Jan Bender

Interactive Simulation of Rigid Body Dynamics in Computer Graphics

Interactive rigid body simulation is an important part of many modern computer tools, which no authoring tool nor game engine can do without. Such high‐performance computer tools open up new possibilities for changing how designers, engineers, modelers and animators work with their design problems. This paper is a self contained state‐of‐the‐art report on the physics, the models, the numerical methods and the algorithms used in interactive rigid body simulation all of which have evolved and matured over the past 20 years. Furthermore, the paper communicates the mathematical and theoretical details in a pedagogical manner. This paper is not only a stake in the sand on what has been done, it also seeks to give the reader deeper insights to help guide their future research.

A Survey on Position-Based Simulation Methods in Computer Graphics

The dynamic simulation of mechanical effects has a long history in computer graphics. The classical methods in this field discretize Newtons second law in a variety of Lagrangian or Eulerian ways, and formulate forces appropriate for each mechanical effect: joints for rigid bodies; stretching, shearing or bending for deformable bodies and pressure, or viscosity for fluids, to mention just a few. In the last years, the class of position‐based methods has become popular in the graphics community. These kinds of methods are fast, stable and controllable which make them well‐suited for use in interactive environments. Position‐based methods are not as accurate as force‐based methods in general but they provide visual plausibility. Therefore, the main application areas of these approaches are virtual reality, computer games and special effects in movies. This state‐of‐the‐art report covers the large variety of position‐based methods that were developed in the field of physically based simulation. We will introduce the concept of position‐based dynamics, present dynamic simulation based on shape matching and discuss data‐driven upsampling approaches. Furthermore, we will present several applications for these methods.

Efficient GPU Data Structures and Methods to Solve Sparse Linear Systems in Dynamics Applications

We present graphics processing unit (GPU) data structures and algorithms to efficiently solve sparse linear systems that are typically required in simulations of multi‐body systems and deformable bodies. Thereby, we introduce an efficient sparse matrix data structure that can handle arbitrary sparsity patterns and outperforms current state‐of‐the‐art implementations for sparse matrix vector multiplication. Moreover, an efficient method to construct global matrices on the GPU is presented where hundreds of thousands of individual element contributions are assembled in a few milliseconds. A finite‐element‐based method for the simulation of deformable solids as well as an impulse‐based method for rigid bodies are introduced in order to demonstrate the advantages of the novel data structures and algorithms. These applications share the characteristic that a major computational effort consists of building and solving systems of linear equations in every time step. Our solving method results in a speed‐up factor of up to 13 in comparison to other GPU methods.

Position-based Methods for the Simulation of Solid Objects in Computer Graphics

The dynamic simulation of solids has a long history in computer graphics. The classical methods in this field are based on the use of forces or impulses to simulate joints between rigid bodies as well as the stretching, shearing and bending stiffness of deformable objects. In the last years the class of position-based methods has become popular in the graphics community. These kinds of methods are fast, unconditionally stable and controllable which make them well-suited for the use in interactive environments. Position-based methods are not as accurate as force based methods in general but they provide visual plausibility. Therefore, the main application areas of these approaches are virtual reality, computer games and special effects in movies. This state of the art report covers the large variety of position-based methods that were developed in the field of deformable solids. We will introduce the concept of position-based dynamics, present dynamic simulation based on shape matching and discuss data-driven approaches. Furthermore, we will present several applications for these methods.

Position-Based Simulation of Continuous Materials

Abstract 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.

Position-based rigid-body dynamics

We propose a position‐based approach for large‐scale simulations of rigid bodies at interactive frame rates. Our method solves positional constraints between rigid bodies and can therefore be seamlessly integrated into other position‐based methods. Interaction of particles and rigid bodies through common constraints enables two‐way coupling with deformables. The method exhibits exceptional performance and stability while being user controllable and easy to implement. Various results demonstrate the practicability of our method for the resolution of collisions, contacts, stacking and joint constraints. Copyright

Divergence-free smoothed particle hydrodynamics

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.

Divergence-Free SPH for Incompressible and Viscous Fluids

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.

Robust real-time deformation of incompressible surface meshes

We introduce an efficient technique for robustly simulating incompressible objects with thousands of elements in real-time. Instead of considering a tetrahedral model, commonly used to simulate volumetric bodies, we simply use their surfaces. Not requiring hundreds or even thousands of elements in the interior of the object enables us to simulate more elements on the surface, resulting in high quality deformations at low computation costs. The elasticity of the objects is robustly simulated with a geometrically motivated shape matching approach which is extended by a fast summation technique for arbitrary triangle meshes suitable for an efficient parallel computation on the GPU. Moreover, we present an oscillation-free and collision-aware volume constraint, purely based on the surface of the incompressible body. The novel heuristic we propose in our approach enables us to conserve the volume, both globally and locally. Our volume constraint is not limited to the shape matching method and can be used with any method simulating the elasticity of an object. We present several examples which demonstrate high quality volume conserving deformations and compare the run-times of our CPU implementation, as well as our GPU implementation with similar methods.

Fast Dynamic Simulation of Multi-Body Systems Using Impulses

A dynamic simulation method for multi-body systems is prese nt d in this paper. The special feature of this method is that it satisfies all given constraints by computing impul ses. In each simulation step the joint states after the step are predicted. In order to obtain valid states after the simu lation step, impulses are computed and applied to the connected bodies. Since a valid joint state is targeted exac tly, there is no drift as the simulation proceeds in time and so no additional stabilisation is required. In previous approaches the impulses for a multi-body system were computed iteratively. Since dependencies between joints w ere not taken into account, the simulation of complex models was slow. A novel method is presented that uses a syste m of linear equations to describe these dependencies. By solving this typically sparse system the required i mpulses are determined. This method allows a very fast simulation of complex multi-body systems.