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Throughout the past decades, the graphics community has spent major resources on the research and development of physics simulators on the mission to computer-generate behaviors achieving outstanding visual effects or to make the virtual world indistinguishable from reality. The variety and impact of recent research based on Smoothed Particle Hydrodynamics (SPH) demonstrates the concept's importance as one of the most versatile tools for the simulation of fluids and solids. With this survey, we offer an overview of the developments and still-active research on physics simulation methodologies based on SPH that has not been addressed in previous SPH surveys. Following an introduction about typical SPH discretization techniques, we provide an overview over the most used incompressibility solvers and present novel insights regarding their relation and conditional equivalence. The survey further covers recent advances in implicit and particle-based boundary handling and sampling techniques. While SPH is best known in the context of fluid simulation we discuss modern concepts to augment the range of simulatable physical characteristics including turbulence, highly viscous matter, deformable solids, as well as rigid body contact handling. Besides the purely numerical approaches, simulation techniques aided by machine learning are on the rise. Thus, the survey discusses recent data-driven approaches and the impact of differentiable solvers on artist control. Finally, we provide context for discussion by outlining existing problems and opportunities to open up new research directions.

In this paper, we present a novel method for the robust handling of static and dynamic rigid boundaries in Smoothed Particle Hydrodynamics (SPH) simulations. We build upon the ideas of the density maps approach which has been introduced recently by Koschier and Bender. They precompute the density contributions of solid boundaries and store them on a spatial grid which can be efficiently queried during runtime. This alleviates the problems of commonly used boundary particles, like bumpy surfaces and inaccurate pressure forces near boundaries. Our method is based on a similar concept but we precompute the volume contribution of the boundary geometry. This maintains all benefits of density maps but offers a variety of advantages which are demonstrated in several experiments. Firstly, in contrast to the density maps method we can compute derivatives in the standard SPH manner by differentiating the kernel function. This results in smooth pressure forces, even for lower map resolutions, such that precomputation times and memory requirements are reduced by more than two orders of magnitude compared to density maps. Furthermore, this directly fits into the SPH concept so that volume maps can be seamlessly combined with existing SPH methods. Finally, the kernel function is not baked into the map such that the same volume map can be used with different kernels. This is especially useful when we want to incorporate common surface tension or viscosity methods that use different kernels than the fluid simulation.

In this paper we introduce a novel micropolar material model for the simulation of turbulent inviscid fluids. The governing equations are solved by using the concept of Smoothed Particle Hydrodynamics (SPH). As already investigated in previous works, SPH fluid simulations suffer from numerical diffusion which leads to a lower vorticity, a loss in turbulent details and finally in less realistic results. To solve this problem we propose a micropolar fluid model. The micropolar fluid model is a generalization of the classical Navier-Stokes equations, which are typically used in computer graphics to simulate fluids. In contrast to the classical Navier-Stokes model, micropolar fluids have a microstructure and therefore consider the rotational motion of fluid particles. In addition to the linear velocity field these fluids also have a field of microrotation which represents existing vortices and provides a source for new ones. However, classical micropolar materials are viscous and the translational and the rotational motion are coupled in a dissipative way. Since our goal is to simulate turbulent fluids, we introduce a novel modified micropolar material for inviscid fluids with a non-dissipative coupling. Our model can generate realistic turbulences, is linear and angular momentum conserving, can be easily integrated in existing SPH simulation methods and its computational overhead is negligible. Another important visual feature of turbulent liquids is foam. Therefore, we present a post-processing method which considers microrotation in the foam particle generation. It works completely automatic and requires only one user-defined parameter to control the amount of foam.

Graphics research on Smoothed Particle Hydrodynamics (SPH) has produced fantastic visual results that are unique across the board of research communities concerned with SPH simulations. Generally, the SPH formalism serves as a spatial discretization technique, commonly used for the numerical simulation of continuum mechanical problems such as the simulation of fluids, highly viscous materials, and deformable solids. Recent advances in the field have made it possible to efficiently simulate massive scenes with highly complex boundary geometries on a single PC. Moreover, novel techniques allow to robustly handle interactions among various materials. As of today, graphics-inspired pressure solvers, neighborhood search algorithms, boundary formulations, and other contributions often serve as core components in commercial software for animation purposes as well as in computer-aided engineering software.

