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

El prop��sito de esta tesis es la implementaci��n de m��todos eficientes de adaptaci��n de mallas basados en ecuaciones adjuntas en el marco de discretizaciones de vol��menes finitos para mallas no estructuradas. La metodolog��a basada en ecuaciones adjuntas optimiza la malla refin��ndola adecuadamente con el objetivo de mejorar la precisi��n de c��lculo de un funcional de salida dado. El funcional suele ser una magnitud escalar de inter��s ingenieril obtenida por post-proceso de la soluci��n, como por ejemplo, la resistencia o la sustentaci��n aerodin��mica. Usualmente, el m��todo de adaptaci��n adjunta est�� basado en una estimaci��n a posteriori del error del funcional de salida mediante un promediado del residuo num��rico con las variables adjuntas, ���Dual Weighted Residual method��� (DWR). Estas variables se obtienen de la soluci��n del problema adjunto para el funcional seleccionado. El procedimiento habitual para introducir este m��todo en c��digos basados en discretizaciones de vol��menes finitos involucra la utilizaci��n 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 m��todo alternativo basado en reformular la estimaci��n del error del funcional en una malla auxiliar m��s basta y utilizar una t��cnica de estimaci��n del error de truncaci��n, denominada _ -estimation, para estimar los residuos que intervienen en el m��todo DWR. Utilizando esta estimaci��n del error se dise��a un algoritmo de adaptaci��n de mallas que conserva los ingredientes b��sicos de la adaptaci��n adjunta est��ndar pero con un coste computacional asociado sensiblemente menor. La metodolog��a de adaptaci��n adjunta est��ndar y la propuesta en la tesis han sido introducidas en un c��digo de vol��menes finitos utilizado habitualmente en la industria aeron��utica Europea. Se ha investigado la influencia de distintos par��metros num��ricos que intervienen en el algoritmo. Finalmente, el m��todo propuesto se compara con otras metodolog��as de adaptaci��n de mallas y su eficiencia computacional se demuestra en una serie de casos representativos de inter��s aeron��utico. 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.

Experimental and numerical study of cw green laser crystallization of a-Si:H thin films

Crystallization and grain growth technique of thin film silicon are among the most promising methods for improving efficiency and lowering cost of solar cells. A major advantage of laser crystallization and annealing over conventional heating methods is its ability to limit rapid heating and cooling to thin surface layers. Laser energy is used to heat the amorphous silicon thin film, melting it and changing the microstructure to polycrystalline silicon (poly-Si) as it cools. Depending on the laser density, the vaporization temperature can be reached at the center of the irradiated area. In these cases ablation effects are expected and the annealing process becomes ineffective. The heating process in the a-Si thin film is governed by the general heat transfer equation. The two dimensional non-linear heat transfer equation with a moving heat source is solve numerically using the finite element method (FEM), particularly COMSOL Multiphysics. The numerical model help to establish the density and the process speed range needed to assure the melting and crystallization without damage or ablation of the silicon surface. The samples of a-Si obtained by physical vapour deposition were irradiated with a cw-green laser source (Millennia Prime from Newport-Spectra) that delivers up to 15 W of average power. The morphology of the irradiated area was characterized by confocal laser scanning microscopy (Leica DCM3D) and Scanning Electron Microscopy (SEM Hitachi 3000N). The structural properties were studied by micro-Raman spectroscopy (Renishaw, inVia Raman microscope).

Application of global numerical procedures to the analysis of shells

Examples of global solutions of the shell equations are presented, such as the ones based on the well known Levy series expansion. Also discussed are some natural extensions of the Levy method as well as the inherent limitations of these methods concerning the shell model assumptions, boundary conditions and geometric regularity. Finally, some open additional design questions are noted mainly related to the simultaneous use in analysis of these global techniques and the local methods (like the finite elements) to finding the optimal shell shape, and to determining the reinforcement layout.

