Alexey Stomakhin

A material point method for snow simulation

Snow is a challenging natural phenomenon to visually simulate. While the graphics community has previously considered accumulation and rendering of snow, animation of snow dynamics has not been fully addressed. Additionally, existing techniques for solids and fluids have difficulty producing convincing snow results. Specifically, wet or dense snow that has both solid- and fluid-like properties is difficult to handle. Consequently, this paper presents a novel snow simulation method utilizing a user-controllable elasto-plastic constitutive model integrated with a hybrid Eulerian/Lagrangian Material Point Method. The method is continuum based and its hybrid nature allows us to use a regular Cartesian grid to automate treatment of self-collision and fracture. It also naturally allows us to derive a grid-based semi-implicit integration scheme that has conditioning independent of the number of Lagrangian particles. We demonstrate the power of our method with a variety of snow phenomena including complex character interactions.

The affine particle-in-cell method

Hybrid Lagrangian/Eulerian simulation is commonplace in computer graphics for fluids and other materials undergoing large deformation. In these methods, particles are used to resolve transport and topological change, while a background Eulerian grid is used for computing mechanical forces and collision responses. Particle-in-Cell (PIC) techniques, particularly the Fluid Implicit Particle (FLIP) variants have become the norm in computer graphics calculations. While these approaches have proven very powerful, they do suffer from some well known limitations. The original PIC is stable, but highly dissipative, while FLIP, designed to remove this dissipation, is more noisy and at times, unstable. We present a novel technique designed to retain the stability of the original PIC, without suffering from the noise and instability of FLIP. Our primary observation is that the dissipation in the original PIC results from a loss of information when transferring between grid and particle representations. We prevent this loss of information by augmenting each particle with a locally affine, rather than locally constant, description of the velocity. We show that this not only stably removes the dissipation of PIC, but that it also allows for exact conservation of angular momentum across the transfers between particles and grid.

Augmented MPM for phase-change and varied materials

In this paper, we introduce a novel material point method for heat transport, melting and solidifying materials. This brings a wider range of material behaviors into reach of the already versatile material point method. This is in contrast to best-of-breed fluid, solid or rigid body solvers that are difficult to adapt to a wide range of materials. Extending the material point method requires several contributions. We introduce a dilational/deviatoric splitting of the constitutive model and show that an implicit treatment of the Eulerian evolution of the dilational part can be used to simulate arbitrarily incompressible materials. Furthermore, we show that this treatment reduces to a parabolic equation for moderate compressibility and an elliptic, Chorin-style projection at the incompressible limit. Since projections are naturally done on marker and cell (MAC) grids, we devise a staggered grid MPM method. Lastly, to generate varying material parameters, we adapt a heat-equation solver to a material point framework.

Reconstruction of missing data in social networks based on temporal patterns of interactions

We discuss a mathematical framework based on a self-exciting point process aimed at analyzing temporal patterns in the series of interaction events between agents in a social network. We then develop a reconstruction model that allows one to predict the unknown participants in a portion of those events. Finally, we apply our results to the Los Angeles gang network. (Some figures may appear in colour only in the online journal)

Energetically consistent invertible elasticity

We provide a smooth extension of arbitrary isotropic hyperelastic energy density functions to inverted configurations. This extension is designed to improve robustness for elasticity simulations with extremely large deformations and is analogous to the extension given to the first Piola-Kirchoff stress in [ITF04]. We show that our energy-based approach is significantly more robust to large deformations than the first Piola-Kirchoff fix. Furthermore, we show that the robustness and stability of a hyperelastic model can be predicted from a characteristic contour, which we call its primary contour. The extension to inverted configurations is defined via extrapolation from a convex threshold surface that lies in the uninverted portion of the principal stretches space. The extended hyperelastic energy density yields continuous stress and unambiguous stress derivatives in all inverted configurations, unlike in [TSIF05]. We show that our invertible energy-density-based approach outperforms the popular hyperelastic corotated model, and we also show how to use the primary contour methodology to improve the robustness of this model to large deformations.

