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We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a non-symmetric linear system which is more computationally challenging to invert. The semi-Lagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.

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... Curvature plays a crucial role in free boundary problems (FBP) [2] for its relation to surface tension in physics [3,4] and its regularization property in optimization. Such FBPs are ubiquitous in the physical sciences and engineering, with a broad range of applications in multiphase flows [5][6][7][8][9][10][11][12], solidification processes [13][14][15][16][17], and biological morphogenesis [18][19][20]. Accurate curvature computations are thus critical to constructing sound FBP models. ...

... where A and ω are the desired amplitude and frequency. Equation (9) gives rise to the level-set function ...

... 8 Element-wise division. 9 Standardization or z-scoring transforms a vector ψ into ψ , which has mean 0 and variance 1. For us, PCA 10 entails a change of basis, where the new axes are better data descriptors defined by the first m ι singular vectors of the training subset's correlation matrix 11 . ...

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver (Larios-C\'{a}rdenas and Gibou (2021)[1]) that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. As in [1], the core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.

... One of the most common solutions to reduce computational cost while preserving small-scale details in high-resolution simulation is using adaptive grids instead of uniform grids. Losasso et al. 13 first introduced octree structure into grid-based simulation with symmetric pressure projection system to capture finer details of fluids. And they further improved the pressure projection to second-order accuracy. ...

... Octree-based adaptive methods have been widely used in fluid simulation. 13,16 Draw lessons from Reference 15, we propose a grid pyramid of pointer arrays instead of tree-based octree structure to increase the data access performance of The sparse and adaptive grid representation (with dotted cells deleted). NB-FLIP, narrow band fluid implicit particle method NB-FLIP. ...

In this article, we propose a novel hybrid framework by combining smoothed particle hydrodynamics and adaptive narrow band fluid implicit particle method (NB‐FLIP) to faithfully model the multiphysical processes involving heat transfer and phase transition, and to precisely simulate the dynamics of condensed droplets moving along intricate objects. We first formulate a governing physical model built upon an improved phase transition model and an augmented on‐surface drop analysis method to achieve realistic condensation effects over intricate hydrophilic/hydrophobic interface. To achieve both high‐fidelity interactions and high‐resolution visual effects, we further develop an adaptive NB‐FLIP solver with octree‐dictated background grid in order to further enhance the performance of our framework. Experimental results have shown that our approach can be used to efficiently and realistically simulate the small‐scale interaction details between condensed drops and complex objects with arbitrary geometry.

... Popinet [36] introduced a second-order projection method on a graded octree grid, that is, a grid where the size ratio of adjacent cells must be 2:1. This work was later extended by Losasso et al. [28] to relax the restrictions on adaptive refinement. Although Popinet [36] discretized the advection term in the Eulerian approach using an upwind method, Losasso et al. [28] used a semi-Lagrangian method by extrapolating the velocities located on the cell faces to the cell vertices. ...

... This work was later extended by Losasso et al. [28] to relax the restrictions on adaptive refinement. Although Popinet [36] discretized the advection term in the Eulerian approach using an upwind method, Losasso et al. [28] used a semi-Lagrangian method by extrapolating the velocities located on the cell faces to the cell vertices. For cases where the viscosity terms are not neglected, Olshanskii et al. [33] and Guittet et al. [19] developed second-order projection methods on a graded and non-graded octree MAC grid, respectively. ...

We present a second-order monolithic method for solving incompressible Navier--Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and pressure variables are located at cell centers. Compact finite difference methods with ghost values are used to discretize the advection and diffusion terms of the velocity. A pressure gradient and divergence operator on the quadtree that use compact stencils are developed. Furthermore, the proposed method is extended to cubical domains with octree grids. Numerical results demonstrate that the method is second-order convergent in $L^\infty$ norms and can handle irregular domains for various Reynolds numbers.

... Using an octree to organize the point cloud is an ideal scheme, because the octree enables very efficient point cloud searching. An octree is a spatial, segmented, hierarchical data structure whose nodes are recursively decomposed into eight sub-nodes until the minimum resolution is reached [31,32], as illustrated in Figure 1. point clouds in the original data. To address these deficiencies, this study constructed the VO-LVV model, an LVV model based on point cloud data, which is suitable for urban scenes. ...

... Using an octree to organize the point cloud is an ideal scheme, because the octree enables very efficient point cloud searching. An octree is a spatial, segmented, hierarchical data structure whose nodes are recursively decomposed into eight sub-nodes until the minimum resolution is reached [31,32], as illustrated in As shown in Figure 1, for vegetation point cloud data, first, a large bounding box surrounding the entire point cloud is generated according to the distribution range of the dataset in the three coordinate directions of X, Y, and Z, and the root node of level 0 of the octree is obtained. Second, the point cloud density in each non-empty node is Remote Sens. 2022, 14, 855 4 of 14 judged according to the set target density threshold. ...

Currently, three-dimensional (3D) point clouds are widely used in the field of remote sensing and mapping, including the measurement of living vegetation volume (LVV) in cities. However, the existing quantitative methods for LVV measurement are mainly based on single tree modeling or on the calculation of single tree species’ growth parameters, which cannot be applied to the many tree species and complex forest layer structures present in urban regions, and thus are unsuitable for broad application. LVV measurement is based primarily on vegetation point cloud data, which can be obtained through many methods and often lack some information, thus posing problems in the use of traditional LVV measurement methods. To address the above problems, this paper proposes a novel LVV estimation model, which combines the voxel measurement method with an organizing point cloud based on an octree structure (we called it VO-LVV), to estimate the LVV of typical vegetation and landforms in cities. The point cloud data of single plants and multiple plants were obtained through preprocessing to verify the improvement in the calculation efficiency and accuracy of the proposed method. The results indicated that the VO-LVV estimation method, compared with the traditional method, enabled substantial efficiency improvement and higher calculation accuracy. Furthermore, the new method can be simultaneously applied to scenarios of single plants and multiple plants, and can be used for the calculation of LVV in areas with various vegetation types in cities.

... Often, "space domain decomposition methods," e.g. Adaptive Mesh Refinement and parallel processing [34,44,45,92] are used in order to resolve microstructure, but "space domain decomposition methods" require more grid points and the number of additional grids points needing to be added is dependent on the size of the microstructure that needs to be resolved. In an extreme scenario, if the ratio of the microstructure length scale to the background mesh length scale is sufficiently small, the performance of the prevailing "space decomposition methods" will exhibit "diminishing returns," i.e. poor scalability. ...

... (44) Figure 25 shows the shape of the initial interface. We assume the region inside the seed is D. In order to verify the "body-fitted property" of our numerical method, we introduced a rotation angle α (45) in order to test for grid orientation effects. The parametric definition of the initial seed, as a function of α, is given below (45): ...

A novel supermesh method that was presented for computing solutions to the multimaterial heat equation in complex stationary geometries with microstructure, is now applied for computing solutions to the Stefan problem involving complex deforming geometries with microstructure. The supermesh is established by combining the underlying (fixed) structured rectangular grid with the (deforming) piecewise linear interfaces reconstructed by the multi-material moment-of-fluid method. The temperature diffusion equation with Dirichlet boundary conditions at interfaces is solved by the linear exact multi-material finite volume method upon the supermesh. The interface propagation equation is discretized using the unsplit cell-integrated semi-Lagrangian method. The level set method is also coupled during this process in order to assist in the initialization of the (transient) provisional velocity field. The resulting method is validated on both canonical and challenging benchmark tests. Algorithm convergence results based on grid refinement are reported. It is found that the new method approximates solutions to the Stefan problem efficiently, compared to traditional approaches, due to the localized finite volume approximation stencil derived from the underlying supermesh. The new deforming boundary supermesh approach enables one to compute many kinds of complex deforming boundary problems with the efficiency properties of a body fitted mesh and the robustness of a “cut-cell” (a.k.a. “embedded boundary” or “immersed”) method.

... Tree-based grid generation (quadtrees in 2D and octrees in 3D) is common in computational sciences [6,11,17,24,28,30,48,56,58] largely due to its simplicity and parallel scalability. The ability to efficiently refine (and coarsen) regions of interest using tree-based data structures have made it possible to deploy them on large-scale multi-physics simulations [2,3,14,24,32,33,37,37,49,53]. Existing algorithms for tree-based grid generation are mainly focused on axis-aligned hierarchical splitting on isotropic domains (i.e., spheres, squares, and cubes). ...

... Tree-based grid generation (quadtrees in 2D and octrees in 3D) is common in computational sciences [6,11,17,24,28,30,48,56,58] largely due to its simplicity and parallel scalability. The ability to efficiently refine (and coarsen) regions of interest using tree-based data structures have made it possible to deploy them on large-scale multi-physics simulations [2,3,14,24,32,33,37,37,49,53]. Existing algorithms for tree-based grid generation are mainly focused on axis-aligned hierarchical splitting on isotropic domains (i.e., spheres, squares, and cubes). ...

Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a `good' adaptively-refined mesh. In this work, we present an efficient novel octree-based adaptive discretization approach capable of carving out arbitrarily shaped void regions from the parent domain: an essential requirement for fluid simulations around complex objects. Carving out objects produces an $\textit{incomplete}$ octree. We develop efficient top-down and bottom-up traversal methods to perform finite element computations on $\textit{incomplete}$ octrees. We validate the framework by (a) showing appropriate convergence analysis and (b) computing the drag coefficient for flow past a sphere for a wide range of Reynolds numbers ($\mathcal{O}(1-10^6)$) encompassing the drag crisis regime. Finally, we deploy the framework on a realistic geometry on a current project to evaluate COVID-19 transmission risk in classrooms.

... Visual simulation of smoke is notoriously difficult due to its highly turbulent nature, resulting in vortices spanning a vast range of space and time scales. As a consequence, simulating the dynamic behavior of smoke realistically requires not only sophisticated nondissipative numerical solvers [Li et al. 2020;Mullen et al. 2009;Qu et al. 2019;Zhang et al. 2015], but also a spatial discretization with sufficiently high resolution to capture fine-scale structures, either uniformly [Kim et al. 2008b;Zehnder et al. 2018] or adaptively [Losasso et al. 2004; Weißmann and Pinkall 2010a; Zhang et al. 2016]. This inevitably makes such direct numerical simulations computationally intensive. ...

... However, creating complex smoke animations requires relatively high resolutions to capture fine details. Unstructured grids [Ando et al. 2013;de Goes et al. 2015;Klingner et al. 2006;Mullen et al. 2009] and adaptive methods, where higher resolutions are used in regions of interest and/or with more fluctuations [Losasso et al. 2004;Setaluri et al. 2014;Zhu et al. 2013] have been proposed to offer increased efficiency -but the presence of smoke turbulence in the entire domain often prevents computational savings in practice. On the other hand, particle methods, e.g, smoothed particle hydrodynamics [Akinci et al. 2012;Becker and Teschner 2007;Desbrun and Gascuel 1996;Ihmsen et al. 2014;Peer et al. 2015;Solenthaler and Pajarola 2009;Winchenbach et al. 2017] and power particles [de Goes et al. 2015] can easily handle adaptive simulations. ...

Simulating turbulent smoke flows with fine details is computationally intensive. For iterative editing or simply faster generation, efficiently upsampling a low-resolution numerical simulation is an attractive alternative. We propose a novel learning approach to the dynamic upsampling of smoke flows based on a training set of flows at coarse and fine resolutions. Our multiscale neural network turns an input coarse animation into a sparse linear combination of small velocity patches present in a precomputed over-complete dictionary. These sparse coefficients are then used to generate a high-resolution smoke animation sequence by blending the fine counterparts of the coarse patches. Our network is initially trained from a sequence of example simulations to both construct the dictionary of corresponding coarse and fine patches and allow for the fast evaluation of a sparse patch encoding of any coarse input. The resulting network provides an accurate upsampling when the coarse input simulation is well approximated by patches present in the training set (e.g., for re-simulation), or simply visually plausible upsampling when input and training sets differ significantly. We show a variety of examples to ascertain the strengths and limitations of our approach and offer comparisons to existing approaches to demonstrate its quality and effectiveness.

... For such flows, a fixed/static grid strategy and/or a constant time-stepping technique are not adequate to capture the relevant length and time scales (and thus the correct dynamics) of the flow. To overcome these difficulties, solvers with adaptive mesh refinement (AMR) have emerged and have been successfully employed in different fields: electrohydrodynamics (López-Herrera et al., 2011), study of atmospheric boundary layers (van Hooft et al., 2018), multiphase flows (Fuster et al., 2009;Losasso et al., 2004), flows in complex geometries (Popinet, 2003;Gibou et al., 2002), turbulence modeling (Schneider and Vasilyev, 2010). In general, AMR solvers aim to distribute available computational resources efficiently over a domain by dynamically refining and coarsening the computational grid in space and time. ...

In this article, we extend our Distributed Lagrange Multiplier/Fictitious Domain method previously implemented on simple regular Cartesian grids to quadtree/octree adaptive grids. The objective is to improve both the accuracy and efficiency of our DLM/FD particle-resolved simulation method by extending its computing capabitilies through dynamic local mesh refinement. The main features of our numerical method, such as a first-order operator splitting time algorithm and a second-order reconstruction of the velocity field close to the boundary of the immersed rigid bodies (of arbitrary shape), are unchanged. We implemented our adaptive DLM/FD algorithm within Basilisk, a parallel platform to solve partial differential equations on dynamic quadtree/octree grids. The quadtree/octree structure of the grid and specific design rules of Basilisk impose a special treatment of some of the operations performed on the grid in the DLM/FD-Uzawa algorithm. The new computational method is then tested and validated on a set of flow configurations including the challenging problem of accurately computing lubrication interaction forces without resorting to using any ad hoc correction. Finally, we illustrate the potential of our code to compute complex particle-laden flow configurations that were not attainable in the past with a DLM/FD algorithm implemented on a simple regular Cartesian grid.

... Fluid simulation in large domains such as oceans could be troublesome with uniform volumetric solvers due to the high computational costs and limited memory capacities. Adaptive methods such as octrees [Aanjaneya et al. 2017;Ando and Batty 2020;Hong 2009;Losasso et al. 2004] are usually dissipative at coarse regions. It is not sure if the water waves could be preserved away from the fine region. ...

The simulation of large open water surface is challenging for a uniform volumetric discretization of the Navier-Stokes equation. The water splashes near moving objects, which height field methods for water waves cannot capture, necessitates high resolution simulation such as the Fluid-Implicit-Particle (FLIP) method. On the other hand, FLIP is not efficient for the long-lasting water waves that propagates to long distances, which requires sufficient depth for correct dispersion relationship. This paper presents a new method to tackle this dilemma through an efficient hybridization of volumetric and surface-based advection-projection discretizations. We design a hybrid time-stepping algorithm that combines a FLIP domain and an adaptively remeshed Boundary Element Method (BEM) domain for the incompressible Euler equations. The resulting framework captures the detailed water splashes near moving objects with FLIP, and produces convincing water waves with correct dispersion relationship at modest additional cost.

... The discretization of equation (13) satisfied by the Hodge variable Φ follows the approach introduced in [75], which is a second-order accurate extension of the discretization of [76]. ...

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the temperature, the flow velocity and pressure, and the interfacial velocity. The approach utilizes level-set methods for sharp and implicit interface tracking, hybrid finite-difference/finite-volume discretizations on adaptive quadtree grids, and a pressure-free projection method for the solution of the incompressible Navier-Stokes equations. The method is first verified with numerical convergence tests using a synthetic solution. Then, computational studies of ice formation on a cylinder in crossflow are performed and provide good quantitative agreement with existing experimental results, reproducing qualitative phenomena that have been observed in past experiments. Finally, we investigate the role of varying Reynolds and Stefan numbers on the emerging interface morphologies and provide new insights around the time evolution of local and average heat transfer at the interface.

... The diffusive effects are treated implicitly for stability, while the non-linear reactive terms are treated explicitly. All diffusive fluxes are approximated using the second-order antisymmetric discretization proposed by Lossaso et al. [33] and used in our previous studies [23,61,9,59,56]. ...

Motivated by experimental observations of asymmetric protein aggregates distributions in dividing yeast cells, we present a conservative finite volume approach for reaction-diffusion systems defined over deforming geometries. The key idea of our approach is to use spatio-temporal control volumes instead of integrating the time-discretized equations in space, as it is common practice. Both our theoretical and computational results demonstrate the convergence of our method and highlight how traditional approaches can lead to inaccurate solutions. We employ this novel approach to investigate the partitioning of protein aggregates in dividing yeast cells, leveraging the flexibility of the level set method to construct realistic biological geometries. Using a simple reaction-diffusion model, we find that spatial heterogeneity in yeast cells during division can alone create asymmetries in the concentration of protein aggregates. Moreover, we find that obstructing intracellular entities, such as nuclei or insoluble protein compartments, amplify these asymmetries, suggesting that they may play an essential role in regulating molecular partitioning. Beyond these findings, our results illustrate the flexibility of our approach and its potential to design realistic predictive tools to explore intracellular bio-mechanisms.

... In general, the grid-based spatial partitioning method, which is widely used in the Eulerian simulation technique, is mainly used to express smoke, water, and fire [19], [20]. Space can be optimized using data structures such as octree [21], [22], [25], but efficiency is not significantly improved because the amount of computation required for octree construction increases when the originally given grid resolution is high. A simple solution to this issue is to reduce the space to multiscale. ...

In this paper, we propose a highly efficient method for synthesizing high-resolution(HR) smoke simulations based on deep learning. A major issue for physics-based HR fluid simulations is that they require large amounts of physical memory and long execution times. In recent years, this issue has been addressed by developing deep-learning-based super-resolution(SR) methods that convert low-resolution(LR) fluid simulation results to HR(High-resolution) versions. However, these methods were not very efficient because they performed operations even in areas with low density or no density. In this paper, we propose a method that can maximize its efficiency by introducing a downscaled and binarized adaptive octree. However, even if it is divided by octree, because the number of nodes increases when the resolution of the simulation space is large, we reduce the size of the space by multiscaling and at the same time perform binarization to preserve the density that may be lost in this process. The octree calculated in this process has a structure similar to that of a multigrid solver, and the octree calculated at coarse resolution is restored to its original size and used for HR expression. Finally, we apply the SR process only to those areas having significant density values. Using the proposed method, the SR process is significantly faster and the memory efficiency is improved. The performance of our method is compared with that of an existing SR method to demonstrate its efficiency.

... Aside from rendering, hierarchical data formats have also been used during simulation to improve performance. Losasso et al. [43] create a method for simulating water and smoke on an unrestricted octree to capture small scale visual detail and allow for efficient solving of the Poisson equation. Popinet [48] uses quadtrees and octrees for a flexible and efficient approach to solving time-dependent incompressible Euler equations. ...

We present an approach for hierarchical super resolution (SR) using neural networks on an octree data representation. We train a hierarchy of neural networks, each capable of 2x upscaling in each spatial dimension between two levels of detail, and use these networks in tandem to facilitate large scale factor super resolution, scaling with the number of trained networks. We utilize these networks in a hierarchical super resolution algorithm that upscales multiresolution data to a uniform high resolution without introducing seam artifacts on octree node boundaries. We evaluate application of this algorithm in a data reduction framework by dynamically downscaling input data to an octree-based data structure to represent the multiresolution data before compressing for additional storage reduction. We demonstrate that our approach avoids seam artifacts common to multiresolution data formats, and show how neural network super resolution assisted data reduction can preserve global features better than compressors alone at the same compression ratios.

... Commonly, quadtrees and octrees were used as to define the spatial hierarchy with node types and subdivision rules varying depending on the application [49,48,20,38]. ...

Within computational continuum mechanics there exists a large category of simulation methods which operate by tracking Lagrangian particles over an Eulerian background grid. These Lagrangian/Eulerian hybrid methods, descendants of the Particle-In-Cell method (PIC), have proven highly effective at simulating a broad range of materials and mechanics including fluids, solids, granular materials, and plasma. These methods remain an area of active research after several decades, and their applications can be found across scientific, engineering, and entertainment disciplines. This thesis presents a GPU driven PIC-like simulation framework created using the Vulkan® API. Vulkan is a cross-platform and open-standard explicit API for graphics and GPU compute programming. Compared to its predecessors, Vulkan offers lower overhead, support for host parallelism, and finer grain control over both device resources and scheduling. This thesis harnesses those advantages to create a programmable GPU compute pipeline backed by a Vulkan adaptation of the SPgrid data-structure and multi-buffered particle arrays. The CPU host system works asynchronously with the GPU to maximize utilization of both the host and device. The framework is demonstrated to be capable of supporting Particle-in-Cell like simulation methods, making it viable for GPU acceleration of many Lagrangian particle on Eulerian grid hybrid methods. This novel framework is the first of its kind to be created using Vulkan® and to take advantage of GPU sparse memory features for grid sparsity.

