Efficient adaptive methods based on adjoint equations and truncation error estimation for unstructured grids

El propósito de esta tesis es la implementación de métodos eficientes de adaptación de mallas basados en ecuaciones adjuntas en el marco de discretizaciones de volúmenes finitos para mallas no estructuradas. La metodología basada en ecuaciones adjuntas optimiza la malla refinándola adecuadamente con el objetivo de mejorar la precisión de cálculo de un funcional de salida dado. El funcional suele ser una magnitud escalar de interés ingenieril obtenida por post-proceso de la solución, como por ejemplo, la resistencia o la sustentación aerodinámica. Usualmente, el método de adaptación adjunta está basado en una estimación a posteriori del error del funcional de salida mediante un promediado del residuo numérico con las variables adjuntas, “Dual Weighted Residual method” (DWR). Estas variables se obtienen de la solución del problema adjunto para el funcional seleccionado. El procedimiento habitual para introducir este método en códigos basados en discretizaciones de volúmenes finitos involucra la utilización de una malla auxiliar embebida obtenida por refinamiento uniforme de la malla inicial. El uso de esta malla implica un aumento significativo de los recursos computacionales (por ejemplo, en casos 3D el aumento de memoria requerida respecto a la que necesita el problema fluido inicial puede llegar a ser de un orden de magnitud). En esta tesis se propone un método alternativo basado en reformular la estimación del error del funcional en una malla auxiliar más basta y utilizar una técnica de estimación del error de truncación, denominada _ -estimation, para estimar los residuos que intervienen en el método DWR. Utilizando esta estimación del error se diseña un algoritmo de adaptación de mallas que conserva los ingredientes básicos de la adaptación adjunta estándar pero con un coste computacional asociado sensiblemente menor. La metodología de adaptación adjunta estándar y la propuesta en la tesis han sido introducidas en un código de volúmenes finitos utilizado habitualmente en la industria aeronáutica Europea. Se ha investigado la influencia de distintos parámetros numéricos que intervienen en el algoritmo. Finalmente, el método propuesto se compara con otras metodologías de adaptación de mallas y su eficiencia computacional se demuestra en una serie de casos representativos de interés aeronáutico. ABSTRACT The purpose of this thesis is the implementation of efficient grid adaptation methods based on the adjoint equations within the framework of finite volume methods (FVM) for unstructured grid solvers. The adjoint-based methodology aims at adapting grids to improve the accuracy of a functional output of interest, as for example, the aerodynamic drag or lift. The adjoint methodology is based on the a posteriori functional error estimation using the adjoint/dual-weighted residual method (DWR). In this method the error in a functional output can be directly related to local residual errors of the primal solution through the adjoint variables. These variables are obtained by solving the corresponding adjoint problem for the chosen functional. The common approach to introduce the DWR method within the FVM framework involves the use of an auxiliary embedded grid. The storage of this mesh demands high computational resources, i.e. over one order of magnitude increase in memory relative to the initial problem for 3D cases. In this thesis, an alternative methodology for adapting the grid is proposed. Specifically, the DWR approach for error estimation is re-formulated on a coarser mesh level using the _ -estimation method to approximate the truncation error. Then, an output-based adaptive algorithm is designed in such way that the basic ingredients of the standard adjoint method are retained but the computational cost is significantly reduced. The standard and the new proposed adjoint-based adaptive methodologies have been incorporated into a flow solver commonly used in the EU aeronautical industry. The influence of different numerical settings has been investigated. The proposed method has been compared against different grid adaptation approaches and the computational efficiency of the new method has been demonstrated on some representative aeronautical test cases.

Approximation Classes for Adaptive Methods

* This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401, the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan- der von Humboldt Foundation.

