- Research topics
Jacobi Method for Distributed Sparse Linear Systems on Multiple GPUs using CUDA
Darmstadt, TU, Master Thesis, 2018
Modern graphics processing units (GPUs) having many-core architecture are capable of accelerating simulation applications based on Parallel Differential Equation (PDE) tremendously. PDE solvers are the backbone of physics-based simulations and are used in wide variety of applications such as Computational Fluid Dynamics (CFD), computer games, augmented and virtual environments. Modern high-performance computing (HPC) architectures are equipped with multiple GPUs. This pushes the development of parallel algorithms towards solving fine resolution problems which are often inherently large for a single GPU. A Jacobi iterative solver is a popular PDE solver for Poisson equation.This thesis focuses on the scalability of a single GPU Jacobi solver over multiple GPUs. Transferring of data between multiple GPUs in a node remains the primary bottleneck and thus prevents from developing an efficient solver. Redesigning a solver which works on a single GPU and scaling it over multiple GPUs is challenging. Such a task would need identifying and exploiting parallelism opportunities and minimizing the communication latency when transferring data from one GPU to another. Existing methods used for multi-GPU communication are analyzed and optimal approaches are suggested for exploiting the inter-GPU memory bandwidth. Different domain decomposition methods are also suggested and the implications on performance are evaluated. Time taken to solve a problem increases with the resolution of the problem on a single GPU. Dividing the problem into smaller parts and distributing them to multiple GPUs could result in faster computation times. The implications of distributing such a problem are evaluated by measuring computation times and communication latency. Thereby, suggesting ways to optimize the solver by increasing the parallelism.
Efficient Methods for Nonlinear Materials for Interactive Deformation Simulations based on Finite Elements
Darmstadt, TU, Master Thesis, 2017
This thesis evolves around the interactive simulation of deformable objects with the finite element method. It is focused on the case where the underlying material model is nonlinear. It is analyzed which parameter influence interactivity and how much detail w.r.t. discretization is possible. Another challenging task is how to handle inversions. In this thesis an approach is developed that efficiently deals with those. Implicit time integration is necessary to guarantee stability with large time steps. Here, nonlinear materials require the solution of a nonlinear system in each time step. There is no guarantee that iterative solvers converge in those systems. For that reason, stabilization methods are developed that improve the convergence. A nonlinear material model for the simulation of a muscle is implemented and optimized. It is compared to a linearized model to gain a better understanding of the impacts of linearization.
The Finite Element Method for Nonlinear Elasticity in the Field of Computer Graphics
Darmstadt, TU, Studienarbeit, 2017
In this survey different approaches for the simulation of nonlinear elastic behavior of objects were analyzed. The necessary physical background to understand where those nonlinearities emerge as well as the fundamentals of the linear and corotational FEM were presented. Then, different solution concepts for the nonlinear equations from the literature were introduced and analyzed w.r.t. computer graphics criteria. The advantages and disadvantages of each approach was shown in a side by side comparison. Furthermore, situations when each approach could find use were discussed.
A Finite-Volume Discretization for Structural Mechanical Problems in Static Equlibrium
Wiesbaden, Hochschule RheinMain, Master Thesis, 2016
This thesis presents a novel and self-consistent finite volume method (FVM) for computational solid mechanics. Starting from a general dynamic approach, based on a balance of force in static equilibrium, we find a displacement based formulation using the general Hooke's law and the kinematic relationship in the form of the Cauchy strain tensor. Thus we provide an all-purpose method that allows to use isotropic, orthotropic and fully anisotropic material, no matter if it is homogeneous or inhomogeneous. This approach can be the basis of various discretization methods. Associated to the FVM used in fluid dynamics, we apply a cell centered method, approximate the integrals using the midpoint rule and use a difference scheme to arrange the cell-to-cell flux and find a linear system of equations. To disassemble the simulation domain, we pass up a complex mesher and use an equidistant and regular grid. Furthermore we implement a basic cut cell method to get a better approximation of the boundary of the domain. For a suitable description of mechanical structures it is required to work with tensors and vector fields. This is in contrast to problems in fluid dynamics, where the approach can be reduced to vectors and scalar fields. The corresponding changes and extensions are introduced and discussed.