... Different techniques to deal with this problem have been developed in the literature, most of them restricted to quadtree meshes (in 2D) or octree (3D) meshes, which are special cases of hierarchical grids represented by data structures quadtree/octree. Despite this restriction, these tree-based data structures are generally good enough and still an adequate choice for adaptive grids and moving borders [22]. Thus, we intend to implement in the present work the transient KBKZ-PSM model through a method of finite differences in hierarchical meshes that employ interpolations using the moving least squares (MLS) method [23]. ...

In this work, we present the implementation and verification of HiGTree-HiGFlow solver (see for numerical simulation of the KBKZ integral constitutive equation. The numerical method proposed herein is a finite difference technique using tree-based grids. The advantage of using hierarchical grids is that they allow us to achieve great accuracy in local mesh refinements. A moving least squares (MLS) interpolation technique is used to adapt the discretization stencil near the interfaces between grid elements of different sizes. The momentum and mass conservation equations are solved by an implicit method and the Chorin projection method is used for decoupling the velocity and pressure. The Finger tensor is calculated using the deformation fields method and a three-node quadrature formula is used to derive an expression for the integral tensor. The results of velocity and stress fields in channel and contraction-flow problems obtained in our simulations show good agreement with numerical and experimental results found in the literature.

... The Adaptive Mesh Refinement techniques (AMR) are now widely used and have since proven their efficiency, whether on 2D or 3D mesh, structured or unstructured mesh, conforming or not conforming mesh, with domain decomposition or not, see e.g. [46,41,28,15]. ...

Numerical solution of Richards’ equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.

... These approaches have recently been improved using neural network-based methods [37], [38], [39], [40]. While the above works employed only uniform grids, nonuniform grids [4], [41], [42] or even tetrahedral meshes [3], [43], [44] have been used to perform adaptive computations. ...

Fluid simulations are often performed using the incompressible Navier-Stokes equations (INSE), leading to sparse linear systems which are difficult to solve efficiently in parallel. Recently, kinetic methods based on the adaptive-central-moment multiple-relaxation-time (ACM-MRT) model have demonstrated impressive capabilities to simulate both laminar and turbulent flows, with quality matching or surpassing that of state-of-the-art INSE solvers. Furthermore, due to its local formulation, this method presents the opportunity for highly scalable implementations on parallel systems such as GPUs. However, an efficient ACM-MRT-based kinetic solver needs to overcome a number of computational challenges, especially when dealing with complex solids inside the fluid domain. In this paper, we present multiple novel GPU optimization techniques to efficiently implement high-quality ACM-MRT-based kinetic fluid simulations in domains containing complex solids. Our techniques include a new communication-efficient data layout, a load-balanced immersed-boundary method, a multi-kernel launch method using a simplified formulation of ACM-MRT calculations to enable greater parallelism, and the integration of these techniques into a parametric cost model to enable automated parameter search to achieve optimal execution performance. We also extended our method to multi-GPU systems to enable large-scale simulations. To demonstrate the state-of-the-art performance and high visual quality of our solver, we present extensive experimental results and comparisons to other solvers.

... Algorithms in [4] and [21] estimate the underlying numerical error and correct it. The numerical deficiencies of the grid structure can also be reduced by using Lagrangian methods [15]. The previously mentioned works improve the visual result of the Navier-Stokes solver by refining the underlying numerical method. ...

In Eulerian methods, the simulation of an incompressible fluid field requires a pressure field solution, which takes a large amount of time and computation resources to solve a large coarse linear system. The pressure solver has two mathematical features. The first is that it obtains the pressure solution from a velocity divergence distribution in high-dimensional space. The second is that the pressure is iteratively solved in the projection step. Based on these two features, we investigate a convolutional-based neural network, which learns to map the fluid quantities to pressure solution iteratively by inferring from multiple grid scales. Our proposed network extracts features from multiple scales and then aligns them to obtain a pressure field in the original resolution. We trim our network structure to be compact and fast and design it to be iterative like to improve performance. Our approach requires less computation cost, while it achieves comparable performance with recently proposed data-driven methods. Our method can easily be parallelized in GPU devices, and we demonstrate its speed-up ability with the fluid field in larger input scenes.

... One can say that a hierarchical grid is a generalization of quadtree and octree. In this sense, the choice of hierarchical grids is convenient to address the problem of flows in complex geometries [2][3][4][5][6]. ...

Tree-based grids bring the advantage of using fast Cartesian discretizations, such as finite differences, and the flexibility and accuracy of local mesh refinement. The main challenge is how to adapt the discretization stencil near the interfaces between grid elements of different sizes, which is usually solved by local high-order geometrical interpolations. Most methods usually avoid this by limiting the mesh configuration (usually to graded quadtree/octree grids), reducing the number of cases to be treated locally. In this work, we employ a moving least squares meshless interpolation technique, allowing for more complex mesh configurations, still keeping the overall order of accuracy. This technique was implemented in the HiG-Flow code to simulate Newtonian, generalized Newtonian and viscoelastic fluids flows. Numerical tests and application to viscoelastic fluid flow simulations were performed to illustrate the flexibility and robustness of this new approach.

... Popinet applied this idea combined with a non-compact finite volume discretization on the Marker-And-Cell (MAC) configuration [31] to the simulation of incompressible fluid flows [65]. Losasso et al. also proposed a compact finite volume solver on Octree for inviscid free surface flows [46], while Min et al. presented a node-based second-order accurate viscous solver [52]. The present work is based on the approach presented in Guittet et al. [30], which solves the viscous Navier-Stokes equations implicitly on the MAC configuration using a Voronoi partition and where the advection part of the momentum equation is discretized along the characteristic curves with a Backward Differentiation Formula (BDF), semi-Lagrangian scheme [73,81]. ...

We introduce an approach for solving the incompressible Navier-Stokes equations on a forest of Octree grids in a parallel environment. The methodology uses the p4est library of Burstedde et al. (2011) [15] for the construction and the handling of forests of Octree meshes on massively parallel distributed machines and the framework of Mirzadeh et al. (2016) [54] for the discretizations on Octree data structures. We introduce relevant additional parallel algorithms and provide performance analyses for individual building bricks and for the full solver. We demonstrate strong scaling for the solver up to 32,768 cores for a problem involving O(6.1×108) computational cells. We illustrate the dynamic adaptive capabilities of our approach by simulating flows past a stationary sphere, flows due to an oscillatory sphere in a closed box and transport of a passive scalar. Without sacrificing accuracy nor spatial resolution in regions of interest, our approach successfully reduces the number of computational cells to (at most) a few percents of uniform grids with equivalent resolution. We also perform a numerical simulation of the turbulent flow in a superhydrophobic channel with unparalleled wall grid resolution in the streamwise and spanwise directions.

... Tree-based grid generation (quadtrees in 2D and octrees in 3D) is common in computational sciences [4,[9][10][11][12][13][14][15][16] largely due to its simplicity and parallel scalability. The ability to efficiently refine (and coarsen) regions of interest using tree-based data structures have made it possible to deploy them on large-scale multi-physics simulations [2][3][4][17][18][19][20][21]. The use of incomplete octrees enables the handling of any arbitrary complex geometries, where the stated geometry is carved out from the domain (Fig. 1). ...

Complex flow simulations are conventionally performed on HPC clusters. However, the limited availability of HPC resources and steep learning curve of executing on traditional supercomputer infrastructure has drawn attention towards deploying flow simulation software on the cloud. We showcase how a complex computational framework -- that can evaluate COVID-19 transmission risk in various indoor classroom scenarios -- can be abstracted and deployed on cloud services. The availability of such cloud-based personalized planning tools can enable educational institutions, medical institutions, public sector workers (courthouses, police stations, airports, etc.), and other entities to comprehensively evaluate various in-person interaction scenarios for transmission risk. We deploy the simulation framework on the Azure cloud framework, utilizing the Dendro-kT mesh generation tool and PETSc solvers. The cloud abstraction is provided by RocketML cloud infrastructure. We compare the performance of the cloud machines with state-of-the-art HPC machine TACC Frontera. Our results suggest that cloud-based HPC resources are a viable strategy for a diverse array of end-users to rapidly and efficiently deploy simulation software.

... To combat this computational complexity, many researchers proposed adaptive simulation methods [MWN*17] which gain speedups and reduce memory by locally reducing simulation detail where it may not be needed. These methods typically use adaptive spatial data structures like octrees [LGF04], warping grids [IWT*18], or meshes [ATW13; BWHT07; WT08; KFCO06] instead of regular grids. Zhu et al. [ZLC*13] adaptively stretch out the grid cells away from a central point of interest, effectively extending the size of the computational domain and reducing detail away from the grid center. ...

This paper proposes a method for simulating liquids in large bodies of water by coupling together a water surface wave simulator with a 3D Navier‐Stokes simulator. The surface wave simulation uses the equivalent sources method (ESM) to efficiently animate large bodies of water with precisely controllable wave propagation behavior. The 3D liquid simulator animates complex non‐linear fluid behaviors like splashes and breaking waves using off‐the‐shelf simulators using FLIP or the level set method with semi‐Lagrangian advection. We combine the two approaches by using the 3D solver to animate localized non‐linear behaviors, and the 2D wave solver to animate larger regions with linear surface physics. We use the surface motion from the 3D solver as boundary conditions for 2D surface wave simulator, and we use the velocity and surface heights from the 2D surface wave simulator as boundary conditions for the 3D fluid simulation. We also introduce a novel technique for removing visual artifacts caused by numerical errors in 3D fluid solvers: we use experimental data to estimate the artificial dispersion caused by the 3D solver and we then carefully tune the wave speeds of the 2D solver to match it, effectively eliminating any differences in wave behavior across the boundary. To the best of our knowledge, this is the first time such a empirically driven error compensation approach has been used to remove coupling errors from a physics simulator. Our coupled simulation approach leverages the strengths of each simulation technique, animating large environments with seamless transitions between 2D and 3D physics.

