Abstract We present the first closed‐form, per‐element eigensystems for 2D deformation energies in the primary domain of spatial coordinates. Previous analyses have only been able to find such eigensystems in proxy coordinates such as the deformation gradient, but as the spatial positions directly correspond to global degrees of freedom, our analysis opens up avenues for novel algorithms that were not possible before. First, we propose a spectral preconditioner that approximates global near‐nullspace modes using local analytical eigenvectors and clusters global eigenvalues using a low‐rank correction. Second, we derive a unified projected Newton formulation that combines the element eigensystem with the mass and damping terms, yielding a superior eigenvalue filter. We test our preconditioner on surface parameterization problems and our projected Newton framework on dynamic simulation examples. The results show significant efficiency gains over previous methods. In particular, our physics‐aware filtering strategy reduces eigenvalue clamping by up to 30×, which in turn decreases the Newton iteration count by roughly 2×.

Automated volumetric meshing of patient-specific heart geometry can help expedite various biomechanics studies, such as post-intervention stress estimation. Prior meshing techniques often neglect important modeling characteristics for successful downstream analyses, especially for thin structures like the valve leaflets. In this work, we present DeepCarve (Deep Cardiac Volumetric Mesh): a novel deformation-based deep learning method that automatically generates patient-specific volumetric meshes with high spatial accuracy and element quality. The main novelty in our method is the use of minimally sufficient surface mesh labels for precise spatial accuracy and the simultaneous optimization of isotropic and anisotropic deformation energies for volumetric mesh quality. Mesh generation takes only 0.13 seconds/scan during inference, and each mesh can be directly used for finite element analyses without any manual post-processing. Calcification meshes can also be subsequently incorporated for increased simulation accuracy. Numerous stent deployment simulations validate the viability of our approach for large-batch analyses. Our code is available at https://github.com/danpak94/Deep-Cardiac-Volumetric-Mesh.

Abstract We present an efficient new method for computing Mandelbrot‐like fractals (Julia sets) that approximate a user‐defined shape. Our algorithm is orders of magnitude faster than previous methods, as it entirely sidesteps the need for a time‐consuming numerical optimization. It is also more robust, succeeding on shapes where previous approaches failed. The key to our approach is a versor‐modulus analysis of fractals that allows us to formulate a novel shape modulus function that directly controls the broad shape of a Julia set, while keeping fine‐grained fractal details intact. Our formulation contains flexible artistic controls that allow users to seamlessly add fractal detail to desired spatial regions, while transitioning back to the original shape in others. No previous approach allows Mandelbrot‐like details to be “painted” onto meshes.

We present a geometric optimisation framework that can recover fold-over free maps from non-injective initial states using popular flip-preventing distortion energies. Since flip-preventing energies are infinite for folded configurations, we propose a new regularisation scheme that shifts the singular values of the deformation gradient. This allow us to re-use many existing algorithms, especially locally injective methods for initially folded maps. Our regularisation is suitable for both singular value- and invariant-based formulations, and systematically contributes multiple stabilisers to the Hessian. In contrast to proxy-based techniques, we maintain second-order convergence. Compact expressions for the energy eigensystems can be obtained for our extended stretch invariants, enabling the use of fast projected Newton solvers. Although spectral shifting in general has no theoretical guarantees that the global minimum is an injection, extensive experiments show that our framework is fast and extremely robust in practice, and capable of generating high-quality maps from severely distorted, degenerate and folded initialisations.

We advocate for the straightforward applications of the Cholesky and the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithms in the context of nonlinear time integration of deformable objects with dynamic collisions. At the beginning of each time step, we form and factor the Hessian matrix, accounting for all internal forces while omitting the implicit cross-coupling terms from the collision forces between multiple dynamic objects or self collisions. Then during the nonlinear solver iterations of the time step, we implicitly update this Hessian with L-BFGS. This approach is simple to implement and can be readily applied to any nonlinear time integration scheme, including higher-order schemes and quasistatics. We show that this approach works well in a wide range of settings involving complex nonlinear materials, including heterogeneity and anisotropy, as well as collisions, including frictional contact and self collisions.

