We propose a two-scale method for particle-based fluids that allocates computing resources to regions of the fluid where complex flow behavior emerges. Our method uses a low- and a high-resolution simulation that run at the same time. While in the coarse simulation the whole fluid is represented by large particles, the fine level simulates only a subset of the fluid with small particles. The subset can be arbitrarily defined and also dynamically change over time to capture complex flows and small-scale surface details. The low- and high-resolution simulations are coupled by including feedback forces and defining appropriate boundary conditions. Our method offers the benefit that particles are of the same size within each simulation level. This avoids particle splitting and merging processes, and allows the simulation of very large resolution differences without any stability problems. The model is easy to implement, and we show how it can be integrated into a standard SPH simulation as well as into the incompressible PCISPH solver. Compared to the single-resolution simulation, our method produces similar surface details while improving the efficiency linearly to the achieved reduction rate of the particle number.
Month: May 2011
Efficient Elasticity for Character Skinning with Contact and Collisions
We present a new algorithm for near-interactive simulation of skeleton driven, high resolution elasticity models. Our methodology is used for soft tissue deformation in character animation. The algorithm is based on a novel discretization of corotational elasticity over a hexahedral lattice. Within this framework we enforce positive definiteness of the stiffness matrix to allow efficient quasistatics and dynamics. In addition, we present a multigrid method that converges with very high efficiency. Our design targets performance through parallelism using a fully vectorized and branch-free SVD algorithm as well as a stable one-point quadrature scheme. Since body collisions, self collisions and soft-constraints are necessary for real-world examples, we present a simple framework for enforcing them. The whole approach is demonstrated in an end-to-end production-level character skinning system.
Efficient Elasticity for Character Skinning with Contact and Collisions
Real-time Large-deformation Substructuring
This paper shows a method to extend 3D nonlinear elasticity model reduction to open-loop multi-level reduced deformable structures. Given a volumetric mesh, we decompose the mesh into several subdomains, build a reduced deformable model for each domain, and connect the domains using inertia coupling. This makes model reduction deformable simulations much more versatile: localized deformations can be supported without prohibitive computational costs, parts can be re-used and precomputation times shortened. Our method does not use constraints, and can handle large domain rigid body motion in addition to large deformations, due to our derivation of the gradient and Hessian of the rotation matrix in polar decomposition. We show real-time examples with multi-level domain hierarchies and hundreds of reduced degrees of freedom.
Real-Time Eulerian Water Simulation Using a Restricted Tall Cell Grid
We present a new Eulerian fluid simulation method, which allows real-time simulations of large scale three dimensional liquids. Such scenarios have hitherto been restricted to the domain of off-line computation. To reduce computation time we use a hybrid grid representation composed of regular cubic cells on top of a layer of tall cells. With this layout water above an arbitrary terrain can be represented without consuming an excessive amount of memory and compute power, while focusing effort on the area near the surface where it most matters. Additionally, we optimized the grid representation for a GPU implementation of the fluid solver. To further accelerate the simulation, we introduce a specialized multigrid algorithm for solving the Poisson equation and propose solver modifications to keep the simulation stable for large time steps. We demonstrate the efficiency of our approach in several real-world scenarios, all running above 30 frames per second on a modern GPU. Some scenes include additional features such as two-way rigid body coupling as well as particle representations of sub-grid detail.
Real-Time Eulerian Water Simulation Using a Restricted Tall Cell Grid
Solid Simulation with Oriented Particles
We propose a new fast and robust method to simulate various types of solid including rigid, plastic and soft bodies as well as one, two and three dimensional structures such as ropes, cloth and volumetric objects. The underlying idea is to use oriented particles that store rotation and spin, along with the usual linear attributes, i.e. position and velocity. This additional information adds substantially to traditional particle methods. First, particles can be represented by anisotropic shapes such as ellipsoids, which approximate surfaces more accurately than spheres. Second, shape matching becomes robust for sparse structures such as chains of particles or even single particles because the undefined degrees of freedom are captured in the rotational states of the particles. Third, the full transformation stored in the particles, including translation and rotation, can be used for robust skinning of graphical meshes and for transforming plastic deformations back into the rest state.
HOT: Hodge-Optimized Triangulations
We introduce Hodge-optimized triangulations (HOT), a family of well-shaped primal-dual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals (“power diagrams”). They allow greater flexibility in the location of dual vertices while keeping primal-dual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.
On the Velocity of an Implicit Surface
In this article we derive an equation for the velocity of an arbitrary time-evolving implicit surface. Strictly speaking, only the normal component of the velocity is unambiguously defined. This is because an implicit surface does not have a unique parametrization. However, by enforcing a constraint on the evolution of the normal field we obtain a unique tangential component. We apply our formulas to surface tracking and to the problem of computing velocity vectors of a motion blurred blobby surface. Other possible applications are mentioned at the end of the article.
Sparse Meshless Models of Complex Deformable Solids
A new method to simulate deformable objects with heterogeneous material properties and complex geometries is presented. Given a volumetric map of the material properties and an arbitrary number of control nodes, a distribution of the nodes is computed automatically, as well as the associated shape functions. Reference frames attached to the nodes are used to apply skeleton subspace deformation across the volume of the objects. A continuum mechanics formulation is derived from the displacements and the material properties. We introduce novel material-aware shape functions in place of the traditional radial basis functions used in meshless frameworks. In contrast with previous approaches, these allow coarse deformation functions to efficiently resolve non-uniform stiffnesses. Complex models can thus be simulated at high frame rates using a small number of control nodes.
Frame-Based Elastic Models
We present a new type of deformable model which combines the realism of physically based continuum mechanics models and the usability of frame-based skinning methods. The degrees of freedom are coordinate frames. In contrast with traditional skinning, frame positions are not scripted but move in reaction to internal body forces. The displacement field is smoothly interpolated using dual quaternion blending. The deformation gradient and its derivatives are computed at each sample point of a deformed object and used in the equations of Lagrangian mechanics to achieve physical realism. This allows easy and very intuitive definition of the degrees of freedom of the deformable object. The meshless discretization allows on-the-fly insertion of frames to create local deformations where needed. We formulate the dynamics of these models in detail and describe some pre-computations that can be used for speed. We show that our method is effective for behaviors ranging from simple unimodal deformations to complex realistic deformations comparable with Finite Element simulations. To encourage its use, the software will be freely available in the simulation platform SOFA.
Example-Based Elastic Materials
We propose an example-based approach for simulating complex elastic material behavior. Supplied with a few poses that characterize a given object, our system starts by constructing a space of prefered deformations by means of interpolation. During simulation, this example manifold then acts as an additional elastic attractor that guides the object towards its space of prefered shapes. Added on top of existing solid simulation codes, this example potential effectively allows us to implement inhomogeneous and anisotropic materials in a direct and intuitive way. Due to its example-based interface, our method promotes an art-directed approach to solid simulation, which we exemplify on a set of practical examples.