A comparison between PML, infinite elements and an iterative BEM as mesh truncation methods for HP self-adaptive procedures in electromagnetics

Finite element hp-adaptivity is a technology that allows for very accurate numerical solutions. When applied to open region problems such as radar cross section prediction or antenna analysis, a mesh truncation method needs to be used. This paper compares the following mesh truncation methods in the context of hp-adaptive methods: Infinite Elements, Perfectly Matched Layers and an iterative boundary element based methodology. These methods have been selected because they are exact at the continuous level (a desirable feature required by the extreme accuracy delivered by the hp-adaptive strategy) and they are easy to integrate with the logic of hp-adaptivity. The comparison is mainly based on the number of degrees of freedom needed for each method to achieve a given level of accuracy. Computational times are also included. Two-dimensional examples are used, but the conclusions directly extrapolated to the three dimensional case.

Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Bayesian adaptive estimation of arbitrary points on a psychometric function

Bayesian adaptive methods have been extensively used in psychophysics to estimate the point at which performance on a task attains arbitrary percentage levels, although the statistical properties of these estimators have never been assessed. We used simulation techniques to determine the small-sample properties of Bayesian estimators of arbitrary performance points, specifically addressing the issues of bias and precision as a function of the target percentage level. The study covered three major types of psychophysical task (yes-no detection, 2AFC discrimination and 2AFC detection) and explored the entire range of target performance levels allowed for by each task. Other factors included in the study were the form and parameters of the actual psychometric function Psi, the form and parameters of the model function M assumed in the Bayesian method, and the location of Psi within the parameter space. Our results indicate that Bayesian adaptive methods render unbiased estimators of any arbitrary point on psi only when M=Psi, and otherwise they yield bias whose magnitude can be considerable as the target level moves away from the midpoint of the range of Psi. The standard error of the estimator also increases as the target level approaches extreme values whether or not M=Psi. Contrary to widespread belief, neither the performance level at which bias is null nor that at which standard error is minimal can be predicted by the sweat factor. A closed-form expression nevertheless gives a reasonable fit to data describing the dependence of standard error on number of trials and target level, which allows determination of the number of trials that must be administered to obtain estimates with prescribed precision.

Infrared image enhancement based on hybrid-domain consideration and data fusion methods

Au cours des dernières décennies, l’effort sur les applications de capteurs infrarouges a largement progressé dans le monde. Mais, une certaine difficulté demeure, en ce qui concerne le fait que les objets ne sont pas assez clairs ou ne peuvent pas toujours être distingués facilement dans l’image obtenue pour la scène observée. L’amélioration de l’image infrarouge a joué un rôle important dans le développement de technologies de la vision infrarouge de l’ordinateur, le traitement de l’image et les essais non destructifs, etc. Cette thèse traite de la question des techniques d’amélioration de l’image infrarouge en deux aspects, y compris le traitement d’une seule image infrarouge dans le domaine hybride espacefréquence, et la fusion d’images infrarouges et visibles employant la technique du nonsubsampled Contourlet transformer (NSCT). La fusion d’images peut être considérée comme étant la poursuite de l’exploration du modèle d’amélioration de l’image unique infrarouge, alors qu’il combine les images infrarouges et visibles en une seule image pour représenter et améliorer toutes les informations utiles et les caractéristiques des images sources, car une seule image ne pouvait contenir tous les renseignements pertinents ou disponibles en raison de restrictions découlant de tout capteur unique de l’imagerie. Nous examinons et faisons une enquête concernant le développement de techniques d’amélioration d’images infrarouges, et ensuite nous nous consacrons à l’amélioration de l’image unique infrarouge, et nous proposons un schéma d’amélioration de domaine hybride avec une méthode d’évaluation floue de seuil amélioré, qui permet d’obtenir une qualité d’image supérieure et améliore la perception visuelle humaine. Les techniques de fusion d’images infrarouges et visibles sont établies à l’aide de la mise en oeuvre d’une mise en registre précise des images sources acquises par différents capteurs. L’algorithme SURF-RANSAC est appliqué pour la mise en registre tout au long des travaux de recherche, ce qui conduit à des images mises en registre de façon très précise et des bénéfices accrus pour le traitement de fusion. Pour les questions de fusion d’images infrarouges et visibles, une série d’approches avancées et efficaces sont proposés. Une méthode standard de fusion à base de NSCT multi-canal est présente comme référence pour les approches de fusion proposées suivantes. Une approche conjointe de fusion, impliquant l’Adaptive-Gaussian NSCT et la transformée en ondelettes (Wavelet Transform, WT) est propose, ce qui conduit à des résultats de fusion qui sont meilleurs que ceux obtenus avec les méthodes non-adaptatives générales. Une approche de fusion basée sur le NSCT employant la détection comprime (CS, compressed sensing) et de la variation totale (TV) à des coefficients d’échantillons clairsemés et effectuant la reconstruction de coefficients fusionnés de façon précise est proposée, qui obtient de bien meilleurs résultats de fusion par le biais d’une pré-amélioration de l’image infrarouge et en diminuant les informations redondantes des coefficients de fusion. Une procédure de fusion basée sur le NSCT utilisant une technique de détection rapide de rétrécissement itératif comprimé (fast iterative-shrinking compressed sensing, FISCS) est proposée pour compresser les coefficients décomposés et reconstruire les coefficients fusionnés dans le processus de fusion, qui conduit à de meilleurs résultats plus rapidement et d’une manière efficace.

