3,268 research outputs found

    (2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

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    We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α\alpha, the long-wavelength parameter β\beta, the transverse wavelength parameter γ\gamma, and the bottom variation parameter δ\delta. We derived the only possible (2+1)-dimensional extensions of the Korteweg-de Vries equation, the fifth-order KdV equation, and the Gardner equation in three special cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev-Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation.Comment: Section 4, with soliton, cnoidal and superposition solutions to (2+1)-dimensional nonlocal KdV equation, added. In section 5 mistakes corrected. In Section 6 mistakes correcte

    Studies on Nonlinear and Dynamic Soil Structure Interaction

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    High-speed trains, excessive loads in moving trucks, and vibrating machines on foundations on soft ground can generate significant vibration and deformation in the subgrade (soil). Better understanding and realistic analysis of the interaction between railway tracks, pavements, and foundations and the supporting soil under moving and dynamic loads is necessary. Experimental investigations are always associated with large costs when simulating the loading conditions. Modeling dynamic soil-structure interaction problems is often associated with a high level of complexity and a large computational effort. Analytical modeling of these problems that results in accurate and reliable prediction of these soil structure interaction problems with a low computational cost and ease of use is a distinct advantage that can supplement the numerical modeling and experimental investigations. In this research, a new computationally efficient but mathematically rigorous semianalytical continuum model is developed for dynamic analysis of beams resting on layered poroelastic nonlinear soil deposit and subjected to dynamic loads. The proposed model is developed in stages in terms of the complexity of simulating the soil behaviour. First, the soil is simulated as a discrete two-parameter foundation in which the soil body is represented by mechanical springs with shear interactions. Subsequently, the soil is simulated as a linear and nonlinear continuum. Finally, the soil is simulated as a linear and nonlinear poroelastic continuum, For the continuum-based analysis, a simplified continuum approach was adopted in which the soil displacement field is expressed as a product of separable variables. The principle of virtual work was applied to obtain the governing differential equations that were solved partly analytically and partly numerically. The semi-analytical approach was found to be significantly faster than the corresponding full blown finite element analysis. A significant contribution of this work is the simulation of the nonlinear and poroelastic response of soil in the semi-analytical framework, which otherwise require elaborate meshing by the users and high computational effort. A nonlinear hyperbolic stress-strain relationship is used to represent the soil nonlinearity. Biot’s poroelastic theory is used to represent the poroelastic behaviour of soil. The nonlinear dynamic, nonlinear consolidation, and nonlinear poroelastic dynamic responses of the beams under moving and oscillating loads are obtained. It is envisaged that the methods developed in this thesis will provide more insights into the dynamic soil structure interaction problem, and will help in developing design aids

    Numerical resolution of the Navier-Stokes equations with parallel programming for the analysis of heat and mass transfer phenomena.

