Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.
-
Upload
joy-reeves -
Category
Documents
-
view
237 -
download
2
Transcript of Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.
![Page 1: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/1.jpg)
Large Steps in Cloth Simulation - SIGGRAPH 98
박 강 수
![Page 2: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/2.jpg)
Cloth Simulation
![Page 3: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/3.jpg)
Issues in Cloth Simulation
Large time steps- stability
Damping forces- oscillation
Constraints- contact or fix conditions
Solving a large sparse linear system- conjugate gradient iteration
![Page 4: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/4.jpg)
Cloth and Mass-Spring Model
Discrete cloth model
![Page 5: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/5.jpg)
Differential Equation of Spring
x : geometric state vector(position)M : mass distribution matrix of cloth E : scalar function of x (internal energy)F : other forces (air-drag, damping, contact)
![Page 6: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/6.jpg)
Simulation Overview
Notation and Geometry
Position of world space
Forces
Planer coordinate
3ixn3x
3ifn3f
),( ii vu
![Page 7: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/7.jpg)
Simulation Overview
Energy and ForcesInternal forces
- Stretch, Shear, Bending Damping forceCombining all forces
iii m/fx
),( xxfMx 1 ),,,,,,,,,()( 222111
33
nnn
nn
mmmmmmmmmdiag
M
M
![Page 8: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/8.jpg)
Simulation Overview
Sparse Matrices
Very sparse system- n particles : n x n matrix- nonzero entry : dense 3x3 matrices
of scalar
Modified conjugate gradient iterative method
![Page 9: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/9.jpg)
Implicit Integration
Explicit forward Euler method
![Page 10: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/10.jpg)
Implicit Integration(cont.)
Implicit backward Euler method
Nonlinear, need iteration
By Taylor series expansion to f, first order approximation
![Page 11: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/11.jpg)
Implicit Integration(cont.)
Implicit backward Euler method
Rewrite this approximated equation,
![Page 12: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/12.jpg)
Implicit Integration(cont.)
Taking the bottom row of below equationand substituting top row yields,
![Page 13: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/13.jpg)
Implicit Integration(cont.)
Letting I denote the identity matrix, and regrouping,
![Page 14: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/14.jpg)
Forces
xf /EThe force f arising from energy E
Impractical approach - Expressing E as a single monolithic function - Taking derivatives
Batter approach - Decompose E into a sum of sparse energy functions
![Page 15: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/15.jpg)
Forces
But decomposing method is not enough. - Sensible damping function problem
Instead, we define vector condition C(x) which is, - Formulating internal behavior - To be zero
Define associated energy
)()(2
)( xCxCx Tc
kE k is a stiffness constant
![Page 16: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/16.jpg)
Forces (Forces & Force Deriva-tives)
Block form of f
Sparse matrix
Derivative matrix K
Sparse, Symmetric Matrix
![Page 17: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/17.jpg)
Forces (Stretch Forces)
Stretch force
u
vUV coordinates
![Page 18: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/18.jpg)
Forces (Stretch Forces)
Stretch force can be measured by
Unstretched condition
![Page 19: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/19.jpg)
Forces (Stretch Forces)
i
j k
ij1 xxx ik xxx 2
ij1 uuu ik uuu 2
ij1 vvv ik vvv 2
Approximate w(u,v) as a linear function over each triangle,
![Page 20: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/20.jpg)
Forces (Stretch Forces)
Stretch energy
1 vu bbUsually, we set
![Page 21: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/21.jpg)
Forces(Shear & Bend)
Shear force Bending force
Idea : Inner productIdea : angle of adjacent triangles
![Page 22: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/22.jpg)
Forces(Damping)
Strong stretch force ⇔ Strong damping force☞ Prevent oscillation
Damping force Eq:
Damping direction Damping strength
![Page 23: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/23.jpg)
Forces(Damping)
Differentiate the damping eq.
Asymmetric, Sparse Matrix
breaks symmetry, so we omitted this term.
![Page 24: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/24.jpg)
Constraints
Unsuitable approaches
- Reduced Coordinates
- Penalty Methods
- Lagrange Multipliers
![Page 25: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/25.jpg)
Constraints(Mass Modification)
xy-plane constraint :
Generalization
![Page 26: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/26.jpg)
Constraints(Mass Modification)
Rewrite previous eq.
z is change in velocityalong the constrained direction
![Page 27: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/27.jpg)
Constraints(Implementation)
Multiply M,
Symmetric(Positive definite)
These two systems have a same solution Δv.
![Page 28: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/28.jpg)
Constraints(Implementation)
W is singu-lar
Two conditions : residual will be zero,
Linear System Ax=b
![Page 29: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/29.jpg)
Modified Conjugate Gradient Method
![Page 30: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/30.jpg)
Collisions(Initiation)
For collision detect, Coherency-based
bounding-box approach is used.
Penalty force(moving positions)
“Jumpy” behavior in local regions.
![Page 31: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/31.jpg)
Collisions(Position Alteration)
Particle’s position in next step
If collisions occur,
Considering collisions,
![Page 32: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/32.jpg)
Results
![Page 33: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/33.jpg)
Results
![Page 34: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/34.jpg)
Results
![Page 35: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/35.jpg)
Results
![Page 36: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/36.jpg)
Thank youQuestions || Com-
ments ?
![Page 37: Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수.](https://reader033.fdocument.pub/reader033/viewer/2022061612/56649ee15503460f94bf1ed8/html5/thumbnails/37.jpg)
Conjugate gradient Method?
Isn’t there more simple ways to implement mass-
spring systems?