A Cut-Cell Geometric Multigrid Poisson Solver for Fluid Simulation
Computer Graphics Forum
Annual Conference of the European Association for Computer Graphics (Eurographics) <36, 2015, Zürich, Switzerland>
We present a novel multigrid scheme based on a cut-cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved rate of convergence compared to previous methods. Existing fluid solvers with voxelized domains can directly benefit from this approach by only modifying the representation of the non-fluid domain. The necessary building blocks are fully parallelizable and can therefore benefit from multi- and many-core architectures.
Deformation Simulation using Cubic Finite Elements and Efficient p-multigrid Methods
Computers & Graphics
We present a novel p-multigrid method for efficient simulation of corotational elasticity with higher-order finite elements. In contrast to other multigrid methods proposed for volumetric deformation, the resolution hierarchy is realized by varying polynomial degrees on a tetrahedral mesh. The multigrid approach can be either used as a direct method or as a preconditioner for a conjugate gradient algorithm. We demonstrate the efficiency of our approach and compare it to commonly used direct sparse solvers and preconditioned conjugate gradient methods. As the polynomial representation is defined w.r.t. the same mesh, the update of the matrix hierarchy necessary for corotational elasticity can be computed efficiently. We introduce the use of cubic finite elements for volumetric deformation and investigate different combinations of polynomial degrees for the hierarchy. We analyze the applicability of cubic finite elements for deformation simulation by comparing analytical results in a static and dynamic scenario and demonstrate our algorithm in dynamic simulations with quadratic and cubic elements. Applying our method to quadratic and cubic finite elements results in a speed-up of up to a factor of 7 for solving the linear system.
Effiziente Simulation von Masse-Feder-Systemen
Darmstadt, TU, Bachelor Thesis, 2015
In dieser Arbeit wird ein Zeitintegrationsverfahren vorgestellt, mit dem die Bewegungsgleichung eines Masse-Feder-Systems durch ein Optimierungsproblem gelöst werden kann. Das zur Lösung des Optimierungsproblems resultierende lineare Gleichungssystem (LGS) besitzt eine konstante Systemmatrix, die einmalig berechnet werden muss. Hierdurch ist in jedem Zeitschritt der Simulation lediglich eine Neuberechnung des Ergebnisvektors des LGS notwendig. Dieses lässt sich mit Hilfe des präkonditionertem konjugiertem Gradienten Verfahren lösen, deren höchster rechnerischer Aufwand in der Berechnung von dünn besetzten Matrix-Vektor- Multiplikationen (SpMV = Sparse Matrix Vector Multiplication) liegt. Daher wird eine spezielle GPU-Datenstruktur (GPU=Graphic Processor Unit) unter Verwendung regulärer Gitter entwickelt, um eine massiv parallele Berechnung der SpMV zu ermöglichen. Dadurch wird gegenüber anderer aktueller GPU-Implementierungen von SpMV eine deutlich höhere Anzahl von Rechenoperationen pro Sekunde erreicht. Weiterhin kann mit dieser speziellen Datenstruktur eine nahezu vollständige Ausnutzung der Speicherbandbreite der Grafikkarte realisiert werden. Diese Arbeit bietet damit einen guten Ansatz um eine effiziente Simulation von Masse-Feder-Systems zu realisieren.
Interactive Physically Based Simulation - Efficient Higher-Order Elements, Multigrid Approaches and Massively Parallel Data Structures
Darmstadt, TU, Diss., 2015
This thesis covers interactive physically based simulation for applications such as computer games or virtual environments. Interactivity, i.e., the option that a user can influence a system, imposes challenging requirements on the simulation algorithms. A simple way to achieve this goal is to drastically limit the resolution in order to guarantee this low computation time. However, with current methods the number of degrees of freedom will be rather low, which results in a low degree of realism. This is due to the fact that not every detail that is important for realistically representing the physical system can be resolved. This thesis contributes to interactive physically based simulation by developing novel methods and data structures. These can be associated with the three pillars of this thesis: more accurate discrete representations, efficient methods for linear systems, and data structures and methods for massively parallel computing. The novel approaches are evaluated in two application areas relevant in computer generated animation: simulation of dynamic volumetric deformation and fluid dynamics. The resulting accelerations allow for a higher degree of realism because the number of elements or the resolution can be significantly increased.