This tutorial covers various aspects of SPH simulations. Governing equations for mechanical phenomena and their SPH discretizations are discussed. Concepts and implementations of core components such as neighborhood search algorithms, pressure solvers, and boundary handling techniques are presented. Implementation hints for the realization of SPH solvers for fluids, elastic solids, and rigid bodies are given. The tutorial combines the introduction of theoretical concepts with the presentation of actual implementations.

In this paper, we present a novel method for the robust handling of static and dynamic rigid boundaries in Smoothed Particle Hydrodynamics (SPH) simulations. We build upon the ideas of the density maps approach which has been introduced recently by Koschier and Bender. They precompute the density contributions of solid boundaries and store them on a spatial grid which can be efficiently queried during runtime. This alleviates the problems of commonly used boundary particles, like bumpy surfaces and inaccurate pressure forces near boundaries. Our method is based on a similar concept but we precompute the volume contribution of the boundary geometry and store it on a grid. This maintains all benefits of density maps but offers a variety of advantages which are demonstrated in several experiments. Firstly, in contrast to the density maps method we can compute derivatives in the standard SPH manner by differentiating the kernel function. This results in smooth pressure forces, even for lower map resolutions, such that precomputation times and memory requirements are reduced by more than two orders of magnitude compared to density maps. Furthermore, this directly fits into the SPH concept so that volume maps can be seamlessly combined with existing SPH methods. Finally, the kernel function is not baked into the map such that the same volume map can be used with different kernels. This is especially useful when we want to incorporate common surface tension or viscosity methods that use different kernels than the fluid simulation.

In this paper we present a novel operator splitting approach for corotated FEM simulations. The deformation energy of the corotated linear material model consists of two additive terms. The first term models stretching in the individual spatial directions and the second term describes resistance to volume changes. By formulating the backward Euler time integration scheme as an optimization problem, we show that the first term is invariant to rotations. This allows us to use an operator splitting approach and to solve both terms individually with different numerical methods. The stretching part is solved accurately with an optimization integrator, which can be done very efficiently because the system matrix is constant over time such that its Cholesky factorization can be precomputed. The volume term is solved approximately by using the compliant constraints method and Gauss-Seidel iterations. Further, we introduce the analytic polar decomposition which allows us to speed up the extraction of the rotational part of the deformation gradient and to recover inverted elements. Finally, this results in an extremely fast and robust simulation method with high visual quality that outperforms standard corotated FEMs by more than two orders of magnitude and even the fast but inaccurate PBD and shape matching methods by more than one order of magnitude without having their typical drawbacks. This enables a very efficient simulation of complex scenes containing more than a million elements.

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.

In this paper we present a robust remeshing-free cutting algorithm on the basis of the eXtended Finite Element Method (XFEM) and fully implicit time integration. One of the most crucial points of the XFEM is that integrals over discontinuous polynomials have to be computed on subdomains of the polyhedral elements. Most existing approaches construct a cut-aligned auxiliary mesh for integration. In contrast, we propose a cutting algorithm that includes the construction of specialized quadrature rules for each dissected element without the requirement to explicitly represent the arising subdomains. Moreover, we solve the problem of ill-conditioned or even numerically singular solver matrices during time integration using a novel algorithm that constrains non-contributing degrees of freedom (DOFs) and introduce a preconditioner that efficiently reuses the constructed quadrature weights. Our method is particularly suitable for fine structural cutting as it decouples the added number of DOFs from the cut's geometry and correctly preserves geometry and physical properties by accurate integration. Due to the implicit time integration these fine features can still be simulated robustly using large time steps. As opposed to this, the vast majority of existing approaches either use remeshing or element duplication. Remeshing based methods are able to correctly preserve physical quantities but strongly couple cut geometry and mesh resolution leading to an unnecessary large number of additional DOFs. Element duplication based approaches keep the number of additional DOFs small but fail at correct conservation of mass and stiffness properties. We verify consistency and robustness of our approach on simple and reproducible academic examples while stability and applicability are demonstrated in large scenarios with complex and fine structural cutting.

