The processes studied by nuclear engineers generally include coupled physics phenomena (Thermal-Hydraulics, Neutronics, Material Mechanics, etc.) and modeling such multiphysics processes numerically can be computationally intensive. A way to reduce the computational burden is to use spatial meshes that are optimally suited for a specific solution; such meshes are obtained through a process known as Adaptive Mesh Refinement (AMR). AMR can be especially useful for modeling multiphysics phenomena by allowing each solution component to be computed on an independent mesh (Multimesh AMR). Using AMR on time dependent problems requires the spatial mesh to change in time as the solution changes in time. Current algorithms presented in the literature address this concern by adapting the spatial mesh at every time step, which can be inefficient. This Thesis proposes an algorithm for saving computational resources by using a spatially adapted mesh for multiple time steps, and only adapting the spatial mesh when the solution has changed significantly. This Thesis explores the mechanisms used to determine when and where to spatially adapt for time dependent, coupled physics problems. The algorithm is implemented using the Deal.ii fiinite element library [1, 2], in 2D and 3D, and is tested on a coupled neutronics and heat conduction problem in 2D. The algorithm is shown to perform better than a uniformly refined static mesh and, in some cases, a mesh that is spatially adapted at every time step.

Implementations of numerical simulations for solving systems of partial differential equations are often not verified and are falsely assumed to work correctly. As a result, the implementations are prone to coding errors that could degrade the accuracy of the solution. In order to ensure that a code is written correctly, rigorous verification of all parts of the code is necessary. Code verification is the task of ascertaining whether a numerical algorithm is solving the governing equations of the problem correctly. If an exact solution existed for the governing equations then verification would be easier but these solutions are rare because of the non-linearity of common Computational Fluid Dynamics (CFD) problems. In the absence of exact solutions, grid refinement studies are the most commonly used methods to verify codes using simulations on a sequence of grids but even these studies have limitations. The Method of Manufactured Solutions (MMS) is a novel and a recently developed technique that verifies the observed order-ofuracy of the implementation of a numerical algorithm. The method is more general and overcomes many of the limitations of the method of exact solutions and grid refinement studies. The central idea is to modify the governing equations and the boundary conditions by adding forcing functions or source terms in order to drive the discrete solution to a prescribed or ``manufactured' solution chosen a priori. A grid convergence study is performed subsequently to determine the observed orders. Two methods of accuracy assessment are presented here - solution accuracy analysis and residual error analysis. The method based on the error in the spatial residual is computationally less expensive and proved to be a valuable debugging tool. In the present work, the Method of Manufactured Solutions (MMS) is implemented on a compressible flow solver that solves the two-dimensional Euler equations on structured grids and an incompressible code that solves the two-dimensional Navier-Stokes equations on unstructured meshes. Exponential functions are used to ``manufacture' steady solutions to the governing equations. Solution and residual error analyses are presented. The influence of grid non-uniformity on the numerical accuracy is studied.

Code verification is the process of ensuring, to the degree possible, that there are no algorithm deficiencies and coding mistakes (bugs) in a scientific computing simulation. In this work, techniques are presented for performing code verification of boundary conditions commonly used in compressible and incompressible Computational Fluid Dynamics (CFD) codes. Using a compressible CFD code, this study assesses the subsonic inflow (isentropic and fixed-mass), subsonic outflow, supersonic outflow, no-slip wall (adiabatic and isothermal), and inviscid slip-wall. The use of simplified curved surfaces is proposed for easier generation of manufactured solutions during the verification of certain boundary conditions involving many constraints. To perform rigorous code verification, general grids with mixed cell types at the verified boundary are used. A novel approach is introduced to determine manufactured solutions for boundary condition verification when the velocity-field is constrained to be divergence-free during the simulation in an incompressible CFD code. Order of accuracy testing using the Method of Manufactured Solutions (MMS) is employed here for code verification of the major components of an open-source, multiphase flow code - MFIX. The presence of two-phase governing equations and a modified SIMPLE-based algorithm requiring divergence-free flows makes the selection of manufactured solutions more involved than for single-phase, compressible flows. Code verification is performed here on 2D and 3D, uniform and stretched meshes for incompressible, steady and unsteady, single-phase and two-phase flows using the two-fluid model of MFIX. In a CFD simulation, truncation error (TE) is the difference between the continuous governing equation and its discrete approximation. Since TE can be shown to be the local source term for the discretization error, TE is proposed as the criterion for determining which regions of the computational mesh should be refined/coarsened. For mesh modification, an error equidistribution strategy to perform r-refinement (i.e., mesh node relocation) is employed. This technique is applied to 1D and 2D inviscid flow problems where the exact (i.e., analytic) solution is available. For mesh adaptation based upon TE, about an order of magnitude improvement in discretization error levels is observed when compared with the uniform mesh. / Ph. D.