Haptiisit ja hapteemit : tapaustutkimus kuurosokean henkilön kosketukseen perustuvan kommunikaation kehityksestä

Haptices and haptemes: A case study of developmental process in touch-based communication of acquired deafblind people This research is the first systematic, longitudinal process and development description of communication using touch and body with an acquired deafblind person. The research consists of observational and analysed written and video materials mainly from two informants´ experiences during period of 14 years. The research describes the adaptation of Social-Haptic methods between a couple, and other informants´ experiences, which have been collated from biographies and through giving national and international courses. When the hearing and sight deteriorates due to having an acquired deafblind condition, communication consists of multi-systematic and adaptive methods. A person`s expressive language, spoken or Sign Language, usually remains unchanged, but the methods of receiving information could change many times during a person s lifetime. Haptices are made from haptemes that determines which regulations are analysed. When defining haptemes the definition, classification and varied meanings of touch were discovered. Haptices include sharing a personal body space, meaning of touch-contact, context and using different communication channels. Communication distances are classified as exact distance, estimated distance and touch distance. Physical distance can be termed as very long, long, medium or very close. Social body space includes the body areas involved in sending and receiving haptices and applying different types of contacts. One or two hands can produce messages by using different hand shapes and orientations. This research classifies how the body can be identified into different areas such as body orientation, varied body postures, body position levels, social actions and which side of the body is used. Spatial body space includes environmental and situational elements. Haptemes of movements are recognised as the direction of movements, change of directions on the body, directions between people, pressure, speed, frequency, size, length, duration, pause, change of rhythm, shape, macro and micro movements. Haptices share multidimensional meanings and emotions. Research describes haptices in different situations enhancing sensory information and functioning also as an independent language. Haptices includes social-haptic confirmation system, social quick messages, body drawing, contact to the people and the environment, guiding and sharing art experiences through movements. Five stages of emotional differentiation were identified as very light, light, medium, heavy and very heavy touch. Haptices give the possibility to share different art, hobby and game experiences. A new communication system development based on the analysis of the research data is classified into different phases. These are experimental initiation, social deconstruction, developing the description of Social-Haptic communication and generalisation of the theory as well as finding and conceptualising the haptices and haptemes. The use and description of haptices is a social innovation, which illustrates the adaptive function of the body and perceptual senses that can be taught to a third party. Keywords: deafblindness, hapteme, haptic, haptices, movement, social-haptic communication, social-haptic confirmation system, tactile, touch

Une approche fréquentielle pratique pour l'échantillonnage adaptatif en espace image