... Multiple works have presented different alternations to the uniform grid that achieve this. Losasso et al. [2004] was the first to introduce a fluid simulation on an octree structure. Museth [2013] introduced OpenVDB, a sparse data structure organized as a tree with a high branching factor. ...

We introduce Dynamic Constrained Grid (DCGrid), a hierarchical and adaptive grid structure for fluid simulation combined with a scheme for effectively managing the grid adaptations. DCGrid is designed to be implemented on the GPU and used in high-performance simulations. Specifically, it allows us to efficiently vary and adjust the grid resolution across the spatial domain and to rapidly evaluate local stencils and individual cells in a GPU implementation. A special feature of DCGrid is that the control of the grid adaption is modeled as an optimization under a constraint on the maximum available memory, which addresses the memory limitations in GPU-based simulation. To further advance the use of DCGrid in high-performance simulations, we complement DCGrid with an efficient scheme for approximating collisions between fluids and static solids on cells with different resolutions. We demonstrate the effectiveness of DCGrid for smoke flows and complex cloud simulations in which terrain-atmosphere interaction requires working with cells of varying resolution and rapidly changing conditions. Finally, we compare the performance of DCGrid to that of alternative adaptive grid structures.

... Free-surface flow simulation has been receiving recurring attention in computer graphics over the past decades for its visually appealing interface dynamics and various small-scale flow details. The level-set method [Fedkiw and Osher 2002;Gibou et al. 2018] along with its many-particle augmentations [Enright et al. 2002;Losasso et al. 2008], data-structure modifications [Houston et al. 2006;Losasso et al. 2006Losasso et al. , 2004, and parallel implementations [Hughes et al. 2007;Mazhar et al. 2013], has been one of the most successful practices for high-resolution water animation. The swirling flow details on the water surface mainly stem from vortical enhancement forces Rasmussen et al. 2003], vortex particles [Selle et al. 2005], or less-dissipative advection schemes [MacCormack 2003] that are applied on the grid solver. ...

We propose a novel Clebsch method to simulate the free-surface vortical flow. At the center of our approach lies a level-set method enhanced by a wave-function correction scheme and a wave-function extrapolation algorithm to tackle the Clebsch method's numerical instabilities near a dynamic interface. By combining the Clebsch wave function's expressiveness in representing vortical structures and the level-set function's ability on tracking interfacial dynamics, we can model complex vortex-interface interaction problems that exhibit rich free-surface flow details on a Cartesian grid. We showcase the efficacy of our approach by simulating a wide range of new free-surface flow phenomena that were impractical for previous methods, including horseshoe vortex, sink vortex, bubble rings, and free-surface wake vortices.

... Fluid Dynamics. Since the seminal work of Stam [1999] many methods have been introduced to improve solvers for two way coupling [Lu et al. 2016;Teng et al. 2016;Zhu and Bridson 2005], based on reduced order methods [Gupta and Narasimhan 2007;Jones et al. 2016;Treuille et al. 2006], Smoothed Particle Hydrodynamics [Ihmsen et al. 2014;Koschier et al. 2019], or with an emphasis on compute and memory eciency [Ferstl et al. 2014;Losasso et al. 2004;McAdams et al. 2010;Setaluri et al. 2014;Zehnder et al. 2018]. Numerous methods focus on even more intricate features of uid ows, for example based on eigenfunctions [Cui et al. 2018], momentum transfer and regional projections [Zhang et al. 2016], style-transfer [Sato et al. 2018b], optimization [Inglis et al. 2017], or based on narrow band representations [Ferstl et al. 2016]. ...

Fig. 1. Three types of thunderstorm supercells simulated with our framework: low precipitation supercell (le), a classical supercell (middle), and a high-precipitation supercell (right). The complex interplay of a number of physical and meteorological phenomena makes simulating clouds a challenging and open research problem. We explore a physically accurate model for simulating clouds and the dynamics of their transitions. We propose rst-principle formulations for computing buoyancy and air pressure that allow us to simulate the variations of atmospheric density and varying temperature gradients. Our simulation allows us to model various cloud types, such as cumulus, stratus, and stratoscu-mulus, and their realistic formations caused by changes in the atmosphere. Moreover, we are able to simulate large-scale cloud super cells-clusters of cumulonimbus formations-that are commonly present during thunderstorms. To enable the ecient exploration of these stormscapes, we propose a lightweight set of high-level parameters that allow us to intuitively explore cloud formations and dynamics. Our method allows us to simulate cloud formations of up to about 20 km ⇥ 20 km extents at interactive rates. We explore the capabilities of physically accurate and yet interactive cloud simulations by showing numerous examples and by coupling our model with atmosphere measurements of real-time weather services to simulate cloud formations in the now. Finally, we quantitatively assess our model with cloud fraction proles, a common measure for comparing cloud types.

... We note that our method is unlikely to work well for dense A since that would likely entail using an impractical number of convolutional layers. An interesting question to consider is how well our method and current models would apply to discrete Poisson matrices arising from non-uniform grids, e.g., quadtrees or octrees [30]. ...

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L^2$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

... However, they suffer from increased numerical dissipation, which can result in noticeable visual artifacts, such as volume loss. These issues can be mitigated (although not completely resolved) through the use of adaptivity [Aanjaneya et al. 2017;Ando et al. 2013;Chentanez and Muller 2011;Ferstl et al. 2014;Losasso et al. 2004;Setaluri et al. 2014], which places more resolution in regions of interest, and less resolution elsewhere. ...

We propose a novel scheme for simulating two-way coupled interactions between nonlinear elastic solids and incompressible fluids. The key ingredient of this approach is a ghost matrix operator-splitting scheme for strongly coupled nonlinear elastica and incompressible fluids through the weak form of their governing equations. This leads to a stable and efficient method handling large time steps under the CFL limit while using a single monolithic solve for the coupled pressure fields, even in the case with highly nonlinear elastic solids. The use of the Material Point Method (MPM) is essential in the designing of the scheme, it not only preserves discretization consistency with the hybrid Lagrangian-Eulerian fluid solver, but also works naturally with our novel interface quadrature (IQ) discretization for free-slip boundary conditions. While traditional MPM suffers from sticky numerical artifacts, our framework naturally supports discontinuous tangential velocities at the solid-fluid interface. Our IQ discretization results in an easy-to-implement, fully particle-based treatment of the interfacial boundary, avoiding the additional complexities associated with intermediate level set or explicit mesh representations. The efficacy of the proposed scheme is verified by various challenging simulations with fluid-elastica interactions.

... These are efficiently implemented with narrow-band methods that track a deforming zero-crossing interface [40]. Additional memory efficiently has been demonstrated with adaptive structures like octree grids [1,22,50], Dynamic Tubular Grids (DT-Grid, based on compressed-row-storage) [37], or tall-cell grids [3,10]. ...

We introduce NeuralVDB, which improves on an existing industry standard for efficient storage of sparse volumetric data, denoted VDB, by leveraging recent advancements in machine learning. Our novel hybrid data structure can reduce the memory footprints of VDB volumes by orders of magnitude, while maintaining its flexibility and only incurring a small (user-controlled) compression errors. Specifically, NeuralVDB replaces the lower nodes of a shallow and wide VDB tree structure with multiple hierarchy neural networks that separately encode topology and value information by means of neural classifiers and regressors respectively. This approach has proven to maximize the compression ratio while maintaining the spatial adaptivity offered by the higher-level VDB data structure. For sparse signed distance fields and density volumes, we have observed compression ratios on the order of $10\times$ to more than $100\times$ from already compressed VDB inputs, with little to no visual artifacts. We also demonstrate how its application to animated sparse volumes can both accelerate training and generate temporally coherent neural networks.

... Curvature is a fundamental geometrical attribute related to minimal surfaces in differential geometry [3] and surface tension in physics [4,5]. In particular, mean curvature plays a crucial role in solving free-boundary problems (FBP) [6,7] with applications in multiphase flows [8][9][10][11][12][13][14], heat conduction and solidification [15][16][17][18][19], wildfire propagation [20], biological systems [21][22][23][24], and computer graphics and vision [25][26][27][28][29][30][31]. In surface-tension modeling, for example, computing curvature accurately at the interface is essential for recovering specific equilibrium solutions to the FBP's partial differential equations (PDE) [5]. ...

We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [arXiv:2201.12342][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction as a preprocessing step and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions.

... Curvature plays a crucial role in free boundary problems (FBP) [1] for its relation to surface tension in physics [2,3] and its regularization property in optimization. Such FBPs are ubiquitous in the physical sciences and engineering, with a broad range of applications in multiphase flows [4][5][6][7][8][9][10][11], solidification processes [12][13][14][15][16], and biological morphogenesis [17][18][19]. Accurate curvature computations are thus critical to constructing sound FBP models. ...

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-Cárdenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.

We introduce a novel liquid simulation approach that combines a spatially adaptive pressure projection solver with the Particle-in-Cell (PIC) method. The solver relies on a generalized version of the Finite Difference (FD) method to approximate the pressure field and its gradients in tree-based grid discretizations, possibly non-graded. In our approach, FD stencils are computed by using meshfree interpolations provided by a variant of Radial Basis Function (RBF), known as RBF-Finite-Difference (RBF-FD). This meshfree version of the FD produces differentiation weights on scattered nodes with high-order accuracy. Our method adapts a quadtree/octree dynamically in a narrow-band around the liquid interface, providing an adaptive particle sampling for the PIC advection step. Furthermore, RBF affords an accurate scheme for velocity transfer between the grid and particles, keeping the system's stability and avoiding numerical dissipation. We also present a data structure that connects the spatial subdivision of a quadtree/octree with the topology of its corresponding dual-graph. Our data structure makes the setup of stencils straightforward, allowing its updating without the need to rebuild it from scratch at each time-step. We show the effectiveness and accuracy of our solver by simulating incompressible inviscid fluids and comparing results with regular PIC-based solvers available in the literature.