Computational Fluid Dynamics (CFD) applications usually involve intensive computations, which can be accelerated through using open accelerators, especially GPUs due to their common use in the scientific computing community. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented numerically correctly, which is called code verification. This dissertation focuses on accelerating research CFD codes on multi-CPUs/GPUs using MPI and OpenACC, as well as the code verification for turbulence model implementation using the method of manufactured solutions and code-to-code comparisons. First, a variety of performance optimizations both agnostic and specific to applications and platforms are developed in order to 1) improve the heterogeneous CPU/GPU compute utilization; 2) improve the memory bandwidth to the main memory; 3) reduce communication overhead between the CPU host and the GPU accelerator; and 4) reduce the tedious manual tuning work for GPU scheduling. Both finite difference and finite volume CFD codes and multiple platforms with different architectures are utilized to evaluate the performance optimizations used. A maximum speedup of over 70 is achieved on 16 V100 GPUs over 16 Xeon E5-2680v4 CPUs for multi-block test cases. In addition, systematic studies of code verification are performed for a second-order accurate finite volume research CFD code. Cross-term sinusoidal manufactured solutions are applied to verify the Spalart-Allmaras and k-omega SST model implementation, both in 2D and 3D. This dissertation shows that the spatial and temporal schemes are implemented numerically correctly. / Doctor of Philosophy / Computational Fluid Dynamics (CFD) is a numerical method to solve fluid problems, which usually requires a large amount of computations. A large CFD problem can be decomposed into smaller sub-problems which are stored in discrete memory locations and accelerated by a large number of compute units. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented correctly, which is called code verification. This dissertation focuses on the CFD code acceleration as well as the code verification for turbulence model implementation. In this dissertation, multiple Graphic Processing Units (GPUs) are utilized to accelerate two CFD codes, considering that the GPU has high computational power and high memory bandwidth. A variety of optimizations are developed and applied to improve the performance of CFD codes on different parallel computing systems. The program execution time can be reduced significantly especially when multiple GPUs are used. In addition, code-to-code comparisons with some NASA CFD codes and the method of manufactured solutions are utilized to verify the correctness of a research CFD code.

The thesis presents xavier, a novel tool-set for model checking of concurrent x86-64 assembly programs, via Partial Order Reduction (POR). xavier{} presents a realistic platform for systematically exploring and analyzing the state-space of concurrent x86 assembly programs, with the aim of detecting bugs via assertion failures in mainstream programs. Recently, a number of state-of-the-art model checking solutions have been introduced to efficiently explore the state-space of concurrent programs, using POR algorithms. However, such solutions are inefficient while analyzing stateful programming languages, such as the x86 assembly language, due to their low level of abstraction. To this end, xavier{} makes two contributions: i) a novel order-sensitivity based POR algorithm, that is applicable to concurrent x86 assembly, ii) an x86 machine-model that can accurately perform relaxed-consistency emulation of concurrent x86 assembly, without the need for any translations. We demonstrate the applicability of xavier{} through an evaluation on several classical mutual-exclusion benchmarks and mainstream benchmarks from the Userspace Read-Copy-Update (URCU) concurrency library, where the benchmarks range from $250-3700$ lines of x86 assembly. The framework is the first that supports systematic model checking of concurrent x86 assembly programs, and the effectiveness of xavier{} is demonstrated by reproducing a concurrency issue of threads accessing intermediate states in the URCU library, which stems from an assumption violation. / Master of Science / Sound verification of multi-threaded programs necessitate a systematic analysis of program state-spaces that result from thread interactions. Consequently, model-checking cite{godefroid1997model, Clarke2018} has been one of the prominent methods used to tackle the verification of multi-threaded programs. However, existing model-checking solutions are inefficient while analyzing stateful programming languages, such as the x86 assembly language, due to the solutions' higher level of abstraction. Therefore, the thesis presents xavier, a novel tool-set and a realistic platform for systematically exploring and analyzing the state-space of mainstream concurrent x86 assembly programs, with the aim of detecting bugs via assertion failures. To this end, xavier{} makes two contributions: i) a novel order-sensitivity based Partial Order Reduction algorithm, which efficiently explores the state space of concurrent x86 assembly, ii) an x86 machine-model that can accurately emulate the execution of concurrent x86 assembly, without the need for any translations. We demonstrate the applicability of xavier{} through an evaluation on several classical mutual-exclusion and mainstream benchmarks from the Userspace Read-Copy-Update (URCU) concurrency library, where the benchmarks range from $250-3700$ lines of x86 assembly. Moreover, we demonstrate the effectiveness of xavier{} by reproducing a concurrency issue in the URCU library, which manifests as a result of an assumption violation.

