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.

Finite volume modelling of an icefield with subglacial lake

Momentum, mass and energy balance laws provide the tools for the study of the evolution of an icefield covering a subglacial lake. The ice is described as a non-Newtonian fluid with a power-law constitutive relationship with temperature- and stress-dependent viscosity (Glen?s law) [1]. The phase transition mechanisms at the air/ice and ice/water interfaces yield moving boundary formulations, and lake hydrodynamics requires equation reduction for treating the turbulence.

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

Interactions between roughness and topography in hydraulic models

Analysis of river flow using hydraulic modelling and its implications in derived environ-mental applications are inextricably connected with the way in which the river boundary shape is represented. This relationship is scale-dependent upon the modelling resolution which in turn determines the importance of a subscale performance of the model and the way subscale (surface and flow) processes are parameterised. Commonly, the subscale behaviour of the model relies upon a roughness parameterisation whose meaning depends on the dimensionality of the hydraulic model and the resolution of the topographic represen¬tation scale. This latter is, in turn, dependent on the resolution of the computational mesh as well as on the detail of measured topographic data. Flow results are affected by this interactions between scale and subscale parameterisation according to the dimensionality approach. The aim of this dissertation is the evaluation of these interactions upon hy¬draulic modelling results. Current high resolution topographic source availability induce this research which is tackled using a suitable roughness approach according to each di¬mensionality with the purpose of the interaction assessment. A 1D HEC-RAS model, a 2D raster-based diffusion-wave model with a scale-dependent distributed roughness parame-terisation and a 3D finite volume scheme with a porosity algorithm approach to incorporate complex topography have been used. Different topographic sources are assessed using a 1D scheme. LiDAR data are used to isolate the mesh resolution from the topographic content of the DEM effects upon 2D and 3D flow results. A distributed roughness parameterisation, using a roughness height approach dependent upon both mesh resolution and topographic content is developed and evaluated for the 2D scheme. Grain-size data and fractal methods are used for the reconstruction of topography with microscale information, required for some applications but not easily available. Sensitivity of hydraulic parameters to this topographic parameterisation is evaluated in a 3D scheme at different mesh resolu¬tions. Finally, the structural variability of simulated flow is analysed and related to scale interactions. Model simulations demonstrate (i) the importance of the topographic source in a 1D models; (ii) the mesh resolution approach is dominant in 2D and 3D simulations whereas in a 1D model the topographic source and even the roughness parameterisation impacts are more critical; (iii) the increment of the sensitivity to roughness parameterisa-tion in 1D and 2D schemes with detailed topographic sources and finer mesh resolutions; and (iv) the topographic content and microtopography impact throughout the vertical profile of computed 3D velocity in a depth-dependent way, whereas 2D results are not affected by topographic content variations. Finally, the spatial analysis shows that the mesh resolution controls high resolution model scale results, roughness parameterisation control 2D simulation results for a constant mesh resolution; and topographic content and micro-topography variations impacts upon the organisation of flow results depth-dependently in a 3D scheme. Resumen La topografía juega un papel fundamental en la distribución del agua y la energía en los paisajes naturales (Beven and Kirkby 1979; Wood et al. 1997). La simulación hidráulica combinada con métodos de medición del terreno por teledetección constituyen una poderosa herramienta de investigación en la comprensión del comportamiento de los flujos de agua debido a la variabilidad de la superficie sobre la que fluye. La representación e incorporación de la topografía en el esquema hidráulico tiene una importancia crucial en los resultados y determinan el desarrollo de sus aplicaciones al campo medioambiental. Cualquier simulación es una simplificación de un proceso del mundo real, y por tanto el grado de simplificación determinará el significado de los resultados simulados. Este razonamiento es particularmente difícil de trasladar a la simulación hidráulica donde aspectos de la escala tan diferentes como la escala de los procesos de flujo y de representación del contorno son considerados conjuntamente incluso en fases de parametrización (e.g. parametrización de la rugosidad). Por una parte, esto es debido a que las decisiones de escala vienen condicionadas entre ellas (e.g. la dimensionalidad del modelo condiciona la escala de representación del contorno) y por tanto interaccionan en sus resultados estrechamente. Y por otra parte, debido a los altos requerimientos numéricos