Numerical solutions obtained through the use of Computational Fluid Dynamics (CFD) are subject to discretization error, which is locally generated by truncation error. The discretization error is extremely difficult to properly estimate and this in turn leads to uncertainty over the quality of the numerical solutions obtained via CFD methods and the engineering functionals computed using these solutions. Adjoint error estimation techniques specifically seek to estimate the error in functionals, but are dependent upon accurate truncation error estimates. This work examines the application of new, single-grid, truncation error estimation procedures to the problem of adjoint error estimation for both the quasi-1D and 2D Euler equations. The new truncation error estimation techniques are based on local reconstructions of the computed solutions and comparisons are made for the quasi-1D study in order to determine the most appropriate solution variables to reconstruct as well as the most appropriate reconstruction method. In addition, comparisons are made between the single-grid truncation error estimates and methods based on uniformally refining or coarsening the underlying numerical mesh on which the computed solutions are obtained. A method based on an refined grid error estimate is shown to work well for a non-isentropic flow for the quasi-1D Euler equations, but all truncation error estimations methods ultimately result in over prediction of functional discretization error in the presence of a shock in 2D. Alternatives to adjoint methods, which can only estimate the error in a single functional for each adjoint solution obtained, are examined for the 2D Euler equations. The defection correction method and error transport equations are capable of locally improving the entire computed solution, allowing for error estimates in multiple functionals. It is found that all three functional discretization error estimates perform similarly for the same truncation error estimate, although the defect correction method is the most costly from a computational viewpoint. Comparisons are made between truncation error and adjoint weighted truncation error based adaptive indicators. For the quasi-1D Euler equations it is found that both methods are competitive, however the truncation error based method is cheaper as a separate adjoint solve is avoided. For the 2D Euler equations, the truncation error estimates on the adapted meshes suffer due to a lack of smooth grid transformations which are used in reconstructing the computed solutions. In order to complete this work, a new CFD code incorporating a variety of best practices from the field of Computer Science is developed as well as a new method of performing code verification using the method of manufactured solutions which is significantly easier to implement than traditional manufactured solution techniques. / Ph. D.

El Método de los Elementos Finitos (MEF) se ha afianzado durante las últimas décadas como una de las técnicas numéricas más utilizadas para resolver una gran variedad de problemas en diferentes áreas de la ingeniería, como por ejemplo, el análisis estructural, análisis térmicos, de fluidos, procesos de fabricación, etc. Una de las aplicaciones donde el método resulta de mayor interés es en el análisis de problemas propios de la Mecánica de la Fractura, facilitando el estudio y evaluación de la integridad estructural de componentes mecánicos, la fiabilidad, y la detección y control de grietas. Recientemente, el desarrollo de nuevas técnicas como el Método Extendido de los Elementos Finitos (XFEM) ha permitido aumentar aún más el potencial del MEF. Dichas técnicas mejoran la descripción de problemas con singularidades, con discontinuidades, etc., mediante la adición de funciones especiales que enriquecen el espacio de la aproximación convencional de elementos finitos. Sin embargo, siempre que se aproxima un problema mediante técnicas numéricas, la solución obtenida presenta discrepancias con respecto al sistema que representa. En las técnicas basadas en la representación discreta del dominio mediante elementos finitos (MEF, XFEM, ...) interesa controlar el denominado error de discretización. En la literatura se pueden encontrar numerosas referencias a técnicas que permiten cuantificar el error en formulaciones convencionales de elementos finitos. No obstante, por ser el XFEM un método relativamente reciente, aún no se han desarrollado suficientemente las técnicas de estimación del error para aproximaciones enriquecidas de elementos finitos. El objetivo de esta Tesis es cuantificar el error de discretización cuando se utilizan aproximaciones enriquecidas del tipo XFEM para representar problemas propios de la Mecánica de la Fractura Elástico Lineal (MFEL), como es el caso del modelado de una grieta. / González Estrada, OA. (2010). Estimación y acotación del error de discretización en el modelado de grietas mediante el método extendido de los elementos finitos [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/7203