Interaktive physikalisch-basierte Simulation auf verteilten Systemen
Darmstadt, TU, Master Thesis, 2015
Diese Masterarbeit behandelt die interaktive physikalisch-basierte Simulation auf verteilten Systemen. Es wird untersucht, wie bestehende Algorithmen und Datenstrukturen für interaktive und nicht-interaktive Simulationen adaptiert werden müssen, um auf verteilten Systemen verwendet zu werden. Ein verteiltes System besteht aus mehreren Grafikkarten, die zusätzlich auf Rechenknoten im Netzwerk verteilt sein können. Als Algorithmen werden das Verfahren der konjugierten Gradienten und das Mehrgitterverfahren untersucht. Von Interesse ist die Skalierbarkeit mit Aussagen über die Laufzeit, Speedup und Effizienz im Zusammenhang mit der Anzahl an Freiheitsgraden.
A p-Multigrid Algorithm using Cubic Finite Elements for Efficient Deformation Simulation
VRIPHYS 14: 11th Workshop in Virtual Reality Interactions and Physical Simulations
International Workshop in Virtual Reality Interaction and Physical Simulations (VRIPHYS) <11, 2014, Bremen, Germany>
We present a novel p-multigrid method for efficient simulation of co-rotational elasticity with higher-order finite elements. In contrast to other multigrid methods proposed for volumetric deformation, the resolution hierarchy is realized by varying polynomial degrees on a tetrahedral mesh. We demonstrate the efficiency of our approach and compare it to commonly used direct sparse solvers and preconditioned conjugate gradient methods. As the polynomial representation is defined w.r.t. the same mesh, the update of the matrix hierarchy necessary for co-rotational elasticity can be computed efficiently. We introduce the use of cubic finite elements for volumetric deformation and investigate different combinations of polynomial degrees for the hierarchy. We analyze the applicability of cubic finite elements for deformation simulation by comparing analytical results in a static scenario and demonstrate our algorithm in dynamic simulations with quadratic and cubic elements. Applying our method to quadratic and cubic finite elements results in speed up of up to a factor of 7 for solving the linear system.
Collision Handling between Rigid and Deformable Bodies with Continuous Penalty Forces
Darmstadt, TU, Bachelor Thesis, 2014
Collision handling has been an active research topic in the area of the physically-based simulation of rigid and deformable bodies for many years. A common approach in interactive environments are discrete penalty forces, computing a repulsion forced based on the penetration at one moment in the time step. They provide low computational costs and good scalability, though they suffer from jitter and instability. Tang et al.  improved the approach of discrete penalty forces and introduced 2012 the continuous penalty forces, continuously accumulating penalty forces along the penetration trajectory over the whole time step. Thereby, the jitter and instability issues are reduced. Although, the continuous penalty forces show artifacts especially for enduring contacts, precluding the simulation of sliding contacts. In this thesis, we present a unified system to handle collisions between rigid and deformable bodies with friction. We modify the integration scheme by Bridson et al.  to handle rigid and deformable bodies, apply a continuous collision detection , handle the detected collisions with continuous penalty forces  and apply a penalty-based friction model . We discuss the artifacts arising from the continuous penalty forces algorithm, examine methods to tackle them and apply the new methods to the continuous penalty forces algorithm. Finally, we analyze the results of the continuous penalty forces algorithm in comparison to discrete penalty forces, evaluate our new algorithm to handle the continuous penalty forces artifacts and inspect further improvements.
Position-based Simulation of Continuous Materials
Computers & Graphics
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.
