Physics-Based Animation

Bohan Wang, Mianlun Zheng, Jernej Barbič

Physically based simulation is often combined with geometric mesh animation to add realistic soft-body dynamics to virtual characters. This is commonly done using constraint-based simulation whereby a soft-tissue simulation is constrained to geometric animation of a subpart (or otherwise proxy representation) of the character. We observe that standard constraint-based simulation suffers from an important flaw that limits the expressiveness of soft-body dynamics. Namely, under correct physics, the frequency and amplitude of soft-tissue dynamics arising from constraints (“inertial amplitude”) are coupled, and cannot be adjusted independently merely by adjusting the material properties of the model. This means that the space of physically based simulations is inherently limited and cannot capture all effects typically expected by computer animators. For example, animators need the ability to adjust the frequency, inertial amplitude, gravity sag and damping properties of the virtual character, independently from each other, as these are the primary visual characteristics of the soft-tissue dynamics. We demonstrate that independence can be achieved by transforming the equations of motion into a non-inertial reference coordinate frame, then scaling the resulting inertial forces, and then converting the equations of motion back to the inertial frame. Such scaling of inertia makes it possible for the animator to set the character’s inertial amplitude independently from frequency. We also provide exact controls for the amount of character’s gravity sag, and the damping properties. In our examples, we use linear blend skinning and pose-space deformation for geometric mesh animation, and the Finite Element Method for soft-body constrained simulation; but our idea of scaling inertial forces is general and applicable to other animation and simulation methods. We demonstrate our technique on several character examples.

Adjustable Constrained Soft-Tissue Dynamics

Tetsuya Takahashi, Christopher Batty

We propose Monolith, a monolithic pressure-viscosity-contact solver for more accurately, robustly, and efficiently simulating non-trivial two-way interactions of rigid bodies with inviscid, viscous, or non-Newtonian liquids. Our solver simultaneously handles incompressibility and (optionally) implicit viscosity integration for liquids, contact resolution for rigid bodies, and mutual interactions between liquids and rigid bodies by carefully formulating these as a single unified minimization problem. This monolithic approach reduces or eliminates an array of problematic artifacts, including liquid volume loss, solid interpenetrations, simulation instabilities, artificial “melting” of viscous liquid, and incorrect slip at liquid-solid interfaces. In the absence of solid-solid friction, our minimization problem is a Quadratic Program (QP) with a symmetric positive definite (SPD) matrix and can be treated with a single Linear Complementarity Problem (LCP) solve. When friction is present, we decouple the unified minimization problem into two subproblems so that it can be effectively handled via staggered projections with alternating LCP solves.We also propose a complementary approach for non-Newtonian fluids which can be seamlessly integrated and addressed during the staggered projections. We demonstrate the critical importance of a contact-aware, unified treatment of fluid-solid coupling and the effectiveness of our proposed Monolith solver in a wide range of practical scenarios.

Monolith: A Monolithic Pressure-Viscosity-Contact Solver for Strong Two-Way Rigid-Rigid Rigid-Fluid Coupling

Yunxin Sun, Tamar Shinar, Craig Schroeder

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.

Effective Time Step Restrictions for explicit MPM simulation

Pengbin Tang, Jonas Zehnder, Stelian Coros, Bernhard Thomaszewski

We present a computational method for designing compliant mechanical systems that exhibit large-amplitude oscillations. The technical core of our approach is an optimization-driven design tool that combines sensitivity analysis for optimization with the Harmonic Balance Method for simulation. By establishing dynamic force equilibrium in the frequency domain, our formulation avoids the major limitations of existing alternatives: it handles nonlinear forces, side-steps any transient process, and automatically produces periodic solutions. We introduce design objectives for amplitude optimization and trajectory matching that enable intuitive high-level authoring of large-amplitude motions. Our method can be applied to many types of mechanical systems, which we demonstrate through a set of examples involving compliant mechanisms, flexible rod networks, elastic thin shell models, and multi-material solids. We further validate our approach by manufacturing and evaluating several physical prototypes.

A Harmonic Balance Approach for Designing Compliant Mechanical Systems with Nonlinear Periodic Motions

Andreas Longva, Fabian Löschner, Tassilo Kugelstadt, José Antonio Fernández-Fernández, Jan Bender

As demands for high-fidelity physics-based animations increase, the need for accurate methods for simulating deformable solids grows. While higher-order finite elements are commonplace in engineering due to their superior approximation properties for many problems, they have gained little traction in the computer graphics community. This may partially be explained by the need for finite element meshes to approximate the highly complex geometry of models used in graphics applications. Due to the additional per-element computational expense of higher-order elements, larger elements are needed, and the error incurred due to the geometry mismatch eradicates the benefits of higher-order discretizations. One solution to this problem is the embedding of the geometry into a coarser finite element mesh. However, to date there is no adequate, practical computational framework that permits the accurate embedding into higher-order elements. We develop a novel, robust quadrature generation method that generates theoretically guaranteed high-quality sub-cell integration rules of arbitrary polynomial accuracy. The number of quadrature points generated is bounded only by the desired degree of the polynomial, independent of the embedded geometry. Additionally, we build on recent work in the Finite Cell Method (FCM) community so as to tackle the severe ill-conditioning caused by partially filled elements by adapting an Additive-Schwarz-based preconditioner so that it is suitable for use with state-of-the-art non-linear material models from the graphics literature. Together these two contributions constitute a general-purpose framework for embedded simulation with higher-order finite elements. We finally demonstrate the benefits of our framework in several scenarios, in which second-order hexahedra and tetrahedra clearly outperform their first-order counterparts.