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    Aquesta tesi analitza mètodes numèrics per resoldre les equacions de Navier-Stokes en dinàmica de fluids computacional (CFD, per les sigles en anglès). La investigació es centra a des- envolupar una visió profunda de diferents mètodes numèrics i la seva aplicació a diversos fenòmens de transport. S’aplica una metodologia pas a pas, que abarca l’anàlisi de volums fi- nits i mètodes espectrals, la validació de models i la verificació de codis a través de l’anàlisi de casos d’estudi de convecció-difusió, flux de fluids i turbulència. La investigació revela l’efecte de diferents esquemes d’aproximació a la solució numèrica i emfatitza la importància d’una representació física precisa juntament amb la solidesa matemàtica. S’examina la convergència del mètode de resolució d’equacions iteratiu pel que fa a la naturalesa de la física de l’estudi, i cal destacar la necessitat de tècniques de relaxació apropiades. A més, s’explora el mètode de passos fraccionats per resoldre el fort acoblament de pressió-velocitat a les equacions de Navier-Stokes, mentre es considera l’addició d’altres fenòmens de transport. L’anàlisi de fluxes turbulents mostra la cascada d’energia a l’espai de Fourier i l’efecte del truncament a causa de la discretització espacial o espectral, abordat per l’aplicació de models simplificats, com ara Large Eddy Simulation (LES), aconseguint una solució aproximada amb un menor cost computacional. A més, s’analitza la implementació de la computació en paral·lel utilitzant l’estàndard MPI, emfatitzant-ne l’escalabilitat i el potencial per abordar les demandes creixents de l’anàlisi CFD en els camps de l’enginyeria. En general, aquesta recerca proporciona informació valuosa sobre els mètodes numèrics per a les equacions de Navier-Stokes, la seva aplicació a CFD i les implicacions pràctiques per als processos d’enginyeriaEsta tesis analiza métodos numéricos para resolver las ecuaciones de Navier-Stokes en dinámica de fluidos computacional (CFD, por sus siglas en Inglés). La investigación se centra en desarrollar una visión profunda de distintos métodos numéricos y su aplicación a diversos fenómenos de transporte. Se aplica una metodología paso a paso, que abarca el análisis de volúmenes finitos y métodos espectrales, validación de modelos y verificación de códigos a través del analisis de casos de estudio de convección-difusión, flujo de fluidos y turbulencia. La investigación revela el efecto de diferentes esquemas de aproximación en la solución numérica y enfatiza la importancia de una representación física precisa junto con la solidez matemática. Se examina la convergencia del método de resolución de equaciones iterativo con respecto a la naturaleza de la física del estudio, destacando la necesidad de técnicas de relajación apropiadas. Además, se explora el método de pasos fraccionados para resolver el fuerte acoplamiento de presión-velocidad en las ecuaciones de Navier-Stokes, mientras se considera la adición de otros fenómenos de transporte. El análisis de flujos turbulentos muestra la cascada de energía en el espacio de Fourier y el efecto del truncamiento debido a la discretización espacial o espectral, abordado por la aplicación de modelos simplificados, como Large Eddy Simulation (LES), logrando una solución aproximada con un menor costo computacional. Además, se analiza la implementación de la computación en paralelo utilizando el estándar MPI, enfatizando su escalabilidad y potencial para abordar las crecientes demandas del análisis CFD en los campos de la ingeniería. En general, esta investigación proporciona información valiosa sobre los métodos numéricos para las ecuaciones de Navier-Stokes, su aplicación a CFD y sus implicaciones prácticas para los procesos de ingenieríaThis thesis analyzes numerical methods for solving the Navier-Stokes equations in computational fluid dynamics (CFD). The research focuses on developing a deep insight into different numerical techniques and their application to various transport phenomena. A step-by-step methodology is applied, encompassing the analysis of finite volume and spectral methods, model validation, and code verification with the study of convection-diffusion, fluid flow, and turbulence study cases. The investigation reveals the effect of different approximation schemes on the numerical solution and emphasizes the importance of accurate physics representation alongside mathematical robustness. The convergence of the numerical solver is examined concerning the nature of the studied physics, highlighting the need for appropriate relaxation techniques. Additionally, the fractional step method is explored to solve the strong pressure-velocity coupling in the Navier-Stokes equations while considering the addition of other transport phenomena. The analysis of turbulent flows showcases the energy cascade in the Fourier space and its truncation effect due to spatial or spectral discretization, addressed by the application of simplified models, such as Large Eddy Simulation (LES), capable of approximating the solution with reduced computational cost. Furthermore, the implementation of parallel computing using the MPI standard is discussed, emphasizing its scalability and potential for addressing the growing demands of CFD analysis in engineering fields. Overall, this research provides valuable insights into numerical methods for the Navier-Stokes equations, their application to CFD, and their practical implications for engineering processe

    Leveraging elasticity theory to calculate cell forces: From analytical insights to machine learning

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    Living cells possess capabilities to detect and respond to mechanical features of their surroundings. In traction force microscopy, the traction of cells on an elastic substrate is made visible by observing substrate deformation as measured by the movement of embedded marker beads. Describing the substrates by means of elasticity theory, we can calculate the adhesive forces, improving our understanding of cellular function and behavior. In this dissertation, I combine analytical solutions with numerical methods and machine learning techniques to improve traction prediction in a range of experimental applications. I describe how to include the normal traction component in regularization-based Fourier approaches, which I apply to experimental data. I compare the dominant strategies for traction reconstruction, the direct method and inverse, regularization-based approaches and find, that the latter are more precise while the former is more stress resilient to noise. I find that a point-force based reconstruction can be used to study the force balance evolution in response to microneedle pulling showing a transition from a dipolar into a monopolar force arrangement. Finally, I show how a conditional invertible neural network not only reconstructs adhesive areas more localized, but also reveals spatial correlations and variations in reliability of traction reconstructions