More and more challenging designs are required everyday in today¿s industries. The traditional trial and error procedure commonly used for mechanical parts design is not valid any more since it slows down the design process and yields suboptimal designs. For structural components, one alternative consists in using shape optimization processes which provide optimal solutions. However, these techniques require a high computational effort and require extremely efficient and robust Finite Element (FE) programs. FE software companies are aware that their current commercial products must improve in this sense and devote considerable resources to improve their codes. In this work we propose to use the Cartesian Grid Finite Element Method, cgFEM as a tool for efficient and robust numerical analysis. The cgFEM methodology developed in this thesis uses the synergy of a variety of techniques to achieve this purpose, but the two main ingredients are the use of Cartesian FE grids independent of the geometry of the component to be analyzed and an efficient hierarchical data structure. These two features provide to the cgFEM technology the necessary requirements to increase the efficiency of the cgFEM code with respect to commercial FE codes. As indicated in [1, 2], in order to guarantee the convergence of a structural shape optimization process we need to control the error of each geometry analyzed. In this sense the cgFEM code also incorporates the appropriate error estimators. These error estimators are specifically adapted to the cgFEM framework to further increase its efficiency. This work introduces a solution recovery technique, denoted as SPR-CD, that in combination with the Zienkiewicz and Zhu error estimator [3] provides very accurate error measures of the FE solution. Additionally, we have also developed error estimators and numerical bounds in Quantities of Interest based on the SPR-CD technique to allow for an efficient control of the quality of the numerical solution. Regarding error estimation, we also present three new upper error bounding techniques for the error in energy norm of the FE solution, based on recovery processes. Furthermore, this work also presents an error estimation procedure to control the quality of the recovered solution in stresses provided by the SPR-CD technique. Since the recovered stress field is commonly more accurate and has a higher convergence rate than the FE solution, we propose to substitute the raw FE solution by the recovered solution to decrease the computational cost of the numerical analysis. All these improvements are reflected by the numerical examples of structural shape optimization problems presented in this thesis. These numerical analysis clearly show the improved behavior of the cgFEM technology over the classical FE implementations commonly used in industry. / Cada d'¿a dise¿nos m'as complejos son requeridos por las industrias actuales. Para el dise¿no de nuevos componentes, los procesos tradicionales de prueba y error usados com'unmente ya no son v'alidos ya que ralentizan el proceso y dan lugar a dise¿nos sub-'optimos. Para componentes estructurales, una alternativa consiste en usar procesos de optimizaci'on de forma estructural los cuales dan como resultado dise¿nos 'optimos. Sin embargo, estas t'ecnicas requieren un alto coste computacional y tambi'en programas de Elementos Finitos (EF) extremadamente eficientes y robustos. Las compa¿n'¿as de programas de EF son conocedoras de que sus programas comerciales necesitan ser mejorados en este sentido y destinan importantes cantidades de recursos para mejorar sus c'odigos. En este trabajo proponemos usar el M'etodo de Elementos Finitos basado en mallados Cartesianos (cgFEM) como una herramienta eficiente y robusta para el an'alisis num'erico. La metodolog'¿a cgFEM desarrollada en esta tesis usa la sinergia entre varias t'ecnicas para lograr este prop'osito, cuyos dos ingredientes principales son el uso de los mallados Cartesianos de EF independientes de la geometr'¿a del componente que va a ser analizado y una eficiente estructura jer'arquica de datos. Estas dos caracter'¿sticas confieren a la tecnolog'¿a cgFEM de los requisitos necesarios para aumentar la eficiencia del c'odigo cgFEM con respecto a c'odigos comerciales. Como se indica en [1, 2], para garantizar la convergencia del proceso de optimizaci'on de forma estructural se necesita controlar el error en cada geometr'¿a analizada. En este sentido el c'odigo cgFEM tambi'en incorpora los apropiados estimadores de error. Estos estimadores de error han sido espec'¿ficamente adaptados al entorno cgFEM para aumentar su eficiencia. En esta tesis se introduce un proceso de recuperaci'on de la soluci'on, llamado SPR-CD, que en combinaci'on con el estimador de error de Zienkiewicz y Zhu [3], da como resultado medidas muy precisas del error de la soluci'on de EF. Adicionalmente, tambi'en se han desarrollado estimadores de error y cotas num'ericas en Magnitudes de Inter'es basadas en la t'ecnica SPR-CD para permitir un eficiente control de la calidad de la soluci'on num'erica. Respecto a la estimaci'on de error, tambi'en se presenta un proceso de estimaci'on de error para controlar la calidad del campo de tensiones recuperado obtenido mediante la t'ecnica SPR-CD. Ya que el campo recuperado es por lo general m'as preciso y tiene un mayor orden de convergencia que la soluci'on de EF, se propone sustituir la soluci'on de EF por la soluci'on recuperada para disminuir as'¿ el coste computacional del an'alisis num'erico. Todas estas mejoras se han reflejado en esta tesis mediante ejemplos num'ericos de problemas de optimizaci'on de forma estructural. Los resultados num'ericos muestran claramente un mejor comportamiento de la tecnolog'¿a cgFEM con respecto a implementaciones cl'asicas de EF com'unmente usadas en la industria. / Nadal Soriano, E. (2014). Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/35620

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

To simulate real world problems modeled by differential equations, it is often not sufficient to  consider and tackle a single equation. Rather, complex phenomena are modeled by several partial dierential equations that are coupled to each other. For example, a heart beat involve electric activity, mechanics of the movement of the walls and valves, as well as blood fow - a true multiphysics problem. There may also be ordinary differential equations modeling the reactions on a cellular level, and these may act on a much finer scale in both space and time. Determining efficient and accurate simulation tools for such multiscalar multiphysics problems is a challenge. The five scientific papers constituting this thesis investigate and present solutions to issues regarding accurate and efficient simulation using adaptive finite element methods. These include handling local accuracy through submodeling, analyzing error propagation in time-dependent  multiphysics problems, developing efficient algorithms for adaptivity in time and space, and deriving error analysis for coupled PDE-ODE systems. In all these examples, the error is analyzed and controlled using the framework of dual-weighted residuals, and the spatial meshes are handled using octree based data structures. However, few realistic geometries fit such grid and to address this issue a discontinuous Galerkin Nitsche method is presented and analyzed.

