Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution. / eSSENCE

Numerical modeling of fluid flow and dissolved species transport in the subsurface is a challenging task, given variability and measurement uncertainty in the physical properties of the rock, the complexities of multi-fluid interaction, and limited computational resources. Nonetheless, this thesis seeks to expand our modeling capabilities in the context of contaminant hydrogeology. We describe the numerical simulator CompFlow Bio and use it to model invasion of a nonaqueous phase liquid (NAPL) contaminant through the vadose zone and below the water table in a fractured porous rock. CompFlow Bio is a three-phase, multicomponent, deterministic numerical model for fluid flow and dissolved species transport; it includes capillary pressure and equilibrium partitioning relationships. We have augmented the model to include randomly generated, axis-aligned, discrete fracture networks (DFNs). The DFN is coupled with the porous medium (PM) to form a single continuum. The domain is discretized using a finite-volume scheme in an unstructured mesh of rectilinear control volumes (CVs). Herein we present the governing equations, unstructured mesh creation scheme, algebraic development of fracture intersection CV elimination, and coupling of PM CVs over a fracture plane to permit asperity contact bridged flow. We include: small scale two-phase water-air and NAPL-water simulations to validate the practice of intersection CV elimination; small scale simulations with water-air, NAPL-water, and NAPL-water-air systems in a grid refinement exercise and to demonstrate the effect of asperity contact bridged flow; intermediate scale 3D simulations of NAPL invading the saturated zone, based on the Smithville, Ontario, site; intermediate scale 2D and 3D simulations of NAPL invading the vadose zone and saturated zone with transient recharge, based on the Santa Susana Field Laboratory site, California. Our findings indicate that: the formulation provides a practical and satisfactory way of modeling three-phase flow in discretely fractured porous rock; numerical error caused by spatial discretization manifests itself as several biases in physical flow processes; that asperity contact is important in establishing target water saturation conditions in the vadose zone; and simulation results are sensitive to relative permeability-saturation-capillary pressure relationships. We suggest a number of enhancements to CompFlow Bio to overcome certain computational limitations.

The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.

The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.

Numerical modeling of fluid flow and dissolved species transport in the subsurface is a challenging task, given variability and measurement uncertainty in the physical properties of the rock, the complexities of multi-fluid interaction, and limited computational resources. Nonetheless, this thesis seeks to expand our modeling capabilities in the context of contaminant hydrogeology. We describe the numerical simulator CompFlow Bio and use it to model invasion of a nonaqueous phase liquid (NAPL) contaminant through the vadose zone and below the water table in a fractured porous rock. CompFlow Bio is a three-phase, multicomponent, deterministic numerical model for fluid flow and dissolved species transport; it includes capillary pressure and equilibrium partitioning relationships. We have augmented the model to include randomly generated, axis-aligned, discrete fracture networks (DFNs). The DFN is coupled with the porous medium (PM) to form a single continuum. The domain is discretized using a finite-volume scheme in an unstructured mesh of rectilinear control volumes (CVs). Herein we present the governing equations, unstructured mesh creation scheme, algebraic development of fracture intersection CV elimination, and coupling of PM CVs over a fracture plane to permit asperity contact bridged flow. We include: small scale two-phase water-air and NAPL-water simulations to validate the practice of intersection CV elimination; small scale simulations with water-air, NAPL-water, and NAPL-water-air systems in a grid refinement exercise and to demonstrate the effect of asperity contact bridged flow; intermediate scale 3D simulations of NAPL invading the saturated zone, based on the Smithville, Ontario, site; intermediate scale 2D and 3D simulations of NAPL invading the vadose zone and saturated zone with transient recharge, based on the Santa Susana Field Laboratory site, California. Our findings indicate that: the formulation provides a practical and satisfactory way of modeling three-phase flow in discretely fractured porous rock; numerical error caused by spatial discretization manifests itself as several biases in physical flow processes; that asperity contact is important in establishing target water saturation conditions in the vadose zone; and simulation results are sensitive to relative permeability-saturation-capillary pressure relationships. We suggest a number of enhancements to CompFlow Bio to overcome certain computational limitations.