Analysis of Meshfree Methods for Lagrangian Fluid-Structure Interaction

In this dissertation a new numerical method for solving Fluid-Structure Interaction (FSI) problems in a Lagrangian framework is developed, where solids of different constitutive laws can suffer very large deformations and fluids are considered to be newtonian and incompressible. For that, we first introduce a meshless discretization based on local maximum-entropy interpolants. This allows to discretize a spatial domain with no need of tessellation, avoiding the mesh limitations. Later, the Stokes flow problem is studied. The Galerkin meshless method based on a max-ent scheme for this problem suffers from instabilities, and therefore stabilization techniques are discussed and analyzed. An unconditionally stable method is finally formulated based on a Douglas-Wang stabilization. Then, a Langrangian expression for fluid mechanics is derived. This allows us to establish a common framework for fluid and solid domains, such that interaction can be naturally accounted. The resulting equations are also in the need of stabilization, what is corrected with an analogous technique as for the Stokes problem. The fully Lagrangian framework for fluid/solid interaction is completed with simple point-to-point and point-to-surface contact algorithms. The method is finally validated, and some numerical examples show the potential scope of applications.

Polyolefin fibre-reinforced concrete : from material behaviour to numerical and design considerations

La presente Tesis proporciona una gran cantidad de informaci��n con respecto al uso de un nuevo y avanzado material polim��rico (con base de poliolefina) especialmente adecuada para ser usada en forma de fibras como adici��n en el hormig��n. Se han empleado fibras de aproximadamente 1 mm de di��metro, longitudes entre 48 y 60 mm y una superficie corrugada. Las prometedoras propiedades de este material (baja densidad, bajo coste, buen comportamiento resistente y gran estabilidad qu��mica) justifican el inter��s en desarrollar el esfuerzo de investigaci��n requerido para demostrar las ventajas de su uso en aplicaciones pr��cticas. La mayor parte de la investigaci��n se ha realizado usando hormig��n autocompactante como matriz, ya que este material es ��ptimo para el relleno de los encofrados del hormig��n, aunque tambi��n se ha empleado hormig��n normal vibrado con el fin de comparar algunas propiedades. Adem��s, el importante desarrollo del hormig��n reforzado con fibras en los ��ltimos a��os, tanto en investigaci��n como en aplicaciones pr��cticas, tambi��n es muestra del gran inter��s que los resultados y consideraciones de dise��o que esta Tesis pueden tener. El material compuesto resultante, Hormig��n Reforzado con Fibras de Poliolefina (HRFP o PFRC por sus siglas inglesas) ha sido exhaustivamente ensayado y estudiado en muchos aspectos. Los resultados permiten establecer c��mo conseguidos los objetivos buscados: -Se han cuantificado las propiedades mec��nicas del PFRC con el fin de demostrar su buen comportamiento en la fase fisurada de elementos estructurales sometidos a tensiones de tracci��n. -Contrastar los resultados obtenidos con las bases propuestas en la normativa existente y evaluar las posibilidades para el uso del PFRC con fin estructural para sustituir el armado tradicional con barras de acero corrugado para determinadas aplicaciones. -Se han desarrollado herramientas de c��lculo con el fin de evaluar la capacidad del PFRC para sustituir al hormig��n armado con las barras habituales de acero. -En base a la gran cantidad de ensayos experimentales y a alguna aplicaci��n real en la construcci��n, se han podido establecer recomendaciones y consejos de dise��o para que elementos de este material puedan ser proyectados y construidos con total fiabilidad. Se presentan, adem��s, resultados prometedores en una nueva l��nea de trabajo en el campo del hormig��n reforzado con fibras combinando dos tipolog��as de fibras. Se combinaron fibras de poliolefina con fibras de acero como refuerzo del mismo hormig��n autocompactante detect��ndose sinergias que podr��an ser la base del uso futuro de esta tecnolog��a de hormig��n. This thesis provides a significant amount of information on the use of a new advanced polymer (polyolefin-based) especially suitable in the form of fibres to be added to concrete. At the time of writing, there is a noteworthy lack of research and knowledge about use as a randomly distributed element to reinforce concrete. Fibres with an approximate 1 mm diameter, length of 48-60 mm, an embossed surface and improved mechanical properties are employed. The promising properties of the polyolefin material (low density, inexpensive, and with good strength behaviour and high chemical stability) justify the research effort involved and demonstrate the advantages for practical purposes. While most of the research has used self-compacting concrete, given that this type of matrix material is optimum in filling the concrete formwork, for comparison purposes standard vibration compacted mixes have also been used. In addition, the interest in fibre-reinforced concrete technology, in both research and application, support the significant interest in the results and considerations provided by the thesis. The resulting composite material, polyolefin fibre reinforced concrete (PFRC) has been extensively tested and studied. The results have allowed the following objectives to be met: -Assessment of the mechanical properties of PFRC in order to demonstrate the good performance in the post-cracking strength for structural elements subjected to tensile stresses. -- Assessment of the results in contrast with the existing structural codes, regulations and test methods. The evaluation of the potential of PFRC to meet the requirements and replace traditional steel-bar reinforcement applications. -Development of numerical tools designed to evaluate the capability of PFRC to substitute, either partially or totally, standard steel reinforcing bars either alone or in conjunction with steel fibres. -Provision, based on the large amount of experimental work and real applications, of a series of guidelines and recommendations for the practical and reliable design and use of PFRC. Furthermore, the thesis also reports promising results about an innovative line in the field of fibre-reinforced concrete: the design of a fibre cocktail to reinforce the concrete by using two types of fibres simultaneously. Polyolefin fibres were combined with steel fibres in self-compacting concrete, identifying synergies that could serve as the base in the future use of fibre-reinforced concrete technology.

