Compressed Neighbour Lists for SPH

We propose a novel compression scheme to store neighbour lists for iterative solvers that employ Smoothed Particle Hydrodynamics (SPH). The compression scheme is inspired by Stream VByte, but uses a non-linear mapping from data to data bytes, yielding memory savings of up to 87%. It is part of a novel variant of the Cell-Linked-List (CLL) concept that is inspired by compact hashing with an improved processing of the cell-particle relations. We show that the resulting neighbour search outperforms compact hashing in terms of speed and memory consumption. Divergence-Free SPH (DFSPH) scenarios with up to 1.3 billion SPH particles can be processed on a 24-core PC using 172 GB of memory. Scenes with more than 7 billion SPH particles can be processed in a Message Passing Interface (MPI) environment with 112 cores and 880 GB of RAM. The neighbour search is also useful for interactive applications. A DFSPH simulation step for up to 0.2 million particles can be computed in less than 40 ms on a 12-core PC.     

Narrow Band FLIP for Liquid Simulations

The Fluid Implicit Particle method (FLIP) for liquid simulations uses particles to reduce numerical dissipation and provide important visual cues for events like complex splashes and small‐scale features near the liquid surface. Unfortunately, FLIP simulations can be computationally expensive, because they require a dense sampling of particles to fill the entire liquid volume. Furthermore, the vast majority of these FLIP particles contribute nothing to the fluid's visual appearance, especially for larger volumes of liquid. We present a method that only uses FLIP particles within a narrow band of the liquid surface, while efficiently representing the remaining inner volume on a regular grid. We show that a naïve realization of this idea introduces unstable and uncontrollable energy fluctuations, and we propose a novel coupling scheme between FLIP particles and regular grid which overcomes this problem. Our method drastically reduces the particle count and simulation times while yielding results that are nearly indistinguishable from regular FLIP simulations. Our approach is easy to integrate into any existing FLIP implementation.     

Extended Narrow Band FLIP for Liquid Simulations

The Fluid Implicit Particle method (FLIP) reduces numerical dissipation by combining particles with grids. To improve performance, the subsequent narrow band FLIP method (NB‐FLIP) uses a FLIP‐based fluid simulation only near the liquid surface and a traditional grid‐based fluid simulation away from the surface. This spatially‐limited FLIP simulation significantly reduces the number of particles and alleviates a computational bottleneck. In this paper, we extend the NB‐FLIP idea even further, by allowing a simulation to transition between a FLIP‐like fluid simulation and a grid‐based simulation in arbitrary locations, not just near the surface. This approach leads to even more savings in memory and computation, because we can concentrate the particles only in areas where they are needed. More importantly, this new method allows us to seamlessly transition to smooth implicit surface geometry wherever the particle‐based simulation is unnecessary. Consequently, our method leads to a practical algorithm for avoiding the noisy surface artifacts associated with particle‐based liquid simulations, while simultaneously maintaining the benefits of a FLIP simulation in regions of dynamic motion.     

Temporal Blending for Adaptive SPH

In this paper, we introduce a fast and consistent smoothed particle hydrodynamics (SPH) technique which is suitable for convection–diffusion simulations of incompressible fluids. We apply our temporal blending technique to reduce the number of particles in the simulation while smoothly changing quantity fields. Our approach greatly reduces the error introduced in the pressure term when changing particle configurations. Compared to other methods, this enables larger integration time‐steps in the transition phase. Our implementation is fully GPU‐based to take advantage of the parallel nature of particle simulations.     

Creating and Preserving Vortical Details in SPH Fluid

We present a new method to create and preserve the turbulent details generated around moving objects in SPH fluid. In our approach, a high‐resolution overlapping grid is bounded to each object and translates with the object. The turbulence formation is modeled by resolving the local flow around objects using a hybrid SPH‐FLIP method. Then these vortical details are carried on SPH particles flowing through the local region and preserved in the global field in a synthetic way. Our method provides a physically plausible way to model the turbulent details around both rigid and deformable objects in SPH fluid, and can efficiently produce animations of complex gaseous phenomena with rich visual details.     

