Dissertations / Theses on the topic 'Moving boundary'

The work presented in this thesis is concerned with the modelling of moving boundary problems, with particular reference to the solution of the problem of the release of a store from beneath an aircraft wing. Chapter 2 presents a two-dimensional unstructured mesh generation procedure for generating grids of three-noded triangular elements about any number of arbitrarily shaped geometries and within an arbitrarily shaped domain. The method combines a Quadtree point generation procedure with a Delaunay triangulation algorithm. The method is used to generate the grids for the moving boundary algorithm. Chapter 3, a moving boundary flow solution algorithm and the corresponding data control structure are presented. The flow solver uses the explicit timestepping procedure of Lohner et al. A multiple grid or grid embedding procedure is used to model the motion of a body relative to another or other stationary bodies. A minor grid encloses the moving body and is allowed to move under a prescribed motion over the grid enclosing the stationary bodies spanning the domain. A number of steady state problems are analysed and a simple store release case is examined. Chapter 4 presents an implicit finite element scheme for the solution of the flow problems using an unstructured computational grid. The algorithm is based upon the centred finite difference scheme of Lerat et al. The governing equations are solved using a Generalized Minimal Residual method, which is related to the Conjugate Gradient method. A number of steady state flow solutions are presented. Finally, the implicit algorithm is incorporated into the moving boundary data structure of Chapter 3 and results for the new scheme are presented.

Based on the no-slip boundary condition for walls at rest for the lattice Boltzmann Bathnagar-Gross-Krook method by J.C.G. Verschaeve [Phys. Rev. 80,036703 (2009)], a no-slip boundary condition for walls with a tangential movement is derived. Numerical tests verify that the present boundary condition is second order accurate and stable for relaxation frequencies close to two.

We investigate the development of interfacial instabilities and singularities that occur in solutions to Darcy-type moving boundary problems. We present a robust numerical scheme which can easily be adapted to a wide range of problems that, to date, have not yet been solved. Using this scheme, we provide insight into how perturbing the geometry of a Hele-Shaw cell can be used to control the development of interfacial patterns. Further, we consider how different physical effects influence the development of a singularity due to an air bubble contracting to a point or breaking up into multiple bubbles.

The concept of moving surface boundary-layer control, as applied to the Joukowsky and NACA airfoils, is investigated through a planned experimental program complemented by theoretical and flow visualization studies. The moving surface was provided by one or two rotating cylinders located at the leading edge, the trailing edge, or the top surface of the airfoil. Three carefully designed two-dimensional models, which provided a wide range of single and twin cylinder configurations, were tested at a subcritical Reynolds number (Re = 4.62 x 10⁴ or Re — 2.31 x 10⁵) in a laminar-flow tunnel over a range of angles of attack and cylinder rotational speeds. The test results suggest that the concept is indeed quite promising and can provide a substantial increase in lift and a delay in stall. The leading-edge rotating cylinder effectively extends the lift curve without substantially affecting its slope. When used in conjunction with a second cylinder on the upper surface, further improvements in the maximum lift and stall angle are possible. The maximum coefficient of lift realized was around 2.22, approximately 2.6 times that of the base airfoil. The maximum delay in stall was to around 45°. In general, the performance improves with an increase in the ratio of cylinder surface speed (Uc) to the free stream speed (U). However, the additional benefit derived progressively diminishes with an increase in Uc/U and becomes virtually negligible for Uc/U > 5. There appears to be an optimum location for the leading-edge-cylinder. Tests with the cylinder at the upper side of the leading edge gave quite promising results. Although the CLmax obtained was a little lower than the two-cylinder configuration (1.95 against 2.22), it offers a major advantage in terms of mechanical simplicity. Performance of the leading-edge-cylinder also depends on its geometry. A scooped configuration appears to improve performance at lower values of Uc/U (Uc/U ≤ 1). However, at higher rates of rotation the free stream is insensitive to the cylinder geometry and there is no particular advantage in using the scooped geometry. A rotating trailing-edge-cylinder affects the airfoil characteristics in a fundamentally different manner. In contrast to the leading-edge-cylinder, it acts as a flap by shifting the CL vs. α plots to the left thus increasing the lift coefficient at smaller angles of attack before stall. For example, at α = 4°, it changed the lift coefficient from 0.35 to 1.5, an increase of 330%. Thus in conjunction with the leading-edge- cylinder, it can provide significant improvements in lift over the entire range of small to moderately high angles of incidence (α ≤ 18°). On the theoretical side, to start with, the simple conformal transformation approach is used to obtain a closed form potential-flow solution for the leading-edge-cylinder configuration. Though highly approximate, the solution does predict correct trends and can be used at a relatively small angle of attack. This is followed by an extensive numerical study of the problem using: • the surface singularity approach including wall confinement and separated flow effects; • a finite-difference boundary-layer scheme to account for viscous corrections; and • an iteration procedure to construct an equivalent airfoil, in accordance with the local displacement thickness of the boundary layer, and to arrive at an estimate for the pressure distribution. Effect of the cylinder is considered either through the concept of slip velocity or a pair of counter-rotating vortices located below the leading edge. This significantly improves the correlation. However, discrepancies between experimental and numerical results do remain. Although the numerical model generally predicts CLmax with a reasonable accuracy, the stall estimate is often off because of an error in the slope of the lift curve. This is partly attributed to the spanwise flow at the model during the wind tunnel tests due to gaps in the tunnel floor and ceiling required for the connections to the externally located model support and cylinder drive motor. However, the main reason is the complex character of the unsteady flow with separation and reattachment, resulting in a bubble, which the present numerical procedure does not model adequately. It is expected that better modelling of the cylinder rotation with the slip velocity depending on a dissipation function, rotation, and angle of attack should considerably improve the situation. Finally, a flow visualization study substantiates, rather spectacularly, effectiveness of the moving surface boundary-layer control and qualitatively confirms complex character of the flow as predicted by the experimental data.
Applied Science, Faculty of
Mechanical Engineering, Department of
Graduate

