In this work, two topics of reservoir simulations are discussed. The first topic is the two-phase compositional flow simulation in hydrocarbon reservoir. The major obstacle that impedes the applicability of the simulation code is the long run time of the simulation procedure, and thus speeding up the simulation code is necessary. Two means are demonstrated to address the problem: parallelism in physical space and the application of sparse grids in parameter space. The parallel code can gain satisfactory scalability, and the sparse grids can remove the bottleneck of flash calculations. Instead of carrying out the flash calculation in each time step of the simulation, a sparse grid approximation of all possible results of the flash calculation is generated before the simulation. Then the constructed surrogate model is evaluated to approximate the flash calculation results during the simulation. The second topic is the wormhole propagation simulation in carbonate reservoir. In this work, different from the traditional simulation technique relying on the Darcy framework, we propose a new framework called Darcy-Brinkman-Forchheimer framework to simulate wormhole propagation. Furthermore, to process the large quantity of cells in the simulation grid and shorten the long simulation time of the traditional serial code, standard domain-based parallelism is employed, using the Hypre multigrid library. In addition to that, a new technique called “experimenting field approach” to set coefficients in the model equations is introduced. In the 2D dissolution experiments, different configurations of wormholes and a series of properties simulated by both frameworks are compared. We conclude that the numerical results of the DBF framework are more like wormholes and more stable than the Darcy framework, which is a demonstration of the advantages of the DBF framework. The scalability of the parallel code is also evaluated, and good scalability can be achieved. Finally, a mixed finite element scheme is proposed for the wormhole simulation.

Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.

<p>Polynomial chaos expansions provide an efficient and robust framework to analyze and quantify uncertainty in computational models. This dissertation explores the use of adaptive sparse grids to reduce the computational cost of determining a polynomial model surrogate while examining and implementing new adaptive techniques.</p><p>Determination of chaos coefficients using traditional tensor product quadrature suffers the so-called curse of dimensionality, where the number of model evaluations scales exponentially with dimension. Previous work used a sparse Smolyak quadrature to temper this dimensional scaling, and was applied successfully to an expensive Ocean General Circulation Model, HYCOM during the September 2004 passing of Hurricane Ivan through the Gulf of Mexico. Results from this investigation suggested that adaptivity could yield great gains in efficiency. However, efforts at adaptivity are hampered by quadrature accuracy requirements.</p><p>We explore the implementation of a novel adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed adaptive pseudo-spectral projection (aPSP) algorithm that is based on a direct application of Smolyak's sparse grid formula, and that allows for the use of arbitrary admissible sparse grids. Such a construction ameliorates the severe restrictions posed by insufficient quadrature accuracy. The adaptive algorithm is tested using an existing simulation database of the HYCOM model during Hurricane Ivan. The {\it a priori} tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling.</p><p>In order to provide a finer degree of resolution control along two distinct subsets of model parameters, we investigate two methods to build polynomial approximations. The two approaches are based with pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids. The control of the error along different subsets of parameters may be needed in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid pseudo-spectral projection is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, adaptive PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error. </p><p>In order to increase efficiency even further, a subsampling technique is developed to allow for local adaptivity within the aPSP algorithm. The local refinement is achieved by exploiting the hierarchical nature of nested quadrature grids to determine regions of estimated convergence. In order to achieve global representations with local refinement, synthesized model data from a lower order projection is used for the final projection. The final subsampled grid was also tested with two more robust, sparse projection techniques including compressed sensing and hybrid least-angle-regression. These methods are evaluated on two sample test functions and then as an {\it a priori} analysis of the HYCOM simulations and the shock-tube ignition model investigated earlier. Small but non-trivial efficiency gains were found in some cases and in others, a large reduction in model evaluations with only a small loss of model fidelity was realized. Further extensions and capabilities are recommended for future investigations.</p> / Dissertation

This thesis addresses the problem of representing few group, homogenised neutron cross sections as a function of state parameters (e.g. burn-up, fuel and moderator temperature, etc.) that describe the conditions in the reactor. The problem is multi-dimensional and the cross section samples, required for building the representation, are the result of expensive transport calculations. At the same time, practical applications require high accuracy. The representation method must therefore be efficient in terms of the number of samples needed for constructing the representation, storage requirements and cross section reconstruction time. Sparse grid methods are proposed for constructing such an efficient representation. Approximation through quasi-regression as well as polynomial interpolation, both based on sparse grids, were investigated. These methods have built-in error estimation capabilities and methods for optimising the representation, and scale well with the number of state parameters. An anisotropic sparse grid integrator based on Clenshaw-Curtis quadrature was implemented, verified and coupled to a pre-existing cross section representation system. Some ways to improve the integrator’s performance were also explored. The sparse grid methods were used to construct cross section representations for various Light Water Reactor fuel assemblies. These reactors have different operating conditions, enrichments and state parameters and therefore pose different challenges to a representation method. Additionally, an example where the cross sections have a different group structure, and were calculated using a different transport code, was used to test the representation method. The built-in error measures were tested on independent, uniformly distributed, quasi-random sample points. In all the cases studied, interpolation proved to be more accurate than approximation for the same number of samples. The primary source of error was found to be the Xenon transient at the beginning of an element’s life (BOL). To address this, the domain was split along the burn-up dimension into “start-up” and “operating” representations. As an alternative, the Xenon concentration was set to its equilibrium value for the whole burn-up range. The representations were also improved by applying anisotropic sampling. It was concluded that interpolation on a sparse grid shows promise as a method for building a cross section representation of sufficient accuracy to be used for practical reactor calculations with a reasonable number of samples. / Thesis (MSc Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

