Kinetic Solvers with Adaptive Mesh in Phase Space

An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for solving multi-dimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a tree of trees data structure. The mesh in r-space is automatically generated around embedded boundaries and dynamically adapted to local solution properties. The mesh in v-space is created on-the-fly for each cell in r-space. Mappings between neighboring v-space trees implemented for the advection operator in configuration space. We have developed new algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive mesh in velocity space: importance sampling, multi-point projection method, and the variance reduction method. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions in a Lorentz gas. New AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce computational cost and memory usage for solving challenging kinetic problems

A new lattice Boltzmann model for interface reactions between immiscible fluids

In this paper, we describe a lattice Boltzmann model to simulate chemical reactions taking place at the interface between two immiscible fluids. The phase-field approach is used to identify the interface and its orientation, the concentration of reactant at the interface is then calculated iteratively to impose the correct reactive flux condition. The main advantages of the model is that interfaces are considered part of the bulk dynamics with the corrective reactive flux introduced as a source/sink term in the collision step, and, as a consequence, the model’s implementation and performance is independent of the interface geometry and orientation. Results obtained with the proposed model are compared to analytical solution for three different benchmark tests (stationary flat boundary, moving flat boundary and dissolving droplet). We find an excellent agreement between analytical and numerical solutions in all cases. Finally, we present a simulation coupling the Shan Chen multiphase model and the interface reactive model to simulate the dissolution of a collection of immiscible droplets with different sizes rising by buoyancy in a stagnant fluid

Simulating anomalous dispersion in porous media using the unstructured lattice Boltzmann method

Flow in porous media is a significant challenge to many computational fluid dynamics methods because of the complex boundaries separating pore fluid and host medium. However, the rapid development of the lattice Boltzmann methods and experimental imaging techniques now allow us to efficiently and robustly simulate flows in the pore space of porous rocks. Here we study the flow and dispersion in the pore space of limestone samples using the unstructured, characteristic based off-lattice Boltzmann method. We use the method to investigate the anomalous dispersion of particles in the pore space. We further show that the complex pore network limits the effectivity by which pollutants in the pore space can be removed by continuous flushing. In the smallest pores, diffusive transport dominates over advective transport and therefore cycles of flushing and no flushing, respectively, might be a more efficient strategy for pollutant removal

A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions

In this paper, we study a lattice Boltzmann method for the advection-diffusion equation with Neumann boundary conditions on general boundaries. A novel mass conservative scheme is introduced for implementing such boundary con- ditions, and is analyzed both theoretically and numerically. Second order convergence is predicted by the theoretical analysis, and numerical investigations show that the convergence is at or close to the predicted rate. The nu- merical investigations include time-dependent problems and a steady-state diffusion problem for computation of effective diffusion coefficients

Numerical analysis of Lattice Boltzmann Methods for the heat equation on a bounded interval

Lattice Boltzmann methods are a promising approach for the numerical solution of fluid-dynamic problems. We consider the one-dimensional Goldstein-Taylor model with the aim to answer some of the questions concerning the numerical analysis of lattice Boltzmann schemes. Discretizations for the solution of the heat equation are presented for a selection of boundary conditions. Stability and convergence of the solutions are proved by employing energy estimates and explicit Fourier representations

Numerical analysis of Lattice Boltzmann Methods for the heat equation on a bounded interval

Lattice Boltzmann methods are a promising approach for the numerical solution of fluid-dynamic problems. We consider the one-dimensional Goldstein-Taylor model with the aim to answer some of the questions concerning the numerical analysis of lattice Boltzmann schemes. Discretizations for the solution of the heat equation are presented for a selection of boundary conditions. Stability and convergence of the solutions are proved by employing energy estimates and explicit Fourier representations

Massiv parallele Simulation von Mehrphasen- und Mehrkomponentenströmungen unter Anwendung des Lattice Boltzmann Verfahrens

This thesis reflects the work mainly performed within the research project FIMOTUM focusing on the determination of transport properties and mechanisms in unsaturated media. The efficient simulation of single- and multiphase flows at the pore scale in highly resolved natural porous media is one of the major topics in this work. For this purpose a simulation kernel which is based on the lattice Boltzmann method (LBM) has been developed and extensively validated. The LBM presented utilizes the Multiple Relaxation Time (MRT) model and fluid/wall boundary conditions of second order accuracy. The model has also been extended to solve multiphase, advection/diffusion and thermal flow problems. Due to the application of an optimized collision model and corresponding boundary conditions, the covered parameter space and the stability of the method could be greatly enhanced. Hence, it was possible to perform simulations in complex geometries at a large scale (2E11+ DoF) which have been obtained with an unprecedented accuracy. A second target of this thesis was the design and implementation of a simulation kernel to perform massively parallel computations with high efficiency. In order to obtain accurate simulation results at reasonable computational effort, a novel grid generation procedure has been developed. The robust and flexible method is based on the decoupling of input geometry and the actual computational grid. It is therefore excellently suited for the grid generation based on natural porous media data sets obtained by CT- or X-ray methods. Aspects concerning the increasing difficulties in pre- and post-processing of large data sets are discussed. Furthermore, special issues in high performance computing environments are highlighted and a tool chain to visualize scientific data in photo-realistic representation is described.Die vorliegende Dissertation gibt im Wesentlichen die Arbeiten wieder, die im Rahmen des FIMOTUM Projektes durchgeführt worden sind, welches sich vornehmlich auf die Untersuchung von Transporteigenschaften in ungesättigten porösen Medien fokussierte. Hierfür wurde ein Software-Prototyp auf Basis der Gitter Boltzmann Methode (LBM) entwickelt und ausführlich validiert. Die vorgestellte LB-Methode basiert auf dem Multiple-Relaxation-Time (MRT) Modell und verwendet Fluid/Wand Randbedingungen mit einer Genauigkeit 2. Ordnung. Das beschriebene Modell wurde zudem für die Simulation von Mehrphasen-, Advektion/Diffusions- und Thermalen Problemen erweitert. Durch die Optimierung des Kollisionsmodells und der entsprechenden Randbedingungen konnte der nutzbare Parameterraum deutlich vergrößert werden, so dass Simulationen in komplexen Geometrien mit mehr als 2.0E+11 Freiheitsgraden möglich wurden. Ein zweites Ziel dieser Arbeit war die Implementierung eines effizienten und hochparallelen Software-Prototypen zur Simulation von fluiddynamischen Problemen. Um möglichst genaue Ergebnisse bei mäßigem Ressourceneinsatz zu erzielen, wurde ein neuartiger Gittergenerierungsprozess entwickelt. Dieses robuste und flexible Verfahren basiert auf der Entkopplung von Eingangsgeometrie und dem eigentlichen Rechengitter. Daher eignet sich dieser Gittergenerator hervorragend für die Erzeugung eines numerischen Gitters aus digitalen Datensätzen natürlicher poröser Medien, wie bspw. Tomographie-Scans. Desweiteren werden, neben allgemeinen Problemen des Hochleistungsrechnens, die zunehmenden Schwierigkeiten bei der Verarbeitung der ständig steigenden Datenmengen im Pre- und Postprocessing diskutiert. Weiterhin wird, unterstützend zur Ergebnisanalyse, eine Prozesskette für die Erzeugung von fotorealistischen Visualisierungen aus Simulationsdaten beschrieben