y computacionales de una representación explícita de alta resolución de los procesos de flujo y discretización de la malla. Además, previo a la modelización hidráulica, la superficie del terreno sobre la que el agua fluye debe ser modelizada y por tanto presenta su propia escala de representación, que a su vez dependerá de la escala de los datos topográficos medidos con que se elabora el modelo. En última instancia, esta topografía es la que determina el comportamiento espacial del flujo. Por tanto, la escala de la topografía en sus fases de medición y modelización (resolución de los datos y representación topográfica) previas a su incorporación en el modelo hidráulico producirá a su vez un impacto que se acumulará al impacto global resultante debido a la escala computacional del modelo hidráulico y su dimensión. La comprensión de las interacciones entre las complejas geometrías del contorno y la estructura del flujo utilizando la modelización hidráulica depende de las escalas consideradas en la simplificación de los procesos hidráulicos y del terreno (dimensión del modelo, tamaño de escala computacional y escala de los datos topográficos). La naturaleza de la aplicación del modelo hidráulico (e.g. habitat físico, análisis de riesgo de inundaciones, transporte de sedimentos) determina en primer lugar la escala del estudio y por tanto el detalle de los procesos a simular en el modelo (i.e. la dimensionalidad) y, en consecuencia, la escala computacional a la que se realizarán los cálculos (i.e. resolución computacional). Esta última a su vez determina, el detalle geográfico con que deberá representarse el contorno acorde con la resolución de la malla computacional. La parametrización persigue incorporar en el modelo hidráulico la cuantificación de los procesos y condiciones físicas del sistema natural y por tanto debe incluir no solo aquellos procesos que tienen lugar a la escala de modelización, sino también aquellos que tienen lugar a un nivel subescalar y que deben ser definidos mediante relaciones de escalado con las variables modeladas explícitamente. Dicha parametrización se implementa en la práctica mediante la provisión de datos al modelo, por tanto la escala de los datos geográficos utilizados para parametrizar el modelo no sólo influirá en los resultados, sino también determinará la importancia del comportamiento subescalar del modelo y el modo en que estos procesos deban ser parametrizados (e.g. la variabilidad natural del terreno dentro de la celda de discretización o el flujo en las direcciones laterales y verticales en un modelo unidimensional). En esta tesis, se han utilizado el modelo unidimensional HEC-RAS, (HEC 1998b), un modelo ráster bidimensional de propagación de onda, (Yu 2005) y un esquema tridimensional de volúmenes finitos con un algoritmo de porosidad para incorporar la topografía, (Lane et al. 2004; Hardy et al. 2005). La geometría del contorno viene definida por la escala de representación topográfica (resolución de malla y contenido topográfico), la cual a su vez depende de la escala de la fuente cartográfica. Todos estos factores de escala interaccionan en la respuesta del modelo hidráulico a la topografía. En los últimos años, métodos como el análisis fractal y las técnicas geoestadísticas utilizadas para representar y analizar elementos geográficos (e.g. en la caracterización de superficies (Herzfeld and Overbeck 1999; Butler et al. 2001)), están promoviendo nuevos enfoques en la cuantificación de los efectos de escala (Lam et al. 2004; Atkinson and Tate 2000; Lam et al. 2006) por medio del análisis de la estructura espacial de la variable (e.g. Bishop et al. 2006; Ju et al. 2005; Myint et al. 2004; Weng 2002; Bian and Xie 2004; Southworth et al. 2006; Pozd-nyakova et al. 2005; Kyriakidis and Goodchild 2006). Estos métodos cuantifican tanto el rango de valores de la variable presentes a diferentes escalas como la homogeneidad o heterogeneidad de la variable espacialmente distribuida (Lam et al. 2004). En esta tesis, estas técnicas se han utilizado para analizar el impacto de la topografía sobre la estructura de los resultados hidráulicos simulados. Los datos de teledetección de alta resolución y técnicas GIS también están siendo utilizados para la mejor compresión de los efectos de escala en modelos medioambientales (Marceau 1999; Skidmore 2002; Goodchild 2003) y se utilizan en esta tesis. Esta tesis como corpus de investigación aborda las interacciones de esas escalas en la modelización hidráulica desde un punto de vista global e interrelacionado. Sin embargo, la estructura y el foco principal de los experimentos están relacionados con las nociones espaciales de la escala de representación en relación con una visión global de las interacciones entre escalas. En teoría, la representación topográfica debe caracterizar la superficie sobre la que corre el agua a una adecuada (conforme a la finalidad y dimensión del modelo) escala de discretización, de modo que refleje los procesos de interés. La parametrización de la rugosidad debe de reflejar los efectos de la variabilidad de la superficie a escalas de más detalle que aquellas representadas explícitamente en la malla topográfica (i.e. escala de discretización). Claramente, ambos conceptos están físicamente relacionados por un

