1. UT DALLAS Erik Jonsson School of Engineering & Computer Science FEARLESS engineering Stable Real-Time Deformations Authors: Matthias Muller, Julie Dorsey, Leonard McMillan, Robert Jagnow, Barbara Cutler Presented by: Yin Yang

2. FEARLESS engineering Outlines Dynamics Deformations Non-Linear Elasticity Linear Elasticity Stiffness Warping Rotation Tensor Field Results

3. FEARLESS engineering Dynamics Deformations Finite element method (FEM, [Bathe, 1982]) –More accurate result obtained –Body Subdivided –Interpolate deformation within basic element Element type(shape) does not matter NOT Linear!

4. FEARLESS engineering Dynamics Deformations Dynamic equilibrium equation [Cook,1981] Defines a coupled system of 3n ODE of position Solve it for each time step!

5. FEARLESS engineering Dynamics Deformations Many solvers have been proposed –Explicit: the status in the next step depend only on the status on the previous –Implicit: A linear system needs to be solved at each step! Euler first order method –handle discontinuities better(i.e. collision) than higher order method [Desbrun et al. 1999]

6. FEARLESS engineering Dynamics Deformations Implicit Euler (Backward Euler) For detailed solver formulation [Witkin and Baraff, 1998]

7. FEARLESS engineering Non-Linear Elasticity Assume no rigid body motion (i.e. only deformation) Original defined on change of length over infinitesimal material Change of SQUARED length –Green-Lagrange strain tensor

8. FEARLESS engineering Non-Linear Elasticity Quadratic stress-displacement relationship A good by product Still not a full second order approximation Why? Only one coefficient (i.e. Young’s modulus) F is no linear!

9. FEARLESS engineering Linear Elasticity Linear elasticity K is Jacobian of F at x = x0 Called Stiffness matrix Constant, can be pre-computed Faster! Even more stable!

10. FEARLESS engineering Linear Elasticity Why not linear? –Large deformation can not get rendered correctly –More precisely… –Linear strain tensor does not change during any rigid body transformation –Rotational deformation!

11. FEARLESS engineering Stiffness Warping Try to use linear tensor E.F for a single tetrahedral element Assume the rotation is known

12. FEARLESS engineering Stiffness Warping Rotate the deformed positions back Compute E.F Rotate back again to the deformed position

13. FEARLESS engineering Stiffness Warping What is rotation? – Use the global rigid rotation [Terzopoulos and Witkin, 1988] acceptable result on stiff material with little rotation bad result on deformable body with large rotation (i.e. Growth of volume) Individual R for each vertex

14. FEARLESS engineering Rotation Tensor Field Rotation is not unique! –Use least square

15. FEARLESS engineering Rotation Tensor Field A simpler, faster method used –Try to compute local orthonormal frame –n1: normalized average of three edges –n2: cross of n1 and a chosen edge –n3: cross of n1 and n2 –Sounds heuristic….

16. FEARLESS engineering Rotation Tensor Field

17. FEARLESS engineering Algorithm

18. FEARLESS engineering Results Claimed that as fast as linear solver!