    Advancements in Fluid Simulation Through Enhanced Conservation Schemes

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    To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study new methods in solving the Navier-Stokes equations using models which have enhanced conservation qualities, in particular, the energy, momentum, and angular momentum conserving (EMAC) scheme. The EMAC scheme has gained popularity in the mathematics community over the past few years as a desirable method to model fluid flow. It has been proven to conserve energy, momentum, angular momentum, helicity, and others. EMAC has also been shown to perform better and maintain accuracy over long periods of time compared to other schemes. We investigate a fully discrete error analysis of EMAC and SKEW. We show that a problematic dependency on the Reynolds number is present in the analysis for SKEW, but not in EMAC under certain conditions. To further explore this concept, we include some numerical experiments designed to highlight these differences in the error analysis. Additionally, we include other projection methods to measure performance. Following this, we introduce a new EMAC variant which applies a differential spatial filter to the EMAC scheme, named EMAC-Reg. Standard models, including EMAC, require especially fine meshes with high Reynold\u27s numbers. This is problematic because the linear systems for 3D flows will be far too large and take an extraordinary amount of time to compute. EMAC-Reg not only performs better on a coarser mesh, but maintains conservation properties as well. Another topic in fluid flow computing that has been gaining recognition is reduced order models. This method uses experimental data to create new models of reduced computational complexity. We introduce the concept of consistency between a full order and a reduced order model, i.e., using the same numerical scheme for the full order and reduced order model. For inconsistency, one could use SKEW in the full order model and then EMAC for the reduced order model. We explore the repercussions of having inconsistency between these two models analytically and experimentally. To obtain a proper linear system from the Navier-Stokes equations, we must solve the nonlinear problem first. We will explore a method used to reduce iteration counts of nonlinear problems, known as Anderson acceleration. We will discuss how we implemented this using the finite element library deal.II \cite{dealII94}, measure the iteration counts and time, and compare against Newton and Picard iterations

    Downscaling Using CDAnet Under Observational and Model Noises: The Rayleigh-Benard Convection Paradigm

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    Efficient downscaling of large ensembles of coarse-scale information is crucial in several applications, such as oceanic and atmospheric modeling. The determining form map is a theoretical lifting function from the low-resolution solution trajectories of a dissipative dynamical system to their corresponding fine-scale counterparts. Recently, a physics-informed deep neural network ("CDAnet") was introduced, providing a surrogate of the determining form map for efficient downscaling. CDAnet was demonstrated to efficiently downscale noise-free coarse-scale data in a deterministic setting. Herein, the performance of well-trained CDAnet models is analyzed in a stochastic setting involving (i) observational noise, (ii) model noise, and (iii) a combination of observational and model noises. The analysis is performed employing the Rayleigh-Benard convection paradigm, under three training conditions, namely, training with perfect, noisy, or downscaled data. Furthermore, the effects of noises, Rayleigh number, and spatial and temporal resolutions of the input coarse-scale information on the downscaled fields are examined. The results suggest that the expected l2-error of CDAnet behaves quadratically in terms of the standard deviations of the observational and model noises. The results also suggest that CDAnet responds to uncertainties similar to the theorized and numerically-validated CDA behavior with an additional error overhead due to CDAnet being a surrogate model of the determining form map

    The generation, propagation, and mixing of oceanic lee waves

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    Lee waves are generated when oceanic flows interact with rough seafloor topography. They extract momentum and energy from the geostrophic flow, causing drag and enhancing turbulent mixing in the ocean interior when they break. Mixing across density surfaces (diapycnal mixing) driven by lee waves and other topographic interaction processes in the abyssal ocean plays an important role in upwelling the densest waters in the global ocean, thus sustaining the lower cell of the meridional overturning circulation. Lee waves are generated at spatial scales that are unresolved by global models, so their impact on the momentum and buoyancy budgets of the ocean through drag and diapycnal mixing must be parameterised. Linear theory is often used to estimate the generation rate of lee waves and to construct global maps of lee wave generation. However, this calculation and subsequent inferences of lee wave mixing rely on several restrictive assumptions. Furthermore, observations suggest that lee wave mixing in the deep ocean is significantly overestimated by this theory. In this thesis, we remove some common assumptions at each stage of the lee wave lifecycle to investigate the reasons for this discrepancy and to motivate and inform future climate model parameterisations. Firstly, we investigate the way that seafloor topography is represented in lee wave parameterisations, finding that typical spectral methods can lead to an overestimate of wave energy flux. Next, we make the case for considering lee waves as a full water column process by modelling the effect of vertically varying background flows and the ocean surface on lee wave propagation. Finally, we take a holistic view of topographic mixing in the abyssal ocean, finding that deep stratified water mass interfaces may modify the nature of the lee wave field, and themselves contribute to mixing and upwelling in the deep ocean through topographic interaction.Open Acces

