1. Interactive Animation of Structured Deformable Objects Mathieu Desbrun Peter Schroder Alan Barr

2. Overview – Main problem Few techniques are currently able to handle the animation in interactive rates so far Both efficiency and stability are desired

3. Overview A stable and efficient algorithm for animating mass-spring system based cloth Focus is the cloth modeling and the interaction between cloth and different environments based on this technique

4. Overview - Outline Principle of implicit integration Integration scheme Post-step modification Cloth simulation in virtual environment Results

5. Implicit integration – 1D Each discrete mass point i of mass m at position x i moves at speed of v i Adjacent points are connected by spring of stiffness k Superscript indices indicate the current step number

6. Implicit integration – 1D The explicit Euler integration looks like Assuming the force F i is constant over a time step

7. Implicit integration – 1D The time step should be inversely proportional to the square root of the stiffness k [Courant condition] System is stable only for small time steps

8. Implicit integration – 1D The implicit integration is Replacing the forces at time t by for forces at time t+dt

9. Implicit integration – 1D How to compute F n+1 without knowing the exact position of next time step? By a first-order approximation

10. Implicit integration – 1D Notice H is the negated Hessian matrix of the system H is CONSTANT (for 1D case) and symmetric

11. Implicit integration – 1D Re-write Put everything together Then we have

12. Implicit integration - Comparison Extra force added to F n as It ’ s used as artificial viscosity: a mass point is influenced by the motion of its neighbors It ’ s proportional to dt and k

13. Implicit integration – Comparison Where acts like ‘ filters ’ The bigger the stiffness k is, the wider filters are. The resulting force on a mass point will take into account more forces around

14. Integration Scheme - Extension Extension 1D to 2D/3D A mass point i is connected to all the other points j with springs of rest length and stiffness k ij

15. Integration Scheme - Extension H is not CONSTANT any more, instead It ’ s a 3n x 3n matrix and needs to be solved at each time step

16. Integration Scheme - Splitting By decomposing the forces into 2 parts, a linear and a nonlinear one: Do approximate integration of two parts

17. Integration Scheme – Linear part From above analysis, it ’ s easy to solve this linear part cause the 3n x 3n Hessian matrix H is CONSTANT

18. Integration Scheme – Nonlinear part The nonlinear part always has the same magnitude between t and t+dt, which means it only rotates An angle error is introduced, which needs to be balanced with a straightforward displacement later

19. Integration Scheme – Nonlinear part After the internal forces have been filtered, compute resulting global torque T as x G is the center of gravity. Because the sum of all internal forces is 0, re-write T as

20. Integration Scheme – Nonlinear part T is supposed to be 0, so we can modify the integration output to balance it. Simply add the correction force on each mass point i This is not a zero-error scheme. In practice the result becomes implausible only for big k or dt: wrinkled mesh

21. Post-step Modification - Motivation Mass-spring is not perfect to model cloth The elongation is proportional to the force applied but the natural force/deformation curve is normally nonlinear

22. Post-step Modification – Implementation A force/deformation ratio threshold for each spring as d max. Iterate over stretched spring and shrink them When to stop? After fixed iteration steps Time is up

23. Post-step Modification – Inverse dynamics process

24. Results