We introduce a fast, expressive method for simulating fluids over radial domains, including discs, spheres, cylinders, ellipses, spheroids, and tori. We do this by generalizing the spectral approach of Laplacian Eigenfunctions, resulting in what we call spiral-spectral fluid simulations. Starting with a set of divergence-free analytical bases for polar and spherical coordinates, we show that their singularities can be removed by introducing a set of carefully selected enrichment functions. Orthogonality is established at minimal cost, viscosity is supported analytically, and we specifically design basis functions that support scalable FFT-based reconstructions. Additionally, we present an efficient way of computing all the necessary advection tensors. Our approach applies to both three-dimensional flows as well as their surface-based, codimensional variants. We establish the completeness of our basis representation, and compare against a variety of existing solvers.

High-resolution fluid simulations are computationally expensive, so many post-processing methods have been proposed to add turbulent details to low-resolution flows. Guiding methods are one promising approach for adding naturalistic, detailed motions as a post-process, but can be inefficient. Thus, we propose a novel, efficient method that formulates fluid guidance as a minimization problem in stream function space. Input flows are first converted into stream functions, and a high resolution flow is then computed via optimization. The resulting problem sizes are much smaller than previous approaches, resulting in faster computation times. Additionally, our method does not require an expensive pressure projection, but still preserves mass. The method is both easy to implement and easy to control, as the user can control the degree of guiding with a single, intuitive parameter. We demonstrate the effectiveness of our method across various examples.

We present a new approach that allows large time steps in dynamic simulations. Our approach, ConJac, is based on condensation, a technique for eliminating many degrees of freedom (DOFs) by expressing them in terms of the remaining degrees of freedom. In this work, we choose a subset of nodes to be dynamic nodes, and apply condensation at the velocity level by defining a linear mapping from the velocities of these chosen dynamic DOFs to the velocities of the remaining quasistatic DOFs. We then use this mapping to derive reduced equations of motion involving only the dynamic DOFs. We also derive a novel stabilization term that enables us to use complex nonlinear material models. ConJac remains stable at large time steps, exhibits highly dynamic motion, and displays minimal numerical damping. In marked contrast to subspace approaches, ConJac gives exactly the same configuration as the full space approach once the static state is reached. ConJac works with a wide range of moderate to stiff materials, supports anisotropy and heterogeneity, handles topology changes, and can be combined with existing solvers including rigid body dynamics.

Abstract Stochastic structural analysis can assess whether a fabricated object will break under real‐world conditions. While this approach is powerful, it is also quite slow, which has previously limited its use to coarse resolutions (e.g., 26 × 34 × 28). We show that this approach can be made asymptotically faster, which in practice reduces computation time by two orders of magnitude, and allows the use of previously‐infeasible resolutions. We achieve this by showing that the probability gradient can be computed in linear time instead of quadratic, and by using a robust new scheme that stabilizes the inertia gradients used by the optimization. Additionally, we propose a constrained restart method that deals with local minima, and a sheathing approach that further reduces the weight of the shape. Together, these components enable the discovery of previously‐inaccessible designs.

Abstract The Baraff‐Witkin [BW98] model has been a popular formulation for cloth for 20 years. However, its relationship to the finite element method (FEM) has always been unclear, because the model resists being written as an isotropic, hyperelastic strain energy. In this paper, we show that this is because the Baraff‐Witkin model is actually a coupled anisotropic strain energy. We show that its stretching term approximates the isotropic As‐Rigid‐As‐Possible (ARAP) energy, and its shearing term is a cross‐fiber coupling energy common in biomechanics. While it has been known empirically for some time that the model can produce indefinite force Jacobians, the conditions under which they occur has never been clear. Our formulation enables a complete eigenanalysis that precisely characterizes exactly when indefiniteness occurs, and leads to fast, analytic, semi‐positive‐definite projection methods. Finally, our analysis suggests a generalized Baraff‐Witkin energy with non‐orthogonal warp and weft directions.

Abstract This paper presents a novel generative model to synthesize fluid simulations from a set of reduced parameters. A convolutional neural network is trained on a collection of discrete, parameterizable fluid simulation velocity fields. Due to the capability of deep learning architectures to learn representative features of the data, our generative model is able to accurately approximate the training data set, while providing plausible interpolated in‐betweens. The proposed generative model is optimized for fluids by a novel loss function that guarantees divergence‐free velocity fields at all times. In addition, we demonstrate that we can handle complex parameterizations in reduced spaces, and advance simulations in time by integrating in the latent space with a second network. Our method models a wide variety of fluid behaviors, thus enabling applications such as fast construction of simulations, interpolation of fluids with different parameters, time re‐sampling, latent space simulations, and compression of fluid simulation data. Reconstructed velocity fields are generated up to 700× faster than re‐simulating the data with the underlying CPU solver, while achieving compression rates of up to 1300×.