On the application of meshfree methods to the nonlinear dynamics of multibody systems

Esta tesis propone una completa formulaci��n termo-mec��nica para la simulaci��n no-lineal de mecanismos flexibles basada en m��todos libres de malla. El enfoque se basa en tres pilares principales: la formulaci��n de Lagrangiano total para medios continuos, la discretizaci��n de Bubnov-Galerkin, y las funciones de forma libres de malla. Los m��todos sin malla se caracterizan por la definici��n de un conjunto de funciones de forma en dominios solapados, junto con una malla de integraci��n de las ecuaciones discretas de balance. Dos tipos de funciones de forma se han seleccionado como representaci��n de las familias interpolantes (Funciones de Base Radial) y aproximantes (M��nimos Cuadrados M��viles). Su formulaci��n se ha adaptado haciendo sus par��metros compatibles, y su ausencia de conectividad predefinida se ha aprovechado para interconectar m��ltiples dominios de manera autom��tica, permitiendo el uso de mallas de fondo no conformes. Se propone una formulaci��n generalizada de restricciones, juntas y contactos, v��lida para s��lidos r��gidos y flexibles, siendo estos ��ltimos discretizados mediante elementos finitos (MEF) o libres de malla. La mayor ventaja de este enfoque reside en que independiza completamente el dominio con respecto de las uniones y acciones externas a cada s��lido, permitiendo su definici��n incluso fuera del contorno. Al mismo tiempo, tambi��n se minimiza el n��mero de ecuaciones de restricci��n necesarias para la definici��n de uniones realistas. Las diversas validaciones, ejemplos y comparaciones detalladas muestran como el enfoque propuesto es gen��rico y extensible a un gran n��mero de sistemas. En concreto, las comparaciones con el MEF indican una importante reducci��n del error para igual n��mero de nodos, tanto en simulaciones mec��nicas, como t��rmicas y termo-mec��nicas acopladas. A igualdad de error, la eficiencia num��rica de los m��todos libres de malla es mayor que la del MEF cuanto m��s grosera es la discretizaci��n. Finalmente, la formulaci��n se aplica a un problema de dise��o real sobre el mantenimiento de estructuras masivas en el interior de un reactor de fusi��n, demostrando su viabilidad en an��lisis de problemas reales, y a su vez mostrando su potencial para su uso en simulaci��n en tiempo real de sistemas no-lineales. A new complete formulation is proposed for the simulation of nonlinear dynamic of multibody systems with thermo-mechanical behaviour. The approach is founded in three main pillars: total Lagrangian formulation, Bubnov-Galerkin discretization, and meshfree shape functions. Meshfree methods are characterized by the definition of a set of shape functions in overlapping domains, and a background grid for integration of the Galerkin discrete equations. Two different types of shape functions have been chosen as representatives of interpolation (Radial Basis Functions), and approximation (Moving Least Squares) families. Their formulation has been adapted to use compatible parameters, and their lack of predefined connectivity is used to interconnect different domains seamlessly, allowing the use of non-conforming meshes. A generalized formulation for constraints, joints, and contacts is proposed, which is valid for rigid and flexible solids, being the later discretized using either finite elements (FEM) or meshfree methods. The greatest advantage of this approach is that makes the domain completely independent of the external links and actions, allowing to even define them outside of the boundary. At the same time, the number of constraint equations needed for defining realistic joints is minimized. Validation, examples, and benchmarks are provided for the proposed formulation, demonstrating that the approach is generic and extensible to further problems. Comparisons with FEM show a much lower error for the same number of nodes, both for mechanical and thermal analyses. The numerical efficiency is also better when coarse discretizations are used. A final demonstration to a real problem for handling massive structures inside of a fusion reactor is presented. It demonstrates that the application of meshfree methods is feasible and can provide an advantage towards the definition of nonlinear real-time simulation models.