Meshes based on triangulation (2D) or tetrahedralization (3D) are widely used in applications such as computer graphics, interpolation, surveying, and terrain modelling. Although the most important use is in numerical methods for the solution of partial differential equations. This solution is use for simulation of complex physical processes, which are described by this equations. The main topic of this thesis is mash generation.

This paper gives an overview of the mathematical and practical tools that can be used in turbo-machinery RANS simulation to realize unstructured mesh adaptation. It first presents the concept of metric and recalls that the hessian of the physical flow properties can become, thanks to small modifications, both a metric and a upper bound of the P1 projection error. The resulting metric is then studied on a simple 2D case. In a second part, the industrial application of this concept is addressed and the tools used to overcome the turbo-machinery specificities are explained. Finally, some 2D and 3D results are presented.

Recent research shows that liquid crystals can be used to report the presence of different types of substances through optical amplication of ligand-receptor binding. In this work, simulations based on a coarse-grained method have been performed to study a class of liquid-crystal-based sensors. A tensor order parameter was used to model the liquid crystalline system and the Beris-Edwards formulation was employed to obtain the time evolution of a liquid crystal medium containing particles. The simulation cases are built using three-dimensional unstructured meshes and the simulation geometries studied include simple models involving spheres as well as detailed modeling for a protein. The dynamics of a liquid crystal medium confined between two solid walls has been studied in the presence of spherical particles and a representative biological macromolecule. Comparisons of steady state and transient solutions from the present study with corresponding results from molecular dynamics based simulations in the literature yield good agreements.

The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. However, new algorithms need to be developed to take advantage of the new computer architecture (in particular, distributed memory clusters). Since the solution of linear systems still demands most of the computational effort in many problems (such as the approximation of partial differential equation models) iterative methods and, in particular, efficient preconditioners need to be developed. In this study, we consider application of incomplete LU (ILU) preconditioners for finite element models to partial differential equations. Since finite elements lead to large, sparse systems, reordering the node numbers can have a substantial influence on the effectiveness of these preconditioners. We study two implementations of the ILU preconditioner: a stucturebased method and a threshold-based method. The main emphasis of the thesis is to test a variety of breadth-first ordering strategies on the convergence properties of the preconditioned systems. These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy. / Master of Science

Multi-core architectures are present throughout a large selection of computing devices from cell phones to super-computers. Parallel applications running on these devices solve bigger problems in a shorter time. Writing those applications is a difficult task for programmers. They need to deal with low-level parallel mechanisms such as data distribution, inter-processor communication, and task placement. The goal of the Standard Template Adaptive Parallel Library (STAPL) is to provide a generic high-level framework to develop parallel applications. One of the first steps of a parallel application is to partition and distribute the data throughout the system. An important data structure for parallel applications to store large amounts of data and model many types of relations is the graph. A mesh, which is a special type of graph, is often used to model a spatial domain in scientific applications. Graph and mesh partitioning has many applications such as VLSI circuit design, parallel task scheduling, and data distribution. Data distribution, significantly impacts the performance of a parallel application. In this thesis, we introduce the STAPL Parallel Graph Partitioner Framework. This framework provides a generic infrastructure to partition arbitrary graphs and meshes and to build customized partitioners. It includes the state of the art parallel k-way multilevel scheme to partition arbitrary graphs, a parallel mesh partitioner with parameterized partition shape, and a customized partitioner used for discrete ordinates particle transport computations. This framework is also part of a generic library, STAPL, allowing the partitioning of the data and development of the whole parallel application to be done in the same environment. We show the user-friendly interface of the framework and its scalability for partitioning different mesh and graph benchmarks on a Cray XE6 system. We also highlight the performance of our customized unstructured mesh partitioner for a discrete ordinates particle transport code. The developed columnar decompositions significantly reduce the execution time of simultaneous sweeps on unstructured meshes.