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Jan Bender and Dan Koschier, Divergence-Free Smoothed Particle Hydrodynamics, In Proceedings of ACM SIGGRAPH / EUROGRAPHICS Symposium on Computer Animation (SCA), 2015

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# Abstract

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.

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Jan Bender, Matthias MÃ¼ller and Miles Macklin, Position-Based Simulation Methods in Computer Graphics, In Tutorial Proceedings of Eurographics, 2015

# Abstract

The physically-based simulation of mechanical effects has been an important research topic in computer graphics for more than two decades. Classical methods in this field discretize Newton's second law and determine different forces to simulate various effects like stretching, shearing, and bending of deformable bodies or pressure and viscosity of fluids, to mention just a few. Given these forces, velocities and finally positions are determined by a numerical integration of the resulting accelerations.

In the last years position-based simulation methods have become popular in the graphics community. In contrast to classical simulation approaches these methods compute the position changes in each simulation step directly, based on the solution of a quasi-static problem. Therefore, position-based approaches are fast, stable and controllable which make them well-suited for use in interactive environments. However, these methods are generally not as accurate as force-based methods but still provide visual plausibility. Hence, the main application areas of position-based simulation are virtual reality, computer games and special effects in movies and commercials.

In this tutorial we first introduce the basic concept of position-based dynamics. Then we present different solvers and compare them with the classical implicit Euler method. We discuss approaches to improve the convergence of these solvers. Moreover, we show how position-based methods are applied to simulate hair, cloth, volumetric deformable bodies, rigid body systems and fluids. We also demonstrate how complex effects like anisotropy or plasticity can be simulated and introduce approaches to improve the performance. Finally, we give an outlook and discuss open problems.

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 Armadillos Elastoplastic Dragon Millipede Pile Different constraints

Jan Bender, Dan Koschier, Patrick Charrier and Daniel Weber, Position-Based Simulation of Continuous Materials, Computers & Graphics 44, 2014

# 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.

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Dan Koschier, Sebastian Lipponer and Jan Bender, Adaptive Tetrahedral Meshes for Brittle Fracture Simulation, In Proceedings of ACM SIGGRAPH / EUROGRAPHICS Symposium on Computer Animation (SCA), 2014, accepted

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# Abstract

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.

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Crispin Deul, Patrick Charrier and Jan Bender, Position-Based Rigid Body Dynamics, In Proceedings of the 27th International Conference on Computer Animation and Social Agents, 2014

# Abstract

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 therefore integrates nicely with 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.

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Jan Bender, Dynamiksimulation in der Computergraphik, Habilitationsschrift, KIT, KIT Scientific Publishing, 2014

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# Abstract

Die physikalisch-basierte Simulation von StarrkÃ¶rpern und deformierbaren FestkÃ¶rpern ist ein wichtiges und aktuelles Forschungsgebiet in der Computergraphik und ein essentieller Bestandteil in vielen Anwendungen, wie z.B. Virtual Prototyping, Computeranimationen, Spiele, Spezialeffekte in Filmen oder Trainingssimulatoren. Dabei stehen oft interaktive Simulationen im Vordergrund, in denen ein Benutzer in Echtzeit mit den simulierten KÃ¶rpern interagieren kann. Dadurch werden hohe Anforderungen an die Geschwindigkeit und StabilitÃ¤t der Simulationsverfahren gestellt.

In dieser Arbeit werden interaktive Simulationsmethoden fÃ¼r MehrkÃ¶rpersysteme, Textilien und inkompressible deformierbare VolumenkÃ¶rper vorgestellt. AuÃŸerdem wird gezeigt, wie die Simulation durch den Einsatz GPU-basierter Methoden deutlich beschleunigt werden kann.

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Jan Bender, Matthias MÃ¼ller, Miguel A. Otaduy, Matthias Teschner and Miles Macklin, A Survey on Position-Based Simulation Methods in Computer Graphics, Computer Graphics Forum 33, 6, 2014

# Abstract

The dynamic simulation of mechanical effects has a long history in computer graphics. The classical methods in this field discretize Newton's 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.

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 Cloth Wrinkle Mesh Armadillos Ducks and tori Hair simulation

Jan Bender, Kenny Erleben, Jeff Trinkle and Erwin Coumans, Interactive Simulation of Rigid Body Dynamics in Computer Graphics, In STAR Proceedings of Eurographics, 2012

# Abstract

Interactive rigid body simulation is an important part of many modern computer tools. No authoring tool nor a game engine can do without. The 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 has evolved and matured over the past 20 years. The paper covers applications and the usage of interactive rigid body simulation.

Besides the mathematical and theoretical details that this paper communicates in a pedagogical manner the paper surveys common practice and reflects on applications of interactive rigid body simulation. The grand merger of interactive and off-line simulation methods is imminent, multi-core is everyman's property. These observations pose future challenges for research which we reflect on. In perspective several avenues for possible future work is touched upon such as more descriptive models and contact point generation problems. This paper is not only a stake in the sand on what has been done, it also seeks to give newcomers practical hands on advices and reflections that can give experienced researchers afterthought for the future.