Wind Tunnel Test and CFD/CAA Analysis on a Scaled Model of a Nose Landing Gear
Greener Aviation. Clean Sky Breakthroughs and Worldwide Status
Conference "Greener Air" <2014, Brussels, Belgium>
In work package 2.2.4 "NLG Low-Noise Enabling Technologies" of the Clean Sky GRA LNC project, the Fraunhofer Institute proposes hubcaps for reducing noise from a nose landing gear (NLG) as the most promising solution. The purpose of this paper is to prove the effect of the hubcaps experimentally and numerically. A simplified and 1:5-scaled model of a NLG was first created by the rapid prototyping technique together with hubcaps that can cover both the outer and inner hub cavities. Noise radiated from various NLG configurations with and without hubcaps were measured during they were placed in the wind tunnel. In the configuration without hubcaps, two major noise peaks in addition to a continuous spectrum were observed in the direction parallel to the wheel axle. When the inner hubcaps were attached to the NLG, the levels of the peaks were significantly reduced. The outer caps have no effects on the noise reduction. Nearly the same noise spectrum as the original no-hubcap configuration was observed. Although the peaks were not clearly observed in the direction perpendicular to the axle, the same noise reduction could be recognized in the inner-hubcap configuration. In the numerical examination, a stationary CFD analysis with a k-\\'0f turbulence model was first performed and a CAA analysis was then carried out based on Lighthill's aeroacoustic analogy after reconstructing a time-varying turbulent flow by a stochastic noise generation and radiation model. In the CAA analysis of the no-hubcap configuration, a strong fluctuation in the right and left inner hub cavities, where pressure is oscillating alternately, was observed. This fluctuation served as a dipole noise source whose direction is parallel to the wheel axle. The simulated spectrum of far field sound pressure in this direction has the peaks corresponding to the ones experimentally observed. In the hubcap configuration, the pressure fluctuation in the inner hubcap cavities was greatly reduced. Because of this, the noise peaks were well depressed. Due to the dipole characteristics of the noise source, no clear peaks were simulated in the far field spectrum in the direction perpendicular to the axle. In conclusion, the effectiveness of the inner hubcaps has been proved in the wind tunnel experiment and confirmed in the numerical analysis. The mechanism of noise reduction by the inner caps has also been clarified.
Efficient GPU Data Structures and Methods to Solve Sparse Linear Systems in Dynamics Applications
Computer Graphics Forum
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.
Fast and Stable Cloth Simulation Based on Multi-Resolution Shape Matching
Computers & Graphics
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.
Enabling Virtual Assembly Training in and beyond the Automotive Industry
Proceedings of the VSMM 2012
International Conference on Virtual Systems and MultiMedia (VSMM) <18, 2012, Milan, Italy>
Virtual assembly training systems show a high potential to complement or even replace physical setups for training of assembly processes in and beyond the automotive industry. The precondition for the breakthrough of virtual training is that it overcomes the problems of former approaches. The paper presents the design approach taken during the development of a game-based, virtual training system for procedural assembly knowledge in the EU-FP7 project VISTRA. One key challenge to address when developing virtual assembly training is the extensive authoring effort for setting up virtual environments. Although knowledge from the product and manufacturing design is available and could be used for virtual training, a concept for integration of this data is still missing. This paper presents the design of a platform which transfers available enterprise data into a unified model for virtual training and thus enables virtual training of workers at the assembly line before the physical prototypes exist. The data requirements and constraints stemming from industrial partners involved in the project will be discussed. A second hurdle for virtual training is the insufficient user integration and acceptance. In this context, the paper introduces an innovative hardware set-up for game-based user interaction, which has been chosen to enhance user involvement and acceptance of virtual training
Energy-Preserving Integrators for Fluid Animation with Discrete Exterior Calculus on Two-Dimensional Meshes
Darmstadt, TU, Bachelor Thesis, 2012
In the last decades a lot of approaches have been developed for implementing computational fluid dynamics (CFD) in the computer graphics community. One of the new approaches in fluid simulations is the discrete exterior calculus (DEC). DEC uses well-centered meshes to describe its integration space. Mullen et al. [MCP+09] introduced 2009 a new integration scheme based on DEC and the Navier-Stokes equations that preserves mass by definition of DEC. His discrete formulation of the Navier-Stokes equations provides full control about viscosity and moreover an almost perfect preservation of kinetic energy. We translate Mullens discretization into two dimensions and extend it to regular girds. We will discuss how to manage non-trivial boundary conditions. Finally we will analyze the results of Mullens approach, and analyze alternative methods to further improve those results.
