Aplicación de Métodos Meshless al Análisis de Problemas de Autovalores

La presente Tesis Doctoral aborda la aplicación de métodos meshless, o métodos sin malla, a problemas de autovalores, fundamentalmente vibraciones libres y pandeo. En particular, el estudio se centra en aspectos tales como los procedimientos para la resolución numérica del problema de autovalores con estos métodos, el coste computacional y la viabilidad de la utilización de matrices de masa o matrices de rigidez geométrica no consistentes. Además, se acomete en detalle el análisis del error, con el objetivo de determinar sus principales fuentes y obtener claves que permitan la aceleración de la convergencia. Aunque en la actualidad existe una amplia variedad de métodos meshless en apariencia independientes entre sí, se han analizado las diferentes relaciones entre ellos, deduciéndose que el método Element-Free Galerkin Method [Método Galerkin Sin Elementos] (EFGM) es representativo de un amplio grupo de los mismos. Por ello se ha empleado como referencia en este análisis. Muchas de las fuentes de error de un método sin malla provienen de su algoritmo de interpolación o aproximación. En el caso del EFGM ese algoritmo es conocido como Moving Least Squares [Mínimos Cuadrados Móviles] (MLS), caso particular del Generalized Moving Least Squares [Mínimos Cuadrados Móviles Generalizados] (GMLS). La formulación de estos algoritmos indica que la precisión de los mismos se basa en los siguientes factores: orden de la base polinómica p(x), características de la función de peso w(x) y forma y tamaño del soporte de definición de esa función. Se ha analizado la contribución individual de cada factor mediante su reducción a un único parámetro cuantificable, así como las interacciones entre ellos tanto en distribuciones regulares de nodos como en irregulares. El estudio se extiende a una serie de problemas estructurales uni y bidimensionales de referencia, y tiene en cuenta el error no sólo en el cálculo de autovalores (frecuencias propias o carga de pandeo, según el caso), sino también en términos de autovectores. This Doctoral Thesis deals with the application of meshless methods to eigenvalue problems, particularly free vibrations and buckling. The analysis is focused on aspects such as the numerical solving of the problem, computational cost and the feasibility of the use of non-consistent mass or geometric stiffness matrices. Furthermore, the analysis of the error is also considered, with the aim of identifying its main sources and obtaining the key factors that enable a faster convergence of a given problem. Although currently a wide variety of apparently independent meshless methods can be found in the literature, the relationships among them have been analyzed. The outcome of this assessment is that all those methods can be grouped in only a limited amount of categories, and that the Element-Free Galerkin Method (EFGM) is representative of the most important one. Therefore, the EFGM has been selected as a reference for the numerical analyses. Many of the error sources of a meshless method are contributed by its interpolation/approximation algorithm. In the EFGM, such algorithm is known as Moving Least Squares (MLS), a particular case of the Generalized Moving Least Squares (GMLS). The accuracy of the MLS is based on the following factors: order of the polynomial basis p(x), features of the weight function w(x), and shape and size of the support domain of this weight function. The individual contribution of each of these factors, along with the interactions among them, has been studied in both regular and irregular arrangement of nodes, by means of a reduction of each contribution to a one single quantifiable parameter. This assessment is applied to a range of both one- and two-dimensional benchmarking cases, and includes not only the error in terms of eigenvalues (natural frequencies or buckling load), but also of eigenvectors

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.

Reduced order adaptive models for systems of PDEs using POD

In this work we propose a method to accelerate time dependent numerical solvers of systems of PDEs that require a high cost in computational time and memory. The method is based on the combined use of such numerical solver with a proper orthogonal decomposition, from which we identify modes, a Galerkin projection (that provides a reduced system of equations) and the integration of the reduced system, studying the evolution of the modal amplitudes. We integrate the reduced model until our a priori error estimator indicates that our approximation in not accurate. At this point we use again our original numerical code in a short time interval to adapt the POD manifold and continue then with the integration of the reduced model. Application will be made to two model problems: the Ginzburg-Landau equation in transient chaos conditions and the two-dimensional pulsating cavity problem, which describes the motion of liquid in a box whose upper wall is moving back and forth in a quasi-periodic fashion. Finally, we will discuss a way of improving the performance of the method using experimental data or information from numerical simulations

Video Prioritization for Unequal Error Protection

We analyze the effect of packet losses in video sequences and propose a lightweight Unequal Error Protection strategy which, by choosing which packet is discarded, reduces strongly the Mean Square Error of the received sequence

ITEM: Inter-Texture Error Measurement for 3D Meshes

We introduce a simple and innovative method to compare any two texture maps, regardless of their sizes, aspect ratios, or even masks, as long as they are both meant to be mapped onto the same 3D mesh. Our system is based on a zero-distortion 3D mesh unwrapping technique which compares two new adapted texture atlases with the same mask but different texel colors, and whose every texel covers the same area in 3D. Once these adapted atlases are created, we measure their difference with ITEM-RMSE, a slightly modified version of the standard RMSE defined for images. ITEM-RMSE is more meaningful and reliable than RMSE because it only takes into account the texels inside the mask, since they are the only ones that will actually be used during rendering. Our method is not only very useful to compare the space efficiency of different texture atlas generation algorithms, but also to quantify texture loss in compression schemes for multi-resolution textured 3D meshes.

