Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第14599号 / 工博第3067号 / 新制||工||1456(附属図書館) / 26951 / UT51-2009-D311 / 京都大学大学院工学研究科都市環境工学専攻 / (主査)教授 大西 有三, 教授 間瀬 肇, 准教授 西山 哲 / 学位規則第4条第1項該当

In science and engineering, many simulations are carried out over domains consisting of multiple materials separated by curves/surfaces. If partial differential equations (PDEs) are used to model these simulations, it usually leads to the so-called interface problems of PDEs whose coefficients are discontinuous. In this dissertation, we consider nonconforming immersed "nite element (IFE) methods and error analysis for interface problems. We "first consider the second order elliptic interface problem with a discontinuous diffusion coefficient. We propose new IFE spaces based on the nonconforming rotated Q1 "finite elements on Cartesian meshes. The degrees of freedom of these IFE spaces are determined by midpoint values or average integral values on edges. We investigate fundamental properties of these IFE spaces, such as unisolvency and partition of unity, and extend well-known trace inequalities and inverse inequalities to these IFE functions. Through interpolation error analysis, we prove that these IFE spaces have optimal approximation capabilities. We use these IFE spaces to develop partially penalized Galerkin (PPG) IFE schemes whose bilinear forms contain penalty terms over interface edges. Error estimation is carried out for these IFE schemes. We prove that the PPG schemes with IFE spaces based on integral-value degrees of freedom have the optimal convergence in an energy norm. Following a similar approach, we prove that the interior penalty discontinuous Galerkin schemes based on these IFE functions also have the optimal convergence. However, for the PPG schemes based on midpoint-value degrees of freedom, we prove that they have at least a sub-optimal convergence. Numerical experiments are provided to demonstrate features of these IFE methods and compare them with other related numerical schemes. We extend nonconforming IFE schemes to the planar elasticity interface problem with discontinuous Lam"e parameters. Vector-valued nonconforming rotated Q1 IFE functions with integral-value degrees of freedom are unisolvent with appropriate interface jump conditions. More importantly, the Galerkin IFE scheme using these vector-valued nonconforming rotated Q1 IFE functions are "locking-free" for nearly incompressible elastic materials. In the last part of this dissertation, we consider potential applications of IFE methods to time dependent PDEs with moving interfaces. Using IFE functions in the discretization in space enables the applicability of the method of lines. Crank-Nicolson type fully discrete schemes are also developed as alternative approaches for solving moving interface problems. / Ph. D.

In the thesis work, a code to solve the two dimensional compressible Euler equations for external flows around arbitrary geometries have been developed. A Cartesianmesh generator is incorporated to the solver. Hence the pre-processing can be performed together with the solution within a single code. The code is written in the C++ programming language and its object oriented capabilities have been exploited to save memory in the data structure developed. The Cartesian mesh is formed by dividing squares successively into its four quadrants. The main advantage of using this type of a mesh is the ability to generate meshes around geometries of arbitrary complexity quickly and to adapt the mesh easily based on the solution. The main disadvantage of this method is that the treatment of the cells that are cut by the geometry. For the solution procedure Roe&rsquo / s method as well as flux vector splitting methods are used for the flux evaluation. The flux vector splitting schemes used are van Leer, AUSM, AUSMD and AUSMV methods. Time discretization is performed using a multi-stage method. To increase the accuracy least squares reconstruction is employed. The code is validated by performing calculations around a NACA0012 airfoil profile. The effect of reconstruction is demonstrated by plotting the pressure coefficient on the airfoil. The distribution obtained using reconstruction is very close to the experimental one while there is a considerable deviation for the case without reconstruction. Also the shock capturing capabilities of different methods have been investigated. In addition the performance of each method is analyzed for flow around an NLR 7301 airfoil with a flap.

Nuclear nonproliferation research and forensics have a need for improved software solutions, particularly in the estimates of the transmutation of nuclear fuel during burnup and depletion. At the same time, parallel computers have become effectively sized to enable full core simulations using highly-detailed 3d mesh models. In this work, the capability for modeling 3d reactor models is researched with PENBURN, a burnup/depletion code that couples to the PENTRAN Parallel Sn Transport Solver and also to the Monte Carlo solver MCNP5 using the multigroup option. This research is computationally focused, but will also compare a subset of results of experimental Pressurized Water Reactor (PWR) burnup spectroscopy data available with a designated BR3 PWR burnup benchmark. Also, this research will analyze large-scale Cartesian mesh models that can be feasibly modeled for 3d burnup, as well as investigate the improvement of finite differencing schemes used in parallel discrete ordinates transport with PENTRAN, in order to optimize runtimes for full core transport simulation, and provide comparative results with Monte Carlo simulations. Also, the research will consider improvements to software that will be parallelized, further improving large model simulation using hybrid OpenMP-MPI. The core simulations that form the basis of this research, utilizing discrete ordinates methods and Monte Carlo methods to drive time and space dependent isotopic reactor production using the PENBURN code, will provide more accurate detail of fuel compositions that can benefit nuclear safety, fuel management, non-proliferation, and safeguards applications.

Cette thèse étudie le raffinement local de maillage à la fois en espace et en temps pour l’équation de l’elastodynamique du second ordre pour le calcul haute performance. L’objectif est de mettre en place des méthodes numériques pour traiter des hétérogénéités de petite taille ayant un impact important sur la propagation des ondes. Nous utilisons une approche par éléments finis de Galerkin discontinus avec pénalisation pour leur flexibilité et facilité de parallélisation. La formulation éléments finis que nous proposons a pour particularité d’être élasto-acoustique, pour pouvoir prendre en compte des hétérogénéités acoustiques de petite taille. Par ailleurs, nous proposons un terme de pénalisation optimisé qui est mieux adapté à l’équation de l’élastodynamique, conduisant en particulier à une meilleure condition CFL. Nous avons aussi amélioré une formulation PML du second ordre pour laquelle nous avons proposé une nouvelle discrétisation temporelle qui rend la formulation plus stable. En tirant parti de la p-adaptivité et des maillages non-conformes des méthodes de Galerkin discontinues combiné à une méthode de pas de temps local, nous avons grandement réduit le coût du raffinement local. Ces méthodes ont été implémentées en C++, en utilisant des techniques de template metaprogramming, au sein d’un code parallèle à mémoire distribuée (MPI) et partagée (OpenMP). Enfin, nous montrons le potentiel de notre approche sur des cas tests de validation et sur des cas plus réalistes avec des milieux présentant des hydrofractures. / This thesis studies local mesh refinement both in time and space for the second order elastodynamic equation in a high performance computing context. The objective is to develop numerical methods to treat small heterogeneities that have global impact on wave propagation. We use an internal penalty discontinuous Galerkin finite element approach for its flexibity and parallelization capabilities. The elasto-acoustic finite element formulation we discuss is elasto-acoustic in order to handle local acoustic heterogeneities. We also propose an optimized penalty term more suited to the elastodynamic equation that results in better CFL condition. We improve a second order PML formulation with an original time discretization that results in a more stable formulation. Using the p-adaptivity and nonconforming mesh capabilities of discontinuous Galerkin methods combined with a local time stepping method, we greatly reduce the high computational cost of local refinements. These methods have been implemented in C++, using template metaprogramming, in a distributed memory (MPI) and shared memory (OpenMP) parallel code. Finally, we show the potential of our methods on validation test cases and on more realistic test cases with medium including hydrofractures.