En synthèse d'images réalistes, l'intensité finale d'un pixel est calculée en estimant une intégrale de rendu multi-dimensionnelle. Une large portion de la recherche menée dans ce domaine cherche à trouver de nouvelles techniques afin de réduire le coût de calcul du rendu tout en préservant la fidelité et l'exactitude des images résultantes. En tentant de réduire les coûts de calcul afin d'approcher le rendu en temps réel, certains effets réalistes complexes sont souvent laissés de côté ou remplacés par des astuces ingénieuses mais mathématiquement incorrectes. Afin d'accélerer le rendu, plusieurs avenues de travail ont soit adressé directement le calcul de pixels individuels en améliorant les routines d'intégration numérique sous-jacentes; ou ont cherché à amortir le coût par région d'image en utilisant des méthodes adaptatives basées sur des modèles prédictifs du transport de la lumière. L'objectif de ce mémoire, et de l'article résultant, est de se baser sur une méthode de ce dernier type[Durand2005], et de faire progresser la recherche dans le domaine du rendu réaliste adaptatif rapide utilisant une analyse du transport de la lumière basée sur la théorie de Fourier afin de guider et prioriser le lancer de rayons. Nous proposons une approche d'échantillonnage et de reconstruction adaptative pour le rendu de scènes animées illuminées par cartes d'environnement, permettant la reconstruction d'effets tels que les ombres et les réflexions de tous les niveaux fréquentiels, tout en préservant la cohérence temporelle.

Equidistributing Meshes with Constraints

Adaptive methods which “equidistribute” a given positive weight function are now used fairly widely for selecting discrete meshes. The disadvantage of such schemes is that the resulting mesh may not be smoothly varying. In this paper a technique is developed for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh. Given a weight function $f \geqq 0$ on an interval $[a,b]$ and constants $c$ and $K$, the method produces a mesh with points $x_0 = a,x_{j + 1} = x_j + h_j ,j = 0,1, \cdots ,n - 1$ and $x_n = b$ such that\[ \int_{xj}^{x_{j + 1} } {f \leqq c\quad {\text{and}}\quad \frac{1} {K}} \leqq \frac{{h_{j + 1} }} {{h_j }} \leqq K\quad {\text{for}}\, j = 0,1, \cdots ,n - 1 . \] A theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given. Examples show that the procedure is effective in practice. Other types of constraints on equidistributing meshes are also discussed. The principal application of the procedure is to the solution of boundary value problems, where the weight function is generally some error indicator, and accuracy and convergence properties may depend on the smoothness of the mesh. Other practical applications include the regrading of statistical data.

The combination of neural estimates in prediction and decision problems

In this dissertation, different ways of combining neural predictive models or neural-based forecasts are discussed. The proposed approaches consider mostly Gaussian radial basis function networks, which can be efficiently identified and estimated through recursive/adaptive methods. Two different ways of combining are explored to get a final estimate – model mixing and model synthesis –, with the aim of obtaining improvements both in terms of efficiency and effectiveness. In the context of model mixing, the usual framework for linearly combining estimates from different models is extended, to deal with the case where the forecast errors from those models are correlated. In the context of model synthesis, and to address the problems raised by heavily nonstationary time series, we propose hybrid dynamic models for more advanced time series forecasting, composed of a dynamic trend regressive model (or, even, a dynamic harmonic regressive model), and a Gaussian radial basis function network. Additionally, using the model mixing procedure, two approaches for decision-making from forecasting models are discussed and compared: either inferring decisions from combined predictive estimates, or combining prescriptive solutions derived from different forecasting models. Finally, the application of some of the models and methods proposed previously is illustrated with two case studies, based on time series from finance and from tourism.