Dans cette thèse, une nouvelle méthode de frontières immergées a été développée pour la simulation d'interaction fluide-structure, appelée la méthode de frontières immergées mobiles (en langage anglo-saxon: MIBM). L'objectif principal de cette nouvelle méthode est de déplacer arbitrairement les solides à géométrie complexe dans un fluide visqueux incompressible, sans remailler le domaine fluide. Cette nouvelle méthode a l'avantage d'imposer la condition de non-glissement à l'interface d'une manière exacte via une force sans introduire des constantes artificielles modélisant la structure rigide. Cet avantage conduit également à la satisfaction de la condition CFL avec un pas de temps plus grand. Pour un calcul précis de la force induite par les frontières mobiles, un système linéaire a été introduit et résolu par la méthode de gradient conjugué. La méthode proposée peut être intégrée facilement dans des solveurs résolvant les équations de Navier-Stokes. Dans ce travail la MIBM a été mise en œuvre en couplage avec un solveur fluide utilisant une méthode de projection adaptée pour obtenir des solutions d'ordre deux en temps et en espace. Le champ de pression a été obtenu par l'équation de Poisson qui a été résolue à l'aide de la méthode du gradient conjugué préconditionné par la méthode multi-grille. La combinaison de ces deux méthodes a permis un gain de temps considérable par rapport aux méthodes classiques de la résolution des systèmes linéaires. De plus le code de calcul développé a été parallélisé sur l'unité graphique GPU équipée de la bibliothèque CUDA pour aboutir à des hautes performances de calcul. Enfin, comme application de nos travaux sur la MIBM, nous avons étudié le couplage "fort" d'interaction fluide-structure (IFS). Pour ce type de couplage, un schéma implicite partitionné a été adopté dans lequel les conditions à l'interface sont satisfaites via un schéma de type "point fixe". Pour réduire le temps de calcul inhérent à cette application, un nouveau schéma de couplage a été proposé pour éviter la résolution de l'équation de Poisson durant les itérations du "point fixe". Cette nouvelle façon de résoudre les problèmes IFS a montré des performances prometteuses pour des systèmes en IFS complexe
In this thesis a novel non-body conforming mesh formulation is developed, called the moving immersed boundary method (MIBM), for the numerical simulation of fluid-structure interaction (FSI). The primary goal is to enable solids of complex shape to move arbitrarily in an incompressible viscous fluid, without fitting the solid boundary motion with dynamic meshes. This novel method enforces the no-slip boundary condition exactly at the fluid-solid interface with a boundary force, without introducing any artificial constants to the rigid body formulation. As a result, large time step can be used in current method. To determine the boundary force more efficiently in case of moving boundaries, an additional moving force equation is derived and the resulting system is solved by the conjugate gradient method. The proposed method is highly portable and can be integrated into any fluid solver as a plug-in. In the present thesis, the MIBM is implemented in the fluid solver based on the projection method. In order to obtain results of high accuracy, the rotational incremental pressure correction projection method is adopted, which is free of numerical boundary layer and is second order accurate. To accelerate the calculation of the pressure Poisson equation, the multi-grid method is employed as a preconditioner together with the conjugate gradient method as a solver. The code is further parallelized on the graphics processing unit (GPU) with the CUDA library to enjoy high performance computing. At last, the proposed MIBM is applied to the study of two-way FSI problem. For stability and modularity reasons, a partitioned implicit scheme is selected for this strongly coupled problem. The interface matching of fluid and solid variables is realized through a fixed point iteration. To reduce the computational cost, a novel efficient coupling scheme is proposed by removing the time-consuming pressure Poisson equation from this fixed point interaction. The proposed method has shown a promising performance in modeling complex FSI system