Rapid CFD für die frühe konzeptionelle Design Phase
NAFEMS Online Magazin
Ein wichtiger Teil des Produktentwicklungszyklus ist die Optimierung der strömungs- oder strukturmechanischen Eigenschaften einer Komponente, die normalerweise in einem iterativen und sehr aufwändigen Prozess stattfindet. Neben der Modifikation, Vereinfachung und des Vernetzens der Bauteilgeometrie, kann die Simulation mitunter Stunden bis Tage dauern. In frühen konzeptionellen Designphasen müssen verschiedene Materialparameter sowie unterschiedliche Geometrien ausprobiert und verglichen werden, um zu einem für das spätere Produkt optimalen Design zu gelangen. Dieser zeitaufwändige Prozess begrenzt deutlich die Anzahl der Möglichkeiten, die analysiert werden können. In dieser Arbeit wird das Framework "Rapid CFD" vorgestellt, das es ermöglicht, schnelle Strömungssimulationen für die frühe konzeptionelle Designphase einzusetzen. Um eine solche Geschwindigkeit zu erreichen, wird die Berechnung und Visualisierung von zweidimensionalen Strömungen in Echtzeit kombiniert. Das ermöglicht die interaktive Modifikation von Parametern und Randbedingungen und damit eine schnelle Analyse und Bewertung von unterschiedlichen Geometrien und eine frühzeitige Optimierung eines Bauteils. Das Framework führt alle Berechnungen auf der Graphikkarte (graphics processing unit - GPU) aus und vermeidet damit das aufwändige Kopieren zwischen CPU- und GPU-Hauptspeicher. Die Berechnungen werden auf einem Standard-Desktop PC ausgeführt, sodass die Simulationsergebnisse im Graphikkartenspeicher bleiben und direkt zur Visualisierung verwendet werden können. Für die Modellierung der Geometrie werden B-Splines verwendet, damit Benutzer lokal die Form durch einzelne Kontrollpunkte modifizieren können. Die Diskretisierung wird ebenfalls auf der GPU ausgeführt. Die Berechnung eines einzelnen Zeitschritts auch für Millionen von Unbekannten wird in Bruchteilen von Sekunden durchgeführt. Die intuitive geometrische Manipulation in Kombination mit der unmittelbaren Visualisierung der Simulationsgrößen wie Druck und Geschwindigkeit ermöglichen die direkte Analyse des Einflusses von Geometrie- und Parameteränderungen. Obwohl diese neuartige Simulationstechnik noch nicht die hohe Präzision konventioneller Simulationen erreicht, ermöglicht diese Technik die Beobachtung von Trends und Tendenzen.
Schnelle Strömungsberechnungen mit GPU
Digital Engineering Magazin
Eine neue Tragfläche entsteht am Computer. Ist ihr Auftrieb tatsächlich besser als bei den herkömmlichen? Eine Computersimulation kann hierüber Aufschluss geben. Konventionelle Simulationen liefern die gewünschten Ergebnisse gewöhnlich erst nach mehreren Stunden oder Tagen. Erst anschließend können Modifikationen an der Geometrie vorgenommen werden, um die Eigenschaften zu verbessern. Ein neues Verfahren liefert nun die ersten Simulationsergebnisse bereits in Echtzeit. Es nutzt die Prozessoren der Grafikkarten (Graphics Processing Unit- GPU) für die notwendigen Berechnungen.