This thesis addresses the problem of representing few group, homogenised neutron cross sections as a function of state parameters (e.g. burn-up, fuel and moderator temperature, etc.) that describe the conditions in the reactor. The problem is multi-dimensional and the cross section samples, required for building the representation, are the result of expensive transport calculations. At the same time, practical applications require high accuracy. The representation method must therefore be efficient in terms of the number of samples needed for constructing the representation, storage requirements and cross section reconstruction time. Sparse grid methods are proposed for constructing such an efficient representation. Approximation through quasi-regression as well as polynomial interpolation, both based on sparse grids, were investigated. These methods have built-in error estimation capabilities and methods for optimising the representation, and scale well with the number of state parameters. An anisotropic sparse grid integrator based on Clenshaw-Curtis quadrature was implemented, verified and coupled to a pre-existing cross section representation system. Some ways to improve the integrator’s performance were also explored. The sparse grid methods were used to construct cross section representations for various Light Water Reactor fuel assemblies. These reactors have different operating conditions, enrichments and state parameters and therefore pose different challenges to a representation method. Additionally, an example where the cross sections have a different group structure, and were calculated using a different transport code, was used to test the representation method. The built-in error measures were tested on independent, uniformly distributed, quasi-random sample points. In all the cases studied, interpolation proved to be more accurate than approximation for the same number of samples. The primary source of error was found to be the Xenon transient at the beginning of an element’s life (BOL). To address this, the domain was split along the burn-up dimension into “start-up” and “operating” representations. As an alternative, the Xenon concentration was set to its equilibrium value for the whole burn-up range. The representations were also improved by applying anisotropic sampling. It was concluded that interpolation on a sparse grid shows promise as a method for building a cross section representation of sufficient accuracy to be used for practical reactor calculations with a reasonable number of samples. / Thesis (MSc Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

Calculations of reactor parameters of interest (such as neutron multiplication factors, decay heat, reaction rates, etc.), are often based on models which are dependent on groupwise neutron cross sections. The uncertainties associated with these neutron cross sections are propagated to the final result of the calculated reactor parameters. There is a need to characterize this uncertainty and to be able to apportion the uncertainty in a calculated reactor parameter to the different sources of uncertainty in the groupwise neutron cross sections, this procedure is known as sensitivity analysis. The focus of this study is the application of a modified global sensitivity analysis technique to calculations of reactor parameters that are dependent on groupwise neutron cross–sections. Sensitivity analysis can help in identifying the important neutron cross sections for a particular model, and also helps in establishing best–estimate optimized nuclear reactor physics models with reduced uncertainties. In this study, our approach to sensitivity analysis will be similar to the variance–based global sensitivity analysis technique, which is robust, has a wide range of applicability and provides accurate sensitivity information for most models. However, this technique requires input variables to be mutually independent. A modification to this technique, that allows one to deal with input variables that are block–wise correlated and normally distributed, is presented. The implementation of the modified technique involves the calculation of multi–dimensional integrals, which can be prohibitively expensive to compute. Numerical techniques specifically suited to the evaluation of multidimensional integrals namely Monte Carlo, quasi–Monte Carlo and sparse grids methods are used, and their efficiency is compared. The modified technique is illustrated and tested on a two–group cross–section dependent problem. In all the cases considered, the results obtained with sparse grids achieved much better accuracy, while using a significantly smaller number of samples. / Thesis (M.Sc. Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2011.

Calculations of reactor parameters of interest (such as neutron multiplication factors, decay heat, reaction rates, etc.), are often based on models which are dependent on groupwise neutron cross sections. The uncertainties associated with these neutron cross sections are propagated to the final result of the calculated reactor parameters. There is a need to characterize this uncertainty and to be able to apportion the uncertainty in a calculated reactor parameter to the different sources of uncertainty in the groupwise neutron cross sections, this procedure is known as sensitivity analysis. The focus of this study is the application of a modified global sensitivity analysis technique to calculations of reactor parameters that are dependent on groupwise neutron cross–sections. Sensitivity analysis can help in identifying the important neutron cross sections for a particular model, and also helps in establishing best–estimate optimized nuclear reactor physics models with reduced uncertainties. In this study, our approach to sensitivity analysis will be similar to the variance–based global sensitivity analysis technique, which is robust, has a wide range of applicability and provides accurate sensitivity information for most models. However, this technique requires input variables to be mutually independent. A modification to this technique, that allows one to deal with input variables that are block–wise correlated and normally distributed, is presented. The implementation of the modified technique involves the calculation of multi–dimensional integrals, which can be prohibitively expensive to compute. Numerical techniques specifically suited to the evaluation of multidimensional integrals namely Monte Carlo, quasi–Monte Carlo and sparse grids methods are used, and their efficiency is compared. The modified technique is illustrated and tested on a two–group cross–section dependent problem. In all the cases considered, the results obtained with sparse grids achieved much better accuracy, while using a significantly smaller number of samples. / Thesis (M.Sc. Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2011.