Navier-Stokes-based methods have been used in computer graphics to simulate liquids, especially water. These physically based methods are computationally intensive, and require rendering the water surface at each step of the simulation process. The rendering of water surfaces requires knowing which 3D grid cells are crossed by the water’s surface, that is, tracking the surface across the cells is necessary. Solutions to water surface tracking and rendering problems exist in literature, but they are either too computationally intensive to be appropriate for real-time scenarios, as is the case of deformable implicit surfaces and ray-tracing, or too application-specific, as is the case of height-fields to simulate and render water mantles (e.g., lakes and oceans). This paper proposes a novel solution to water surface tracking that does not compromise the overall simulation performance. This approach differs from previous solutions in that it directly classifies and annotates the density of each 3D grid cell as either water, air, or water-air (i.e., water surface), opening the opportunity for easily reconstructing the water surface at an interactive frame rate.

The accuracy of advection has a great influence on the visual effect of fluid simulation. Constrained interpolation profile (CIP) method has been an important advection scheme because of its third‐order accuracy and the fact that it only needs to be performed over a compact stencil, but extending it to high‐dimensional advection equations is not easy, because it involves complex calculations and large memory overheads, and is usually unstable. In this article, we propose a stable and efficient three‐dimensional (3D) CIP scheme which can maintain high accuracy but requires low computation and memory cost. We first construct an efficient two‐dimensional (2D) CIP scheme based on dimensional splitting and local Taylor expansions, and then propose an effective way to extend it for 3D applications without decreasing the computational accuracy or affecting the stability. The experimental results show the advantages of our method over the state‐of‐the‐art advection schemes. A stable and efficient 3D CIP‐based advection scheme, which maintains third‐order of accuracy with low computation and memory cost, was proposed for high‐quality fluid animation. We first construct an efficient 2D CIP scheme based on dimensional splitting and local Taylor expansions, and then propose an effective way to extend it for 3D applications without decreasing the computational accuracy or affecting the stability. Top: Simulated with our method. Bottom: (Left to right) Linear, BFECC, MCIP, USCIP, our method.

We propose to enhance the capability of standard free-surface flow simulators with efficient support for immersed bubbles through two new models: constraint-based bubbles and affine fluid regions. Unlike its predecessors, our constraint-based model entirely dispenses with the need for advection or projection inside zero-density bubbles, with extremely modest additional computational overhead that is proportional to the surface area of all bubbles. This surface-only approach is easy to implement, realistically captures many familiar bubble behaviors, and even allows two or more distinct liquid bodies to correctly interact across completely unsimulated air. We augment this model with a per-bubble volume-tracking and correction framework to minimize the cumulative effects of gradual volume drift. To support bubbles with non-zero densities, we propose a novel reduced model for an irregular fluid region with a single pointwise incompressible affine vector field. This model requires only 11 interior velocity degrees of freedom per affine fluid region in 3D, and correctly reproduces buoyant, stationary, and sinking behaviors of a secondary fluid phase with non-zero density immersed in water. Since the pressure projection step in both the above schemes is a slightly modified Poisson-style system, we propose novel Multigrid-based preconditioners for Conjugate Gradients for fast numerical solutions of our new discretizations. Furthermore, we observe that by enforcing an incompressible affine vector field over a coalesced set of grid cells, our reduced model is effectively an irregular coarse super-cell. This offers a convenient and flexible adaptive coarsening strategy that integrates readily with the standard staggered grid approach for fluid simulation, yet supports coarsened regions that are arbitrary voxelized shapes, and provides an analytically divergence-free interior. We demonstrate its effectiveness with a new adaptive liquid simulator whose interior regions are coarsened into a mix of tiles with regular and irregular shapes.

The “procedural” approach to animating ocean waves is the dominant algorithm for animating larger bodies of water in interactive applications as well as in off‐line productions — it provides high visual quality with a low computational demand. In this paper, we widen the applicability of procedural water wave animation with an extension that guarantees the satisfaction of boundary conditions imposed by terrain while still approximating physical wave behavior. In combination with a particle system that models wave breaking, foam, and spray, this allows us to naturally model waves interacting with beaches and rocks. Our system is able to animate waves at large scales at interactive frame rates on a commodity PC.

We present an algorithmically efficient and parallelized domain decomposition based approach to solving Poisson’s equation on irregular domains. Our technique employs the Schur complement method, which permits a high degree of parallel efficiency on multicore systems. We create a novel Schur complement preconditioner which achieves faster convergence, and requires less computation time and memory. This domain decomposition method allows us to apply different linear solvers for different regions of the flow. Subdomains with regular boundaries can be solved with an FFT-based Fast Poisson Solver. We can solve systems with 1,024 ³ degrees of freedom, and demonstrate its use for the pressure projection step of incompressible liquid and gas simulations. The results demonstrate considerable speedup over preconditioned conjugate gradient methods commonly employed to solve such problems, including a multigrid preconditioned conjugate gradient method.

Simulating liquid phenomena utilizing Eulerian frameworks is challenging, since highly energetic flows often induce severe topological changes, creating thin and complex liquid surfaces. Thus, capturing structures that are small relative to the grid size become intractable, since continually increasing the resolution will scale sub-optimally due to the pressure projection step. Previous methods successfully relied on using higher resolution grids for tracking the liquid surface implicitly; however this technique comes with drawbacks. The mismatch of pressure samples and surface degrees of freedom will cause artifacts such as hanging blobs and permanent kinks at the liquid-air interface. In this paper, we propose an extended cut-cell method for handling liquid structures that are smaller than a grid cell. At the core of our method is a novel iso-surface Poisson Solver, which converges with second-order accuracy for pressure values while maintaining attractive discretization properties such as symmetric positive definiteness. Additionally, we extend the iso-surface assumption to be also compatible with surface tension forces. Our results show that the proposed method provides a novel framework for handling arbitrarily small splashes that can also correctly interact with objects embodied by complex geometries.

In this article, we propose an improved refinement process for the simulation of incompressible low-viscosity turbulent flows using Smoothed Particle Hydrodynamics, under adaptive volume ratios of up to 1 : 1, 000, 000. We derive a discretized objective function, which allows us to generate ideal refinement patterns for any kernel function and any number of particles a priori without requiring intuitive initial user-input. We also demonstrate how this objective function can be optimized online to further improve the refinement process during simulations by utilizing a gradient descent and a modified evolutionary optimization. Our investigation reveals an inherent residual refinement error term, which we smooth out using improved and novel methods. Our improved adaptive method is able to simulate adaptive volume ratios of 1 : 1, 000, 000 and higher, even under highly turbulent flows, only being limited by memory consumption. In general, we achieve more than an order of magnitude greater adaptive volume ratios than prior work.

Fluid simulations are often performed using the incompressible Navier-Stokes equations (INSE), leading to sparse linear systems which are difficult to solve efficiently in parallel. Recently, kinetic methods based on the adaptive-central-moment multiple-relaxation-time (ACM-MRT) model have demonstrated impressive capabilities to simulate both laminar and turbulent flows, with quality matching or surpassing that of state-of-the-art INSE solvers. Furthermore, due to its local formulation, this method presents the opportunity for highly scalable implementations on parallel systems such as GPUs. However, an efficient ACM-MRT-based kinetic solver needs to overcome a number of computational challenges, especially when dealing with complex solids inside the fluid domain. In this paper, we present multiple novel GPU optimization techniques to efficiently implement high-quality ACM-MRT-based kinetic fluid simulations in domains containing complex solids. Our techniques include a new communication-efficient data layout, a load-balanced immersed-boundary method, a multi-kernel launch method using a simplified formulation of ACM-MRT calculations to enable greater parallelism, and the integration of these techniques into a parametric cost model to enable automated prameter search to achieve optimal execution performance. We also extended our method to multi-GPU systems to enable large-scale simulations. To demonstrate the state-of-the-art performance and high visual quality of our solver, we present extensive experimental results and comparisons to other solvers.

La création de paysages virtuels réalistes est un défi important de l'informatique graphique. Le but de cette thèse est d'amplifier un terrain de grande taille et d'ajouter du dynamisme en animant des phénomènes pseudopériodiques. Nous nous sommes alors intéressés aux scènes comprenant des rivières: la modélisation de ce type de scène inclut la génération de la géométrie du réseau hydraulique mais aussi l'animation du mouvement de l'eau correspondante. Bien que ces deux éléments aient été étudiés individuellement, il n'existe pas de méthode pour générer de manière concomitante le réseau et l'animation. Afin de répondre à cette problématique, nous proposons un nouveau modèle de représentation de rivière permettant l'animation de la surface de l'eau ainsi qu'une méthode de génération procédurale pour sculpter et animer un réseau cohérent de rivières sur un terrain existant. Notre modèle de rivière utilise les arbres de construction hiérarchiques de terrain pour former la géométrie du lit de la rivière et nous proposons un nouveau modèle appelé arbre d'écoulement pour représenter la surface de l'eau. Les feuilles de l'arbre, sont des primitives de flux à support compact spatialement localisées et variables dans le temps. Les fonctions utilisées représentent des caractéristiques spécifiques comme des ondulations, des turbulences ou encore des tourbillons. Elles sont procédurales et paramétrées afin de générer de la variété. Les nœuds internes de l'arbre sont des opérateurs de combinaison permettant de former la surface finale de l'eau. Ce modèle permet de créer une animation temps réel de l'eau sur des rivières de plusieurs kilomètres de long tout en comportant du détail localement. Nous proposons également une méthode de génération de rivière qui utilise l'élévation d'un terrain de base pour obtenir le réseau d'écoulement. Il creuse ensuite le lit des rivières dans le terrain et génère de manière concomitante l'animation de la surface de l'eau. Les caractéristiques, telles que la largeur, la profondeur et la forme du lit de la rivière, ainsi que l'élévation et l'écoulement de la surface du fluide, sont fonction du terrain et du type de la rivière. Le lit de la rivière est creusé dans le terrain en combinant des primitives d'élévation à support compact échantillonnées le long de la trajectoire de la rivière. La génération permet la production d'une vaste gamme de formes de rivières, allant des rivières à méandres dans les plaines aux rapides de montagnes. Le modèle permet une édition interactive et intuitive des trajectoires des rivières mais aussi du choix des primitives. Notre méthode de création de rivière permet d'amplifier un grand terrain en sculptant un lit de rivière précis mais aussi de créer l'animation de la surface de l'eau correspondante

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the temperature, the flow velocity and pressure, and the interfacial velocity. The approach utilizes level-set methods for sharp and implicit interface tracking, hybrid finite-difference/finite-volume discretizations on adaptive quadtree grids, and a pressure-free projection method for the solution of the incompressible Navier-Stokes equations. The method is first verified with numerical convergence tests using a synthetic solution. Then, computational studies of ice formation on a cylinder in crossflow are performed and provide good quantitative agreement with existing experimental results, reproducing qualitative phenomena that have been observed in past experiments. Finally, we investigate the role of varying Reynolds and Stefan numbers on the emerging interface morphologies and provide new insights around the time evolution of local and average heat transfer at the interface.