Crack mechanical failure in lithium niobate crystal under ion irradiation; novel simulation by extended finite elements

Swift heavy ion irradiation (ions with mass heavier than 15 and energy exceeding MeV/amu) transfer their energy mainly to the electronic system with small momentum transfer per collision. Therefore, they produce linear regions (columnar nano-tracks) around the straight ion trajectory, with marked modifications with respect to the virgin material, e.g., phase transition, amorphization, compaction, changes in physical or chemical properties. In the case of crystalline materials the most distinctive feature of swift heavy ion irradiation is the production of amorphous tracks embedded in the crystal. Lithium niobate is a relevant optical material that presents birefringence due to its anysotropic trigonal structure. The amorphous phase is certainly isotropic. In addition, its refractive index exhibits high contrast with those of the crystalline phase. This allows one to fabricate waveguides by swift ion irradiation with important technological relevance. From the mechanical point of view, the inclusion of an amorphous nano-track (with a density 15% lower than that of the crystal) leads to the generation of important stress/strain fields around the track. Eventually these fields are the origin of crack formation with fatal consequences for the integrity of the samples and the viability of the method for nano-track formation. For certain crystal cuts (X and Y), these fields are clearly anisotropic due to the crystal anisotropy. We have used finite element methods to calculate the stress/strain fields that appear around the ion-generated amorphous nano-tracks for a variety of ion energies and doses. A very remarkable feature for X cut-samples is that the maximum shear stress appears on preferential planes that form +/-45º with respect to the crystallographic planes. This leads to the generation of oriented surface cracks when the dose increases. The growth of the cracks along the anisotropic crystal has been studied by means of novel extended finite element methods, which include cracks as discontinuities. In this way we can study how the length and depth of a crack evolves as function of the ion dose. In this work we will show how the simulations compare with experiments and their application in materials modification by ion irradiation.

Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose.

Simulación numérica de un problema de contaminación atmosférica

La ecuación en derivadas parciales de advección difusión con reacción química es la base de los modelos de dispersión de contaminantes en la atmósfera, y los diferentes métodos numéricos empleados para su resolución han sido objeto de amplios estudios a lo largo de su desarrollo. En esta Tesis se presenta la implementación de un nuevo método conservativo para la resolución de la parte advectiva de la ecuación en derivadas parciales que modela la dispersión de contaminantes dentro del modelo mesoescalar de transporte químico CHIMERE. Este método está basado en una técnica de volúmenes finitos junto con una interpolación racional. La ventaja de este método es la conservación exacta de la masa transportada debido al empleo de la ley de conservación de masas. Para ello emplea una formulación de flujo basado en el cálculo de la integral ponderada dentro de cada celda definida para la discretización del espacio en el método de volúmenes finitos. Los resultados numéricos obtenidos en las simulaciones realizadas (implementando el modelo conservativo para la advección en el modelo CHIMERE) se han comparado con los datos observados de concentración de contaminantes registrados en la red de estaciones de seguimiento y medición distribuidas por la Península Ibérica. Los datos estadísticos de medición del error, la media normalizada y la media absoluta normalizada del error, presentan valores que están dentro de los rangos propuestos por la EPA para considerar el modelo preciso. Además, se introduce un nuevo método para resolver la parte advectivadifusiva de la ecuación en derivadas parciales que modeliza la dispersión de contaminantes en la atmósfera. Se ha empleado un método de diferencias finitas de alto orden para resolver la parte difusiva de la ecuación de transporte de contaminantes junto con el método racional conservativo para la parte advectiva en una y dos dimensiones. Los resultados obtenidos de la aplicación del método a diferentes situaciones incluyendo casos académicos y reales han sido comparados con la solución analítica de la ecuación de advección-difusión, demostrando que el nuevo método proporciona un resultado preciso para aproximar la solución. Por último, se ha desarrollado un modelo completo que contempla los fenómenos advectivo y difusivo con reacción química, usando los métodos anteriores junto con una técnica de diferenciación regresiva (BDF2). Esta técnica consiste en un método implícito multipaso de diferenciación regresiva de segundo orden, que nos permite resolver los problemas rígidos típicos de la química atmosférica, modelizados a través de sistemas de ecuaciones diferenciales ordinarias. Este método hace uso de la técnica iterativa Gauss- Seidel para obtener la solución de la parte implícita de la fórmula BDF2. El empleo de la técnica de Gauss-Seidel en lugar de otras técnicas comúnmente empleadas, como la iteración por el método de Newton, nos proporciona rapidez de cálculo y bajo consumo de memoria, ideal para obtener modelos operativos para la resolución de la cinética química atmosférica. ABSTRACT Extensive research has been performed to solve the atmospheric chemicaladvection- diffusion equation and different numerical methods have been proposed. This Thesis presents the implementation of an exactly conservative method for the advection equation in the European scale Eulerian chemistry transport model CHIMERE based on a rational interpolation and a finite volume algorithm. The advantage of the method is that the cell-integrated average is predicted via a flux formulation, thus the mass is exactly conserved. Numerical results are compared with a set of observation registered at some monitoring sites in Spain. The mean normalized bias and the mean normalized absolute error present values that are inside the range to consider an accurate model performance. In addition, it has been introduced a new method to solve the advectiondiffusion equation. It is based on a high-order accurate finite difference method to solve de diffusion equation together with a rational interpolation and a finite volume to solve the advection equation in one dimension and two dimensions. Numerical results obtained from solving several problems include academic and real atmospheric problems have been compared with the analytical solution of the advection-diffusion equation, showing that the new method give an efficient algorithm for solving such problems. Finally, a complete model has been developed to solve the atmospheric chemical-advection-diffusion equation, adding the conservative method for the advection equation, the high-order finite difference method for the diffusion equation and a second-order backward differentiation formula (BDF2) to solve the atmospheric chemical kinetics. The BDF2 is an implicit, second order multistep backward differentiation formula used to solve the stiff systems of ordinary differential equations (ODEs) from atmospheric chemistry. The Gauss-Seidel iteration is used for approximately solving the implicitly defined BDF solution, giving a faster tool than the more commonly used iterative modified Newton technique. This method implies low start-up costs and a low memory demand due to the use of Gauss-Seidel iteration.