Interactive Deformable Models with Quadratic Bases in Bernstein-Bézier-Form
The Visual Computer
Computer Graphics International (CGI) <29, 2011, Ottawa, Canada>
We present a physically based interactive simulation technique for de formable objects. Our method models the geometry as well as the displacements using quadratic basis functions in Bernstein-Bézier form on a tetrahedral finite element mesh. The Bernstein-Bézier formulation yields significant advantages compared to approaches using the monomial form. The implementation is simplified, as spatial derivatives and integrals of the displacement field are obtained analytically avoiding the need for numerical evaluations of the elements' stiffness matrices. We introduce a novel traversal accounting for adjacency in order to accelerate the reconstruction of the global matrices. We show that our proposed method can compensate the additional effort introduced by the co-rotational formulation to a large extent. We validate our approach on several models and demonstrate new levels of accuracy and performance in comparison to current state-of-the-art.
Rapid CFD for the Early Conceptual Design Phase
The Integration of CFD into the Product Development Process
Seminar the Integration of CFD into the Product Development Process <2011, Wiesbaden, Germany>
An important step of the product development is the optimization of the components' physical behavior, which is usually done in a costly iterative process. Besides the modification, simplification, and (re-) meshing of the component's geometry, simulating its behavior can take hours or even days. In the early conceptual design phase, different material properties and shapes need to be tested and compared, in order to optimally design the component. Nonetheless, time consuming simulations limit the realm of possibilities. We have developed a framework for enabling rapid Computational Fluid Dynamics (CFD) for the early conceptual design phase. In order to achieve this, we combine the computation and visualization of 2D fluid flow in real time with the modification of fluid parameters, boundary conditions and geometry. This allows for the rapid assessment and analysis of different shapes and therefore the optimization of the component. Our framework is completely based on graphic processing units (GPUs), i.e., all computations are performed on the GPU avoiding costly memory transfers between graphic hardware and CPU memory. The computations are performed on a single desktop PC, thus the simulation results can reside in GPU memory and can directly be visualized. B-Spline curves are used for modelling the geometry and the user can interactively modify it by means of inserting and moving control points or applying local smooth deformations, with the corresponding rapid update of the discretization on the GPU. Computing one single time step is performed in fractions of a second, even if the fluid flow is modelled with about one million degrees of freedom. The fast geometric manipulation combined with the direct visualization of quantities like velocity or pressure field allows for an immediate feedback of shape or parameter changes. Although fast simulations do not yet achieve the high precision compared to conventional simulations, their results are suitable for analyzing trends.
Splines auf Tetraederpartitionen für physikalisch basierte Deformationssimulation
Darmstadt, TU, Bachelor Thesis, 2011
Bei der Simulation von elastischen Deformationen mit Hilfe der Finiten-Elemente-Methode gibt es zwei Möglichkeiten die Genauigkeit der Approximation zu verbessern. Zum Einen die Erhöhung der Anzahl der Elemente, zum Anderen die Erhöhung des Polynomgrades der Basisfunktionen. Durch die Erhöhung des Polynomgrades steigt die Anzahl der Freiheitsgrade, was wiederum zu längeren Berechnungszeiten führt. Beschrieben wird die elastische Deformation von Körpern durch Partielle Differentialgleichungen (PDGLn) mit glatten Lösungen. Dies ermöglicht die Approximation der PDGLn mit den Basisfunktionen von Splines, die weniger Freiheitsgrade benötigen, da sie wegen ihrer Stetigkeitseigenschaften per Definition glatte Funktionen beschreiben. Die Simulation deformierbarer Materialien mit Splinefunktionen wird vorgestellt und die dadurch erreichte Genauigkeit und Performanz wird ermittelt und im Vergleich zur Finiten-Elemente-Methode diskutiert.
Two-dimensional Circulation-preserving Fluid Simulation with Discrete Exterior Calculus
Darmstadt, TU, Bachelor Thesis, 2011
The development of efficient and stable fluid simulations is a challenging task in computer graphics. Elcott et al.  describe an approach, based on Discrete Exterior Calculus for simulating the fluid flow. A vorticity based formulation of the incompressible Navier-Stokes equations is used, resulting in a mass-conserving representation of the velocity field by definition. This approach preserves vorticity at a discrete level, resulting in a visually more realistic fluid flow. We extend this approach to regular grids in two dimensions. So, we avoid computationally expensive mesh constructions. We discuss non-trivial boundary conditions and arbitrary topologies. The vorticity conservation properties are compared with the classical mesh based approach of Elcott et. al. We especially analyze the corresponding pressure fields near the boundaries of inner objects.
