1. By Tiantian Liu et al (SIGGRAPH 2017) Presented By Anirudh Ganesh
Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials
By Tiantian Liu et al (SIGGRAPH 2017)
Presented By Anirudh Ganesh

2. Motivation Real Time Simulation focused
Contrast with Offline application
Hence they want to create a fast enough method for real-time for hyper-elastic materials that is more generalizable than the existing methods

3. Existing Methods (Historic)
Baraff and Witkin 1998
Cloth dynamics with Euler which it solved using 1 Newton iteration
Goldenthal et al 2007
extention to simulate in-extensible cloth
Touriner et al 2015
support for stiff materials robustly
All the above use Newton’s Method hence cannot be run in real time

4. Existing Methods (Modern)
PBD gives a superfast solution for softbody simulation
Position based dynamics, Muller at al 2007
Unified particle physics for real-time applications, Macklin et al 2014
Fast simulation, but material property degrades over iterations

5. Existing Work (Projective Dynamics)
Fast Simulation of Mass-Spring Systems, Liu et al 2013
Shape-Up: Shaping Discrete Geometry with Projections, Bouazis et al 2014
ADMM ⊇ Projective Dynamics: Fast Simulation of General Constitutive Models, Narain et al 2016

6. Existing Work (Chebyshev Methods)
Huamin Wang A Chebyshev semi-iterative approach for accelerating projective and position-based dynamics.
Huamin Wang and Yin Yang Descent Methods for Elastic Body Simulation on the GPU. ACM Transactions on Graphics (SIGGRAPH Asia).
Can be seen as a Quasi Newton method

7. Quasi Newton Method For a traditional newton method, Compute gradient
Compute Hessian
Step is then computed using inverse of Hessian times the gradient
For a quasi newton method,
Instead of using Hessian, we try to approximate it using a spd matrix.

8. Projective Dynamics Projective Dynamics Energy is defined as
Ei(x) = min pi∈Mi E˜ i(x, pi), E˜ i(x, z) = kGix − zk 2 F
Then the total energy is E(x) = X i wiEi(x),
When trying to minimize it, we can get the solution as
x ∗ = (M/h2 + L) −1 (Jp + My/h2 )

9. Limitations of Projective Dynamics
Energy needs to be in quadratic distance form.
This is not the case in most hyper-elastic material. For eg, Neo Hookian, Co- Rotated linear elasticity or Kirchoff cannot be represented as a quadratic distance.

10. Proposed Solution
Approximate p in terms of x so that it becomes function of x
Then take gradient of it using chain rule
∇g(x) = 1 h2 M(x − y) + Lx − Jp(x)
Jacobian is 0 for Projective Dynamics kind of energy.
So replacing this in Newton’s equation yields,
(M/h2 +L) −1∇g(x) = x − (M/h2 +L) −1 (Jp(x) +My/h2 )
Which can be solved as
x* = −(M/h2 + L) −1∇g(x)

11. Generalizing the proposed solution
In last step we assumed J is 0, but this is only true for Projective Dynamics style of Energy.
In order to generalize we bring it in Valanis-Landel form then we approximate the stiffness of the material as the slope of best linear approximation of
f(σ1) = ∂Ψ / ∂σ1 σ2=1,σ3=1 = a’ 0 (σ1) + 2b’ 0 (σ1) + c’ 0 (σ1)
Also in projective dynamics the line search parameter is always 1, but this also needs to be generalised. In order to do this, they perform a backtracking line search starting from 2 and dividing it by 2 each iteration till it converges.

12. Results

13. Convergence Rates (compared to Chebyshev)

14. Convergence Rate (comparison of various L BFGS settings)

15. Convergence (compared to other methods)