Global Stability Analysis of a Compressible Turbulent Flow around a High-Lift Configuration

The purpose of this work is to analyze a complex high lift configuration for which significant regions of separated flow are present. Current state of the art methods have some diffculty to predict the origin and the progression of this separated flow when increasing the angle of attack. The mechanisms responsible for the maximum lift limit on multi-element wing con?gurations are not clear; this stability analysis could help to understand the physics behind the phenomenon and to find a relation between the flow separation and the instability onset. The methodology presented herein consists in the computation of a steady base flow solution based on a finite volume discretization and a proposal of the solution for a generalized eigenvalue problem corresponding to the perturbed and linearized problem. The eigenvalue problem has been solved with the Arnoldi iterative method, one of the Krylov subspace projection methods. The described methodology was applied to the NACA0012 test case in subsonic and in transonic conditions and, finally, for the first time to the authors knowledge, on an industrial multi-component geometry, such as the A310 airfoil, in order to identify low frequency instabilities related to the separation. One important conclusion is that for all the analyzed geometries, one unstable mode related to flow separation appears for an angle of attack greater than the one correspondent to the maximum lift coe?cient condition. Finally, an adjoint study was carried out in order to evaluate the receptivity and the structural sensitivity of the geometries, giving an indication of the domain region that could be modified resulting in the biggest change of the flowfield.

Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile

In this work, various turbulent solutions of the two-dimensional (2D) and three-dimensional compressible Reynolds averaged Navier?Stokes equations are analyzed using global stability theory. This analysis is motivated by the onset of flow unsteadiness (Hopf bifurcation) for transonic buffet conditions where moderately high Reynolds numbers and compressible effects must be considered. The buffet phenomenon involves a complex interaction between the separated flow and a shock wave. The efficient numerical methodology presented in this paper predicts the critical parameters, namely, the angle of attack and Mach and Reynolds numbers beyond which the onset of flow unsteadiness appears. The geometry, a NACA0012 profile, and flow parameters selected reproduce situations of practical interest for aeronautical applications. The numerical computation is performed in three steps. First, a steady baseflow solution is obtained; second, the Jacobian matrix for the RANS equations based on a finite volume discretization is computed; and finally, the generalized eigenvalue problem is derived when the baseflow is linearly perturbed. The methodology is validated predicting the 2D Hopf bifurcation for a circular cylinder under laminar flow condition. This benchmark shows good agreement with the previous published computations and experimental data. In the transonic buffet case, the baseflow is computed using the Spalart?Allmaras turbulence model and represents a mean flow where the high frequency content and length scales of the order of the shear-layer thickness have been averaged. The lower frequency content is assumed to be decoupled from the high frequencies, thus allowing a stability analysis to be performed on the low frequency range. In addition, results of the corresponding adjoint problem and the sensitivity map are provided for the first time for the buffet problem. Finally, an extruded three-dimensional geometry of the NACA0012 airfoil, where all velocity components are considered, was also analyzed as a Triglobal stability case, and the outcoming results were compared to the previous 2D limited model, confirming that the buffet onset is well detected.