Non-failure analysis for logic programs

We provide a method whereby, given mode and (upper approximation) type information, we can detect procedures and goals that can be guaranteed to not fail (i.e., to produce at least one solution or not termínate). The technique is based on an intuitively very simple notion, that of a (set of) tests "covering" the type of a set of variables. We show that the problem of determining a covering is undecidable in general, and give decidability and complexity results for the Herbrand and linear arithmetic constraint systems. We give sound algorithms for determining covering that are precise and efiicient in practice. Based on this information, we show how to identify goals and procedures that can be guaranteed to not fail at runtime. Applications of such non-failure information include programming error detection, program transiormations and parallel execution optimization, avoiding speculative parallelism and estimating lower bounds on the computational costs of goals, which can be used for granularity control. Finally, we report on an implementation of our method and show that better results are obtained than with previously proposed approaches.

Modelización de costas rocosas acantiladas

El retroceso de las costas acantiladas es un fenómeno muy extendido sobre los litorales rocosos expuestos a la incidencia combinada de los procesos marinos y meteorológicos que se dan en la franja costera. Este fenómeno se revela violentamente como movimientos gravitacionales del terreno esporádicos, pudiendo causar pérdidas materiales y/o humanas. Aunque el conocimiento de estos riesgos de erosión resulta de vital importancia para la correcta gestión de la costa, el desarrollo de modelos predictivos se encuentra limitado desde el punto de vista geomorfológico debido a la complejidad e interacción de los procesos de desarrollo espacio-temporal que tienen lugar en la zona costera. Los modelos de predicción publicados son escasos y con importantes inconvenientes: a) extrapolación, extienden la información de registros históricos; b) empíricos, sobre registros históricos estudian la respuesta al cambio de un parámetro; c) estocásticos, determinan la cadencia y magnitud de los eventos futuros extrapolando las distribuciones de probabilidad extraídas de catálogos históricos; d) proceso-respuesta, de estabilidad y propagación del error inexplorada; e) en Ecuaciones en Derivadas Parciales, computacionalmente costosos y poco exactos. La primera parte de esta tesis detalla las principales características de los modelos más recientes de cada tipo y, para los más habitualmente utilizados, se indican sus rangos de aplicación, ventajas e inconvenientes. Finalmente como síntesis de los procesos más relevantes que contemplan los modelos revisados, se presenta un diagrama conceptual de la recesión costera, donde se recogen los procesos más influyentes que deben ser tenidos en cuenta, a la hora de utilizar o crear un modelo de recesión costera con el objetivo de evaluar la peligrosidad (tiempo/frecuencia) del fenómeno a medio-corto plazo. En esta tesis se desarrolla un modelo de proceso-respuesta de retroceso de acantilados costeros que incorpora el comportamiento geomecánico de materiales cuya resistencia a compresión no supere los 5 MPa. El modelo simula la evolución espaciotemporal de un perfil-2D del acantilado que puede estar formado por materiales heterogéneos. Para ello, se acoplan la dinámica marina: nivel medio del mar, cambios en el nivel medio del lago, mareas y oleaje; con la evolución del terreno: erosión, desprendimiento rocoso y formación de talud de derrubios. El modelo en sus diferentes variantes es capaz de incluir el análisis de la estabilidad geomecánica de los materiales, el efecto de los derrubios presentes al pie del acantilado, el efecto del agua subterránea, la playa, el run-up, cambios en el nivel medio del mar o cambios (estacionales o interanuales) en el nivel medio de la masa de agua (lagos). Se ha estudiado el error de discretización del modelo y su propagación en el tiempo a partir de las soluciones exactas para los dos primeros periodos de marea para diferentes aproximaciones numéricas tanto en tiempo como en espacio. Los resultados obtenidos han permitido justificar las elecciones que minimizan el error y los métodos de aproximación más adecuados para su posterior uso en la modelización. El modelo ha sido validado frente a datos reales en la costa de Holderness, Yorkshire, Reino Unido; y en la costa norte del lago Erie, Ontario, Canadá. Los resultados obtenidos presentan un importante avance en los modelos de recesión costera, especialmente en su relación con las condiciones geomecánicas del medio, la influencia del agua subterránea, la verticalización de los perfiles rocosos y su respuesta ante condiciones variables producidas por el cambio climático (por ejemplo, nivel medio del mar, cambios en los niveles de lago, etc.). The recession of coastal cliffs is a widespread phenomenon on the rocky shores that are exposed to the combined incidence of marine and meteorological processes that occur in the shoreline. This phenomenon is revealed violently and occasionally, as gravitational movements of the ground and can cause material or human losses. Although knowledge of the risks of erosion is vital for the proper management of the coast, the development of cliff erosion predictive models is limited by the complex interactions between environmental processes and material properties over a range of temporal and spatial scales. Published prediction models are scarce and present important drawbacks: extrapolation, that extend historical records to the future; empirical, that based on historical records studies the system response against the change in one parameter; stochastic, that represent of cliff behaviour based on assumptions regarding the magnitude and frequency of events in a probabilistic framework based on historical records; process-response, stability and error propagation unexplored; PDE´s, highly computationally expensive and not very accurate. The first part of this thesis describes the main features of the latest models of each type and, for the most commonly used, their ranges of application, advantages and disadvantages are given. Finally as a synthesis of the most relevant processes that include the revised models, a conceptual diagram of coastal recession is presented. This conceptual model includes the most influential processes that must be taken into account when using or creating a model of coastal recession to evaluate the dangerousness (time/frequency) of the phenomenon to medium-short term. A new process-response coastal recession model developed in this thesis has been designed to incorporate the behavioural and mechanical characteristics of coastal cliffs which are composed of with materials whose compressive strength is less than 5 MPa. The model simulates the spatial and temporal evolution of a cliff-2D profile that can consist of heterogeneous materials. To do so, marine dynamics: mean sea level, waves, tides, lake seasonal changes; is coupled with the evolution of land recession: erosion, cliff face failure and associated protective colluvial wedge. The model in its different variants can include analysis of material geomechanical stability, the effect of debris present at the cliff foot, groundwater effects, beach and run-up effects, changes in the mean sea level or changes (seasonal or inter-annual) in the mean lake level. Computational implementation and study of different numerical resolution techniques, in both time and space approximations, and the produced errors are exposed and analysed for the first two tidal periods. The results obtained in the errors analysis allow us to operate the model with a configuration that minimizes the error of the approximation methods. The model is validated through profile evolution assessment at various locations of coastline retreat on the Holderness Coast, Yorkshire, UK and on the north coast of Lake Erie, Ontario, Canada. The results represent an important stepforward in linking material properties to the processes of cliff recession, in considering the effect of groundwater charge and the slope oversteeping and their response to changing conditions caused by climate change (i.e. sea level, changes in lakes levels, etc.).

