Physics-Based Animation

In this paper we introduce a new approach for the embedding of linear elastic deformable models. Our technique results in significant improvements in the efficient physically based simulation of highly detailed objects. First, our embedding takes into account topological details, that is, disconnected parts that fall into the same coarse element are simulated independently. Second, we account for the varying material properties by computing stiffness and interpolation functions for coarse elements which accurately approximate the behavior of the embedded material. Finally, we also take into account empty space in the coarse embeddings, which provides a better simulation of the boundary. The result is a straightforward approach to simulating complex deformable models with the ease and speed associated with a coarse regular embedding, and with a quality of detail that would only be possible at much finer resolution.

Preserving Topology and Elasticity for Embedded Deformable Models

We propose an approach for efficiently simulating elastic objects made of non-homogeneous, non-isotropic materials. Based on recent developments in homogenization theory, a methodology is introduced to approximate a deformable object made of arbitrary fine structures of various linear elastic materials with a dynamically-similar coarse model. This coarsening of the material properties allows for simulation of fine, heterogeneous structures on very coarse grids while capturing the proper dynamics of the original dynamical system, thus saving orders of magnitude in computational time. Examples including inhomogeneous and/or anisotropic materials can be realistically simulated in realtime using a numerically-coarsened model made of a few mesh elements.

Numerical Coarsening of Inhomogeneous Elastic Materials

Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive-time stepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energy-preserving Harlow-Welch / Crank-Nicolson scheme to simplicial grids.

Energy-Preserving Integrators for Fluid Animation

Smoothed Particle Hydrodynamics (SPH) is a powerful technique for the animation of natural phenomena. While early SPH approaches in Computer Graphics have mainly been concerned with liquids or gases, recent research also focuses on the dynamics of deformable solids using SPH. In this paper, we present a novel corotational SPH formulation for deformable solids. The rigid body modes are extracted from the deformation field which allows to use a linear strain tensor. In contrast to previous rotationally invariant meshless approaches, we show examples using coplanar and collinear particle data sets. The presented approach further allows for a unified meshfree representation of deformable solids and fluids. This enables the animation of sophisticated phenomena, such as phase transitions. The versatility and the efficiency of the presented SPH scheme for deformable solids is illustrated in various experiments.

Co-rotated SPH for Deformable Solids

This paper provides a formal connexion between springs and continuum mechanics in the context of onedimensional
and two-dimensional elasticity. In a first stage, the equivalence between tensile springs and the finite element
discretization of stretching energy on planar curves is established. Furthermore, when considering a quadratic strain function of stretch, we introduce a new type of springs called tensile biquadratic springs. In a second stage, we extend this equivalence to non-linear membranes (St Venant-Kirchhoff materials) on triangular meshes leading to triangular biquadratic and quadratic springs. Those tensile and angular springs produce isotropic deformations parameterized by Young modulus and Poisson ratios on unstructured meshes in an efficient and simple way. For a specific choice of the Poisson ratio, 0.3, we show that regular spring-mass models may be used realistically to simulate a membrane behavior. Finally, the different spring formulations are tested in pure traction and cloth simulation experiments.

Triangular Springs for Modeling Nonlinear Membranes

This article describes a computationally efficient formulation and an algorithm for tetrahedral finite-element simulation of elastic objects subject to Saint Venant-Kirchhoff (StVK) material law. The number of floating point operations required by the algorithm is in the range of 15% to 27% for computing the vertex forces from a given set of vertex positions, and 27% to 38% for the tangent stiffness matrix, in comparison to a well-optimized algorithm directly derived from the conventional Total Lagrangian formulation. In the new algorithm, the data is associated with edges and tetrahedron-sharing edge-pairs (TSEPs), as opposed to tetrahedra, to avoid redundant computation. Another characteristic of the presented formulation is that it reduces to that of a spring-network model by simply ignoring all the TSEPs. The technique is demonstrated through an interactive application involving haptic interaction, being combined with a linearized implicit integration technique employing a preconditioned conjugate gradient method.

An Edge-Based Computationally Efficient Formulation of Saint-Venant Kirchhoff Tetrahedral Finite Elements

This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their state-space geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary user-specified constraints.

Lie Group Integrators for Animation and Control of Vehicles

Thin elastic rods such as cables, phone coils, tree branches, or hair, are common objects in the real world but computing their dynamics accurately remains challenging. The recent Super-Helix model, based on the discrete equations of Kirchhoff for a piecewise helical rod, is one of the most promising models for simulating non-stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom – or equivalently to a low number of variations in curvature along the mean curve. This paper proposes a new, recursive scheme for the dynamics of a Super-Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone’s algorithm, we exploit the recursive kinematics of a Super-Helix to propagate elements inertias from the free end to the clamped end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree-like structures of Super-Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements.

Linear Time Super-Helices

We present Continuum-based Strain Limiting (CSL) – a new method for limiting deformations in physically-based cloth simulations. Recent developments have led to methods which excel at simulating nearly inextensible materials, but the efficient simulation of general biphasic textiles and their anisotropic behavior remains challenging. Other approaches use softer materials and enforce limits on edge elongations, leading to discretization-dependent behavior. Moreover, they offer no explicit control over shearing and stretching unless specifically aligned meshes are used, which makes them less attractive for practical animation of anisotropic textiles. Based on a continuum deformation measure, our method allows accurate deformation control using individual thresholds for all types of strain. We impose deformation limits element-wise and cast the problem as a system of linear equations. We show how to further improve efficiency using an approximate formulation. CSL can be combined with any type of cloth simulator and, as a velocity filter, integrates seamlessly into standard collision handling frameworks.

Continuum-based Strain Limiting

We present a new Eulerian-Lagrangian method for physics-based simulation of fluid flow, which includes automatic generation of sub-scale spray and bubbles. The Marker Level Set method is used to provide a simple geometric criterion for free marker generation. A filtering method, inspired from Weber number thresholding, further controls the free marker generation (in a physics-based manner). Two separate models are used, one for sub-scale droplets, the other for sub-scale bubbles. Droplets are evolved in a Newtonian manner, using a density extension drag force field, while bubbles are evolved using a model based on Stokes’ Law. We show that our model for sub-scale droplet and bubble dynamics is simple to couple with a full (macro-scale) Navier-Stokes two-phase flow model and is quite powerful in its applications. Our animations include coarse grained multiphase features interacting with fine scale multiphase features.

Simulation of Two-Phase Flow with Sub-Scale Droplets and Bubble Effects