The estimation of truncation error by tau-estimation revisited

The aim of this paper was to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local truncation error using the @t-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error.

Global Linear Instability at the Dawn of its 4th Decade: A List of Challenges (A Practical Guide on how to Contain the Euphoria and Avoid the Oversell)

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic threedimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions.1 The theory addresses flows developing in complex geometries, in which the parallel or weakly nonparallel basic flow approximation invoked by classic linear stability theory does not hold. As such, global linear theory is called to fill the gap in research into stability and transition in flows over or through complex geometries. Historically, global linear instability has been (and still is) concerned with solution of multi-dimensional eigenvalue problems; the maturing of non-modal linear instability ideas in simple parallel flows during the last decade of last century2–4 has given rise to investigation of transient growth scenarios in an ever increasing variety of complex flows. After a brief exposition of the theory, connections are sought with established approaches for structure identification in flows, such as the proper orthogonal decomposition and topology theory in the laminar regime and the open areas for future research, mainly concerning turbulent and three-dimensional flows, are highlighted. Recent results obtained in our group are reported in both the time-stepping and the matrix-forming approaches to global linear theory. In the first context, progress has been made in implementing a Jacobian-Free Newton Krylov method into a standard finite-volume aerodynamic code, such that global linear instability results may now be obtained in compressible flows of aeronautical interest. In the second context a new stable very high-order finite difference method is implemented for the spatial discretization of the operators describing the spatial BiGlobal EVP, PSE-3D and the TriGlobal EVP; combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

MicroRaman Spectroscopy of Si Nanowires: Influence of Size

Si Nanowires (NWs) were studied by Raman microspectroscopy. The Raman spectrum of the NWs reveals important thermal effects, which broaden and shift the one phonon Raman bands. The low thermal conductivity of the NWs and the low thermal dissipation are responsible for the temperature enhancement in the NW under the excitation with the laser beam. We have modeled, using finite element methods, the interaction between the laser beam and the NWs. The Raman spectrum of Si NWs is interpreted in terms of the temperature induced by the laser beam excitation, in correlation with finite element methods (fem) for studying the interaction between the laser beam and the NWs.

A new boundary condition solved with B.I.E.M.

Among the classical operators of mathematical physics the Laplacian plays an important role due to the number of different situations that can be modelled by it. Because of this a great effort has been made by mathematicians as well as by engineers to master its properties till the point that nearly everything has been said about them from a qualitative viewpoint. Quantitative results have also been obtained through the use of the new numerical techniques sustained by the computer. Finite element methods and boundary techniques have been successfully applied to engineering problems as can be seen in the technical literature (for instance [ l ] , [2], [3] . Boundary techniques are especially advantageous in those cases in which the main interest is concentrated on what is happening at the boundary. This situation is very usual in potential problems due to the properties of harmonic functions. In this paper we intend to show how a boundary condition different from the classical, but physically sound, is introduced without any violence in the discretization frame of the Boundary Integral Equation Method. The idea will be developed in the context of heat conduction in axisymmetric problems but it is hoped that its extension to other situations is straightforward. After the presentation of the method several examples will show the capabilities of modelling a physical problem.

Comparison of Mesh Adaptation Using the Adjoint Methodology and Truncation Error Estimates

Mesh adaptation based on error estimation has become a key technique to improve th eaccuracy o fcomputational-fluid-dynamics computations. The adjoint-based approach for error estimation is one of the most promising techniques for computational-fluid-dynamics applications. Nevertheless, the level of implementation of this technique in the aeronautical industrial environment is still low because it is a computationally expensive method. In the present investigation, a new mesh refinement method based on estimation of truncation error is presented in the context of finite-volume discretization. The estimation method uses auxiliary coarser meshes to estimate the local truncation error, which can be used for driving an adaptation algorithm. The method is demonstrated in the context of two-dimensional NACA0012 and three-dimensional ONERA M6 wing inviscid flows, and the results are compared against the adjoint-based approach and physical sensors based on features of the flow field.

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two(and periodic in one)or all three spatial directions.After a brief exposition of the theory,some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton–Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers