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# Snow Business: Computational Elastoplasticity in the Movies and Beyond

Joseph Teran, University of California, Los Angeles.
The following is a short introduction to the American Mathematical Society Invited Address, to be presented at the upcoming 2018 SIAM Annual Meeting (AN18) in Portland, Ore., from July 9-13. Look for feature articles by other AN18 invited speakers introducing the topics of their talks in future issues.

Over the past two decades, visual effects have come to rely on a wide range of numerical methods for partial differential equations (PDEs). Be it the crashing water waves in Disney’s Moana or falling snow in Frozen, audiences now expect the computer-generated world to look and move like the real thing. The demand for realism is so high that it is impractical—or impossible—for animators to reproduce the dynamics of everyday materials like clothing, water, sand, snow, or hair without using the laws of physics that dictate their motion.

The governing physics is expressed with PDEs derived from classical continuum mechanics (e.g., the Navier-Stokes equations for water). The PDEs are highly nonlinear and involve geometrically-complicated domains, like the upper body of the character in Figure 1a and the snow under Anna’s feet in Figure 1b. Given these constraints, one can only solve the equations with numerical approximation and scientific computing. Techniques from computational fluid dynamics, like particle-in-cell [1-2, 4] and the finite element method for elastic solids [6], are now commonplace in the production of blockbuster movies for these reasons.

Many everyday materials behave elastically for a wide range of strains, but plastically upon approaching nonphysical stresses. Common examples include metals and granular materials like sand, snow, mud, and dirt. One can even describe frictional contact as a plastic constraint on states of stresses that arise during contact. In continuum mechanics, the Cauchy stress $$\mathbf{\sigma}$$ is defined by the relation between internal surfaces of contact with normals $$\mathbf{n}$$ and the contact force per unit area (or traction) $$\mathbf{t}=\mathbf{\sigma}\mathbf{n}=\mathbf{t}_\tau+(\mathbf{n}\cdot\mathbf{\sigma}\mathbf{n})\mathbf{n}$$. When the contact force must obey Coulomb friction, the tangential component $$\mathbf{t}_\tau$$ of the force must be smaller in magnitude than a coefficient of friction $$c_F$$ times the normal component $$-\mathbf{n}\cdot\mathbf{\sigma}\mathbf{n}$$:

$|\mathbf{t}_\tau| \leq -c_F \mathbf{n}\cdot\mathbf{\sigma}\mathbf{n}. \tag1$

Figure 1. Simulation of hyperelastic and plastic materials. 1a. The finite element method simulates soft elastic tissue on a Disney movie character. The discrete simulation mesh is shown in blue. Image courtesy of [6]. 1b. The material point method simulates the snow under Anna’s feet as an elastoplastic granular material in Disney’s Frozen. Image courtesy of [7]. 1c. One can model frictional contact between layers of clothing with a discretized elastoplastic continuum model. Images courtesy of [3].

For example, one can derive the Mohr-Coulomb and Drucker-Prager plastic yield conditions [5] for granular materials by simply applying this Coulomb friction condition to a hyperelastic constitutive model (see Figure 1b). We have recently shown that even clothing can be simulated from a continuum view, where Coulomb friction during contact places a constraint on the types of stresses that are physical (see Figure 1c). The material point method [8] is key to translating these continuum descriptions of plasticity physics into discretized approximations that one can use for visual effects. This technique is a generalization of the particle-in-cell approach [1, 2] to history-dependent materials, and is not used for a broad range of materials whose physics is naturally described by elastoplasticity.

I will discuss these aspects and more during my talk at the 2018 SIAM Annual Meeting.

References
[1] Brackbill, J., & Ruppel, H. (1986). FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J. Comp. Phys., 65, 314-343.
[2] Harlow, F. (1964). The particle-in-cell method for numerical solution of problems in fluid dynamics. Meth. Comp. Phys., 3, 319-343.
[3] Jiang, C., Gast, T., & Teran, J. (2017). Anisotropic elastoplasticity for cloth, knit and hair frictional contact. ACM Trans. Graph., 36(4) 152:1-152:14.
[4] Jiang, C., Schroeder, C., Selle, A., Teran, J., & Stomakhin, A. (2015). The affine particle-in-cell method. ACM Trans. Graph., 34(4), 51:1-51:10.
[5] Klar, G., Gast, T., Pradhana, A., Fu, C., Schroeder, C., Jiang, C., & Teran, J. (2016). Drucker-prager elastoplasticity for sand animation. ACM Trans. Graph., 35(4),103:1-103:12.
[6] McAdams, A., Zhu, Y., Selle, A., Empey, M., Tamstorf, R., Teran, J., & Sifakis, E. (2011). Efficient elasticity for character skinning with contact and collisions. ACM Trans. Graph., 30(4), 37:1- 37:12.
[7] Stomakhin, A., Schroeder, C., Chai, L., Teran, J., & Selle, A. (2013). A material point method for snow simulation. ACM Trans. Graph., 32(4), 102:1-102:10.
[8] Sulsky, D., Chen, Z., & Schreyer, H. (1994). A particle method for history-dependent materials. Comp. Meth. App. Mech. Eng., 118(1), 179-196.

 Joseph Teran is a professor in the Department of Mathematics at the University of California, Los Angeles.