The simulation of large open water surface is challenging using a uniform volumetric discretization of the Navier-Stokes equations. Simulating water splashes near moving objects, which height field methods for water waves cannot capture, necessitates high resolutions. Such simulations can be carried out using the Fluid-Implicit-Particle (FLIP) method. However, the FLIP method is not efficient for the long-lasting water waves that propagate to long distances, which require sufficient depth for a correct dispersion relationship. This paper presents a new method to tackle this dilemma through an efficient hybridization of volumetric and surface-based advection-projection discretizations. We design a hybrid time-stepping algorithm that combines a FLIP domain and an adaptively remeshed Boundary Element Method (BEM) domain for the incompressible Euler equations. The resulting framework captures the detailed water splashes near moving objects with the FLIP method, and produces convincing water waves with correct dispersion relationships at modest additional costs.

Dans ce mémoire, nous proposons de répondre au problème de suivi de surface déformable à l'aide d'une technique de remaillage fondée sur les diagrammes de Voronoï restreints. Ces derniers offrent une une partition d'un domaine surfacique permettant entre autre d'optimiser la répartition d'un ensemble d'échantillons sur ce domaine et d'en définir une triangulation remarquable : la triangulation de Delaunay restreinte. Cette solution de remaillage n'est efficace que lorsque certaines conditions sont réunies. Aussi, le premier travail a consisté à mettre en oeuvre une technique d'analyse des configurations de cellules et une méthode de correction automatique permettant de générer une nouvelle approximation homéomorphe au domaine initial quelque soit l'échantillonnage donné. Un second travail propose d'améliorer la proximité entre le maillage initial et le résultat du remaillage : il s'agit de minimiser l'erreur d'approximation qui est ici exprimée sous forme de différences locales de volume. Ces deux outils sont associés à une stratégie d'échantillonnage qui permet de maintenir une densité d'échantillonnage constante tout au long de la déformation et propose ainsi une nouvelle méthode de suivi d'une surface libre pour la simulation d'écoulement de fluidesincompressibles.

We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more accurately. The core of our hybrid framework is a switching mechanism that relies on well established numerical techniques to gauge curvature. If the curvature magnitude is larger than a resolution-dependent threshold, it uses a neural network to yield a better approximation. Our networks are multilayer perceptrons fitted to synthetic data sets composed of sinusoidal- and circular-interface samples at various configurations. To reduce data set size and training complexity, we leverage the problem's characteristic symmetry and build our models on just half of the curvature spectrum. These savings lead to a powerful inference system able to outperform any of its numerical or neural component alone. Experiments with stationary, smooth interfaces show that our hybrid solver is notably superior to conventional numerical methods in coarse grids and along steep interface regions. Compared to prior research, we have observed outstanding gains in precision after training the regression model with data pairs from more than a single interface type and transforming data with specialized input preprocessing. In particular, our findings confirm that machine learning is a promising venue for reducing or removing mass loss in the level-set method.

We present a second-order monolithic method for solving incompressible Navier–Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and pressure variables are located at cell centers. Compact finite difference methods with ghost values are used to discretize the advection and diffusion terms of the velocity. A pressure gradient and divergence operator on the quadtree that use compact stencils are developed. Furthermore, the proposed method is extended to cubical domains with octree grids. Numerical results demonstrate that the method is second-order convergent in L∞ norms and can handle irregular domains for various Reynolds numbers.

The Hodge decomposition, that is an important feature of incompressible fluid flows, is orthogonal and the projection taking its incompressible component is therefore stable. The decomposition is implemented by solving the Poisson equation. In order to simulate incompressible fluid flows in a stable manner, it is desired to utilize a Poisson solver that attains the orthogonality of the Hodge decomposition in a discrete level.
When a Poisson solver induces the orthogonality, its associated linear system is necessarily symmetric. With this regard, the symmetric Poisson solvers [9], [8] by Losasso et al. are more advantageous not only to efficiently solving the linear system but also to stably simulating fluid flows than nonsymmetric ones. Their numerical solutions were empirically observed to be first and second order accurate, respectively. One may expect that each of their numerical gradients has convergence order that is one less than that of its numerical solution.
However, we in this work show that super-convergence holds true with both Poisson solvers. Rigorous analysis is presented to prove that the difference is one half, not one between the convergence orders of numerical solution and gradient in both solvers. The analysis is then validated with numerical results. We furthermore show that both Poisson solvers, being symmetric, indeed satisfy the orthogonal property in the discrete level and yield stable implementations of the Hodge decomposition in octree grids.

Multigrid methods are quite efficient for solving the pressure Poisson equation in simulations of incompressible flow. However, for viscous liquids, geometric multigrid turned out to be less efficient for solving the variational viscosity equation. In this contribution, we present an Unsmoothed Aggregation Algebraic MultiGrid (UAAMG) method with a multi-color Gauss-Seidel smoother, which consistently solves the variational viscosity equation in a few iterations for various material parameters. Moreover, we augment the OpenVDB data structure with Intel SIMD intrinsic functions to perform sparse matrix-vector multiplications efficiently on all multigrid levels. Our framework is 2.0 to 14.6 times faster compared to the state-of-the-art adaptive octree solver in commercial software for the large-scale simulation of both non-viscous and viscous flow. The code is available at http://computationalsciences.org/publications/shao-2022-multigrid.html.

Real-life multiphase flows exhibit a number of complex and visually appealing behaviors, involving bubbling, wetting, splashing, and glugging. However, most state-of-the-art simulation techniques in graphics can only demonstrate a limited range of multiphase flow phenomena, due to their inability to handle the real water-air density ratio and to the large amount of numerical viscosity introduced in the flow simulation and its coupling with the interface. Recently, kinetic-based methods have achieved success in simulating large density ratios and high Reynolds numbers efficiently; but their memory overhead, limited stability, and numerically-intensive treatment of coupling with immersed solids remain enduring obstacles to their adoption in movie productions. In this paper, we propose a new kinetic solver to couple the incompressible Navier-Stokes equations with a conservative phase-field equation which remedies these major practical hurdles. The resulting two-phase immiscible fluid solver is shown to be efficient due to its massively-parallel nature and GPU implementation, as well as very versatile and reliable because of its enhanced stability to large density ratios, high Reynolds numbers, and complex solid boundaries. We highlight the advantages of our solver through various challenging simulation results that capture intricate and turbulent air-water interaction, including comparisons to previous work and real footage.

We propose a new adaptive liquid simulation framework that achieves highly detailed behavior with reduced implementation complexity. Prior work has shown that spatially adaptive grids are efficient for simulating large-scale liquid scenarios, but in order to enable adaptivity along the liquid surface these methods require either expensive boundary-conforming (re-)meshing or elaborate treatments for second order accurate interface conditions. This complexity greatly increases the difficulty of implementation and maintainability, potentially making it infeasible for practitioners. We therefore present new algorithms for adaptive simulation that are comparatively easy to implement yet efficiently yield high quality results. First, we develop a novel staggered octree Poisson discretization for free surfaces that is second order in pressure and gives smooth surface motions even across octree T-junctions, without a power/Voronoi diagram construction. We augment this discretization with an adaptivity-compatible surface tension force that likewise supports T-junctions. Second, we propose a moving least squares strategy for level set and velocity interpolation that requires minimal knowledge of the local tree structure while blending near-seamlessly with standard trilinear interpolation in uniform regions. Finally, to maximally exploit the flexibility of our new surface-adaptive solver, we propose several novel extensions to sizing function design that enhance its effectiveness and flexibility. We perform a range of rigorous numerical experiments to evaluate the reliability and limitations of our method, as well as demonstrating it on several complex high-resolution liquid animation scenarios.

Simulation of natural phenomena is one of the important research fields in computer graphics. In particular, clouds play an important role in creating images of outdoor scenes. Fluid simulation is effective in creating realistic clouds because clouds are the visualization of atmospheric fluid. In this paper, we propose a simulation technique, based on a numerical solution of the partial differential equation of the atmospheric fluid model, for creating animated cumulus and cumulonimbus clouds with features formed by turbulent vortices.

Realistically animated fluids can add substantial realism to interactive applications such as virtual surgery simulators or computer games. In this paper we propose an interactive method based on Smoothed Particle Hydrodynamics (SPH) to simulate fluids with free surfaces. The method is an extension of the SPH-based technique by Desbrun to animate highly deformable bodies. We gear the method towards fluid simulation by deriving the force density fields directly from the Navier-Stokes equation and by adding a term to model surface tension effects. In contrast to Eulerian grid-based approaches, the particle-based approach makes mass conservation equations and convection terms dispensable which reduces the complexity of the simulation. In addition, the particles can directly be used to render the surface of the fluid. We propose methods to track and visualize the free surface using point splatting and marching cubes-based surface reconstruction. Our animation method is fast enough to be used in interactive systems and to allow for user interaction with models consisting of up to 5000 particles.