Numerical Error Prediction and its applications in CFD using tau-estimation

Nowadays, Computational Fluid Dynamics (CFD) solvers are widely used within the industry to model fluid flow phenomenons. Several fluid flow model equations have been employed in the last decades to simulate and predict forces acting, for example, on different aircraft configurations. Computational time and accuracy are strongly dependent on the fluid flow model equation and the spatial dimension of the problem considered. While simple models based on perfect flows, like panel methods or potential flow models can be very fast to solve, they usually suffer from a poor accuracy in order to simulate real flows (transonic, viscous). On the other hand, more complex models such as the full Navier- Stokes equations provide high fidelity predictions but at a much higher computational cost. Thus, a good compromise between accuracy and computational time has to be fixed for engineering applications. A discretisation technique widely used within the industry is the so-called Finite Volume approach on unstructured meshes. This technique spatially discretises the flow motion equations onto a set of elements which form a mesh, a discrete representation of the continuous domain. Using this approach, for a given flow model equation, the accuracy and computational time mainly depend on the distribution of nodes forming the mesh. Therefore, a good compromise between accuracy and computational time might be obtained by carefully defining the mesh. However, defining an optimal mesh for complex flows and geometries requires a very high level expertize in fluid mechanics and numerical analysis, and in most cases a simple guess of regions of the computational domain which might affect the most the accuracy is impossible. Thus, it is desirable to have an automatized remeshing tool, which is more flexible with unstructured meshes than its structured counterpart. However, adaptive methods currently in use still have an opened question: how to efficiently drive the adaptation ? Pioneering sensors based on flow features generally suffer from a lack of reliability, so in the last decade more effort has been made in developing numerical error-based sensors, like for instance the adjoint-based adaptation sensors. While very efficient at adapting meshes for a given functional output, the latter method is very expensive as it requires to solve a dual set of equations and computes the sensor on an embedded mesh. Therefore, it would be desirable to develop a more affordable numerical error estimation method. The current work aims at estimating the truncation error, which arises when discretising a partial differential equation. These are the higher order terms neglected in the construction of the numerical scheme. The truncation error provides very useful information as it is strongly related to the flow model equation and its discretisation. On one hand, it is a very reliable measure of the quality of the mesh, therefore very useful in order to drive a mesh adaptation procedure. On the other hand, it is strongly linked to the flow model equation, so that a careful estimation actually gives information on how well a given equation is solved, which may be useful in the context of _ -extrapolation or zonal modelling. The following work is organized as follows: Chap. 1 contains a short review of mesh adaptation techniques as well as numerical error prediction. In the first section, Sec. 1.1, the basic refinement strategies are reviewed and the main contribution to structured and unstructured mesh adaptation are presented. Sec. 1.2 introduces the definitions of errors encountered when solving Computational Fluid Dynamics problems and reviews the most common approaches to predict them. Chap. 2 is devoted to the mathematical formulation of truncation error estimation in the context of finite volume methodology, as well as a complete verification procedure. Several features are studied, such as the influence of grid non-uniformities, non-linearity, boundary conditions and non-converged numerical solutions. This verification part has been submitted and accepted for publication in the Journal of Computational Physics. Chap. 3 presents a mesh adaptation algorithm based on truncation error estimates and compares the results to a feature-based and an adjoint-based sensor (in collaboration with Jorge Ponsín, INTA). Two- and three-dimensional cases relevant for validation in the aeronautical industry are considered. This part has been submitted and accepted in the AIAA Journal. An extension to Reynolds Averaged Navier- Stokes equations is also included, where _ -estimation-based mesh adaptation and _ -extrapolation are applied to viscous wing profiles. The latter has been submitted in the Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. Keywords: mesh adaptation, numerical error prediction, finite volume Hoy en día, la Dinámica de Fluidos Computacional (CFD) es ampliamente utilizada dentro de la industria para obtener información sobre fenómenos fluidos. La Dinámica de Fluidos Computacional considera distintas modelizaciones de las ecuaciones fluidas (Potencial, Euler, Navier-Stokes, etc) para simular y predecir las fuerzas que actúan, por ejemplo, sobre una configuración de aeronave. El tiempo de cálculo y la precisión en la solución depende en gran