    Circulation Statistics in Homogeneous and Isotropic Turbulence

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    This is the committee version of a Thesis presented to the PostGrad Program in Physics of the Physics Institute of the Federal University of Rio de Janeiro (UFRJ), as a necessary requirement for the title of Ph.D. in Science (Physics). The development of the Vortex Gas Model (VGM) introduces a novel statistical framework for describing the characteristics of velocity circulation. In this model, the underlying foundations rely on the statistical attributes of two fundamental constituents. The first is a GMC field that governs intermittent behavior and the second constituent is a Gaussian Free field responsible for the partial polarization of the vortices in the gas. The model is revisited in a more sophisticated language, where volume exclusion among vortices is addressed. These additions were subsequently validated through numerical simulations of turbulent Navier-Stokes equations. This revised approach harmonizes with the multifractal characteristics exhibited by circulation statistics, offering a compelling elucidation for the phenomenon of linearization of the statistical circulation moments, observed in recent numerical simulation. In the end, a field theoretical approach, known as Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) functional method is carried out in the context of circulation probability density function. This approach delves into the realm of extreme circulation events, often referred to as Instantons, through two distinct methodologies: The First investigates the linear solutions and, by a renormalization group argument a time-rescaling symmetry is discussed. Secondly, a numerical strategy is implemented to tackle the nonlinear instanton equations in the axisymmetric approximation. This approach addresses the typical topology exhibited by the velocity field associated with extreme circulation events.Comment: Ph.D. Thesis - preliminary versio

    Discovering Causal Relations and Equations from Data

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    Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws and principles that are invariant, robust and causal explanations of the world has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventional studies in the system under study. With the advent of big data and the use of data-driven methods, causal and equation discovery fields have grown and made progress in computer science, physics, statistics, philosophy, and many applied fields. All these domains are intertwined and can be used to discover causal relations, physical laws, and equations from observational data. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of Physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for observational causal and equation discovery, point out connections, and showcase a complete set of case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is being revolutionised with the efficient exploitation of observational data, modern machine learning algorithms and the interaction with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems.Comment: 137 page

    Análisis numérico de morfodinámica en perfiles de playa durante eventos episódicos con CFD

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    Beaches are highly valuable assets from an economical, social and environmental perspective, and understanding them is fundamental to develop effective strategies for coastal management. Morphodynamic processes have a fundamental role in driving the evolution of beaches, being responsible for the morphology changes associated to wave action. However, the ways in which these processes are generated are not sufficiently understood. This thesis contributes to expanding the knowledge on the interplay between hydrodynamics and morphology that bring about morphodynamic processes and what are their main drivers. To tackle this objective, a CFD numerical model for cross-shore morphodynamics is developed and validated, and results from it are used as a basis for a detailed analysis of the main cross-shore morphodynamic processes.Las playas son entornos de alto valor económico, social y ambiental. Comprender su funcionamiento es fundamental para desarrollar estrategias de gestión efectivas. Los llamados procesos morfodinámicos juegan un papel clave en su comportamiento, dado que son los responsables de los cambios morfológicos derivados de la acción del oleaje. Sin embargo, existe una falta de conocimiento acerca de cómo éstos se producen. Esta tesis contribuye a expandir el conocimiento en estos procesos, concretamente durante la ocurrencia de eventos episódicos, analizando las interacciones entre hidrodinámica y morfología que los origina, así como sus principales condicionantes. Para ello, se desarrolla y valida un modelo numérico CFD que permite reproducir los principales procesos morfodinámicos en perfiles de playa. Los resultados obtenidos de este modelo se emplean como base para un análisis detallado de procesos morfodinámicos en perfiles de playa
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