We present an analysis of anisotropic hyperelasticity, specifically transverse isotropy, that obtains closed-form expressions for the eigendecompositions of many common energies. We then use these to build fast and concise Newton implementations. We leverage our analysis in two separate applications. First, we show that existing anisotropic energies are not inversion-safe, and contain spurious stable states under large deformation. We then propose a new anisotropic strain invariant that enables the formulation of a novel, robust, and inversion-safe energy. The new energy fits completely within our analysis, so closed-form expressions are obtained for its eigensystem as well. Secondly, we use our analysis to rehabilitate badly-conditioned finite elements. Using this method, we can robustly simulate large deformations even when a mesh contains degenerate, zero-volume elements. We accomplish this by swapping the badly-behaved isotropic direction with a well-behaved anisotropic term. We validate our approach on a variety of examples.

Many strategies exist for optimizing non-linear distortion energies in geometry and physics applications, but devising an approach that achieves the convergence promised by Newton-type methods remains challenging. In order to guarantee the positive semi-definiteness required by these methods, a numerical eigendecomposition or approximate regularization is usually needed. In this article, we present analytic expressions for the eigensystems at each quadrature point of a wide range of isotropic distortion energies. These systems can then be used to project energy Hessians to positive semi-definiteness analytically. Unlike previous attempts, our formulation provides compact expressions that are valid both in 2D and 3D, and does not introduce spurious degeneracies. At its core, our approach utilizes the invariants of the stretch tensor that arises from the polar decomposition of the deformation gradient. We provide closed-form expressions for the eigensystems for all these invariants, and use them to systematically derive the eigensystems of any isotropic energy. Our results are suitable for geometry optimization over flat surfaces or volumes, and agnostic to both the choice of discretization and basis function. To demonstrate the efficiency of our approach, we include comparisons against existing methods on common graphics tasks such as surface parameterization and volume deformation.

The Laplacian Eigenfunction method for fluid simulation, which we refer to as Eigenfluids , introduced an elegant new way to capture intricate fluid flows with near-zero viscosity. However, the approach does not scale well, as the memory cost grows prohibitively with the number of eigenfunctions. The method also lacks generality, because the dynamics are constrained to a closed box with Dirichlet boundaries, while open, Neumann boundaries are also needed in most practical scenarios. To address these limitations, we present a set of analytic eigenfunctions that supports uniform Neumann and Dirichlet conditions along each domain boundary, and show that by carefully applying the discrete sine and cosine transforms, the storage costs of the eigenfunctions can be made completely negligible. The resulting algorithm is both faster and more memory-efficient than previous approaches, and able to achieve lower viscosities than similar pseudo-spectral methods. We are able to surpass the scalability of the original Laplacian Eigenfunction approach by over two orders of magnitude when simulating rectangular domains. Finally, we show that the formulation allows forward scattering to be directed in a way that is not possible with any other method.

Generating realistic fluid simulations remains computationally expensive, and animators can expend enormous effort trying to achieve a desired motion. To reduce such costs, several methods have been developed in which high-resolution turbulence is synthesized as a post process. Since global motion can then be obtained using a fast, low-resolution simulation, less effort is needed to create a realistic animation with the desired behavior. While much research has focused on accelerating the low-resolution simulation, the problem controlling the behavior of the turbulent, high-resolution motion has received little attention. In this paper, we show that style transfer methods from image editing can be adapted to transfer the turbulent style of an existing fluid simulation onto a new one. We do this by extending example-based image synthesis methods to handle velocity fields using a combination of patch-based and optimization-based texture synthesis. This approach allows us to take into account the incompressibility condition, which we have found to be a important factor during synthesis. Using our method, a user can easily and intuitively create high-resolution fluid animations that have a desired turbulent motion.