Higher-Order Finite Elements for Embedded Simulation

David A.B. Hyde, Steven W. Gagniere, Alan Marquez-Razon, Joseph Teran

We present an updated Lagrangian discretization of surface tension forcesfor the simulation of liquids with moderate to extreme surface tension effects.The potential energy associated with surface tension is proportional to the surface area of the liquid. We design discrete forces as gradients of this energy with respect to the motion of the fluid over a time step. We show that this naturally allows for inversion of the Hessian of the potential energy required with the use of Newton’s method to solve the systems of nonlinear equations associated with implicit time stepping. The rotational invariance of the surface tension energy makes it non-convex and we define a definiteness fix procedure as in [Teran et al. 2005]. We design a novel level-set-based boundary quadrature technique to discretize the surface area calculation in our energy based formulation. Our approach works most naturally with Particle-In-Cell [Harlow 1964] techniques and we demonstrate our approach with a weakly incompressible model for liquid discretized with the Material Point Method [Sulsky et al.1994]. We show that our approach is essential for allowing efficient implicit numerical integration in the limit of high surface tension materials like liquid metals.

An Implicit Updated Lagrangian Formulation for Liquids with Large Surface Energy

Zahra Forootaninia, Rahul Narain

We propose a simple and efficient method for guiding an Eulerian smoke simulation to match the behavior of a specified velocity field, such as a low-resolution animation of the same scene, while preserving the rich, turbulent details arising in the simulated fluid. Our method works by simply combining the high-frequency component of the simulated fluid velocity with the low-frequency component of the input guiding field. In contrast to previous work, we show that it is essential to use ideal low-pass and high-pass filters in the frequency domain, in order to avoid artifacts resulting from loss of small-scale details over time.We demonstrate our method on many scenes including those with static and moving obstacles, and show that it produces high-quality results with very little computational overhead.

Frequency-Domain Smoke Guiding

Xiao-Song Chen, Chen-Feng Li, Geng-Chen Cao, Yun-Tao Jiang and Shi-Min Hu

In physically based-based animation, pure particle methods are popular due to their simple data structure, easy implementation, and convenient parallelization. As a pure particle-based method and using Galerkin discretization, the Moving Least Square Reproducing Kernel Method (MLSRK) was developed in engineering computation as a general numerical tool for solving PDEs. The basic idea of Moving Least Square (MLS) has also been used in computer graphics to estimate d formation gradient for deformable solids. Based on these previous studies, we propose a multiphase MLSRK framework that animates complex and coupled fluids and solids in a unified manner. Specifically, we use the Cauchy momentum equation and phase field model to uniformly capture the momentum balance and phase evolution/interaction in a multiphase system, and systematically formulate the MLSRK discretization to support general multiphase constitutive models. A series of animation examples are presented to demonstrate the performance of our new multiphase MLSRK framework,including hyperelastic, elastoplastic, viscous, fracturing and multiphase coupling behaviours etc.

A Moving Least Square Reproducing Kernel Particle Method for Unified Multiphase Continuum Simulation

Moritz Geilinger, David Hahn, Jonas Zehnder, Moritz Bächer, Bernhard Thomaszewski, Stelian Coros

We present a differentiable dynamics solver that is able to handle fric-tional contact for rigid and deformable objects within a unified framework.Through a principled mollification of normal and tangential contact forces,our method circumvents the main difficulties inherent to the non-smooth nature of frictional contact. We combine this new contact model with fully-implicit time integration to obtain a robust and efficient dynamics solver that is analytically differentiable. In conjunction with adjoint sensitivity analysis, our formulation enables gradient-based optimization with adaptive trade-offs between simulation accuracy and smoothness of objective function landscapes. We thoroughly analyse our approach on a set of simulation examples involving rigid bodies, visco-elastic materials, and coupled multi-body systems. We furthermore showcase applications of our differentiable simulator to parameter estimation for deformable objects, motion planning for robotic manipulation, trajectory optimization for compliant walking robots, as well as efficient self-supervised learning of control policies

ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact

Yuwei Xiao, Szeyu Chan, Siqi Wang, Bo Zhu, Xubo Yang

Enabling adaptivity on a uniform Cartesian grid is challenging due to its highly structured grid cells and axis-aligned grid lines. In this paper, we propose a new grid structure – the adaptive staggered-tilted (AST) grid –to conduct adaptive fluid simulations on a regular discretization. The key mechanics underpinning our new grid structure is to allow the emergence of a new set of tilted grid cells from the nodal positions on a background uniform grid. The original axis-aligned cells, in conjunction with the populated axis-tilted cells, jointly function as the geometric primitives to enable adaptivity on a regular spatial discretization. By controlling the states of the tilted cells both temporally and spatially, we can dynamically evolve the adaptive discretizations on an Eulerian domain. Our grid structure preserves almost all the computational merits of a uniform Cartesian grid, including he cache-coherent data layout, the easiness for parallelization, and the existence of high-performance numerical solvers. Further, our grid structure can be integrated into other adaptive grid structures, such as an Octree or a sparsely populated grid, to accommodate the T-junction-free hierarchy. We demonstrate the efficacy of our AST grid by showing examples of large-scale incompressible flow simulation in domains with irregular boundaries

An Adaptive Staggered-Tilted Grid for Incompressible Flow Simulation