medida de los modelos utilizados, así como de la dimensión espacial del problema considerado. Mientras que modelos simples basados en flujos perfectos, como modelos de flujos potenciales, se pueden resolver rápidamente, por lo general aducen de una baja precisión a la hora de simular flujos reales (viscosos, transónicos, etc). Por otro lado, modelos más complejos tales como el conjunto de ecuaciones de Navier-Stokes proporcionan predicciones de alta fidelidad, a expensas de un coste computacional mucho más elevado. Por lo tanto, en términos de aplicaciones de ingeniería se debe fijar un buen compromiso entre precisión y tiempo de cálculo. Una técnica de discretización ampliamente utilizada en la industria es el método de los Volúmenes Finitos en mallas no estructuradas. Esta técnica discretiza espacialmente las ecuaciones del movimiento del flujo sobre un conjunto de elementos que forman una malla, una representación discreta del dominio continuo. Utilizando este enfoque, para una ecuación de flujo dado, la precisión y el tiempo computacional dependen principalmente de la distribución de los nodos que forman la malla. Por consiguiente, un buen compromiso entre precisión y tiempo de cálculo se podría obtener definiendo cuidadosamente la malla, concentrando sus elementos en aquellas zonas donde sea estrictamente necesario. Sin embargo, la definición de una malla óptima para corrientes y geometrías complejas requiere un nivel muy alto de experiencia en la mecánica de fluidos y el análisis numérico, así como un conocimiento previo de la solución. Aspecto que en la mayoría de los casos no está disponible. Por tanto, es deseable tener una herramienta que permita adaptar los elementos de malla de forma automática, acorde a la solución fluida (remallado). Esta herramienta es generalmente más flexible en mallas no estructuradas que con su homóloga estructurada. No obstante, los métodos de adaptación actualmente en uso todavía dejan una pregunta abierta: cómo conducir de manera eficiente la adaptación. Sensores pioneros basados en las características del flujo en general, adolecen de una falta de fiabilidad, por lo que en la última década se han realizado grandes esfuerzos en el desarrollo numérico de sensores basados en el error, como por ejemplo los sensores basados en el adjunto. A pesar de ser muy eficientes en la adaptación de mallas para un determinado funcional, este último método resulta muy costoso, pues requiere resolver un doble conjunto de ecuaciones: la solución y su adjunta. Por tanto, es deseable desarrollar un método numérico de estimación de error más asequible. El presente trabajo tiene como objetivo estimar el error local de truncación, que aparece cuando se discretiza una ecuación en derivadas parciales. Estos son los términos de orden superior olvidados en la construcción del esquema numérico. El error de truncación proporciona una información muy útil sobre la solución: es una medida muy fiable de la calidad de la malla, obteniendo información que permite llevar a cabo un procedimiento de adaptación de malla. Está fuertemente relacionado al modelo matemático fluido, de modo que una estimación precisa garantiza la idoneidad de dicho modelo en un campo fluido, lo que puede ser útil en el contexto de modelado zonal. Por último, permite mejorar la precisión de la solución resolviendo un nuevo sistema donde el error local actúa como término fuente (_ -extrapolación). El presenta trabajo se organiza de la siguiente manera: Cap. 1 contiene una breve reseña de las técnicas de adaptación de malla, así como de los métodos de predicción de los errores numéricos. En la primera sección, Sec. 1.1, se examinan las estrategias básicas de refinamiento y se presenta la principal contribución a la adaptación de malla estructurada y no estructurada. Sec 1.2 introduce las definiciones de los errores encontrados en la resolución de problemas de Dinámica Computacional de Fluidos y se examinan los enfoques más comunes para predecirlos. Cap. 2 está dedicado a la formulación matemática de la estimación del error de truncación en el contexto de la metodología de Volúmenes Finitos, así como a un procedimiento de verificación completo. Se estudian varias características que influyen en su estimación: la influencia de la falta de uniformidad de la malla, el efecto de las no linealidades del modelo matemático, diferentes condiciones de contorno y soluciones numéricas no convergidas. Esta parte de verificación ha sido presentada y aceptada para su publicación en el Journal of Computational Physics. Cap. 3 presenta un algoritmo de adaptación de malla basado en la estimación del error de truncación y compara los resultados con sensores de featured-based y adjointbased (en colaboración con Jorge Ponsín del INTA). Se consideran casos en dos y tres dimensiones, relevantes para la validación en la industria aeronáutica. Este trabajo ha sido presentado y aceptado en el AIAA Journal. También se incluye una extensión de estos métodos a las ecuaciones RANS (Reynolds Average Navier- Stokes), en donde adaptación de malla basada en _ y _ -extrapolación son aplicados a perfiles con viscosidad de alas. Este último trabajo se ha presentado en los Actas de la Institución de Ingenieros Mecánicos, Parte G: Journal of Aerospace Engineering. Palabras clave: adaptación de malla, predicción del error numérico, volúmenes finitos