Hybrid Simulation of Miscible Mixing with Viscous Fingering

By modeling mass transfer phenomena, we simulate solids and liquids dissolving or changing to other substances. We also deal with the very small‐scale phenomena that occur when a fluid spreads out at the interface of another fluid. We model the pressure at the interfaces between fluids with Darcy's Law and represent the viscous fingering phenomenon in which a fluid interface spreads out with a fractal‐like shape. We use hybrid grid‐based simulation and smoothed particle hydrodynamics (SPH) to simulate intermolecular diffusion and attraction using particles at a computable scale. We have produced animations showing fluids mixing and objects dissolving.     

An Efficient Hybrid Incompressible SPH Solver with Interface Handling for Boundary Conditions

We propose a hybrid smoothed particle hydrodynamics solver for efficientlysimulating incompressible fluids using an interface handling method for boundary conditions in the pressure Poisson equation. We blend particle density computed with one smooth and one spiky kernel to improve the robustness against both fluid–fluid and fluid–solid collisions. To further improve the robustness and efficiency, we present a new interface handling method consisting of two components: free surface handling for Dirichlet boundary conditions and solid boundary handling for Neumann boundary conditions. Our free surface handling appropriately determines particles for Dirichlet boundary conditions using Jacobi‐based pressure prediction while our solid boundary handling introduces a new term to ensure the solvability of the linear system. We demonstrate that our method outperforms the state‐of‐the‐art particle‐based fluid solvers.     

A Parallel SPH Implementation on Multi‐Core CPUs

This paper presents a parallel framework for simulating fluids with the Smoothed Particle Hydrodynamics (SPH) method. For low computational costs per simulation step, efficient parallel neighbourhood queries are proposed and compared. To further minimize the computing time for entire simulation sequences, strategies for maximizing the time step and the respective consequences for parallel implementations are investigated. The presented experiments illustrate that the parallel framework can efficiently compute large numbers of time steps for large scenarios. In the context of neighbourhood queries, the paper presents optimizations for two efficient instances of uniform grids, that is, spatial hashing and index sort. For implementations on parallel architectures with shared memory, the paper discusses techniques with improved cache‐hit rate and reduced memory transfer. The performance of the parallel implementations of both optimized data structures is compared. The proposed solutions focus on systems with multiple CPUs. Benefits and challenges of potential GPU implementations are only briefly discussed.     

Hydraulic Erosion Using Smoothed Particle Hydrodynamics

This paper presents a new technique for modification of 3D terrains by hydraulic erosion. It efficiently couples fluid simulation using a Lagrangian approach, namely the Smoothed Particle Hydrodynamics (SPH) method, and a physically-based erosion model adopted from an Eulerian approach. The eroded sediment is associated with the SPH particles and is advected both implicitly, due to the particle motion, and explicitly, through an additional velocity field, which accounts for the sediment transfer between the particles. We propose a new donor-acceptor scheme for the explicit advection in SPH. Boundary particles associated to the terrain are used to mediate sediment exchange between the SPH particles and the terrain itself. Our results show that this particle-based method is efficient for the erosion of dense, large, and sparse fluid. Our implementation provides interactive results for scenes with up to 25,000 particles.     

Realtime Two‐Way Coupling of Meshless Fluids and Nonlinear FEM

In this paper, we present a novel method to couple Smoothed Particle Hydrodynamics (SPH) and nonlinear FEM to animate the interaction of fluids and deformable solids in real time. To accurately model the coupling, we generate proxy particles over the boundary of deformable solids to facilitate the interaction with fluid particles, and develop an efficient method to distribute the coupling forces of proxy particles to FEM nodal points. Specifically, we employ the Total Lagrangian Explicit Dynamics (TLED) finite element algorithm for nonlinear FEM because of many of its attractive properties such as supporting massive parallelism, avoiding dynamic update of stiffness matrix computation, and efficient solver. Based on a predictor‐corrector scheme for both velocity and position, different normal and tangential conditions can be realized even for shell‐like thin solids. Our coupling method is entirely implemented on modern GPUs using CUDA. We demonstrate the advantage of our two‐way coupling method in computer animation via various virtual scenarios.     

