scientist
Autodesk, Inc.
Toronto, Canada
Towards a Unified Dynamics Solver forComputer Graphics
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Motivation
Ruysdael (1628-1682)
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Too many Solvers
rigid bodies cloth fluids
Solvers have to interact
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Unified Solver
Two-way interaction
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Unified Solver
King Dome, Seattle, March 26, 2000
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History
Simple idea : 2000Fooling around (palm demo) : 2001First prototypes : 2003-2005First real implementation : 2005First public demo : 2006Released in MAYA 8.5 (nCloth) : 2007nParticles : 2008
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Nucleus
Nucleus API MAYA
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Nucleus
Nucleus
MAYA
~ 100 files ~ 40,000 files
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Approach
General Shape Model
Stable Dynamics
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Shape Model
1 Theorem (Brouwer 1910)
Every continuous mapping can be approximated
by a piece-wise linear simplicial map.
“Simplicial Approximation Theorem”
Fundamental theorem of CG modeling
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Shape Model
1 Theorem
Let K and L be complexes; let K be ¯nite.
Given a continuous map h:jKj ! jLjthere is an N such that h has a simplicial
approximation f : sdNK ! L.
“Simplicial Approximation Theorem”
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Simplicial?
0-simplex 1-simplex 2-simplex 3-simplex
k-simplex
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Simplicial Complex
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Simplicial Complex
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Simplicial Complex
Dover: $2.75
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Shape Model
10
3
2
5
67
4
8
9
2 edges:-(8,9) +(7,8)
7 triangles:+(0,1,4), +(1,2,5)-(0,3,4), -(1,4,5)+(3,4,7), +(4,5,6),+(4,6,7)
Definition purely topological
(i,j,k) = -(i,k,j)
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Simplicial Complex
Single class for all primitives:
class simplex {
int k;
int sign;
int vertex[k+1];
int child[k+1];
int n_parents;
int parent[n_parents];
};
Code not specific for a primitive: simple
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Dynamics
x(t) = (x1(t); ¢ ¢ ¢ ; xN(t)) 2R3N
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Dynamics
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Newton’s Law
Äx = ¡rf(x) + fe
x(0) = x0
_x(0) = v0
E = 12_x2 + f(x)¡ fe ¢ x
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Isaac Newton
Isaac Newton's Principia 1687
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Simple Example
x
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Spring
Äx=¡x
x(0) = x0
_x(0) = v0
E = 12_x2 + 1
2x2
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Spring
Äx=¡x
z(0) = z0
z = x+ i _x 2C
_z =¡i z
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Spring
z(0) = z0
_z =¡i z
z(t) = e¡it z(0)
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A Thought…
R6n ¡!C3n
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Spring
_x
x
jz(t)j2 = jz(0)j2
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Spring
_x
x
_z =¡i z
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Explicit Solver
_x
x
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Explicit Solver
_x
x
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Implicit Solver
_x
x
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Explicit Solver
1h
¡zn ¡ zn¡1
¢= ¡i zn¡1
zn = zn¡1 ¡ ih zn¡1
zn =¡1 + h2
¢e¡ih zn¡1
zn =¡1 + h2
¢ne¡inh z0
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Implicit Solver
1h
¡zn ¡ zn¡1
¢= ¡i zn
(1 + ih) zn = zn¡1
zn = 11+h2
e¡ih zn¡1
zn = 1(1+h2)n
e¡inh z0
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Symplectic Solver
_x
x
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Symplectic Solver
_x
x
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Symplectic Solver
_x
x
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Symplectic Solver
_x
x
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Symplectic?
“Official Definition”:
“Plaiting or joining together; -- said of a bone next above thequadrate in the mandibular suspensorium of many fishes,which unites together the other bones of the suspensorium”.
Fish bones?
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Symplectic?
In math: Hermann Weyl (1930’s) : com-plex sym-plectic
Com : Latin rootSym : Greek root
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Symplectic?
_x
x
_x
x
A
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Symplectic?
µ_x1
x1
¶=
µ1 ¡hh 1¡ h2
¶µ_x0
x0
¶
A
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Simple Idea
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Simple Idea
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Nucleus
Strategy:
- Use constraints- Implicit on velocity- Explicit on position
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Nucleus
C (x+ v+ ±v) = 0
±vSolve for:
v = v+ ±vx= x+v
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Deformations
1 2 3
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Deformations1-simplex 2-simplex 3-simplex
1 3 6
1/2
1/3
3
1/2
12
6
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How To Solve?