Optimization Integrator for Large Time Steps

Practical time steps in todays state-of-the-art simulators typically rely on Newtons method to solve large systems of nonlinear equations. In practice, this works well for small time steps but is unreliable at large time steps at or near the frame rate, particularly for difficult or stiff simulations. We show that recasting backward Euler as a minimization problem allows Newtons method to be stabilized by standard optimization techniques with some novel improvements of our own. The resulting solver is capable of solving even the toughest simulations at the 24Hz frame rate and beyond. We show how simple collisions can be incorporated directly into the solver through constrained minimization without sacrificing efficiency. We also present novel penalty collision formulations for self collisions and collisions against scripted bodies designed for the unique demands of this solver. Finally, we show that these techniques improve the behavior of Material Point Method (MPM) simulations by recasting it as an optimization problem.

A material point method for viscoelastic fluids, foams and sponges

We present a new Material Point Method (MPM) for simulating viscoelastic fluids, foams and sponges. We design our discretization from the upper convected derivative terms in the evolution of the left Cauchy-Green elastic strain tensor. We combine this with an Oldroyd-B model for plastic flow in a complex viscoelastic fluid. While the Oldroyd-B model is traditionally used for viscoelastic fluids, we show that its interpretation as a plastic flow naturally allows us to simulate a wide range of complex material behaviors. In order to do this, we provide a modification to the traditional Oldroyd-B model that guarantees volume preserving plastic flows. Our plasticity model is remarkably simple (foregoing the need for the singular value decomposition (SVD) of stresses or strains). Lastly, we show that implicit time stepping can be achieved in a manner similar to [Stomakhin et al. 2013] and that this allows for high resolution simulations at practical simulation times.

The material point method for simulating continuum materials

Simulating the physical behaviors of deformable objects and fluids has been an important topic in computer graphics. While the Lagrangian Finite Element Method (FEM) is widely used for elasto-plastic solids, it usually requires additional computational components in the case of large deformation, mesh distortion, fracture, self-collision and coupling between materials. Often, special solvers and strategies need to be developed for a particular problem. Recently, the hybrid Eulerian/Lagrangian Material Point Method (MPM) was introduced to the graphics community. It uses a continuum description of the governing equations and utilizes user-controllable elasto-plastic constitutive models. The hybrid nature of MPM allows using a regular Cartesian grid to automate treatment of self-collision and fracture. Like other particle methods such as Smoothed Particle Hydrodynamics (SPH), topology change is easy due to the lack of explicit connectivity between Lagrangian particles. Furthermore, MPM allows a grid-based implicit integration scheme that has conditioning independent of the number of Lagrangian particles. MPM also provides a unified particle simulation framework similar to Position Based Dynamics (PBD) for easy coupling of different materials. The power of MPM has been demonstrated in a number of recent papers for simulating various materials including elastic objects, snow, lava, sand and viscoelastic fluids. It is also highly integrated into the production framework of Walt Disney Animation Studios and has been used in featured animations including Frozen, Big Hero 6 and Zootopia.

A second order virtual node algorithm for Navier-Stokes flow problems with interfacial forces and discontinuous material properties

We present a numerical method for the solution of the Navier-Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergence-free velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semi-Lagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L^~ and L^2) and first order for the pressure (in L^~, second order in L^2). Our temporal discretization is also second order accurate.

Fluxed animated boundary method

We present a novel approach to guiding physically based particle simulations using boundary conditions. Unlike commonly used ad hoc particle techniques for adding and removing the material from a simulation, our approach is principled by utilizing the concept of volumetric flux. Artists are provided with a simple yet powerful primitive called a fluxed animated boundary (FAB), allowing them to specify a control shape and a material flow field. The system takes care of enforcing the corresponding boundary conditions and necessary particle reseeding. We show how FABs can be used artistically or physically. Finally, we demonstrate production examples that show the efficacy of our method.