Local Poisson SPH For Viscous Incompressible Fluids

Enforcing fluid incompressibility is one of the time‐consuming aspects in SPH. In this paper, we present a local Poisson SPH (LPSPH) method to solve incompressibility for particle based fluid simulation. Considering the pressure Poisson equation, we first convert it into an integral form, and then apply a discretization to convert the continuous integral equation to a discretized summation over all the particles in the local pressure integration domain determined by the local geometry. To control the approximation error, we further integrate our local pressure solver into the predictive‐corrective framework to avoid the computational cost of solving a pressure Poisson equation globally. Our method can effectively eliminate the large density deviations mainly caused by the solid boundary treatment and free surface topological change, and show advantage of a higher convergence rate over the predictive‐corrective incompressible SPH (PCISPH).     

A Physically Consistent Implicit Viscosity Solver for SPH Fluids

In this paper, we present a novel physically consistent implicit solver for the simulation of highly viscous fluids using the Smoothed Particle Hydrodynamics (SPH) formalism. Our method is the result of a theoretical and practical in‐depth analysis of the most recent implicit SPH solvers for viscous materials. Based on our findings, we developed a list of requirements that are vital to produce a realistic motion of a viscous fluid. These essential requirements include momentum conservation, a physically meaningful behavior under temporal and spatial refinement, the absence of ghost forces induced by spurious viscosities and the ability to reproduce complex physical effects that can be observed in nature. On the basis of several theoretical analyses, quantitative academic comparisons and complex visual experiments we show that none of the recent approaches is able to satisfy all requirements. In contrast, our proposed method meets all demands and therefore produces realistic animations in highly complex scenarios. We demonstrate that our solver outperforms former approaches in terms of physical accuracy and memory consumption while it is comparable in terms of computational performance. In addition to the implicit viscosity solver, we present a method to simulate melting objects. Therefore, we generalize the viscosity model to a spatially varying viscosity field and provide an SPH discretization of the heat equation.     

Robust and Efficient Wave Simulations on Deforming Meshes

The goal of this paper is to enable the interactive simulation of phenomena such as animated fluid characters. While full 3D fluid solvers achieve this with control algorithms, these 3D simulations are usually too costly for real‐time environments. In order to achieve our goal, we reduce the problem from a three‐ to a two‐dimensional one, and make use of the shallow water equations to simulate surface waves that can be solved very efficiently. In addition to a low runtime cost, stability is likewise crucial for interactive applications. Hence, we make use of an implicit time integration scheme to obtain a robust solver. To ensure a low energy dissipation, we apply an Implicit Newmark time integration scheme. We propose a general formulation of the underlying equations that is tailored towards the use with an Implicit Newmark integrator. Furthermore, we gain efficiency by making use of a direct solver. Due to the generality of our formulation, the fluid simulation can be coupled interactively with arbitrary external forces, such as forces caused by inertia or collisions. We will discuss the properties of our algorithm, and demonstrate its robustness with simulations on strongly deforming meshes.     

Boundary Handling at Cloth–Fluid Contact

We present a robust and efficient method for the two‐way coupling between particle‐based fluid simulations and infinitesimally thin solids represented by triangular meshes. Our approach is based on a hybrid method that combines a repulsion force approach with a continuous intersection handling to guarantee that no penetration occurs. Moreover, boundary conditions for the tangential component of the fluid's velocity are implemented to model the different slip conditions. The proposed method is particularly useful for dynamic surfaces, like cloth and thin shells. In addition, we demonstrate how standard fluid surface reconstruction algorithms can be modified to prevent the calculated surface from intersecting close objects. For both the two‐way coupling and the surface reconstruction, we take into account that the fluid can wet the cloth. We have implemented our approach for the bidirectional interaction between liquid simulations based on Smoothed Particle Hydrodynamics (SPH) and standard mesh‐based cloth simulation systems.     

