Many natural phenomena which occur in the realm of visual computing and computational physics, like the dynamics of cloth, fibers, fluids, and solids as well as collision scenarios are described by stiff Hamiltonian equations of motion, i.e. differential equations whose solution spectra simultaneously contain extremely high and low frequencies. This usually impedes the development of physically accurate and at the same time efficient integration algorithms. We present a straightforward computationally oriented introduction to advanced concepts from classical mechanics. We provide an easy to understand step-by-step introduction from variational principles over the Euler-Lagrange formalism and the Legendre transformation to Hamiltonian mechanics. Based on such solid theoretical foundations, we study the underlying geometric structure of Hamiltonian systems as well as their discrete counterparts in order to develop sophisticated structure preserving integration algorithms to efficiently perform high fidelity simulations.
Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.
We devise a method for the accurate simulation of wound healing and skin deformation. This is based on adequate formulations modeling the underlying biological processes. Cell movements and proliferation are described by a biochemical model whereas a biomechanical model covers effects like wound contraction and the influence of the healing process on the surrounding skin. The resulting simulation framework is very efficient and can be used with realistic input parameters like those measured in biochemistry and biophysics. The accurate behavior of our approach is shown by reproducing regenerative healing processes as well as specific effects such as anisotropic wound contraction, scarring and scab formation. Its efficiency and robustness is illustrated on a broad spectrum of complex examples.
We devise a physically based approach to the accurate simulation of stiff fibers like human hair, wool, or yarn. For that we describe fibers as three-dimensional coupled oscillator networks. The application of special analytical mapping expressions allows us to mimic the existence of Young's and shear modulus in the oscillator network so that real material parameters can be used. For the efficient numerical treatment of the stiff equations of motion of the system a Damped Exponential Time Integrator (DETI) is introduced. This type of integrator is able to take large time steps during the solution process of the stiff system while sustaining stability. It also handles Rayleigh damping analytically by employing the closed-form solution of the fully damped harmonic oscillator. We validate the fiber model against the outcome obtained by solving the special Cosserat theory of rods. Moreover, we demonstrate the efficiency of our approach on some complex fiber assemblies like human hair and fiber meshes. Compared to established methods we reach a significant speed up and at the same time achieve highly accurate results.
Natürliche Phänomene und technische Anwendungen, die im Kontext ihrer Simulation auf sogenannte „steife” Cauchyprobleme führen, sind allgegenwärtig: die Dynamik von molekularen Strukturen, Fasern, Geweben und deformierbaren Objekten sind nur wenige Beispiele. Ihre stabile Integration erfordert häufig unverhältnismäßig kleine Zeitschrittweiten, was eine effiziente Simulation erschwert oder in vielen Fällen sogar unmöglich macht. Diesbezüglich besteht aus numerischer Sicht ein Bedarf an Integrationsalgorithmen, die lange Zeitschrittweiten handhaben können und damit effiziente und gleichzeitig physikalisch akkurate Simulationen ermöglichen. Unter Berücksichtigung der physikalischen Modellierung der spezifischen Phänomene werden in der vorliegenden Abhandlung strukturerhaltende semianalytische Integrationsalgorithmen entworfen und bezüglich ihrer Praktikabilität im Kontext realer Simulationen eingesetzt und evaluiert.
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of the solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.
We investigate numerical integration of ordinary differential equations (ODEs) for Hamiltonian Monte Carlo (HMC). High-quality integration is crucial for designing efficient and effective proposals for HMC. While the standard method is leapfrog (Stömer-Verlet) integration, we propose the use of an exponential integrator, which is robust to stiff ODEs with highly-oscillatory components. This oscillation is difficult to reproduce using leapfrog integration, even with carefully selected integration parameters and preconditioning. Concretely, we use a Gaussian distribution approximation to segregate stiff components of the ODE. We integrate this term analytically for stability and account for deviation from the approximation using variation of constants. We consider various ways to derive Gaussian approximations and conduct extensive empirical studies applying the proposed "exponential HMC" to several benchmarked learning problems. We compare to state-of-the-art methods for improving leapfrog HMC and demonstrate the advantages of our method in generating many effective samples with high acceptance rates in short running times.
To be able to take into account a multitude of physical effects, high fidelity simulations are nowadays of growing interest for analyzing and synthesizing visual data. In contrast to most numerical simulations in engineering, local accuracy is secondary to the global visual plausibility. Global accuracy can be achieved by preserving the geometric nature and physical quantities of the simulated systems for which reason geometric integration algorithms like symplectic methods are often considered as a natural choice. Additionally, if the underlying phenomena behaves numerically stiff, a non-geometric nature comes into play requiring for strategies to capture different timescales accurately.
In this contribution, a hybrid semi-analytical, semi-numerical Gautschi-type exponential integrator for modeling and design applications is presented. It is based on the idea to handle strong forces through analytical expressions to allow for long-term stability in stiff cases. By using an appropriate set of analytical filter functions, this explicit scheme is symplectic as well as time-reversible. It is further parallelizable exploiting the power of up-to-date hardware. To demonstrate its applicability in the field of visual computing, various examples including collision scenarios and molecular modeling are presented.
This work presents analytic-numeric approaches for solving Cauchy problems in the context of physical simulations of stiff elastodynamic scenarios. Classical explicit numerical integration schemes have the shortcoming that step sizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to non-physical behavior. To overcome these specific detriments, an appropriate class of integrators that solves the stiff part of the governing equations by employing a closed-form solution is introduced. With these techniques, up to three orders of magnitude greater time steps can be handled compared to conventional methods, and, at the same time, a tremendous increase in overall long-term stability is achieved. This advantageous behavior is demonstrated across a broad spectrum of complex models that include deformable solids, trusses, and textiles, including damping, collision responses, friction, and non-linear material behavior. To realize an efficient and physically accurate simulation of fiber-based systems such as human hair, wool infills, and brushes, an appropriate approach for the physically accurate simulation of densely packed fiber assemblies is presented. This model works on the level of single fibers and considers effects such as adhesion, anisotropic frictional contacts, and Coulomb far-field interactions. Each fiber is modeled as a three-dimensional network of coupled oscillators in which the order of the particles is such that analytical expressions for the mapping of well-known mechanical quantities such as Young’s modulus and shear modulus can be derived. Non-linear material behavior is modeled by introducing support forces. This process allows for large deformations of single fibers without the need for more complicated models such as Kirchhoff rods. The efficiency of this concept in proven with a detailed scientific validation, which demonstrates that this method is capable of correctly capturing all the effects that occur in real fiber assemblies.
We devise a new algorithm for the extraction of vine leaf veins. Our method performs a directional edge tracing on the responses of appropriate adaptive Gabor filters in order to extract the network of the main veins. The respective curvature vectors are used for the classification of different cultivars using support vector machines. We evaluate the advantageous behavior and the robustness of our approach on a test set consisting of 150 light transmitted images of different vine leaves.