Physics-Based Animation

Simon Duenser, Bernhard Thomaszewski, Roi Poranne, Stelian Coros

Many flexible structures are characterized by a small number of compliant modes, i.e., large deformation paths that can be traversed with little mechanical effort, whereas resistance to other deformations is much stiffer. Predicting the compliant modes for a given flexible structure, however, is challenging. While linear eigenmodes capture the small-deformation behavior, they quickly divert into states of unrealistically high energy for larger displacements. Moreover, they are inherently unable to predict nonlinear phenomena such as buckling, stiffening, multistability, and contact. To address this limitation, we propose Nonlinear Compliant Modes—a physically-principled extension of linear eigenmodes for large-deformation analysis. Instead of constraining the entire structure to deform along a given eigenmode, our method only prescribes the projection of the system’s state onto the linear mode while all other degrees of freedom follow through energy minimization. We evaluate the potential of our method on a diverse set of flexible structures, ranging from compliant mechanisms to topology-optimized joints and structured materials. As validated through experiments on physical prototypes, our method correctly predicts a broad range of nonlinear effects that linear eigenanalysis fails to capture.

Nonlinear Compliant Modes for Large-Deformation Analysis of Flexible Structures

Alvin Shi, Theodore Kim

We analyze a wide class of penalty energies used for contact response through the lens of a reduced frame. Applying our analysis to both spring-based and barrier-based energies, we show that we can obtain closed-form, analytic eigensystems that can be used to guarantee positive semidefiniteness in implicit solvers. Our approach is both faster than direct numerical methods, and more robust than approximate methods such as Gauss-Newton. Over the course of our analysis, we investigate physical interpretations for two separate notions of length. Finally, we showcase the stability of our analysis on challenging strand, cloth, and volume scenarios with large timesteps on the order of 1/40 s.

A Unified Analysis of Penalty-Based Collision Energies

Haomiao Wu, Theodore Kim

Angle-based energies appear in numerous physics-based simulation models, including thin-shell bending and isotropic elastic strands. We present a generic analysis of these energies that allows us to analytically filter the negative eigenvalues of the second derivative (Hessian), which is critical for stable, implicit time integration. While these energies are usually formulated in terms of angles and positions, we propose an abstract edge stencil that succinctly parameterizes the edge deformation, and allows us to derive generic, closed-form analytical expressions for the energy eigensystems. The resultant eigenvectors have straightforward geometric interpretations.We demonstrate that our method is readily applicable to a variety of 2D and 3D angle-based elastic energies, including both cloth and strands, and is up to 7x faster than numerical eigendecomposition.

An Eigenanalysis of Angle-Based Deformation Energies

Alvin Shi, Haomiao Wu, Jarred Parr, A.M. Darke, Theodore Kim

We present an isotropic, hyperelastic model specifically designed for the efficient simulation of tightly coiled hairs whose curl radii approach 5 mm. Our model is robust to large bends and torsions, even when they appear at the scale of the strand discretization. The terms of our model are consistently quadratic with respect to their primary variables, do not require per-edge frames or any parallel transport operators, and can efficiently take large timesteps on the order of 1/30 of a second. Additionally, we show that it is possible to obtain fast, closed-form eigensystems for all the terms in the energy. Our eigenanalysis is sufficiently generic that it generalizes to other models. Our entirely vertex-based formulation integrates naturally with existing finite element codes, and we demonstrate its efficiency and robustness in a variety of scenarios.

Lifted Curls: A Model for Tightly Coiled Hair Simulation

Li Huang, Fan Yang, Chendi Wei, Yuju (Edwin) Chen, Chun Yuan, Ming Gao

This paper introduces a hair simulator optimized for real-time applications, including console and cloud gaming, avatar live-streaming, and metaverse environments. We view the collisions between strands as a mechanism to preserve the overall volume of the hair and adopt explicit Material Point Method (MPM) to resolve the strand-strand collision. For simulating single-strand behavior, a semi-implicit Discrete Elastic Rods (DER) model is used. We build upon a highly efficient GPU MPM framework recently presented by Fei et al. [2021b] and propose several schemes to largely improve the performance of building and solving the semi-implicit DER systems on GPU. We demonstrate the efficiency of our pipeline by a few practical scenes that achieve up to 260 frames-per-second (FPS) with more than two thousand simulated strands on Nvidia GeForce RTX 3080.

Towards Realtime: A Hybrid Physics-based Method for Hair Animation on GPU

Paul Zhang, Zoë Marschner, Justin Solomon, Rasmus Tamstorf

Sum-of-Squares Programming (SOSP) has recently been introduced to graphics as a unified way to address a large set of difficult problems involving higher order primitives. Unfortunately, a challenging aspect of this approach is the computational cost—especially for problems involving multiple geometries like collision detection. In this paper, we present techniques to reduce the cost of SOSP significantly. We use these improvements to speed up difficult problems like collision detection between Bézier triangles by as much as 300×. In addition, motivated by hair bundle simulation, we present SOSP based collision detection on the tapered cubic cylinder. We also present an algebraic formulation of rigid body motion enabling SOSP based collision detection for curved geometries and trajectories simultaneously. While these new formulations are complex, our speedups make them feasible. These advances improve the applicability of SOSP based collision detection and enable the continued progress of higher-order geometry processing.