Energy-entropy-momentum time integration methods for coupled smooth dissipative problems

Esta tesis aborda la formulaci��n, an��lisis e implementaci��n de m��todos num��ricos de integraci��n temporal para la soluci��n de sistemas disipativos suaves de dimensi��n finita o infinita de manera que su estructura continua sea conservada. Se entiende por dichos sistemas aquellos que involucran acoplamiento termo-mec��nico y/o efectos disipativos internos modelados por variables internas que siguen leyes continuas, de modo que su evoluci��n es considerada suave. La din��mica de estos sistemas est�� gobernada por las leyes de la termodin��mica y simetr��as, las cuales constituyen la estructura que se pretende conservar de forma discreta. Para ello, los sistemas disipativos se describen geom��tricamente mediante estructuras metripl��cticas que identifican claramente las partes reversible e irreversible de la evoluci��n del sistema. As��, usando una de estas estructuras conocida por las siglas (en ingl��s) de GENERIC, la estructura disipativa de los sistemas es identificada del mismo modo que lo es la Hamiltoniana para sistemas conservativos. Con esto, m��todos (EEM) con precisi��n de segundo orden que conservan la energ��a, producen entrop��a y conservan los impulsos lineal y angular son formulados mediante el uso del operador derivada discreta introducido para asegurar la conservaci��n de la Hamiltoniana y las simetr��as de sistemas conservativos. Siguiendo estas directrices, se formulan dos tipos de m��todos EEM basados en el uso de la temperatura o de la entrop��a como variable de estado termodin��mica, lo que presenta importantes implicaciones que se discuten a lo largo de esta tesis. Entre las cuales cabe destacar que las condiciones de contorno de Dirichlet son naturalmente impuestas con la formulaci��n basada en la temperatura. Por ��ltimo, se validan dichos m��todos y se comprueban sus mejores prestaciones en t��rminos de la estabilidad y robustez en comparaci��n con m��todos est��ndar. This dissertation is concerned with the formulation, analysis and implementation of structure-preserving time integration methods for the solution of the initial(-boundary) value problems describing the dynamics of smooth dissipative systems, either finite- or infinite-dimensional ones. Such systems are understood as those involving thermo-mechanical coupling and/or internal dissipative effects modeled by internal state variables considered to be smooth in the sense that their evolutions follow continuos laws. The dynamics of such systems are ruled by the laws of thermodynamics and symmetries which constitutes the structure meant to be preserved in the numerical setting. For that, dissipative systems are geometrically described by metriplectic structures which clearly identify the reversible and irreversible parts of their dynamical evolution. In particular, the framework known by the acronym GENERIC is used to reveal the systems' dissipative structure in the same way as the Hamiltonian is for conserving systems. Given that, energy-preserving, entropy-producing and momentum-preserving (EEM) second-order accurate methods are formulated using the discrete derivative operator that enabled the formulation of Energy-Momentum methods ensuring the preservation of the Hamiltonian and symmetries for conservative systems. Following these guidelines, two kind of EEM methods are formulated in terms of entropy and temperature as a thermodynamical state variable, involving important implications discussed throughout the dissertation. Remarkably, the formulation in temperature becomes central to accommodate Dirichlet boundary conditions. EEM methods are finally validated and proved to exhibit enhanced numerical stability and robustness properties compared to standard ones.

