Neural Stress Fields for Reduced-order Elastoplasticity and Fracture

We propose a hybrid neural network and physics framework for reduced-order modeling of elastoplasticity and fracture. State-of-the-art scientific computing models like the Material Point Method (MPM) faithfully simulate large-deformation elastoplasticity and fracture mechanics. However, their long runtime and large memory consumption render them unsuitable for applications constrained by computation time and memory usage, e.g., virtual reality. To overcome these barriers, we propose a reduced-order framework. Our key innovation is training a low-dimensional manifold for the Kirchhoff stress field via an implicit neural representation. This low-dimensional neural stress field (NSF) enables efficient evaluations of stress values and, correspondingly, internal forces at arbitrary spatial locations. In addition, we also train neural deformation and affine fields to build low-dimensional manifolds for the deformation and affine momentum fields. These neural stress, deformation, and affine fields share the same low-dimensional latent space, which uniquely embeds the high-dimensional simulation state. After training, we run new simulations by evolving in this single latent space, which drastically reduces the computation time and memory consumption. Our general continuum-mechanics-based reduced-order framework is applicable to any phenomena governed by the elastodynamics equation. To showcase the versatility of our framework, we simulate a wide range of material behaviors, including elastica, sand, metal, non-Newtonian fluids, fracture, contact, and collision. We demonstrate dimension reduction by up to 100,000X and time savings by up to 10X

Variational Bonded Discrete Element Method with Manifold Optimization

This paper proposes a novel approach that combines variational integration with the bonded discrete element method (BDEM) to achieve faster and more accurate fracture simulations. The approach leverages the efficiency of implicit integration and the accuracy of BDEM in modeling fracture phenomena. We introduce a variational integrator and a manifold optimization approach utilizing a nullspace operator to speed up the solving of quaternion-constrained systems. Additionally, the paper presents an element packing and surface reconstruction method specifically designed for bonded discrete element methods. Results from the experiments prove that the proposed method offers 2.8 to 12 times faster state-of-the-art methods

Model reduction for the material point method via an implicit neural representation of the deformation map

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate $\textit{dynamics}$ for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ -- with negligible errors

Q-NET: A Network for Low-dimensional Integrals of Neural Proxies

Many applications require the calculation of integrals of multidimensional functions. A general and popular procedure is to estimate integrals by averaging multiple evaluations of the function. Often, each evaluation of the function entails costly computations. The use of a \emph{proxy} or surrogate for the true function is useful if repeated evaluations are necessary. The proxy is even more useful if its integral is known analytically and can be calculated practically. We propose the use of a versatile yet simple class of artificial neural networks -- sigmoidal universal approximators -- as a proxy for functions whose integrals need to be estimated. We design a family of fixed networks, which we call Q-NETs, that operate on parameters of a trained proxy to calculate exact integrals over \emph{any subset of dimensions} of the input domain. We identify transformations to the input space for which integrals may be recalculated without resampling the integrand or retraining the proxy. We highlight the benefits of this scheme for a few applications such as inverse rendering, generation of procedural noise, visualization and simulation. The proposed proxy is appealing in the following contexts: the dimensionality is low ($<10$D); the estimation of integrals needs to be decoupled from the sampling strategy; sparse, adaptive sampling is used; marginal functions need to be known in functional form; or when powerful Single Instruction Multiple Data/Thread (SIMD/SIMT) pipelines are available for computation.Comment: 11 pages (including appendix and references

Efficient Deformations Using Custom Coordinate Systems

Physics-based deformable object simulations have been playing an increasingly important role in 3D computer graphics. They have been adopted for humanoid character animations as well as special effects such as fire and explosion. However, simulations of large, complex systems can consume large amounts of computation and mostly remain offline, which prohibits their use for interactive applications.We present several highly efficient schemes for deformable object simulation using custom spatial coordinate systems. Our choices span the spectrum of subspace to full space and both Lagrangian and Eulerian viewpoints.Subspace methods achieve massive speedups over their “full space” counterparts by drastically reducing the degrees of freedom involved in the simulation. A long standing difficulty in subspace simulation is incorporating various non-linearities. They introduce expensive computational bottlenecks and quite often cause novel deformations that are outside the span of the subspace.We address these issues in articulated deformable body simulations from a Lagrangian viewpoint. We remove the computational bottleneck of articulated self-contact handling by deploying a pose-space cubature scheme, a generalization of the standard “cubature” approximation. To handle novel deformations caused by arbitrary external collisions, we introduce a generic approach called subspace condensation, which activates full space simulation on the fly when an out-of-basis event is encountered. Our proposed frameworkefficiently incorporates various non-linearities and allows subspace methods to be used in cases where they previously would not have been considered.Deformable solids can interact not only with each other, but also with fluids. Wedesign a new full space method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. No-slip boundary conditions are automatically satisfied by imposing a global divergence-free condition. We are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method