A study was performed to compare two different methods for generating surrogate models for the analysis and design of airfoils. Initial research was performed to compare the accuracy of surrogate models for predicting the lift and drag of an airfoil with data collected from highidelity simulations using a modern CFD code along with lower-order models using a panel code. This was followed by an evaluation of the Class Shape Trans- formation (CST) method for parameterizing airfoil geometries as a prelude to the use of surrogate models for airfoil design optimization and the implementation of software to use CST to modify airfoil shapes as part of the airfoil design process. Optimization routines were coupled with surrogate modeling techniques to study the accuracy and efficiency of the surrogate models to produce optimal airfoil shapes. Finally, the results of the current research are summarized, and suggestions are made for future research.

Ce travail se compose de deux parties relevant d'applications industrielles différentes. <br />La première traite d'une antenne spatiale réseau active. <br />Il faut d'abord calculer les lois d'alimentation pour satisfaire les contraintes de rayonnement. Nous transformons un problème avec de nombreux minima locaux en un problème d'optimisation convexe, dont l'optimum est le minimum global du problème initial, en utilisant le principe de conservation de l'énergie. <br />Nous résolvons ensuite un problème d'optimisation topologique: il faut réduire le nombre d'éléments rayonnants (ER). Nous appliquons une décomposition en valeurs singulières à l'ensemble des modules optimaux relaxés, puis un algorithme de type gradient topologique décide les regroupements entre ER élémentaires. <br /><br />La deuxième partie porte sur une simulation type boîte noire d'un accident chimique. <br />Nous effectuons une étude de fiabilité et de sensibilité suivant un grand nombre de paramètres (probabilités de défaillance, point de conception, et paramètres influents). Sans disposer du gradient, nous utilisons un modèle réduit. <br />Dans un premier cas test nous avons comparé les réseaux neuronaux et la méthode d'interpolation sur grille éparse Sparse Grid (SG). Les SG sont une technique émergente: grâce à leur caractère hiérarchique et un algorithme adaptatif, elles deviennent particulièrement efficaces pour les problèmes réels (peu de variables influentes). <br />Elles sont appliquées à un cas test en plus grande dimension avec des améliorations spécifiques (approximations successives et seuillage des données). <br />Dans les deux cas, les algorithmes ont donné lieu à des logiciels opérationnels.

Ce travail présente un développement et une analyse des approches numériques déterministes et probabilistes efficaces pour les équations aux dérivées partielles avec des coefficients et données aléatoires. On s'intéresse au problème d'écoulement stationnaire avec des données aléatoires. Une méthode de projection dans le cas unidimensionnel est présentée, permettant de calculer efficacement la moyenne de la solution. Nous utilisons la méthode de collocation anisotrope des grilles clairsemées. D'abord, un indicateur de l'erreur satisfaisant une borne supérieure de l'erreur est introduit, il permet de calculer les poids d'anisotropie de la méthode. Ensuite, nous démontrons une amélioration de l'erreur a priori de la méthode. Elle confirme l'efficacité de la méthode en comparaison avec Monte-Carlo et elle sera utilisée pour accélérer la méthode par l'extrapolation de Richardson. Nous présentons aussi une analyse numérique d'une méthode probabiliste pour quantifier la migration d'un contaminant dans un milieu aléatoire. Nous considérons le problème d'écoulement couplé avec l'équation d'advection-diffusion, où on s'intéresse à la moyenne de l'extension et de la dispersion du soluté. Le modèle d'écoulement est discrétisée par une méthode des éléments finis mixtes, la concentration du soluté est une densité d'une solution d'une équation différentielle stochastique, qui sera discrétisée par un schéma d'Euler. Enfin, on présente une formule explicite de la dispersion et des estimations de l'erreur a priori optimales. / This work presents a development and an analysis of an effective deterministic and probabilistic approaches for partial differential equation with random coefficients and data. We are interesting in the steady flow equation with stochastic input data. A projection method in the one-dimensional case is presented to compute efficiently the average of the solution. An anisotropic sparse grid collocation method is also used to solve the flow problem. First, we introduce an indicator of the error satisfying an upper bound of the error, it allows us to compute the anisotropy weights of the method. We demonstrate an improvement of the error estimation of the method which confirms the efficiency of the method compared with Monte Carlo and will be used to accelerate the method using the Richardson extrapolation technique. We also present a numerical analysis of one probabilistic method to quantify the migration of a contaminant in random media. We consider the previous flow problem coupled with the advection-diffusion equation, where we are interested in the computation of the mean extension and the mean dispersion of the solute. The flow model is discretized by a mixed finite elements method and the concentration of the solute is a density of a solution of the stochastic differential equation, this latter will be discretized by an Euler scheme. We also present an explicit formula of the dispersion and an optimal a priori error estimates.