Depth Averaged Models for Fast Landslide Propagation: Mathematical, Rheological and Numerical Aspects

This paper presents an overview of depth averaged modelling of fast catastrophic landslides where coupling of solid skeleton and pore fluid (air and water) is important. The first goal is to show how Biot-Zienkiewicz models can be applied to develop depth integrated, coupled models. The second objective of the paper is to consider a link which can be established between rheological and constitutive models. Perzyna��s viscoplasticity can be considered a general framework within which rheological models such as Bingham and cohesive frictional fluids can be derived. Among the several alternative numerical models, we will focus here on SPH which has not been widely applied by engineers to model landslide propagation. We propose an improvement, based on combining Finite Difference meshes associated to SPH nodes to describe pore pressure evolution inside the landslide mass. We devote a Section to analyze the performance of the models, considering three sets of tests and examples which allows to assess the model performance and limitations: (i) Problems having an analytical solution, (ii) Small scale laboratory tests, and (iii) Real cases for which we have had access to reliable information

Adaptation Strategies for Discontinuous Galerkin Spectral Element Methods by means of Truncation Error Estimations

In this work a p-adaptation (modification of the polynomial order) strategy based on the minimization of the truncation error is developed for high order discontinuous Galerkin methods. The truncation error is approximated by means of a truncation error estimation procedure and enables the identification of mesh regions that require adaptation. Three truncation error estimation approaches are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. Fine solutions, which are obtained by enriching the polynomial order, are required to solve the numerical problem with adequate accuracy. For the three truncation error estimation methods the former needs time converged solutions, while the last two rely on non-converged solutions, which lead to faster computations. Based on these truncation error estimation methods, algorithms for mesh adaptation were designed and tested. Firstly, an isotropic adaptation approach is presented, which leads to equally distributed polynomial orders in different coordinate directions. This first implementation is improved by incorporating a method to extrapolate the truncation error. This results in a significant reduction of computational cost. Secondly, the employed high order method permits the spatial decoupling of the estimated errors and enables anisotropic p-adaptation. The incorporation of anisotropic features leads to meshes with different polynomial orders in the different coordinate directions such that flow-features related to the geometry are resolved in a better manner. These adaptations result in a significant reduction of degrees of freedom and computational cost, while the amount of improvement depends on the test-case. Finally, this anisotropic approach is extended by using error extrapolation which leads to an even higher reduction in computational cost. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. The main result is that the two quasi-a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of a factor of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively. RESUMEN En este trabajo se ha desarrollado una estrategia de adaptaci��n-p (modificaci��n del orden polin��mico) para m��todos Galerkin discontinuo de alto orden basada en la minimizaci��n del error de truncaci��n. El error de truncaci��n se estima utilizando el m��todo tau-estimation. El estimador permite la identificaci��n de zonas de la malla que requieren adaptaci��n. Se distinguen tres t��cnicas de estimaci��n: a posteriori, quasi a priori y quasi a priori con correci��n. Todas las estrategias requieren una soluci��n obtenida en una malla fina, la cual es obtenida aumentando de manera uniforme el orden polin��mico. Sin embargo, mientras que el primero requiere que esta soluci��n est�� convergida temporalmente, el resto utiliza soluciones no convergidas, lo que se traduce en un menor coste computacional. En este trabajo se han dise��ado y probado algoritmos de adaptaci��n de malla basados en m��todos tau-estimation. En primer lugar, se presenta un algoritmo de adaptacin is��tropo, que conduce a discretizaciones con el mismo orden polin��mico en todas las direcciones espaciales. Esta primera implementaci��n se mejora incluyendo un m��todo para extrapolar el error de truncaci��n. Esto resulta en una reducci��n significativa del coste computacional. En segundo lugar, el m��todo de alto orden permite el desacoplamiento espacial de los errores estimados, permitiendo la adaptaci��n anisotropica. Las mallas obtenidas mediante esta t��cnica tienen distintos ��rdenes polin��micos en cada una de las direcciones espaciales. La malla final tiene una distribuci��n ��ptima de ��rdenes polin��micos, los cuales guardan relaci��n con las caracter��sticas del flujo que, a su vez, depenen de la geometr��a. Estas t��cnicas de adaptaci��n reducen de manera significativa los grados de libertad y el coste computacional. Por ��ltimo, esta aproximaci��n anisotropica se extiende usando extrapolaci��n del error de truncaci��n, lo que conlleva un coste computational a��n menor. Las estrategias se verifican y se comparan en t��minors de precisi��n y coste computacional utilizando las ecuaciones de Euler y Navier Stokes. Los dos m��todos quasi a priori consiguen una reducci��n significativa del coste computacional en comparaci��n con aumento uniforme del orden polin��mico. En concreto, para una capa l��mite viscosa, obtenemos una mejora en tiempo de computaci��n de 6.6 y 7.6 respectivamente, para las aproximaciones quasi-a priori y quasi-a priori con correcci��n.

