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

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

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

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

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

Existing Work (Chebyshev Methods) Huamin Wang. 2015. A Chebyshev semi-iterative approach for accelerating projective and position-based dynamics. Huamin Wang and Yin Yang. 2016. Descent Methods for Elastic Body Simulation on the GPU. ACM Transactions on Graphics (SIGGRAPH Asia). Can be seen as a Quasi Newton method

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.

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 )

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.

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)

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.

Results https://www.youtube.com/watch?v=7o3g-rG1qYg

Convergence Rates (compared to Chebyshev)

Convergence Rate (comparison of various L BFGS settings)

Convergence (compared to other methods)