Nonlinear hyperelastic energies play a key role in capturing the fleshy appearance of virtual characters. Real-world, volume-preserving biological tissues have Poisson’s ratios near 1/2, but numerical simulation within this regime is notoriously challenging. In order to robustly capture these visual characteristics, we present a novel version of Neo-Hookean elasticity. Our model maintains the fleshy appearance of the Neo-Hookean model, exhibits superior volume preservation, and is robust to extreme kinematic rotations and inversions. We obtain closed-form expressions for the eigenvalues and eigenvectors of all of the system’s components, which allows us to directly project the Hessian to semipositive definiteness, and also leads to insights into the numerical behavior of the material. These findings also inform the design of more sophisticated hyperelastic models, which we explore by applying our analysis to Fung and Arruda-Boyce elasticity. We provide extensive comparisons against existing material models.

Abstract We present an interactive technique for generating realistic lightning. Our method captures the main characteristics of the dielectric breakdown model, a physical model for lightning formation. Our algorithm uses a distance‐based approximation to quickly compute the electric potentials of different charge types. In particular, we use a rational function in lieu of summed potentials to better produce interesting lightning patterns. We also propose to use the waypoints commonly available in many game scenes to guide lightning shapes in complex scenes. We found that our algorithm is 2 times faster than the state‐of‐the art method with better controls on lighting shapes, and can generate realistic lightning shapes interactively.

We propose a method to simulate the rich, scale-dependent dynamics of water waves. Our method preserves the dispersion properties of real waves, yet it supports interactions with obstacles and is computationally efficient. Fundamentally, it computes wave accelerations by way of applying a dispersion kernel as a spatially variant filter, which we are able to compute efficiently using two core technical contributions. First, we design novel, accurate, and compact pyramid kernels which compensate for low-frequency truncation errors. Second, we design a shadowed convolution operation that efficiently accounts for obstacle interactions by modulating the application of the dispersion kernel. We demonstrate a wide range of behaviors, which include capillary waves, gravity waves, and interactions with static and dynamic obstacles, all from within a single simulation.

We present a new method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. Using the recently developed Eulerian Solids approach [Levin et al. 2011], we are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method. Eulerian Solids have previously been integrated using explicit schemes, but we develop an implicit scheme that allows large time steps to be taken. The in-compressibility condition is satisfied in both the solid and the fluid, which has the added benefit of simplifying collision handling.

Abstract We present the first 3D algorithm capable of answering the question: what would a Mandelbrot‐like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi‐resolution strategies. The resulting shapes are not invariant under common operations such as translation, and instead undergo intricate, non‐linear transformations.

We present a method to increase the apparent resolution of particle-based liquid simulations. Our method first outputs a dense, temporally coherent, regularized point set from a coarse particle-based liquid simulation. We then apply a surface-only Lagrangian wave simulation to this high-resolution point set. We develop novel methods for seeding and simulating waves over surface points, and use them to generate high-resolution details. We avoid error-prone surface mesh processing, and robustly propagate waves without the need for explicit connectivity information. Our seeding strategy combines a robust curvature evaluation with multiple bands of seeding oscillators, injects waves with arbitrarily fine-scale structures, and properly handles obstacle boundaries. We generate detailed fluid surfaces from coarse simulations as an independent post-process that can be applied to most particle-based fluid solvers.

Subspace deformable body simulations can be very fast, but can behave unrealistically when behaviors outside the prescribed subspace such as novel external collisions, are encountered. We address this limitation by presenting a fast, flexible new method that allows full space computation to be activated in the neighborhood of novel events while the rest of the body still computes in a subspace. We achieve this using a method we call subspace condensation , a variant on the classic static condensation precomputation. However, instead of a precomputation, we use the speed of subspace methods to perform the condensation at every frame. This approach allows the full space regions to be specified arbitrarily at runtime, and forms a natural two-way coupling with the subspace regions. While condensation is usually only applicable to linear materials, the speed of our technique enables its application to non-linear materials as well. We show the effectiveness of our approach by applying it to a variety of articulated character scenarios.

We present an efficient new subspace method for simulating the self-contact of articulated deformable bodies, such as characters. Self-contact is highly structured in this setting, as the limited space of possible articulations produces a predictable set of coherent collisions. Subspace methods can leverage this coherence, and have been used in the past to accelerate the collision detection stage of contact simulation. We show that these methods can be used to accelerate the entire contact computation, and allow self-contact to be resolved without looking at all of the contact points . Our analysis of the problem yields a broader insight into the types of non-linearities that subspace methods can efficiently approximate, and leads us to design a pose-space cubature scheme. Our algorithm accelerates self-contact by up to an order of magnitude over other subspace simulations, and accelerates the overall simulation by two orders of magnitude over full-rank simulations. We demonstrate the simulation of high resolution (100K -- 400K elements) meshes in self-contact at interactive rates (5.8 -- 50 FPS).