Numerical vorticity creation based on impulse conservation.

The problem of creating solenoidal vortex elements to satisfy no-slip boundary conditions in Lagrangian numerical vortex methods is solved through the use of impulse elements at walls and their subsequent conversion to vortex loops. The algorithm is not uniquely defined, due to the gauge freedom in the definition of impulse; the numerically optimal choice of gauge remains to be determined. Two different choices are discussed, and an application to flow past a sphere is sketched.

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis���Mor�� condition for q-superlinear convergence. Simple numerical examples illustrate the results.

A compact difference scheme for numerical solutions of second order dual-phase-lagging models of microscale heat transfer

Dual-phase-lagging (DPL) models constitute a family of non-Fourier models of heat conduction that allow for the presence of time lags in the heat flux and the temperature gradient. These lags may need to be considered when modeling microscale heat transfer, and thus DPL models have found application in the last years in a wide range of theoretical and technical heat transfer problems. Consequently, analytical solutions and methods for computing numerical approximations have been proposed for particular DPL models in different settings. In this work, a compact difference scheme for second order DPL models is developed, providing higher order precision than a previously proposed method. The scheme is shown to be unconditionally stable and convergent, and its accuracy is illustrated with numerical examples.

A Finite Element Numerical Algorithm for Modelling and Data Fitting in Complex Systems

Numerical modelling methodologies are important by their application to engineering and scientific problems, because there are processes where analytical mathematical expressions cannot be obtained to model them. When the only available information is a set of experimental values for the variables that determine the state of the system, the modelling problem is equivalent to determining the hyper-surface that best fits the data. This paper presents a methodology based on the Galerkin formulation of the finite elements method to obtain representations of relationships that are defined a priori, between a set of variables: y = z(x1, x2,...., xd). These representations are generated from the values of the variables in the experimental data. The approximation, piecewise, is an element of a Sobolev space and has derivatives defined in a general sense into this space. The using of this approach results in the need of inverting a linear system with a structure that allows a fast solver algorithm. The algorithm can be used in a variety of fields, being a multidisciplinary tool. The validity of the methodology is studied considering two real applications: a problem in hydrodynamics and a problem of engineering related to fluids, heat and transport in an energy generation plant. Also a test of the predictive capacity of the methodology is performed using a cross-validation method.

Numerical and experimental investigation of structural stiffness influence on ratcheting convection cell in granular soils under cyclic loading

Lateral cyclic loaded structures in granular soils can lead to an accumulation of irreversible strains by changing their mechanical response (densification) and forming a closed convective cell in the upper layer of the bedding. In the present thesis the convective cell dimension, formation and grain migration inside this closed volume have been studied and presented in relation to structural stiffness and different loads. This relation was experimentally investigated by applying a cyclic lateral force to a scaled flexible vertical element embedded in dry granular soil. The model was monitored with a camera in order to derive the displacement field by means of the PIV technique. Modelling large soil deformation turns out to be difficult, using mesh-based methods. Consequently, a mesh-free approach (DEM) was chosen in order to investigate the granular flow with the aim of extracting interesting micromechanical information. In both the numerical and experimental analyses the effect of different loading magnitudes and different dimensions of the vertical element were considered. The main results regarded the different development, shape and dimensions of the convection cell and the surface settlements. Moreover, the Discrete Element Method has proven to give satisfactory results in the modelling of large deformation phenomena such as the ratcheting convective cell.

Error estimation and iterative improvement for the numerical solution of operator equations /

Bibliography: p. 85-87.