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

Exploring the use of skeletal tracking for cheaper motion graphs and on-set decision making in free-viewpoint video production

In free-viewpoint video (FVV), the motion and surface appearance of a real-world performance is captured as an animated mesh. While this technology can produce high-fidelity recreations of actors, the required 3D reconstruction step has substantial processing demands. This means FVV experiences are currently expensive to produce, and the processing delay means on-set decisions are hampered by a lack of feedback. This work explores the possibility of using RGB-camera-based skeletal tracking to reduce the amount of content that must be 3D reconstructed, as well as aiding on-set decision making. One particularly relevant application is in the construction of Motion Graphs, where state-of-the-art techniques require large amounts of content to be 3D reconstructed before a graph can be built, resulting in large amounts of wasted processing effort. Here, we propose the use of skeletons to assess which clips of FVV content to process, resulting in substantial cost savings with a limited impact on performance accuracy. Additionally, we explore how this technique could be utilised on set to reduce the possibility of requiring expensive reshoots

Numerical methods and stochastic simulation algorithms for reaction-drift-diffusion systems

In recent years, there has been increased awareness that stochasticity in chemical reactions and diffusion of molecules can have significant effects on the outcomes of intracellular processes, particularly given the low copy numbers of many proteins and mRNAs present in a cell. For such molecular species, the number and locations of molecules can provide a more accurate and detailed description than local concentration. In addition to diffusion, drift in the movements of molecules can play a key role in the dynamics of intracellular processes, and can often be modeled as arising from potential fields. Examples of sources of drift include active transport, variations in chemical potential, material heterogeneities in the cytoplasm, and local interactions with subcellular structures. This dissertation presents a new numerical method for simulating the stochastically varying numbers and locations of molecular species undergoing chemical reactions and drift-diffusion. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for reaction-diffusion systems and the Wang—Peskin—Elston lattice discretization of the Fokker—Planck equation that describes drift-diffusion processes in which the drift arises from potential fields. In the FPKMC method, each molecule is enclosed within a "protective domain," either by itself or with a small number of other molecules. To sample when a molecule leaves its protective domain or a reaction occurs, the original FPKMC method relies on analytic solutions of one- and two-body diffusion equations within the protective domains, and therefore cannot be used in situations with non-constant drift. To allow for such drift in our new method (hereafter Dynamic Lattice FPKMC or DL-FPKMC), each molecule undergoes a continuous-time random walk on a lattice within its protective domain, and the lattices change adaptively over time. One of the most commonly used spatial models for stochastic reaction-diffusion systems is the Smoluchowski diffusion-limited reaction (SDLR) model. The DL-FPKMC method generates convergent realizations of an extension of the SDLR model that includes drift from potentials. We present detailed numerical results demonstrating the convergence and accuracy of our method for various types of potentials (smooth, discontinuous, and constant). We also present several illustrative applications of DL-FPKMC, including examples motivated by cell biology

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes