Abstract In this paper, we present a novel network‐based approach for reconstructing signed distance functions from fluid particles. The method uses a weighting kernel to transfer particles to a regular grid, which forms the input to a convolutional neural network. We propose a regression‐based regularization to reduce surface noise without penalizing high‐curvature features. The reconstruction exhibits improved spatial surface smoothness and temporal coherence compared with existing state of the art surface reconstruction methods. The method is insensitive to particle sampling density and robustly handles thin features, isolated particles, and sharp edges.

In this paper, we propose a neural network-based approach for learning to represent the behavior of plastic solid materials ranging from rubber and metal to sand and snow. Unlike elastic forces such as spring forces, these plastic forces do not result from the positional gradient of any potential energy, imposing great challenges on the stability and flexibility of their simulation. Our method effectively resolves this issue by learning a generalizable plastic energy whose derivative closely matches the analytical behavior of plastic forces. Our method, for the first time, enables the simulation of a wide range of arbitrary elasticity-plasticity combinations using time step-independent, unconditionally stable optimization-based time integrators. We demonstrate the efficacy of our method by learning and producing challenging 2D and 3D effects of metal, sand, and snow with complex dynamics.

Abstract In this paper we analyze the stability of the explicit material point method (MPM). We focus on PIC, APIC, and CPIC transfers using quadratic and cubic splines in two and three dimensions. We perform a fully three‐dimensional Von Neumann stability analysis to study the behavior within the bulk of a material. This reveals the relationship between the sound speed, CFL number, and actual time step restriction and its dependence on discretization options. We note that boundaries are generally less stable than the interior, with stable time steps generally decreasing until the limit when particles become isolated. We then analyze the stability of a single particle to derive a novel time step restriction that stabilizes simulations at their boundaries. Finally, we show that for explicit MPM with APIC or CPIC transfers, there are pathological cases where growth is observed at arbitrarily small time steps sizes. While these cases do not necessarily pose a problem for practical usage, they do suggest that a guarantee of stability may be theoretically impossible and that necessary but not sufficient time step restrictions may be a necessary and practical compromise.

Abstract Time steps for explicit MPM simulation in computer graphics are often selected by trial and error due to the challenges in automatically selecting stable time step sizes. Our time integration scheme uses time step restrictions that take into account forces, collisions, and even grid‐to‐particle transfers calculated near the end of the time step. We propose a novel set of time step restrictions that allow a time step to be selected that is stable, efficient to compute, and not too far from optimal. We derive the general solution for the sound speed in nonlinear isotropic hyperelastic materials, which we use to enforce the classical CFL time step restriction. We identify a single‐particle instability in explicit MPM integration and propose a corresponding time step restriction in the fluid case. We also propose a reflection‐based boundary condition for domain walls that supports separation and accurate Coulomb friction while preventing particles from penetrating the domain walls.

We propose a novel penalty force to enforce contacts with accurate Coulomb friction. The force is compatible with fully-implicit time integration and the use of optimization-based integration. The contact force is quite general. In addition to processing collisions between deformable objects, the force can be used to couple rigid bodies to deformable objects or the material point method. The force naturally leads to stable stacking without drift over time, even when solvers are not run to convergence. The force leads to an asymmetrical system, and we provide a practical solution for handling these.

We simulate sand dynamics using an elastoplastic, continuum assumption. We demonstrate that the Drucker-Prager plastic flow model combined with a Hencky-strain-based hyperelasticity accurately recreates a wide range of visual sand phenomena with moderate computational expense. We use the Material Point Method (MPM) to discretize the governing equations for its natural treatment of contact, topological change and history dependent constitutive relations. The Drucker-Prager model naturally represents the frictional relation between shear and normal stresses through a yield stress criterion. We develop a stress projection algorithm used for enforcing this condition with a non-associative flow rule that works naturally with both implicit and explicit time integration. We demonstrate the efficacy of our approach on examples undergoing large deformation, collisions and topological changes necessary for producing modern visual effects.

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.

Practical time steps in today's state-of-the-art simulators typically rely on Newton's 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 Newton's 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 [Formula: see text] 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.

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.

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.

Abstract We propose a series of techniques for hybridizing implicit and semi‐implicit time integration methods in a manner that retains much of the speed of the implicit method without sacrificing all of the higher quality vibrations one obtains with methods that handle elastic forces explicitly. We propose our scheme in the context of asynchronous methods, where different parts of the mesh are evolved at different time steps. Whereas traditional asynchronous methods evolve each element independently, we partition all of our elements into two groups: one group evolved at the frame rate using a fully implicit scheme, and another group which takes a number of substeps per frame using a scheme that is implicit on damping forces and explicit on the elastic forces. This allows for a straightforward coupling between the implicit and semi‐implicit methods at frame boundaries for added stability. As has been stressed by various authors, asynchronous schemes take some of the pressure off of mesh generation, allowing time evolution to remain efficient even in the face of sliver elements. Finally, we propose a force distributing projection method which allows one to redistribute the forces felt on boundaries between implicit and semi‐implicit regions of the mesh in a manner that yields improved visual quality.

In this paper, we propose a method designed to allow creatures to actively respond to a fluid environment. We explore various objective functions in order to determine ways to direct the behavior of our creatures. Our proposed method works in conjunction with generalized body forces as well as both one-way and two-way coupled fluid forces. As one might imagine, interesting behaviors can be derived from minimizing and maximizing both drag and lift as well as minimizing the effort that a creature's internal actuators exert. A major application for our work is the automatic specification of secondary motions, for example, certain joints can be animated, while others are automatically solved for in order to satisfy the objective function.

We propose a numerical method for modeling highly deformable nonlinear incompressible solids that conserves the volume locally near each node in a finite element mesh. Our method works with arbitrary constitutive models, is applicable to both passive and active materials (e.g. muscles), and works with simple tetrahedra without the need for multiple quadrature points or stabilization techniques. Although simple linear tetrahedra typically suffer from locking when modeling incompressible materials, our method enforces incompressibility per node (in a one-ring), and we demonstrate that it is free from locking. We correct errors in volume without introducing oscillations by treating position and velocity in separate implicit solves. Finally, we propose a novel method for treating both object contact and self-contact as linear constraints during the incompressible solve, alleviating issues in enforcing multiple possibly conflicting constraints.

Creating surfaces with intricate small-scale features (microgeometry) and detail is an important task in geometric modeling and computer graphics. We present a model processing method capable of producing a wide variety of complex surface features based on displacement mapping and stochastic geometry. The latter is a branch of mathematics that analyzes and characterizes the statistical properties of spatial structures. The technique has been incorporated into an interactive modeling environment that supports the design of stochastic microgeometries. Additionally a tool has been developed that provides random exploration of the technique's entire parameter space by generating sample microgeometry over a broad range of values. We demonstrate the effectiveness of our technique by creating diverse, complex surface structures for a variety of geometric models, e.g. arrowheads, candy bars, busts, planets and coral.