A detailed code verification study of an unstructured finite volume Computational Fluid Dynamics (CFD) code is performed. The Method of Manufactured Solutions is used to generate exact solutions for the Euler and Navier-Stokes equations to verify the correctness of the code through order of accuracy testing. The verification testing is performed on different mesh types which include triangular and quadrilateral elements in 2D and tetrahedral, prismatic, and hexahedral elements in 3D. The requirements of systematic mesh refinement are discussed, particularly in regards to unstructured meshes. Different code options verified include the baseline steady state governing equations, transport models, turbulence models, boundary conditions and unsteady flows. Coding mistakes, algorithm inconsistencies, and mesh quality sensitivities uncovered during the code verification are presented. In recent years, there has been significant work on the development of algorithms for the compressible Navier-Stokes equations on unstructured grids. One of the challenging tasks during the development of these algorithms is the formulation of consistent and accurate diffusion operators. The robustness and accuracy of diffusion operators depends on mesh quality. A survey of diffusion operators for compressible CFD solvers is conducted to understand different formulation procedures for diffusion fluxes. A patch-wise version of the Method of Manufactured Solutions is used to test the accuracy of selected diffusion operators. This testing of diffusion operators is limited to cell-centered finite volume methods which are formally second order accurate. These diffusion operators are tested and compared on different 2D mesh topologies to study the effect of mesh quality (stretching, aspect ratio, skewness, and curvature) on their numerical accuracy. Quantities examined include the numerical approximation errors and order of accuracy associated with face gradient reconstruction. From the analysis, defects in some of the numerical formulations are identified along with some robust and accurate diffusion operators. / Ph. D.

A poluição sonora é um problema central de uma grande diversidade de aplicações industriais. Na engenharia, podemos citar diversos casos que geram ruído, como exemplos os trens, automóveis, rotores de helicópteros e o ruído aerodinâmico das aeronaves, o qual se divide em ruído gerado pelos motores a jato e a estruturas da aeronave. No presente momento o ruído dos motores aeronáuticos, principalmente os jatos, atingiu níveis de ruídos semelhantes às estruturas da aeronave, como por exemplo, eslates, flaps e trens de pouso. Desta forma, as autoridades de transporte aéreo estão exigindo também redução no ruído das estruturas. O presente trabalho apresenta a verificação das potencialidades e limitações do software CFD++, programa este adquirido pela EMBRAER e inserido como parte do projeto Aeronave Silenciosa, para assim poder compreender de uma melhor maneira o fenômeno da aeroacústica, e deste modo, poder contribuir para a redução do ruído externo das aeronaves. Para verificar as potencialidades e limitações do CFD++, foi proposto investigar o mecanismo de som do eslate. Tal fenômeno é devido ao deslocamento da camada limite no intradorso do eslate a partir de onde se desenvolve a camada de mistura, foco do presente trabalho. / Noise pollution is a central problem of a wide variety of industrial applications. In engineering, cite several cases that generate noise, as examples trains, automobiles, rotors of helicopters and the noise generated by aircraft, which is divided into noise generated by jet engines and airframe. At present the noise of aircraft engines, largely the jets reached noise levels similar structures, such as slat, flaps and landing gear. Thus, the air transport authorities are also demanding a reduction in noise of the structures of airframe. This report presents the verification of potentialities and limitations of CFD++, a program acquired by EMBRAER and inserted as part of the Silent Aircraft, so they can understand better how the phenomenon of aeroacoustics, and thus able to reduce contribute external noise from aircraft. To check the potentialities and limitations of CFD++, was proposed to investigate the mechanism of sound generated by the slat. This phenomenon is due to the displacement of the boundary layer on the lower surface of the slat from which the mixed layer develops. The mixing layer is the focus of this work.