Sum-of-Squares Collision Detection for Curved Shapes and Paths

Alexandre Mercier-Aubin, Paul G. Kry

We present a method to improve the computation time of thin shell simulations by using adaptive rigidification to reduce the number of degrees of freedom. Our method uses a discretization independent metric for bending rates, and we derive a membrane strain rate to curvature rate equivalence that permits the use of a common threshold. To improve accuracy, we enhance the elastification oracle by considering both membrane and bending deformation to determine when to rigidify or elastify. Furthermore, we explore different approaches that are compatible with the previous work on adaptive rigidifcation and enhance the accuracy of the elastification on new contacts without increasing the computational overhead. Additionally, we propose a scaling approach that reduces the conditioning issues that arise from mixing rigid and elastic bodies in the same model.

Adaptive Rigidification of Discrete Shell

Tetsuya Takahashi, Christopher Batty

Efficiently solving large-scale box-constrained convex quadratic programs (QPs) is an important computational challenge in physical simulation. We propose a new multilevel preconditioning scheme based on the active-set method and combine it with modified proportioning with reduced gradient projections (MPRGP) to efficiently solve such QPs arising from pressure Poisson equations with non-negative pressure constraints in fluid animation. Our method employs a purely algebraic multigrid method to ensure the solvability of the coarser level systems and to merge only algebraically-connected components, thereby avoiding performance degradation of the preconditioner. We present a filtering scheme to efficiently apply our multilevel preconditioning only to unconstrained subsystems of the pressure Poisson system while reusing the hierarchy constructed per simulation step. We demonstrate the effectiveness of our method over previous approaches in various examples.

A Multilevel Active-Set Preconditioner for Box-Constrained Pressure Poisson Solvers

Haozhe Su, Xuan Li, Tao Xue, Chenfanfu Jiang, Mridul Aanjaneya

We present a generalized constitutive model for versatile physics simulation of inviscid fluids, Newtonian viscosity, hyperelasticity, viscoplasticity, elastoplasticity, and other physical effects that arise due to a mixture of these behaviors. The key ideas behind our formulation are the design of a generalized Kirchhoff stress tensor that can describe hyperelasticity, Newtonian viscosity and inviscid fluids, and the use of pre-projection and post-correction rules for simulating material behaviors that involve plasticity, including elastoplasticity and viscoplasticity. We show how our generalized Kirchhoff stress tensor can be coupled together into a generalized constitutive model that allows the simulation of diverse material behaviors by only changing parameter values. We present several side-by-side comparisons with physics simulations for specific constitutive models to show that our generalized model produces visually similar results. More notably, our formulation allows for inverse learning of unknown material properties directly from data using differentiable physics simulations. We present several 3D simulations to highlight the robustness of our method, even with multiple different materials. To the best of our knowledge, our approach is the first to recover the knowledge of unknown material properties without making explicit assumptions about the data.

A Generalized Constitutive Model for Versatile MPM Simulation and Inverse Learning with Differentiable Physics

Fabian Löschner, José Antonio Fernández-Fernández, Stefan Rhys Jeske, Andreas Longva, Jan Bender

We explore micropolar materials for the simulation of volumetric deformable solids. In graphics, micropolar models have only been used in the form of one-dimensional Cosserat rods, where a rotating frame is attached to each material point on the one-dimensional centerline. By carrying this idea over to volumetric solids, every material point is associated with a microrotation, an independent degree of freedom that can be coupled to the displacement through a material’s strain energy density. The additional degrees of freedom give us more control over bending and torsion modes of a material. We propose a new orthotropic micropolar curvature energy that allows us to make materials stiff to bending in specific directions. For the simulation of dynamic micropolar deformables we propose a novel incremental potential formulation with a consistent FEM discretization that is well suited for the use in physically-based animation. This allows us to easily couple micropolar deformables with dynamic collisions through a contact model inspired from the Incremental Potential Contact (IPC) approach. For the spatial discretization with FEM we discuss the challenges related to the rotational degrees of freedom and propose a scheme based on the interpolation of angular velocities followed by quaternion time integration at the quadrature points. In our evaluation we validate the consistency and accuracy of our discretization approach and demonstrate several compelling use cases for micropolar materials. This includes explicit control over bending and torsion stiffness, deformation through prescription of a volumetric curvature field and robust interaction of micropolar deformables with dynamic collisions.

Micropolar Elasticity in Physically-Based Animation