 Crash test Newton's cradle Roof on impact Jinngine Temple Silo

Crispin Deul and Jan Bender, Physically-Based Character Skinning, Virtual Reality Interactions and Physical Simulations (VRIPhys), 2013

# Abstract

In this paper we present a novel multi-layer model for physically-based character skinning. In contrast to geometric approaches which are commonly used in the field of character skinning, physically-based methods can simulate secondary motion effects. Furthermore, these methods can handle collisions and preserve the volume of the model without the need of an additional post-process. Physically-based approaches are computationally more expensive than geometric methods but they provide more realistic results. Recent works in this area use finite element simulations to model the elastic behavior of skin. These methods require the generation of a volumetric mesh for the skin shape in a pre-processing step. It is not easy for an artist to model the different elastic behaviors of muscles, fat and skin using a volumetric mesh since there is no clear assignment between volume elements and tissue types. For our novel multi-layer model the mesh generation is very simple and can be performed automatically. Furthermore, the model contains a layer for each kind of tissue. Therefore, the artist can easily control the elastic behavior by adjusting the stiffness parameters for muscles, fat and skin. We use shape matching with oriented particles and a fast summation technique to simulate the elastic behavior of our skin model and a position-based constraint enforcement to handle collisions, volume conservation and the coupling of the skeleton with the deformable model. Position-based methods have the advantage that they are fast, unconditionally stable, controllable and provide visually plausible results.

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 linear blend skinning our approach

Nikolas Schmitt, Martin Knuth, Jan Bender and Arjan Kuijper, Multilevel Cloth Simulation using GPU Surface Sampling, Virtual Reality Interactions and Physical Simulations (VRIPhys), 2013

# Abstract

Today most cloth simulation systems use triangular mesh models. However, regular grids allow many optimizations as connectivity is implicit, warp and weft directions of the cloth are aligned to grid edges and distances between particles are equal. In this paper we introduce a cloth simulation that combines both model types. All operations that are performed on the CPU use a low-resolution triangle mesh while GPU-based methods are performed efficiently on a high-resolution grid representation. Both models are coupled by a sampling operation which renders triangle vertex data into a texture and by a corresponding projection of texel data onto a mesh. The presented scheme is very flexible and allows individual components to be performed on different architectures, data representations and detail levels. The results are combined using shader programs which causes a negligible overhead. We have implemented CPU-based collision handling and a GPU-based hierarchical constraint solver to simulate systems with more than 230k particles in real-time.

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Jan Bender, Daniel Weber and Raphael Diziol, Fast and stable cloth simulation based on multi-resolution shape matching, Computers & Graphics 37, 8, 2013

# Abstract

We present an efficient and unconditionally stable method which allows the deformation of very complex stiff cloth models in real-time. This method is based on a shape matching approach which uses edges and triangles as 1D and 2D regions to simulate stretching and shearing resistance. Previous shape matching approaches require large overlapping regions to simulate stiff materials. This unfortunately also affects the bending behavior of the model. Instead of using large regions, we introduce a novel multi-resolution shape matching approach to increase only the stretching and shearing stiffness. Shape matching is performed for each level of the multi-resolution model and the results are propagated from one level to the next one. To preserve the fine wrinkles of the cloth on coarse levels of the hierarchy we present a modified version of the original shape matching method. The introduced method for cloth simulation can perform simulations in linear time and has no numerical damping. Furthermore, we show that multi-resolution shape matching can be performed efficiently on the GPU.

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Jan Bender and Crispin Deul, Adaptive cloth simulation using corotational finite elements, Computers & Graphics 37, 7, 2013

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# Abstract

In this article we introduce an efficient adaptive cloth simulation method which is based on a reversible $\sqrt{3}$-refinement of corotational finite elements. Our novel approach can handle arbitrary triangle meshes and is not restricted to regular grid meshes which are required by other adaptive methods. Most previous works in the area of adaptive cloth simulation use discrete cloth models like mass-spring systems in combination with a specific subdivision scheme. However, if discrete models are used, the simulation does not converge to the correct solution as the mesh is refined. Therefore, we introduce a cloth model which is based on continuum mechanics since continuous models do not have this problem. We use a linear elasticity model in combination with a corotational formulation to achieve a high performance. Furthermore, we present an efficient method to update the sparse matrix structure after a refinement or coarsening step. The advantage of the $\sqrt{3}$-subdivision scheme is that it generates high quality meshes while the number of triangles increases only by a factor of 3 in each refinement step. However, the original scheme was not intended for the use in an interactive simulation and only defines a mesh refinement. In this article we introduce a combination of the original refinement scheme with a novel coarsening method to realize an adaptive cloth simulation with high quality meshes. The proposed approach allows an efficient mesh adaption and therefore does not cause much overhead. We demonstrate the significant performance gain which can be achieved with our adaptive simulation method in several experiments including a complex garment simulation.

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Jan Bender, Matthias MÃ¼ller, Miguel A. Otaduy and Matthias Teschner, Position-based Methods for the Simulation of Solid Objects in Computer Graphics, In STAR Proceedings of Eurographics, 2013

# Abstract

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.

# Images

 Cloth Wrinkle Mesh Armadillos Ducks and tori Hair simulation

Daniel Weber, Jan Bender, Markus Schnoes, Andre Stork and Dieter Fellner, Efficient GPU data structures and methods to solve sparse linear systems in dynamics applications, In Computer Graphics Forum, Computer Graphics Forum 32, 1, 2013

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# Abstract

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.

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# Images

 Bridge Cloth Armadillo Pensatore

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