Large Steps in Cloth Simulation - SIGGRAPH 98

1. Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수

2. Cloth Simulation

3. 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

4. Cloth and Mass-Spring Model Discrete cloth model

5. 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)

6. Simulation Overview Notation and Geometry Position of world space Forces Planer coordinate

7. Simulation Overview Energy and Forces Internal forces - Stretch, Shear, Bending Damping force Combining all forces

8. 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

9. Implicit Integration Explicit forward Euler method

10. Implicit Integration(cont.) Implicit backward Euler method Nonlinear, need iteration By Taylor series expansion to f, first order approximation

11. Implicit Integration(cont.) Implicit backward Euler method Rewrite this approximated equation,

12. Implicit Integration(cont.) Taking the bottom row of below equation and substituting top row yields,

13. Implicit Integration(cont.) Letting I denote the identity matrix, and regrouping,

14. Forces The 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

15. 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 k is a stiffness constant

16. Forces (Forces & Force Derivatives) Block form of f Sparse matrix Derivative matrix K Sparse, Symmetric Matrix

17. Forces (Stretch Forces) Stretch force UV coordinates

18. Forces (Stretch Forces) Stretch force can be measured by Unstretched condition

19. Forces (Stretch Forces) i j k Approximate w(u,v) as a linear function over each triangle,

20. Forces (Stretch Forces) Stretch energy Usually, we set

21. Forces(Shear & Bend) Shear force Bending force Idea : angle of adjacent triangles Idea : Inner product

22. Forces(Damping) Strong stretch force ⇔ Strong damping force ☞ Prevent oscillation Damping direction Damping strength Damping force Eq:

23. Forces(Damping) Differentiate the damping eq. Asymmetric, Sparse Matrix breaks symmetry, so we omitted this term.

24. Constraints Unsuitable approaches - Reduced Coordinates - Penalty Methods - Lagrange Multipliers

25. Constraints(Mass Modification) xy-plane constraint : Generalization

26. Constraints(Mass Modification) Rewrite previous eq. z is change in velocity along the constrained direction

27. Constraints(Implementation) Multiply M, Symmetric(Positive definite) These two systems have a same solution Δv.

28. Constraints(Implementation) W is singular Linear System Ax=b Two conditions : residual will be zero,

29. Modified Conjugate Gradient Method

30. Collisions(Initiation) For collision detect, Coherency-based bounding-box approach is used. Penalty force(moving positions) “Jumpy” behavior in local regions.

31. Collisions(Position Alteration) Particle’s position in next step If collisions occur, Considering collisions,

32. Results

33. Results

34. Results

35. Results

36. Thank you Questions || Comments ?

37. Conjugate gradient Method? Isn’t there more simple ways to implement mass-spring systems?