We present a comprehensive methodology for realistically animating liquid phenomena. Our approach unifies existing computer graphics techniques for simulating fluids and extends them by incorporating more complex behavior. It is based on the Navier–Stokes equations which couple momentum and mass conservation to completely describe fluid motion. Our starting point is an environment containing an arbitrary distribution of fluid, and submerged or semisubmerged obstacles. Velocity and pressure are defined everywhere within this environment and updated using a set of finite difference expressions. The resulting vector and scalar fields are used to drive a height field equation representing the liquid surface. The nature of the coupling between obstacles in the environment and free variables allows for the simulation of a wide range of effects that were not possible with previous computer graphics fluid models. Wave effects such as reflection, refraction, and diffraction, as well as rotational effects such as eddies, vorticity, and splashing are a natural consequence of solving the system. In addition, the Lagrange equations of motion are used to place buoyant dynamic objects into a scene and track the position of spray and foam during the animation process. Typical disadvantages to dynamic simulations such as poor scalability and lack of control are addressed by assuming that stationary obstacles align with grid cells during the finite difference discretization, and by appending terms to the Navier–Stokes equations to include forcing functions. Free surfaces in our system are represented as either a collection of massless particles in 2D, or a height field which is suitable for many of the water rendering algorithms presented by researchers in recent years.

We present a new method for animating water based on a simple, rapid and stable solution of a set of partial differential equations resulting from an approximation to the shallow water equations. The approximation gives rise to a version of the wave equation on a height-field where the wave velocity is proportional to the square root of the depth of the water. The resulting wave equation is then solved with an alternating-direction implicit method on a uniform finite-difference grid. The computational work required for an iteration consists mainly of solving a simple tridiagonal linear system for each row and column of the height field. A single iteration per frame suffices in most cases for convincing animation.Like previous computer-graphics models of wave motion, the new method can generate the effects of wave refraction with depth. Unlike previous models, it also handles wave reflections, net transport of water and boundary conditions with changing topology. As a consequence, the model is suitable for animating phenomena such as flowing rivers, raindrops hitting surfaces and waves in a fish tank as well as the classic phenomenon of waves lapping on a beach. The height-field representation prevents it from easily simulating phenomena such as breaking waves, except perhaps in combination with particle-based fluid models. The water is rendered using a form of caustic shading which simulates the refraction of illuminating rays at the water surface. A wetness map is also used to compute the wetting and drying of sand as the water passes over it.

This paper presents Kizamu, a computer-based sculpting system for creating digital characters for the entertainment industry. Kizamu incorporates a blend of new algorithms, significant technical advances, and novel user interaction paradigms into a system that is both powerful and unique.To meet the demands of high-end digital character design, Kizamu addresses three requirements posed to us by a major production studio. First, animators and artists want digital clay — a medium with the characteristics of real clay and the advantages of being digital. Second, the system should run on standard hardware at interactive rates. Finally, the system must accept and generate standard 3D representations thereby enabling integration into an existing animation production pipeline.At the heart of the Kizamu system are Adaptively Sampled Distance Fields (ADFs), a volumetric shape representation with the characteristics required for digital clay. In this paper, we describe the system and present the major research advances in ADFs that were required to make Kizamu a reality.

In this paper, we introduce techniques for animating explosions and their effects. The primary effect of an explosion is a disturbance that causes a shock wave to propagate through the surrounding medium. The disturbance determines the behavior of nearly all other secondary effects seen in explosion. We simulate the propagation of an explosion through the surrounding air using a computational fluid dynamics model based on the equations for compressible, viscous flow. To model the numerically stable formation of shocks along blast wave fronts, we employ an integration method that can handle steep pressure gradients without introducing inappropriate damping. The system includes two-way coupling between solid objects and surrounding fluid. Using this technique, we can generate a variety of effects including shaped explosive charges, a projectile propelled from a chamber by an explosion, and objects damaged by a blast. With appropriate rendering techniques, our explosion model can be used to create such visual effects as fireballs, dust clouds, and the refraction of light caused by a blast wave.

We describe a method for controlling smoke simulations through user-specified keyframes. To achieve the desired behavior, a continuous quasi-Newton optimization solves for appropriate "wind" forces to be applied to the underlying velocity field throughout the simulation. The cornerstone of our approach is a method to efficiently compute exact derivatives through the steps of a fluid simulation. We formulate an objective function corresponding to how well a simulation matches the user's keyframes, and use the derivatives to solve for force parameters that minimize this function. For animations with several keyframes, we present a novel multipleshooting approach. By splitting large problems into smaller overlapping subproblems, we greatly speed up the optimization process while avoiding certain local minima.

We present a progressive encoding technique specifically designed for complex isosurfaces. It achieves better rate distortion performance than all standard mesh coders, and even improves on all previous single rate isosurface coders. Our novel algorithm handles isosurfaces with or without sharp features, and deals gracefully with high topologic and geometric complexity. The inside/outside function of the volume data is progressively transmitted through the use of an adaptive octree, while a local frame based encoding is used for the fine level placement of surface samples. Local patterns in topology and local smoothness in geometry are exploited by context-based arithmetic encoding, allowing us to achieve an average of 6.10 bits per vertex (b/v) at very low distortion. Of this rate only 0.65 b/v are dedicated to connectivity data: this improves by 24% over the best previous single rate isosurface encoder.

A methodology for controlling fluid animations is developed using the concept of an embedded controller. A controller acts as an interface between the animator and a general tool for calculating three dimensional fluid flow. The major contribution of this paper is that for the first time, it is possible for computer graphics animators, to specify and control a three dimensional fluid animation, without knowledge of the underlying equations or the method used to solve them. In addition the technique is stable, physically accurate, and can be integrated with other animation tools that deal with dynamic objects. To illustrate the method, animations of moving objects fountains, and explosions, together with the straightforward control functions that are used to create them, are presented

We present a progressive encoding technique specifically designed for complex isosurfaces. It achieves better rate distortion performance than all standard mesh coders, and even improves on all previous single rate isosurface coders. Our novel algorithm handles isosurfaces with or without sharp features, and deals gracefully with high topologic and geometric complexity. The inside/outside function of the volume data is progressively transmitted through the use of an adaptive octree, while a local frame based encoding is used for the fine level placement of surface samples. Local patterns in topology and local smoothness in geometry are exploited by context-based arithmetic encoding, allowing us to achieve an average of 6.10 bits per vertex (b/v) at very low distortion. Of this rate only 0.65 b/v are dedicated to connectivity data: this improves by 24% over the best previous single rate isosurface encoder.

We present a graph-based strategy for representing the computational domain for embedded boundary discretizations of conservation-law PDE's. The representation allows recursive generation of coarse-grid geometry representations suitable for multigrid and adaptive mesh refinement calculations. Using this scheme, we implement a simple multigrid V-cycle relaxation algorithm to solve the linear elliptic equations arising from a blockstructured adaptive discretization of Poisson's equation over an arbitrary two-dimensional domain. We demonstrate that the resulting solver is robust to a wide range of twodimensional geometries, and performs as expected for multigrid-based schemes, exhibiting O (N log N) scaling with system size, N . 1 Introduction In the Embedded Boundary (EB) approach to discretizing PDE's in complex geometries, the physical domain is embedded completely within a larger uniform mesh. The bulk of the data underlying an EB discretization utilizes rectangular indexing, and on...

This paper describes a new animation technique for modeling the turbulent rotational motion that occurs when a hot gas interacts with solid objects and the surrounding medium. The method is especially useful for scenes involving swirling steam, rolling or billowing smoke, and gusting wind. It can also model gas motion due to fans and heat convection. The method combines specialized forms of the equations of motion of a hot gas with an efficient method for solving volumetric differential equations at low resolutions. Particular emphasis is given to issues of computational efficiency and ease-of-use of the method by an animator. We present the details of our model, together with examples illustrating its use. Keywords: Animation, Convection, Gaseous Phenomena, Gas Simulations, Physics-Based Modeling, Steam, Smoke, Turbulent Flow. 1 Introduction The turbulent motion of smoke and steam has always inspired interest amongst graphics researchers. The problem of modeling the complex inter-r...

Building animation tools for fluid-like motions is an important and challenging problem with many applications in computer graphics. The use of physics-based models for fluid flow can greatly assist in creating such tools. Physical models, unlike key frame or procedural based techniques, permit an animator to almost effortlessly create interesting, swirling fluid-like behaviors. Also, the interaction of flows with objects and virtual forces is handled elegantly. Until recently, it was believed that physical fluid models were too expensive to allow real-time interaction. This was largely due to the fact that previous models used unstable schemes to solve the physical equations governing a fluid. In this paper, for the first time, we propose an unconditionally stable model which still produces complex fluid-like flows. As well, our method is very easy to implement. The stability of our model allows us to take larger time steps and therefore achieve faster simulations. We have used our model in conjuction with advecting solid textures to create many fluid-like animations interactively in two- and three-dimensions.

In this paper, we propose a new approach to numerical smoke simulation for computer graphics applications. The method proposed here exploits physics unique to smoke in order to design a numerical method that is both fast and efficient on the relatively coarse grids traditionally used in computer graphics applications (as compared to the much finer grids used in the computational fluid dynamics literature). We use the inviscid Euler equations in our model, since they are usually more appropriate for gas modeling and less computationally intensive than the viscous NavierStokes equations used by others. In addition, we introduce a physically consistent vorticity confinement term to model the small scale rolling features characteristic of smoke that are absent on most coarse grid simulations. Our model also correctly handles the interaction of smoke with moving objects. Keywords: Smoke, computational fluid dynamics, Navier-Stokes equations, Euler equations, Semi-Lagrangian methods, stable fluids, vorticity confinement, participating media 1

This paper presents an overview of parallel algorithms and their implementations for solving large sparse linear systems which arise in scientific and engineering applications. Preconditioners constitute the most important ingredient in solving such systems. As will be seen, the most common preconditioners used for sparse linear systems adapt domain decomposition concepts to the more general framework of “distributed sparse linear systems”. Variants of Schwarz procedures and Schur complement techniques are discussed. We also report on our own experience in the parallel implementation of a fairly complex simulation of solid-liquid flows.

We present a new shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.Our approach gives us considerable flexibility in the choice of local shape functions, and in particular we can accurately represent sharp features such as edges and corners by selecting appropriate shape functions. An error-controlled subdivision leads to an adaptive approximation whose time and memory consumption depends on the required accuracy. Due to the separation of local approximation and local blending, the representation is not global and can be created and evaluated rapidly. Because our surfaces are described using implicit functions, operations such as shape blending, offsets, deformations and CSG are simple to perform.