Estimación del error de discretización con principios variacionales multicampo: I - Elasticidad

Desarrollos recientes para encajar dentro de un marco variacional la llamada Formulación Libre sugieren la posibilidad de introducir un nuevo tipo de estimador de error para cálculos por elementos finitos. Este estimador se basa en una diferencia entre ciertos funcionales multicampo, que toman el mismo valor para la solución exacta del problema. En el presente articulo, dividido en dos partes, se introduce la formulación del estimador para problemas de elasticidad y de flexión de placas según las hipótesis clásicas de Kirchhoff. Se presentan también algunos ejeinplos para dar idea de los comportamientos numéricos observados.

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

Neighborhood-corrected interface discontinuity factors for multi-group pin-by-pin diffusion calculations for LWR

Performing three-dimensional pin-by-pin full core calculations based on an improved solution of the multi-group diffusion equation is an affordable option nowadays to compute accurate local safety parameters for light water reactors. Since a transport approximation is solved, appropriate correction factors, such as interface discontinuity factors, are required to nearly reproduce the fully heterogeneous transport solution. Calculating exact pin-by-pin discontinuity factors requires the knowledge of the heterogeneous neutron flux distribution, which depends on the boundary conditions of the pin-cell as well as the local variables along the nuclear reactor operation. As a consequence, it is impractical to compute them for each possible configuration; however, inaccurate correction factors are one major source of error in core analysis when using multi-group diffusion theory. An alternative to generate accurate pin-by-pin interface discontinuity factors is to build a functional-fitting that allows incorporating the environment dependence in the computed values. This paper suggests a methodology to consider the neighborhood effect based on the Analytic Coarse-Mesh Finite Difference method for the multi-group diffusion equation. It has been applied to both definitions of interface discontinuity factors, the one based on the Generalized Equivalence Theory and the one based on Black-Box Homogenization, and for different few energy groups structures. Conclusions are drawn over the optimal functional-fitting and demonstrative results are obtained with the multi-group pin-by-pin diffusion code COBAYA3 for representative PWR configurations.

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.

Automatic mesh generation processor for 3-D parabolic boundary discretization

An automatic Mesh Generation Preprocessor for BE Programs with a considerable of capabilities has been developed. This program allows almost any kind of geometry and tipology to be defined with a small amount of external data, and with an important approximation of the boundary geometry. Also the error checking possibility is very important for a fast comprobation of the results.