Physics-Based Animation

Zhendong Wang, Yin Yang, Huamin Wang

In this paper, we address two limitations of dihedral angle based discrete bending (DAB) models, i.e. the indefiniteness of their energy Hessian and their vulnerability to geometry degeneracies. To tackle the indefiniteness issue, we present novel analytic expressions for the eigensystem of a DAB energy Hessian. Our expressions reveal that DAB models typically have positive, negative, and zero eigenvalues, with four of each, respectively. By using these expressions, we can efficiently project an indefinite DAB energy Hessian as positive semi-definite analytically. To enhance the stability of DAB models at degenerate geometries, we propose rectifying their indefinite geometric stiffness matrix by using orthotropic geometric stiffness matrices with adaptive parameters calculated from our analytic eigensystem. Among the twelve motion modes of a dihedral element, our resulting Hessian for DAB models retains only the desirable bending modes, compared to the undesirable altitude-changing modes of the exact Hessian with original geometric stiffness, all modes of the Gauss-Newton approximation without geometric stiffness, and no modes of the projected Hessians with inappropriate geometric stiffness. Additionally, we suggest adjusting the compression stiffness according to the Kirchhoff-Love thin plate theory to avoid over-compression. Our method not only ensures the positive semi-definiteness but also avoids instability caused by large bending forces at degenerate geometries. To demonstrate the benefit of our approaches, we show comparisons against existing methods on the simulation of cloth and thin plates in challenging examples.

Stable Discrete Bending by Analytic Eigensystem and Adaptive Orthotropic Geometric Stiffness

Han Yan, Bo Ren

Multiple fluid simulation has raised wide research interest in recent years. Despite the impressive successes of current works, simulation of scenes containing mixing or unmixing of high-density-ratio phases using particle-based discretizations still remains a challenging task. In this paper, we propose a peridynamic mixture-model theory that stably handles high-density-ratio multi-fluid simulations. With assistance of novel scalar-valued volume flow states, a particle based discretization scheme is proposed to calculate all the terms in the multi-phase Navier-Stokes equations in an integral form, We also design a novel mass updating strategy for enhancing phase mass conservation and reducing particle volume variations under high density ratio settings in multi-fluid simulations. As a result, we achieve significantly stabler simulations in mixture-model multi-fluid simulations involving mixing and unmixing of high density ratio phases. Various experiments and comparisons demonstrate the effectiveness of our approach.

High Density Ratio Multi-fluid Simulation with Peridynamics

Xingqiao Li*, Xingyu Ni*, Bo Zhu, Bin Wang, and Baoquan Chen (* = joint first authors)

This paper presents a novel level-set method that combines gradient augmentation and reference mapping to enable high-fidelity interface tracking and surface tension flow simulation, preserving small-scale volumes and interface features comparable to the grid size. At the center of our approach is a novel reference-map algorithm to concurrently convect level-set values and gradients, both of which are crucial for reconstructing a dynamic surface exhibiting small-scale volumes. In addition, we develop a full pipeline for the new level-set scheme by incorporating a novel extrapolation algorithm and an enhanced reinitialization procedure into our reference-map method. We test our algorithm by simulating complex surface tension flow phenomena such as raindrop collision, merging, and splashing. We also showcase the efficacy of our approach by performing validation tests and comparing it to a broad range of existing level-set algorithms.

GARM-LS: A Gradient-Augmented Reference-Map Method for Level-Set Fluid Simulation

Zhehao Li, Qingyu Xu, Xiaohan Ye, Bo Ren, Ligang Liu

Differentiable physics simulation has shown its efficacy in inverse design
problems. Given the pervasiveness of the diverse interactions between fluids and solids in life, a differentiable simulator for the inverse design of the motion of rigid objects in two-way fluid-rigid coupling is also demanded. There are two main challenges to develop a differentiable two-way fluid-solid coupling simulator for rigid body control tasks: the ubiquitous, discontinuous contacts in fluid-solid interactions, and the high computational cost of gradient formulation due to the large number of degrees of freedom (DoF) of fluid dynamics. In this work, we propose a novel differentiable SPH-based two-way fluid-rigid coupling simulator to address these challenges. Our purpose is to provide a differentiable simulator for SPH which incorporates a unified representation for both fluids and solids using particles. However, naively differentiating the forward simulation of the particle system encounters gradient explosion issues. We investigate the instability in differentiating the SPH-based fluid-rigid coupling simulator and present a feasible gradient computation scheme to address its differentiability. In addition, we also propose an efficient method to compute the gradient of fluid-rigid coupling without incurring the high computational cost of differentiating the entire high-DoF fluid system. We show the efficacy, scalability, and extensibility of our method in various challenging rigid body control tasks with diverse fluid-rigid interactions and multi-rigid contacts, achieving up to an order of magnitude speedup in optimization compared to baseline methods in experiments.

DiffFR: Differentiable SPH-based Fluid-Rigid Coupling for Rigid Body Control

Huanyu Chen, Danyong Zhao, Jernej Barbič

Capturing material properties of real-world elastic solids is both challenging and highly relevant to many applications in computer graphics, robotics and related fields. We give a non-intrusive, in-situ and inexpensive approach to measure the nonlinear elastic energy density function of man-made materials and biological tissues. We poke the elastic object with 3d-printed rigid cylinders of known radii, and use a precision force meter to record the contact force as a function of the indentation depth, which we measure using a force meter stand, or a novel unconstrained laser setup. We model the 3D elastic solid using the Finite Element Method (FEM), and elastic energy using a compressible Valanis-Landel material that generalizes Neo-Hookean materials by permitting arbitrary tensile behavior under large deformations. We then use optimization to fit the nonlinear isotropic elastic energy so that the FEM contact forces and indentations match their measured real-world counterparts. Because we use carefully designed cubic splines, our materials are accurate in a large range of stretches and robust to inversions, and are therefore “animation-ready” for computer graphics applications. We demonstrate how to exploit radial symmetry to convert the 3D elastostatic contact problem to the mathematically equivalent 2D problem, which vastly accelerates optimization. We also greatly improve the theory and robustness of stretch-based elastic materials, by giving a simple and elegant formula to compute the tangent stiffness matrix, with rigorous proofs and singularity handling. We also contribute the observation that volume compressibility can be estimated by poking with rigid cylinders of different radii, which avoids optical cameras and greatly simplifies experiments. We validate our method by performing full 3D simulations using the optimized materials and confirming that they match real-world forces, indentations and real deformed 3D shapes. We also validate it using a “Shore 00” durometer, a standard device for measuring material hardness.

Capturing Animation-Ready Isotropic Materials Using Systematic Poking

Qiqin Le, Yitong Deng, Jiamu Bu, Bo Zhu, Tao Du

We present a computational framework for simulating deformable surfaces with second-order triangular finite elements. Our method develops numerical schemes for discretizing stretching, shearing, and bending energies of deformable surfaces in a second-order finite-element setting. In particular, we introduce a novel discretization scheme for approximating mean curvatures on a curved triangle mesh. Our framework also integrates a virtual-node finite-element scheme that supports two-way coupling between cut-cell rods without expensive remeshing. We compare our approach with traditional simulation methods using linear and higher-order finite elements and demonstrate its advantages in several challenging settings, such as low-resolution meshes, anisotropic triangulation, and stiff materials. Finally, we showcase several applications of our framework in cloth simulation, mixed Origami and Kirigami, and biologically-inspired soft wing simulation.

Second-Order Finite Elements for Deformable Surfaces

Yitong Deng, Hong-Xing Yu, Diyang Zhang, Jiajun Wu, Bo Zhu

We introduce Neural Flow Maps, a novel simulation method bridging the emerging paradigm of implicit neural representations with fluid simulation based on the theory of flow maps, to achieve state-of-the-art simulation of inviscid fluid phenomena. We devise a novel hybrid neural field representation, Spatially Sparse Neural Fields (SSNF), which fuses small neural networks with a pyramid of overlapping, multi-resolution, and spatially sparse grids, to compactly represent long-term spatiotemporal velocity fields at high accuracy. With this neural velocity buffer in hand, we compute long-term, bidirectional flow maps and their Jacobians in a mechanistically symmetric manner, to facilitate drastic accuracy improvement over existing solutions. These long-range, bidirectional flow maps enable high advection accuracy with low dissipation, which in turn facilitates high-fidelity incompressible flow simulations that manifest intricate vortical structures. We demonstrate the efficacy of our neural fluid simulation in a variety of challenging simulation scenarios, including leapfrogging vortices, colliding vortices, vortex reconnections, as well as vortex generation from moving obstacles and density differences. Our examples show increased performance over existing methods in terms of energy conservation, visual complexity, adherence to experimental observations, and preservation of detailed vortical structures.

Fluid Simulation on Neural Flow Maps

Jiahao Wen, Jernej Barbič

Kirchhoff-Love shells are commonly used in many branches of engineering, including in computer graphics, but have so far been simulated only under limited nonlinear material options. We derive the Kirchhoff-Love thin-shell mechanical energy for an arbitrary 3D volumetric hyperelastic material, including isotropic materials, anisotropic materials, and materials whereby the energy includes both even and odd powers of the principal stretches. We do this by starting with any 3D hyperelastic material, and then analytically computing the corresponding thin-shell energy limit. This explicitly identifies and separates in-plane stretching and bending terms, and avoids numerical quadrature. Thus, in-plane stretching and bending are shown to originate from one and the same process (volumetric elasticity of thin objects), as opposed to from two separate processes as done traditionally in cloth simulation. Because we can simulate materials that include both even and odd powers of stretches, we can accommodate standard mesh distortion energies previously employed for 3D solid simulations, such as Symmetric ARAP and Co-rotational materials. We relate the terms of our energy to those of prior work on Kirchhoff-Love thin-shells in computer graphics that assumed small in-plane stretches, and demonstrate the visual difference due to the presence of our exact stretching and bending terms. Furthermore, our formulation allows us to categorize all distinct hyperelastic Kirchhoff-Love thin-shell energies. Specifically, we prove that for Kirchhoff-Love thin-shells, the space of all hyperelastic materials collapses to two-dimensional hyperelastic materials. This observation enables us to create an interface for the design of thin-shell Kirchhoff-Love mechanical energies, which in turn enables us to create thin-shell materials that exhibit arbitrary stiffness profiles under large deformations.

Kirchhoff-Love Shells with Arbitrary Hyperelastic Materials

Zeshun Zong, Xuan Li, Minchen Li, Maurizio M. Chiaramonte, Wojciech Matusik, Eitan Grinspun, Kevin Carlberg, Chenfanfu Jiang, Peter Yichen Chen

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 elasto-dynamics 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 100,000× and time savings by up to 10×.

Neural Stress Fields for Reduced-order Elastoplasticity and Fracture

Ziyin Qu, Minchen Li, Yin Yang, Chenfanfu Jiang, Fernando de Goes

We present a novel hybrid Lagrangian/Eulerian method for simulating inelastic flows that generates high-quality particle distributions with adaptive volumes. At its core, our approach integrates an updated Lagrangian time discretization of continuum mechanics with the Power Particle-In-Cell geometric representation of deformable materials. As a result, we obtain material points described by optimized density kernels that precisely track the varying particle volumes both spatially and temporally. For efficient CFL-rate simulations, we also propose an implicit time integration for our system using a non-linear Gauss-Seidel solver inspired by X-PBD, viewing Eulerian nodal velocities as primal variables. We demonstrate the versatility of our method with simulations of mesoscale bubbles, sands, liquid, and foams.

Power Plastics: A Hybrid Lagrangian/Eulerian Solver for Mesoscale Inelastic Flows