A poluição sonora é um problema central de uma grande diversidade de aplicações industriais. Na engenharia, podemos citar diversos casos que geram ruído, como exemplos os trens, automóveis, rotores de helicópteros e o ruído aerodinâmico das aeronaves, o qual se divide em ruído gerado pelos motores a jato e a estruturas da aeronave. No presente momento o ruído dos motores aeronáuticos, principalmente os jatos, atingiu níveis de ruídos semelhantes às estruturas da aeronave, como por exemplo, eslates, flaps e trens de pouso. Desta forma, as autoridades de transporte aéreo estão exigindo também redução no ruído das estruturas. O presente trabalho apresenta a verificação das potencialidades e limitações do software CFD++, programa este adquirido pela EMBRAER e inserido como parte do projeto Aeronave Silenciosa, para assim poder compreender de uma melhor maneira o fenômeno da aeroacústica, e deste modo, poder contribuir para a redução do ruído externo das aeronaves. Para verificar as potencialidades e limitações do CFD++, foi proposto investigar o mecanismo de som do eslate. Tal fenômeno é devido ao deslocamento da camada limite no intradorso do eslate a partir de onde se desenvolve a camada de mistura, foco do presente trabalho. / Noise pollution is a central problem of a wide variety of industrial applications. In engineering, cite several cases that generate noise, as examples trains, automobiles, rotors of helicopters and the noise generated by aircraft, which is divided into noise generated by jet engines and airframe. At present the noise of aircraft engines, largely the jets reached noise levels similar structures, such as slat, flaps and landing gear. Thus, the air transport authorities are also demanding a reduction in noise of the structures of airframe. This report presents the verification of potentialities and limitations of CFD++, a program acquired by EMBRAER and inserted as part of the Silent Aircraft, so they can understand better how the phenomenon of aeroacoustics, and thus able to reduce contribute external noise from aircraft. To check the potentialities and limitations of CFD++, was proposed to investigate the mechanism of sound generated by the slat. This phenomenon is due to the displacement of the boundary layer on the lower surface of the slat from which the mixed layer develops. The mixing layer is the focus of this work.

L'étude numérique est un outil de recherche qui est devenu incontournable, en particulier pour la compréhension et le contrôle des systèmes complexes. La simulation des plasmas de fusion par confinement magnétique s'inscrit parfaitement dans cette démarche. Les larges rapports d'échelle en temps et espace, la nature chaotique des plasmas et les très fortes anisotropies imposent l'utilisation de méthodes numériques avancées. C'est dans ce cadre que les deux volets de ma thèse s'inscrivent.Le premier volet est l’originalité de ma thèse, la mise en place la méthode PoPe, une procédure générale de vérification de codes et réduction de modèles. Le principe de cette méthode est de déterminer les équations qui ont permis de générer un ensemble de données : si les données sont issues d'un code de simulation, retrouver ces équations et les comparer au modèle théoriquement implémenté est équivalent à vérifier le code. La précision de la procédure permet de caractériser l'erreur commise jusqu'à retrouver l'ordre des schémas numériques employés, même en régime chaotique.Le second volet de ma thèse se consacre à l’étude du transport turbulent qui détermine la performance des plasmas de fusion. L’étude du transport sous forme d’avalanches dans un modèle de bord fluide est entreprise en quantifiant l’impact du chaos sur l’auto-organisation. Pour un modèle cinétique restreint aux instabilités basse fréquence, la capacité de se bloquer dans deux régimes exclusifs, l’un isolant, l’autre conducteur, est étudiée. Ce modèle est amélioré pour permettre des relaxations entre ces deux états. Pour ces modèles fluide et cinétique, des modèles réduits obtenus avec la méthode PoPe sont proposés. / Numerical analysis is now a key component of research, especially for the understanding and the control of complex systems. Simulations of magnetic confinement plasmas fall within this approach. One of the difficulties of this field is the wide range of spatial scales, time scales, the chaotic nature of plasmas and the strong anisotropies require advanced numerical methods. Each of the two parts of my thesis takes place in this frame of numerical simulation and fusion plasmas.The first part of my thesis is dedicated to the method PoPe, a general method for code verification and model reduction. The principle of this method is to determine the equations which have generated a set of data. If the data was produced by a simulation tool, finding these equations and comparing them to the ones theoretically implemented is equivalent to verifying this simulation tool. The accuracy of this procedure allows to characterize the numerical error and to recover the order of each numerical scheme used.The second part of my thesis deals with the study of turbulent transport which determines the efficiency of fusion plasma. The chaotic avalanches of a fluid model are studied considering the impact of the chaos on the self-organization. For a kinetic model restricted to the low frequency instabilities, the ability to block itself in two regimes, one insulating and the other conducting, is studied. Upgrades of this model are undertaken in order to introduce the possibility of relaxations between the two previous states. For both the fluid and the kinetic model, reduce models are proposed thank to the PoPe method.

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