We present a fast and stable system for animating materials that melt, flow, and solidify. Examples of real-world materials that exhibit these phenomena include melting candles, lava flow, the hardening of cement, icicle formation, and limestone deposition. We animate such phenomena by physical simulation of fluids --- in particular the incompressible viscous Navier-Stokes equations with free surfaces, treating solid and nearly-solid materials as very high viscosity fluids. The computational method is a modification of the Marker-and-Cell (MAC) algorithm in order to rapidly simulate fluids with variable and arbitrarily high viscosity. This allows the viscosity of the material to change in space and time according to variation in temperature, water content, or any other spatial variable, allowing different locations in the same continuous material to exhibit states ranging from the absolute rigidity or slight bending of hardened wax to the splashing and sloshing of water. We create detailed polygonal models of the fluid by splatting particles into a volumetric grid and we render these models using ray tracing with sub-surface scattering. We demonstrate the method with examples of several viscous materials including melting wax and sand drip castles.

We present a general algorithm for the study of the evolution of interfaces in growth processes based on the level set method, using the narrow band and the fast marching approximations, applied on a dynamically adaptive grid. A novel construction of a dynamical adaptive grid is presented. One important feature is that we establish a controllable finite-width region of the highest resolution straddling the physically important boundary. This high-resolution stripe provides a better description of the important physical variables. An efficient and robust coupling of a hierarchical space description structure with the state-of-the-art level set method is established. Copyright © 2003 John Wiley & Sons, Ltd.

We present a new method for physically based modeling and interactive-rate simulation of 3D fluids in computer graphics. By solving the 2D Navier-Stokes equations using a computational fluid dynamics method, we map the surface into 3D using the corresponding pressures in the fluid flow field. The method achieves realistic interactive-rate fluid simulation by solving the physical governing laws of fluids but avoiding the extensive 3D fluid dynamics computation. Unlike previous computer graphics fluid models, our approach can simulate many different fluid behaviors by changing the internal or external boundary conditions. It can model different kinds of fluids by varying the Reynolds number. It can also simulate objects moving or floating in fluids. In addition, we can visualize the animation of the fluid flow field, the streakline of a flow field, and the blending of fluids of different colors. Our model can serve as a testbed to simulate many other fluid phenomena which have never been successfully modeled previously in computer graphics.

We present a new adaptive numerical scheme for solving parabolic PDEs in Cartesian geometry. Applying a finite volume discretization with explicit time integration, both of second order, we employ a fully adaptive multiresolution scheme to represent the solution on locally refined nested grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption strategy to advance the grid in time and to follow the time evolution of the solution directly exploits the multiresolution representation. Applying this new method to several test problems in one, two and three space dimensions, like convection–diffusion, viscous Burgers and reaction–diffusion equations, we show its second-order accuracy and demonstrate its computational efficiency.

A second-order-accurate finite difference discretization of the incompressible Navier–Stokes is presented that discretely conserves mass, momentum, and kinetic energy (in the inviscid limit) in space and time. The method is thus completely free of numerical dissipation and potentially well suited to the direct numerical simulation or large-eddy simulation of turbulent flow. The method uses a staggered arrangement of velocity and pressure on a structured Cartesian grid and retains its discrete conservation properties for both uniform and nonuniform gird spacing. The predicted conservation properties are confirmed by inviscid simulations on both uniform and nonuniform grids. The capability of the method to resolve turbulent flow is demonstrated by repeating the turbulent channel flow simulations of H. Choi and P. Moin (1994, J. Comput. Phys.113, 1), where the effect of computational time step on the computed turbulence was investigated. The present fully conservative scheme achieved turbulent flow solutions over the entire range of computational time steps investigated (Δt+=Δtu2τ/ν=0.4 to 5.0). Little variation in statistical turbulence quantities was observed up to Δt+=1.6. The present results differ significantly from those reported by Choi and Moin, who observed significant discrepancies in the turbulence statistics above Δt+=0.4 and the complete laminarization of the flow at and above Δt+=1.6.

A fast modular numerical method for solving general moving interface problems is presented. It simplifies code development by providing a black-box solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problem-independent velocity extension, and a second-order semi-Lagrangian time stepping scheme which reduces numerical error by exact evaluation of the signed distance function. Adaptive quadtree meshes are used to concentrate computational effort on the interface, so the method moves an N-element interface in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. Numerical results show that the method computes accurate viscosity solutions to a wide variety of difficult geometric moving interface problems involving merging, anisotropy, faceting, nonlocality, and curvature.

In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second-order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second-order-accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second-order accuracy of this numerical method. In addition, we use our approach to formulate a second-order-accurate symmetric implicit time discretization of the heat equation on irregular domains. Then we briefly consider Stefan problems.

An adaptive mesh projection method for the time-dependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volume-of-fluid approach. Second-order convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.

In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.

We present a coupled level set/volume-of-fluid method for computing growth and collapse of vapor bubbles. The liquid is assumed incompressible and the vapor is assumed to have constant pressure in space. Second order algorithms are used for finding “mass conserving” extension velocities, for discretizing the local interfacial curvature and also for the discretization of the cell-centered projection step. Convergence studies are given that demonstrate this second order accuracy. Examples are provided that apply to cavitating bubbles.

Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semi-Lagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effort on the interface, so the methods move an interface withNdegrees of freedom inO(NlogN) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. The methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging, anisotropy, faceting, and curvature.

An adaptive method based on the idea of multiple component grids for the solution of hyperbolic partial differential equations using finite difference techniques is presented. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. The approach is recursive in that fine grids can contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, error estimation procedure, and the data structures, and conclude with numerical examples in one and two space dimensions.

Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents efficient algorithms for this “redistancing” problem. The algorithms use quadtrees and triangulation to compute global approximate signed distance functions. A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants. Given a polygonal interface with N elements, our algorithms run in O(N) space and O(N log N) time. Two-dimensional numerical results show they are highly efficient in practice.

The aim of this work is the development of an automatic, adaptive mesh refinement strategy for solving hyperbolic conservation laws in two dimensions. There are two main difficulties in doing this. The first problem is due to the presence of discontinuities in the solution and the effect on them of discontinuities in the mesh. The second problem is how to organize the algorithm to minimize memory and CPU overhead. This is an important consideration and will continue to be important as more sophisticated algorithms that use data structures other than arrays are developed for use on vector and parallel computers.

In this paper, we present an efficient semi-Lagrangian based particle level set method for the accurate capturing of interfaces. This method retains the robust topological properties of the level set method with- out the adverse effects of numerical dissipation. Both the level set method and the particle level set method typically use high order accurate numerical discretizations in time and space, e.g. TVD Runge–Kutta and HJ-WENO schemes. We demonstrate that these computationally expensive schemes are not required. Instead, fast, low order accurate numerical schemes suffice. That is, the addition of particles to the level set method not only removes the difficulties associated with numerical diffusion, but also alleviates the need for computationally expensive high order accurate schemes. We use an efficient, first order accurate semi-Lagrangian advection scheme coupled with a first order accurate fast marching method to evolve the level set function. To accurately track the underlying flow characteristics, the particles are evolved with a second order accurate method. Since we avoid complex high order accurate numerical methods, extending the algorithm to arbitrary data structures becomes more feasible, and we show preliminary results obtained with an octree-based adaptive mesh.

We present a numerical method using the level set approach for solving incompressible two-phase flow with surface tension. In the level set approach, the free surface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the free surface, with the advection of the level set function, which is smooth. In addition, the free surface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier–Stokes equations, in order to achieve higher resolution of the free surface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

We present a new fluid animation technique in which liquid and gas interact with each other, using the exampleof bubbles rising in water. In contrast to previous studies which only focused on one fluid, our system considersboth the liquid and the gas simultaneously. In addition to the flowing motion, the interactions between liquid andgas cause buoyancy, surface tension, deformation and movement of the bubbles. For the natural manipulationof topological changes and the removal of the numerical diffusion, we combine the volume-of-fluid method andthe front-tracking method developed in the field of computational fluid dynamics. Our minimum-stress surfacetension method enables this complementary combination. The interfaces are constructed using the marching cubesalgorithm. Optical effects are rendered using vertex shader techniques.
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Animation

Abstract In this paper we describe a method for modeling and rendering dynamic behavior of fluids withsplashes and foam. A particle system is built into a fluid simulation system to represent an ocean wavecresting and spraying over another object. We use the Cubic Interpolated Propagation (CIP) method asthe fluid solver. The CIP method can solve liquid and gas together in the framework of fluid dynamicsand has high accuracy in the case of relatively coarse grids. This enables us to simulate the fluids in ashort time and describe the motion of splashes in the air that is associated with the liquid motion well.The foam floating on the water also can be described using the particle system. We integrate the rigidbody simulation with the fluid and particle system to create sophisticated scenes including splashes andfoam. We construct state change rules that are used with the particle system. This controls the generation,vanishing and transition rule of splashes and foam. The transition rule makes the seamless connection betweena splash and foam. We employed a fast volume rendering method with scattering effect for particles.One of the important features of our method is the combination of fast simulation and rendering techniques,which provides dynamic and realistic scenes in a short time.

We propose an adaptive approach for the fast reconstruction of isosurfaces from regular volume data at arbitrary levels of detail. The algorithm has been designed to enable real-time navigation through complex structures while providing user-adjustable resolution levels. Since adaptive on-the-fly reconstruction and rendering is performed from a hierarchical octree representation of the volume data, the method does not depend on preprocessing with respect to a specific isovalue, thus the user can browse interactively through the set of all possible isosurfaces. Special attention is paid to the fixing of cracks in the surface where the adaptive reconstruction level changes and to the efficient estimation of the isosurface's curvature.

A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired.

A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

We present serial and parallel algorithms for solving a system of
equations that arises from the discretization of the Hamilton-Jacobi
equation associated to a trajectory optimization problem of the
following type. A vehicle starts at a prespecified point x<sub>o</sub>
and follows a unit speed trajectory x(t) inside a region in ℛ<sup>m
</sup> until an unspecified time T that the region is exited. A
trajectory minimizing a cost function of the form
∫<sub>0</sub><sup>T</sup> r(x(t))dt+q(x(T)) is sought. The
discretized Hamilton-Jacobi equation corresponding to this problem is
usually solved using iterative methods. Nevertheless, assuming that the
function r is positive, we are able to exploit the problem structure and
develop one-pass algori