Figure 1: Dropping viscoelastic balls in an Eulerian fluid simulation. Invisible geometry is quickly deleted, while the visible surfaces retain their details even after translating through the air and splashing on the ground. We present a method for accurately tracking the moving surface of deformable materials in a manner that gracefully handles topological changes. We employ a Lagrangian surface tracking method, and we use a triangle mesh for our surface representation so that fine features can be retained. We make topological changes to the mesh by first identifying merging or splitting events at a particular grid resolution, and then locally creating new pieces of the mesh in the affected cells using a standard isosurface creation method. We stitch the new, topologically simplified portion of the mesh to the rest of the mesh at the cell boundaries. Our method detects and treats topological events with an emphasis on the preservation of detailed features, while simultaneously simplifying those portions of the material that are not visible. Our surface tracker is not tied to a particular method for simulating deformable materials. In particular, we show results from two significantly different simulators: a Lagrangian FEM simulator with tetrahedral elements, and an Eulerian grid-based fluid simulator. Although our surface tracking method is generic, it is particularly well-suited for simulations that exhibit fine surface details and numerous topological events. Highlights of our results include merging of viscoplastic materials with complex geometry, a taffy-pulling animation with many fold and merge events, and stretching and slicing of stiff plastic material.

Figure 1: Real-time control ensures fixed simulation outcome regardless of runtime user forces: First: the rest configuration of the “T”-shape structure and the two target balls. Second: reference motion from an external simulator; the two ends of the “T ” impact the two balls. Third: user-perturbed real-time simulation, without control. The two ends miss the target. Fourth: controlled user-perturbed real-time simulation, with gentle control forces, tracks the reference motion and successfully impacts the target. The perturbation force load (green arrow; applied 1/5 through the simulation, only in the third and fourth motion) pushes the “T ” in the opposite direction of motion. Recent advances have brought real-time physically based simulation within reach, but simulations are still difficult to control in real time. We present interactive simulations of passive systems such as deformable solids or fluids that are not only fast, but also directable: they follow given input trajectories while simultaneously reacting to user input and other unexpected disturbances. We achieve such directability using a real-time controller that runs in tandem with a real-time physically based simulation. To avoid stiff and overcontrolled systems where the natural dynamics are overpowered, the injection of control forces has to be minimized. This search for gentle forces can be made tractable in real-time by linearizing the system dynamics around the input trajectory, and then using a time-varying linear quadratic regulator to build the controller. We show examples of controlled complex deformable solids and fluids, demonstrating that our approach generates a requested fixed outcome for reasonable user inputs, while simultaneously providing runtime motion variety.

Figure 1: When the level set is advected by the BFECC [Dupont and Liu 2003] method, the simulation of a rising bubble produces volume loss (top). When the proposed volume control method is used, the volume of bubble is preserved regardless of the length of the simulation (bottom). From left to right, each column shows the bubble at t = 0, 0.0625, 0.125, 0.25, 0.5, and 10.0 second. The image on the far right shows a foam structure obtained after raising more than 400 bubbles. Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time. 1

Figure 1: Our method allows us to efficiently simulate complex surface tension phenomena such as this crown splash. The small scales are handled with our surface approach, while the larger scales are computed with the Eulerian simulation. For the shown simulation, our method requires only 22.3 seconds per frame on average. We present an approach to simulate flows driven by surface tension based on triangle meshes. Our method consists of two simulation layers: the first layer is an Eulerian method for simulating surface tension forces that is free from typical strict time step constraints. The second simulation layer is a Lagrangian finite element method that simulates sub-grid scale wave details on the fluid surface. The surface wave simulation employs an unconditionally stable, symplectic time integration method that allows for a high propagation speed due to strong surface tension. Our approach can naturally separate the grid- and sub-grid scales based on a volumepreserving mean curvature flow. As our model for the sub-grid dynamics enforces a local conservation of mass, it leads to realistic pinch off and merging effects. In addition to this method for simulating dynamic surface tension effects, we also present an efficient non-oscillatory approximation for capturing damped surface tension behavior. These approaches allow us to efficiently simulate complex phenomena associated with strong surface tension, such as Rayleigh-Plateau instabilities and crown splashes, in a short amount of time.