The papers in this special section present expanded versions of three of the best papers from the 12th Annual ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), which was held in Anaheim, CA from July 19 to 21, 2013. Now in its dozenth year, SCA has become firmly established as the premiere forum for innovations in the software and technology of computer animation.

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 difficulty 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 advection. 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 propose a method of increasing the apparent spatial resolution of an existing liquid simulation. Previous approaches to this “up-resing” problem have focused on increasing the turbulence of the underlying velocity field. Motivated by measurements in the free surface turbulence literature, we observe that past certain frequencies, it is sufficient to perform a wave simulation directly on the liquid surface, and construct a reduced-dimensional surface-only simulation. We sidestep the considerable problem of generating a surface parameterization by employing an embedding technique known as the Closest Point Method (CPM) that operates directly on a 3D extension field. The CPM requires 3D operators, and we show that for surface operators with no natural 3D generalization, it is possible to construct a viable operator using the inverse Abel transform . We additionally propose a fast, frozen core closest point transform, and an advection method for the extension field that reduces smearing considerably. Finally, we propose two turbulence coupling methods that seed the high-resolution wave simulation in visually expected regions.

The nonlinear and nonstationary nature of Navier-Stokes equations produces fluid flows that can be noticeably different in appearance with subtle changes. In this paper, we introduce a method that can analyze the intrinsic multiscale features of flow fields from a decomposition point of view, by using the Hilbert-Huang transform method on 3D fluid simulation. We show how this method can provide insights to flow styles and help modulate the fluid simulation with its internal physical information. We provide easy-to-implement algorithms that can be integrated with standard grid-based fluid simulation methods and demonstrate how this approach can modulate the flow field and guide the simulation with different flow styles. The modulation is straightforward and relates directly to the flow's visual effect, with moderate computational overhead.

In this extended version of our Symposium on Computer Animation paper, we describe a domain-decomposition method to simulate articulated deformable characters entirely within a subspace framework. We have added a parallelization and eigendecomposition performance analysis, and several additional examples to the original symposium version. The method supports quasistatic and dynamic deformations, nonlinear kinematics and materials, and can achieve interactive time-stepping rates. To avoid artificial rigidity, or “locking,” associated with coupling low-rank domain models together with hard constraints, we employ penaltybased coupling forces. The multidomain subspace integrator can simulate deformations efficiently, and exploits efficient subspace-only evaluation of constraint forces between rotated domains using a novel Fast Sandwich Transform (FST). Examples are presented for articulated characters with quasistatic and dynamic deformations, and interactive performance with hundreds of fully coupled modes. Using our method, we have observed speedups of between 3 and 4 orders of magnitude over full-rank, unreduced simulations.

Finite element simulations of nonlinear deformable models are computationally costly, routinely taking hours or days to compute the motion of detailed meshes. Dimensional model reduction can make simulations orders of magnitude faster, but is unsuitable for general deformable body simulations because it requires expensive precomputations, and it can suppress motion that lies outside the span of a pre-specified low-rank basis. We present an online model reduction method that does not have these limitations. In lieu of precomputation, we analyze the motion of the full model as the simulation progresses, incrementally building a reduced-order nonlinear model, and detecting when our reduced model is capable of performing the next timestep. For these subspace steps, full-model computation is "skipped" and replaced with a very fast (on the order of milliseconds) reduced order step. We present algorithms for both dynamic and quasistatic simulations, and a "throttle" parameter that allows a user to trade off between faster, approximate previews and slower, more conservative results. For detailed meshes undergoing low-rank motion, we have observed speedups of over an order of magnitude with our method.

We perform a detailed flop and bandwidth analysis of Jos Stam's Stable Fluids algorithm on the CPU, GPU, and Cell. In all three cases, we find that the algorithm is bandwidth bound, with the cores sitting idle up to 96% of the time. Knowing this, we propose two modifications to accelerate the algorithm. First, a Mehrstellen discretization for the pressure solver which reduces the running time of the solver by a third. Second, a static caching scheme that eliminates roughly 99% of the random lookups in the advection stage. We observe a 2x speedup in the advection stage using this scheme. Both modifications apply equally well to all three architectures.

We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an r-dimensional deformation subspace with O(r) cubature points, our method is able to evaluate subspace forces at O(r(2)) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration. CR CATEGORIES: I.6.8 [Simulation and Modeling]: Types of Simulation-Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Physically based modeling G.1.4 [Mathematics of Computing]: Numerical Analysis-Quadrature and Numerical Differentiation.

We present a novel wavelet method for the simulation of fluids at high spatial resolution. The algorithm enables large- and small-scale detail to be edited separately, allowing high-resolution detail to be added as a post-processing step. Instead of solving the Navier-Stokes equations over a highly refined mesh, we use the wavelet decomposition of a low-resolution simulation to determine the location and energy characteristics of missing high-frequency components. We then synthesize these missing components using a novel incompressible turbulence function, and provide a method to maintain the temporal coherence of the resulting structures. There is no linear system to solve, so the method parallelizes trivially and requires only a few auxiliary arrays. The method guarantees that the new frequencies will not interfere with existing frequencies, allowing animators to set up a low resolution simulation quickly and later add details without changing the overall fluid motion.

Abstract Turing first theorized that many biological patterns arise through the processes of reaction and diffusion. Subsequently, reaction‐diffusion systems have been studied in many fields, including computer graphics. We first show that for visual simulation purposes, reaction‐diffusion equations can be made unconditionally stable using a variety of straightforward methods. Second, we propose an anisotropy embedding that significantly expands the space of possible patterns that can be generated. Third, we show that by adding an advection term, the simulation can be coupled to a fluid simulation to produce visually appealing flows. Fourth, we couple fast marching methods to our anisotropy embedding to create a painting interface to the simulation. Unconditional stability is maintained throughout, and our system runs at interactive rates. Finally, we show that on the Cell processor, it is possible to implement reaction‐diffusion on top of an existing fluid solver with no significant performance impact. Copyright © 2007 John Wiley & Sons, Ltd.

We present a fast method for simulating, animating, and rendering lightning using adaptive grids. The "dielectric breakdown model" is an elegant algorithm for electrical pattern formation that we extend to enable animation of lightning. The simulation can be slow, particularly in 3D, because it involves solving a large Poisson problem. Losasso et al. recently proposed an octree data structure for simulating water and smoke, and we show that this discretization can be applied to the problem of lightning simulation as well. However, implementing the incomplete Cholesky conjugate gradient (ICCG) solver for this problem can be daunting, so we provide an extensive discussion of implementation issues. ICCG solvers can usually be accelerated using "Eisenstat's trick," but the trick cannot be directly applied to the adaptive case. Fortunately, we show that an "almost incomplete Cholesky" factorization can be computed so that Eisenstat's trick can still be used. We then present a fast rendering method based on convolution that is competitive with Monte Carlo ray tracing but orders of magnitude faster, and we also show how to further improve the visual results using jittering.

We present a novel technique for synthesizing textures over dynamically changing fluid surfaces. We use both image textures as well as bump maps as example inputs. Image textures can enhance the rendering of the fluid by either imparting realistic appearance to it or by stylizing it, whereas bump maps enable the generation of complex micro-structures on the surface of the fluid that may be very difficult to synthesize using simulation. To generate temporally coherent textures over a fluid sequence, we transport texture information, i.e. color and local orientation, between free surfaces of the fluid from one time step to the next. This is accomplished by extending the texture information from the first fluid surface to the 3D fluid domain, advecting this information within the fluid domain along the fluid velocity field for one time step, and interpolating it back onto the second surface -- this operation, in part, uses a novel vector advection technique for transporting orientation vectors. We then refine the transported texture by performing texture synthesis over the second surface using our "surface texture optimization" algorithm, which keeps the synthesized texture visually similar to the input texture and temporally coherent with the transported one. We demonstrate our novel algorithm for texture synthesis on dynamically evolving fluid surfaces in several challenging scenarios.

We present a physically-based method for animating and rendering lightning and other electric arcs. For the simulation, we present the dielectric breakdown model, an elegant formulation of electrical pattern formation. We then extend the model to animate a sustained, 'dancing' electrical arc, by using a simplified Helmholtz equation for propagating electromagnetic waves. For rendering, we use a convolution kernel to produce results competitive with Monte Carlo ray tracing. Lastly, we present user parameters for manipulation of the simulation patterns.