Error estimation must be used to find the accuracy of a designed classifier, an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics. This dissertation is concerned with the analytical formulation of the joint distribution of the true error of misclassification and two of its commonly used estimators, resubstitution and leave-one-out, as well as their marginal and mixed moments, in the context of the Linear Discriminant Analysis (LDA) classification rule. In the first part of this dissertation, we obtain the joint sampling distribution of the actual and estimated errors under a general parametric Gaussian assumption. Exact results are provided in the univariate case and an accurate approximation is obtained in the multivariate case. We show how these results can be applied in the computation of conditional bounds and the regression of the actual error, given the observed error estimate. In practice the unknown parameters of the Gaussian distributions, which figure in the expressions, are not known and need to be estimated. Using the usual maximum-likelihood estimates for such parameters and plugging them into the theoretical exact expressions provides a sample-based approximation to the joint distribution, and also sample-based methods to estimate upper conditional bounds. In the second part of this dissertation, exact analytical expressions for the bias, variance, and Root Mean Square (RMS) for the resubstitution and leave-one-out error estimators in the univariate Gaussian model are derived. All probabilistic characteristics of an error estimator are given by the knowledge of its joint distribution with the true error. Partial information is contained in their mixed moments, in particular, their second mixed moment. Marginal information regarding an error estimator is contained in its marginal moments, in particular, its mean and variance. Since we are interested in estimator accuracy and wish to use the RMS to measure that accuracy, we desire knowledge of the second-order moments, marginal and mixed, with the true error. In the multivariate case, using the double asymptotic approach with the assumption of knowing the common covariance matrix of the Gaussian model, analytical expressions for the first moments, second moments, and mixed moment with the actual error for the resubstitution and leave-one-out error estimators are derived. The results provide accurate small sample approximations and this is demonstrated in the present situation via numerical comparisons. Application of the results is discussed in the context of genomics.

Accuracy of the finite difference sensitivity calculations are improved by calculating the optimum finite difference interval sizes. In an aerodynamic inverse design algorithm, a compressor cascade geometry is perturbed by shape functions and finite differences sensitivity derivatives of the flow variables are calculated with respect to the base geometry flow variables. Sensitivity derivatives are used in an optimization code and a new airfoil is designed verifying given design characteristics. Accurate sensitivities are needed for optimization process. In order to find the optimum finite difference interval size, a method is investigated. Convergence error estimation techniques in iterative solutions and second derivative estimations are investigated to facilitate this method. For validation of the method, analytical sensitivity calculations of Euler equations are used and several applications are performed. Efficiency of the finite difference sensitivity calculations is improved by parallel computing. Finite difference sensitivity calculations are independent tasks in an inverse aerodynamic design algorithm and can be computed separately. Sensitivity calculations are performed on parallel processors and computing time is decreased.

Use of imaging systems for performing various situational awareness tasks in militaryand commercial settings has a long history. There is increasing recognition,however, that a much better job can be done by developing non-traditional opticalsystems that exploit the task-specific system aspects within the imager itself. Insome cases, a direct consequence of this approach can be real-time data compressionalong with increased measurement fidelity of the task-specific features. In others,compression can potentially allow us to perform high-level tasks such as direct trackingusing the compressed measurements without reconstructing the scene of interest.In this dissertation we present novel advancements in feature-specific (FS) imagersfor large field-of-view surveillence, and estimation of temporal object-scene changesutilizing the compressive imaging paradigm. We develop these two ideas in parallel.In the first case we show a feature-specific (FS) imager that optically multiplexesmultiple, encoded sub-fields of view onto a common focal plane. Sub-field encodingenables target tracking by creating a unique connection between target characteristicsin superposition space and the target's true position in real space. This isaccomplished without reconstructing a conventional image of the large field of view.System performance is evaluated in terms of two criteria: average decoding time andprobability of decoding error. We study these performance criteria as a functionof resolution in the encoding scheme and signal-to-noise ratio. We also includesimulation and experimental results demonstrating our novel tracking method. Inthe second case we present a FS imager for estimating temporal changes in the objectscene over time by quantifying these changes through a sequence of differenceimages. The difference images are estimated by taking compressive measurementsof the scene. Our goals are twofold. First, to design the optimal sensing matrixfor taking compressive measurements. In scenarios where such sensing matrices arenot tractable, we consider plausible candidate sensing matrices that either use theavailable <italic>a priori</italic> information or are non-adaptive. Second, we develop closed-form and iterative techniques for estimating the difference images. We present results to show the efficacy of these techniques and discuss the advantages of each.

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.