C (x+ v+ ±v) = 0
Non-linear…
One constraint at a timeLinearizeBFGS
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One Constraint
fk(t) = Ck (x+ v+ t dk) = 0
fk(t) ¼ fk(0) + t f0
k(0) = 0
t = ¡fk(0)=f0
k(0)
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What about ?
Search Direction
dk
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Search Direction
Bridson, Marino and Fedkiw SCA 2003
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Linearize
rC (x+ v) ±v = ¡C (x+ v)
Au= b
LSQR, CGLS
3n£m
AATv = b u = ATv
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BFGS
C (x+ v+ ±v)
rC (x+ v+ ±v)±v
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One ConstraintUpdate velocity
Sequential vs Parallel
Gauss-Seidel Jacobi
Better convergence Slower convergence
Bias No bias
Hard to parallelize Easy to parallelize
Multigrid ?
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level 1 level 2level 0
level 4 level 5level 3
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parallel / no multigrid
sequential / no multigrid
parallel / with multigrid
sequential / multigrid
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Stretch/Compression
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Shear
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Bending
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Collisions
penalty vs space-time
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Space-Time
time
space (1D)
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Space-Time
time
space (1D)
dt
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Space-Time
time
space (1D)a0 b0
a1b1 V1 > 0
V0 < 0
Vt = at ¡ bt
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Space-Time
time
space (1D)a0 b0
a1b1
t = V0=(V0 ¡ V1)
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Space-Time
elastic
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Space-Time
inelastic
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Space-Time
friction
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Space-Time (2D)
time
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Space-Time (2D)
time
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Space-Time (2D)
Necessary but not sufficient
time
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Space-Time (3D)
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Space-Time (3D)
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Summary
Find such that
If stop
Check if primitives overlap at
If yes handle collision
V0 ¢ V1 > 0
Vt = 0t
t
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Higher Dimensions?
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Thickness
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Thickness
Quadratic (2)
Quartic (4)
Sextic (6)
Sextic (6)
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Roots of a Polynomial
t = 12a
¡b§
pb2 ¡ 4ac
¢b2 >> 4ac
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Roots of a Polynomial
t1 =2cq
q =¡b¡ sgn(b)pb2¡ 4ac
t2 =q2a
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Roots of a Polynomial
No stable formulas for degrees 3 and 4.
No formulas for degree > 4 (Abel + Galois)
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Roots of a Polynomial
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Roots of a Polynomial
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Roots of a Polynomial
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Many Primitives
Use hierarchical data structures for speed
AABB tree (simple), actually kDOP
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Use hierarchical data structures for speed
Many Primitives
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Use hierarchical data structures for speed
Many Primitives
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Use hierarchical data structures for speed
Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Etc. Expensive in General
Many Primitives
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Our approach:Iterate over entire time stepUntil all collisions resolved.
Avoids lockups and Zeno’s paradox
…
Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Many Primitives
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Demos
1D demo (t key for space-time)
3D demo
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Torture Tests
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Torture Tests
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Torture Tests
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Torture Tests
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Torture Tests
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Pressure Model
PV (x+v+ ±v) = cM
PMV
: pressure: mass: volume
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No pressure
Pressure Model
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Volume conservation
Pressure Model
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Under pressure
Pressure Model
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Under a lot of pressure
Pressure Model
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Air tightness + Pump rate
Pressure Model
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Battle of the constraints
General Solver
Collision
Self-Collision
Bend
Stretch
Shear
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General Solver
Collisions/stretch Stretch/collisions
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Bend (9)
Shear (7)
Stretch (26)
Self-Collisions (7)
Collisions (6)
General Solver v1.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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General Solver
not interleaved interleaved
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Bend (9)
Shear (7)
Stretch (26)
Self-Collisions (7)
Collisions (6)
General Solver v2.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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Bend
Shear
Stretch
Self-Collisions
Custom
Collisions
General Solver1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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Demos
Curve demo
Cloth demo
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nParticles
New in MAYA 2009
Extends exisiting particles
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nParticles
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nParticles
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Solids
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More Cloth Examples
ballerina
Rigid bodies
Cloth drop
Inflatable Girl Inflatable Guy
ballerina 2
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Duncan on the AREA
http://area.autodesk.com/index.php/blogs_duncan/tag_list/welcome/
Brain
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Future Work
Other nThings
Improve Collisions
Improve Constraint Solver
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The End
Thank You