Advanced Hybrid Particle‐Grid Method with Sub‐Grid Particle Correction

This paper proposes a novel hybrid particle‐grid approach to liquid simulation, which uses the fluid‐implicit‐particle (FLIP) method to resolve the liquid motion and a grid‐based particle correction method to complement FLIP. The correction process addresses the high‐frequency errors in FLIP ensuring that the particles are properly distributed. The proposed approach enables the corrective procedure to avoid directly processing the particle relationships and supports flexible corrective forces. The proposed technique effectively and efficiently improves the distribution of the particles and therefore enhances the overall simulation quality. The experimental results confirm that the technique is able to conserve the liquid volume and to produce dynamic surface motions, thin liquid sheets, and smooth surfaces without disturbing artifacts such as bumpy noise.     

Incompressible SPH using the Divergence‐Free Condition

In this paper, we present a novel SPH framework to simulate incompressible fluid that satisfies both the divergence‐ free condition and the density‐invariant condition. In our framework, the two conditions are applied separately. First, the divergence‐free condition is enforced when solving the momentum equation. Later, the density‐invariant condition is applied after the time integration of the particle positions. Our framework is a purely Lagrangian approach so that no auxiliary grid is required. Compared to the previous density‐invariant based SPH methods, the proposed method is more accurate due to the explicit satisfaction of the divergence‐free condition. We also propose a modified boundary particle method for handling the free‐slip condition. In addition, two simple but effective methods are proposed to reduce the particle clumping artifact induced by the density‐invariant condition.     

Mixing Fluids and Granular Materials

Fluid animations in computer graphics show interactions with various kinds of objects. However, fluid flowing through a granular material such as sand is still not possible within current frameworks. In this paper, we present the simulation of fine granular materials interacting with fluids. We propose a unified Smoothed Particle Hydrodynamics framework for the simulation of both fluid and granular material. The granular volume is simulated as a continuous material sampled by particles. By incorporating previous work on porous flow in this simulation framework we are able to fully couple fluid and sand. Fluid can now percolate between sand grains and influence the physical properties of the sand volume. Our method demonstrates various new effects such as dry soil transforming into mud pools by rain or rigid sand structures being eroded by waves.     

Fast Particle‐based Visual Simulation of Ice Melting

The visual simulation of natural phenomena has been widely studied. Although several methods have been proposed to simulate melting, the flows of meltwater drops on the surfaces of objects are not taken into account. In this paper, we propose a particle‐based method for the simulation of the melting and freezing of ice objects and the interactions between ice and fluids. To simulate the flow of meltwater on ice and the formation of water droplets, a simple interfacial tension is proposed, which can be easily incorporated into common particle‐based simulation methods such as Smoothed Particle Hydrodynamics. The computations of heat transfer, the phase transition between ice and water, the interactions between ice and fluids, and the separation of ice due to melting are further accelerated by implementing our method using CUDA. We demonstrate our simulation and rendering method for depicting melting ice at interactive frame‐rates.     

An Efficient Boundary Handling with a Modified Density Calculation for SPH

We propose a new boundary handling method for smoothed particle hydrodynamics (SPH). Previous approaches required the use of boundary particles to prevent particles from sticking to the boundary. We address this issue by correcting the fundamental equations of SPH with the integration of a kernel function. Our approach is able to directly handle triangle mesh boundaries without the need for boundary particles. We also show how our approach can be integrated into a position‐based fluid framework.     

An Implicit SPH Formulation for Incompressible Linearly Elastic Solids

We propose a novel smoothed particle hydrodynamics (SPH) formulation for deformable solids. Key aspects of our method are implicit elastic forces and an adapted SPH formulation for the deformation gradient that—in contrast to previous work—allows a rotation extraction directly from the SPH deformation gradient. The proposed implicit concept is entirely based on linear formulations. As a linear strain tensor is used, a rotation‐aware computation of the deformation gradient is required. In contrast to existing work, the respective rotation estimation is entirely realized within the SPH concept using a novel formulation with incorporated kernel gradient correction for first‐order consistency. The proposed implicit formulation and the adapted rotation estimation allow for significantly larger time steps and higher stiffness compared to explicit forms. Performance gain factors of up to one hundred are presented. Incompressibility of deformable solids is accounted for with an ISPH pressure solver. This further allows for a pressure‐based boundary handling and a unified processing of deformables interacting with SPH fluids and rigids. Self‐collisions are implicitly handled by the pressure solver.