Semi-Lagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFL-based time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semi-Lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi-Lagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressible-style pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had. 1.

Figure 1: Our method easily scales from an inexpensive, coarse fluid simulation to a detailed turbulent wake. The base solver, shown to the left uses only 32×8×32 cells. The remaining pictures show the influence of our turbulence model, with a varying number of particles from 250k to 1M and 4M from left to right. For the simulation with 1M particles we achieve 15 frames per second on average, including rendering. While the amount of detail directly depends on the number of particles used, the overall flow remains consistent. It is usually difficult to resolve the fine details of turbulent flows, es-pecially when targeting real-time applications. We present a novel, scalable turbulence method that uses a realistic energy model and an efficient particle representation that allows for the accurate and robust simulation of small-scale detail. We compute transport of turbulent energy using a complete two-equation k–ε model with accurate production terms that allows us to capture anisotropic tur-bulence effects, which integrate smoothly into the base flow. We only require a very low grid resolution to resolve the underlying base flow. As we offload complexity from the fluid solver to the particle system, we can control the detail of the simulation easily by adjusting the number of particles, without changing the large scale behavior. In addition, no computations are wasted on areas that are not visible. We demonstrate that due to the design of our algorithm it is highly suitable for massively parallel architectures, and is able to generate detailed turbulent simulations with millions of particles at high framerates.

Figure 1: An efficient subspace re-simulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster. We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing high-resolution fluid simulation. We show how to analyze the results of an existing high-resolution simulation, discover an efficient reduced approximation, and use it to quickly “re-simulate ” novel variations of the original dynamics. Prior subspace methods have had diffi-culty re-simulating the original input dynamics because they lack efficient means of handling semi-Lagrangian advection methods. We show that multi-dimensional cubature schemes can be applied to this and other advection methods, such as MacCormack advec-tion. The remaining pressure and diffusion stages can be written as a single matrix-vector multiply, so as with previous subspace methods, no matrix inversion is needed at runtime. We additionally propose a novel importance sampling-based fitting algorithm that asymptotically accelerates the precomputation stage, and show that the Iterated Orthogonal Projection method can be used to elegantly incorporate moving internal boundaries into a subspace simulation. In addition to efficiently producing variations of the original input, our method can produce novel, abstract fluid motions that we have not seen from any other solver.

We describe an interactive system featuring fluid-driven animation that responds to moving objects. Our system includes a GPUaccelerated Eulerian fluid solver that is suited for real-time use because it is unconditionally stable, takes constant calculation time per frame, and provides good visual fidelity. We dynamically translate the fluid simulation domain to track a user-controlled object. The fluid motion is visualized via its effects on particles which respond to the calculated fluid velocity field, but which are not constrained to stay within the bounds of the simulation domain. As particles leave the simulation domain, they seamlessly transition to purely particle-based motion, obscuring the point at which the fluid simulation ends. We additionally describe a hardware-accelerated volume rendering system that treats the particles as participating media and can render effects such as smoke, dust, or mist. Taken together, these components can be used to add fluid-driven effects to an interactive system without enforcing constraints on user motion, and without visual artifacts resulting from the finite extents of Eulerian fluid simulation methods.

Service-oriented simulation framework: An overview and unifying methodology

We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity and velocity using a basis of global functions defined over the entire simulation domain. We show that choosing Laplacian eigenfunctions for this basis provides benefits, including correspondence with spatial scales of vorticity and precise energy control at each scale. We perform Galerkin projection of the Navier-Stokes equations to derive a time evolution equation in the space of basis coefficients. Our method admits closed form solutions on simple domains but can also be implemented efficiently on arbitrary meshes.