A Geometric Multigrid Method for Simulating Deformable Models on Unstructured, Non-nested Mesh Hierarchies
Darmstadt, TU, Bachelor Thesis, 2010
In order to accelerate the solution of linear systems, multigrid methods utilize different levels of discretizations. Transfer operations between different levels are usually straightforward, since most geometric multigrid solver is embedded in structured problem domains. However, the multigrid method in this thesis makes use of barycentric coordinates to cope with unstructured problems. Thereby, the approach is applied to a Finite Element framework simulating deformable models on both linear and quadratic tetrahedral meshes.
Iterative SLE Solvers over a CPU-GPU Platform
Proceedings 2010 12th IEEE International Conference on High Performance Computing and Communications
IEEE International Conference on High Performance Computing and Communications (HPCC) <12, 2010, Melbourne, Australia>
GPUs (Graphics Processing Units) have become one of the main co-processors that contributed to desktops towards high performance computing. Together with multi-core CPUs, a powerful heterogeneous execution platform is built for massive calculations. To improve application performance and explore this heterogeneity, a distribution of workload in a balanced way over the PUs (Processing Units) plays an important role for the system. However, this problem faces challenges since the cost of a task at a PU is non-deterministic and can be influenced by several parameters not known a priori, like the problem size domain. We present a comparison of iterative SLE (Systems of Linear Equations) solvers, used in many scientific and engineering applications, over a heterogeneous CPU-GPUs platform and characterize scenarios where the solvers obtain better performances. A new technique to improve memory access on matrix vector multiplication used by SLEs on GPUs is described and compared to standard implementations for CPU and GPUs. Such timing profiling is analyzed and break-even points based on the problem sizes are identified for this implementation, pointing whether our technique is faster to use GPU instead of CPU. Preliminary results show the importance of this study applied to a real-time CFD (Computational Fluid Dynamics) application with geometry modification.
Interactive Deformable Models with Quadratic Bases in Bernstein-Bézier Form
We present a physically based interactive simulation technique for deformable objects with curved boundary surfaces. Our method models the object geometry as well as the displacements using quadratic basis functions in Bernstein-Bézier-form on a tetrahedral finite element grid. The Bernstein-Bézier formulation yields significant advantages compared to approaches with linear and quadratic bases in monomial form. Spatial derivatives and integrals of the displacement field are obtained analytically avoiding the need for numerical evaluations of the elements' stiffness matrices. We validate our approach on several different models and demonstrate state-of-the-art results in terms of accuracy and perfomance.
Trivariate Bernstein-Bézier-Techniken für Finite Elemente zur interaktiven Simulation von Deformationen
Darmstadt, TU, Diplomarbeit, 2008
Diese Arbeit befasst sich mit der physikalisch basierten, dynamischen Simulation von deformierbaren Materialien. Dafür wird der zu animierende Körper durch Diskretisierung in eine Menge von Tetraedern zerlegt und die Eigenschaften durch die Angabe von Materialparametern festgelegt. Die Bewegungsgesetze werden aus der Approximation der partiellen Differentialgleichungen der Kontinuumsmechanik abgeleitet. Die resultierenden gewöhnlichen Differentialgleichungen modellieren die Dynamik eines Mehrteilchensystems, deren Lösung ein physikalisch plausibles Bewegungsschema ergibt. In dieser Diplomarbeit wird untersucht, in wie weit eine Modellierung des Deformationsfeldes mit polynomialen Interpolationsfunktionen in Bernstein-Bézier-Basis von Vorteil ist. Dabei soll überprüft werden, ob sich Vereinfachungen für das Aufstellen der Gleichungen ergeben und dadurch eine Beschleunigung der Berechnung erreicht werden kann.