In this paper we present an hp-adaptive algorithm to generate discrete higher-order polynomial Signed Distance Fields (SDFs) on axis-aligned hexahedral grids from manifold polygonal input meshes. Using an orthonormal polynomial basis, we efficiently fit the polynomials to the underlying signed distance function on each cell. The proposed error-driven construction algorithm is globally adaptive and iteratively refines the SDFs using either spatial subdivision (h-refinement) following an octree scheme or by cell-wise adaption of the polynomial approximation's degree (p-refinement). We further introduce a novel decision criterion based on an error-estimator in order to decide whether to apply p- or h-refinement. We demonstrate that our method is able to construct more accurate SDFs at significantly lower memory consumption compared to previous approaches. While the cell-wise polynomial approximation will result in highly accurate SDFs, it can not be guaranteed that the piecewise approximation is continuous over cell interfaces. Therefore, we propose an optimization-based post-processing step in order to weakly enforce continuity. Finally, we apply our generated SDFs as collision detector to the physically-based simulation of geometrically highly complex solid objects in order to demonstrate the practical relevance and applicability of our method.

In this paper we present a novel Smoo­thed Particle Hy­dro­dy­na­mics (SPH) method for the efficient and stable simulation of incompressible fluids. The most efficient SPH-based approaches enforce incompressibility either on position or velocity level. However, the continuity equation for incompressible flow demands to maintain a constant density and a divergence-free velocity field. We propose a combination of two novel implicit pressure solvers enforcing both a low volume compression as well as a divergence-free velocity field. While a compression-free fluid is essential for realistic physical behavior, a divergence-free velocity field drastically reduces the number of required solver iterations and increases the stability of the simulation significantly. Thanks to the improved stability, our method can handle larger time steps than previous approaches. This results in a substantial performance gain since the computationally expensive neighborhood search has to be performed less frequently. Moreover, we introduce a third optional implicit solver to simulate highly viscous fluids which seamlessly integrates into our solver framework. Our implicit viscosity solver produces realistic results while introducing almost no numerical damping. We demonstrate the efficiency, robustness and scalability of our method in a variety of complex simulations including scenarios with millions of turbulent particles or highly viscous materials.

In this paper we introduce a novel micropolar material model for the simulation of turbulent inviscid fluids. The governing equations are solved by using the concept of Smoothed Particle Hydrodynamics (SPH). As already investigated in previous works, SPH fluid simulations suffer from numerical diffusion which leads to a lower vorticity, a loss in turbulent details and finally in less realistic results. To solve this problem we propose a micropolar fluid model. The micropolar fluid model is a generalization of the classical Navier-Stokes equations, which are typically used in computer graphics to simulate fluids. In contrast to the classical Navier-Stokes model, micropolar fluids have a microstructure and therefore consider the rotational motion of fluid particles. In addition to the linear velocity field these fluids also have a field of microrotation which represents existing vortices and provides a source for new ones. However, classical micropolar materials are viscous and the translational and the rotational motion are coupled in a dissipative way. Since our goal is to simulate turbulent fluids, we introduce a novel modified micropolar material for inviscid fluids with a non-dissipative coupling. Our model can generate realistic turbulences, is linear and angular momentum conserving, can be easily integrated in existing SPH simulation methods and its computational overhead is negligible.

In this paper, we present the novel concept of density maps for robust handling of static and rigid dynamic boundaries in fluid simulations based on Smoothed Particle Hydrodynamics (SPH). In contrast to the vast majority of existing approaches, we use an implicit discretization for a continuous extension of the density field throughout solid boundaries. Using the novel representation we enhance accuracy and efficiency of density and density gradient evaluations in boundary regions by computationally efficient lookups into our density maps. The map is generated in a preprocessing step and discretizes the density contribution in the boundary's near-field. In consequence of the high regularity of the continuous boundary density field, we use cubic Lagrange polynomials on a narrow-band structure of a regular grid for discretization. This strategy not only removes the necessity to sample boundary surfaces with particles but also decouples the particle size from the number of sample points required to represent the boundary. Moreover, it solves the ever-present problem of particle deficiencies near the boundary. In several comparisons we show that the representation is more accurate than particle samplings, especially for smooth curved boundaries. We further demonstrate that our approach robustly handles scenarios with highly complex boundaries and even outperforms one of the most recent sampling based techniques.

In this paper we propose a novel method to construct hierarchical $hp$-adaptive Signed Distance Fields (SDFs). We discretize the signed distance function of an input mesh using piecewise polynomials on an axis-aligned hexahedral grid. Besides spatial refinement based on octree subdivision to refine the cell size (h), we hierarchically increase each cell's polynomial degree (p) in order to construct a very accurate but memory-efficient representation. Presenting a novel criterion to decide whether to apply h- or p-refinement, we demonstrate that our method is able to construct more accurate SDFs at significantly lower memory consumption than previous approaches. Finally, we demonstrate the usage of our representation as collision detector for geometrically highly complex solid objects in the application area of physically-based simulation.

We present a new method for particle based fluid simulation, using a combination of Projective Dynamics and Smoothed Particle Hydrodynamics (SPH). The Projective Dynamics framework allows the fast simulation of a wide range of constraints. It offers great stability through its implicit time integration scheme and is parallelizable in large parts, so that it can make use of modern multi core CPUs. Yet existing work only uses Projective Dynamics to simulate various kinds of soft bodies and cloth. We are the first ones to incorporate fluid simulation into the Projective Dynamics framework. Our proposed fluid constraints are derived from SPH and seamlessly integrate into the existing method. Furthermore, we adapt the solver to handle the constantly changing constraints that appear in fluid simulation. We employ a highly parallel matrix-free conjugate gradient solver, and thus do not require expensive matrix factorizations.

In this paper we introduce an efficient and stable implicit SPH method for the physically-based simulation of incompressible fluids. In the area of computer graphics the most efficient SPH approaches focus solely on the correction of the density error to prevent volume compression. However, the continuity equation for incompressible flow also demands a divergence-free velocity field which is neglected by most methods. Although a few methods consider velocity divergence, they are either slow or have a perceivable density fluctuation.

Our novel method uses an efficient combination of two pressure solvers which enforce low volume compression (below 0.01%) and a divergence-free velocity field. This can be seen as enforcing incompressibility both on position level and velocity level. The first part is essential for realistic physical behavior while the divergence-free state increases the stability significantly and reduces the number of solver iterations. Moreover, it allows larger time steps which yields a considerable performance gain since particle neighborhoods have to be updated less frequently. Therefore, our divergence-free SPH (DFSPH) approach is significantly faster and more stable than current state-of-the-art SPH methods for incompressible fluids. We demonstrate this in simulations with millions of fast moving particles.

We present a method for the adaptive simulation of brittle fracture of solid objects based on a novel reversible tetrahedral mesh refinement scheme. The refinement scheme preserves the quality of the input mesh to a large extent, it is solely based on topological operations, and does not alter the boundary, i.e. any geometric feature. Our fracture algorithm successively performs a stress analysis and increases the resolution of the input mesh in regions of high tensile stress. This results in an accurate location of crack origins without the need of a general high resolution mesh which would cause high computational costs throughout the whole simulation. A crack is initiated when the maximum tensile stress exceeds the material strength. The introduced algorithm then proceeds by iteratively recomputing the changed stress state and creating further cracks. Our approach can generate multiple cracks from a single impact but effectively avoids shattering artifacts. Once the tensile stress decreases, the mesh refinement is reversed to increase the performance of the simulation. We demonstrate that our adaptive method is robust, scalable and computes highly realistic fracture results.

We introduce a novel fast and robust simulation method for deformable solids that supports complex physical effects like lateral contraction, anisotropy or elastoplasticity. Our method uses a continuum-based formulation to compute strain and bending energies for two- and three-dimensional bodies. In contrast to previous work, we do not determine forces to reduce these potential energies, instead we use a position-based approach. This combination of a continuum-based formulation with a position-based method enables us to keep the simulation algorithm stable, fast and controllable while providing the ability to simulate complex physical phenomena lacking in former position-based approaches. We demonstrate how to simulate cloth and volumetric bodies with lateral contraction, bending, plasticity as well as anisotropy and proof robustness even in case of degenerate or inverted elements. Due to the continuous material model of our method further physical phenomena like fracture or viscoelasticity can be easily implemented using already existing approaches. Furthermore, a combination with other geometrically motivated methods is possible.