The primary objective of this thesis is to develop a general computational framework to perform large scale moving boundary problems in fluid mechanics. The interactions of moving entities with fluid flow are common to numerous engineering and biomedical applications. The novel computational platform developed comprises of a) an efficient fluid flow solver b) an accurate and easily implemented unified formulation to capture the interactions of the moving bodies with the flow and c) parallel execution capability to enable large scale computations. The above features are formulated and implemented in a computer code, ELAFINT3D. The current thesis demonstrates the accuracy, efficiency and robustness of this framework. The performance of ELAFINT3D on distributed memory systems is also presented. Finally, this framework is employed to simulate a series of large scale, three-dimensional moving boundary problems involving complex interfacial motions and flow phenomena. These numerical experiments establish the strengths of the current tool.

In this thesis, a simple moving boundary technique has been suggested, implemented and verified. The technique may be considered as a generalization of the well-known "ghost" cell approach for boundary condition implementation. According to the proposed idea, the moving body does not appear on the computational grid and is allowed to move over the grid. The impermeable wall boundary condition is enforced by assigning proper gasdynamic values at the grid nodes located inside the moving body close to its boundaries (ghost nodes). The reflection principle taking into account the velocity of the boundaries assigns values at the ghost nodes. The new method does not impose any particular restrictions on the geometry, deformation and law of motion of the moving body.
The developed technique is rather general and can be used with virtually any finite-volume or finite-difference scheme, since the modifications of the schemes themselves are not required. In the present study the proposed technique has been incorporated into a one-dimensional non-adaptive Euler code and a two-dimensional locally adaptive unstructured Euler code.
It is shown that the new approach is conservative with the order of approximation near the moving boundaries. To reduce the conservation error, it is beneficial to use the method in conjunction with local grid adaptation.
The technique is verified for a number of one and two dimensional test cases with analytical solutions. It is applied to the problem of supersonic inlet starting via variable geometry approach. At first, a classical starting technique of changing exit area by a moving wedge is numerically simulated. Then, the feasibility of some novel ideas such as a collapsing frontal body and "tractor-rocket" are explored.

Flow control is a key factor in optimizing the performance of any vehicle moving through fluids. Particularly, in aerodynamics there are many potential benefits for implementing synthetic jets to achieve aircraft designs with less moving parts, uper- maneuverability, and separation control for fuel economy. Piezoelectric synthetic jets are of special interest because of their lightweight and low power consumption. Numerous publications on such jets are available. Actuator properties and boundary conditions relevant to this particular application however are often overlooked. The focus of this project is to numerically model synthetic jets in quiescent air to study the influence of cavity geometry and boundary conditions of the piezoelectric diaphragm on jet velocity. Numerical simulation is performed for two synthetic jet cavities of different height and orifice diameter. The numerical modeling utilizes a turbulent RNG κ – ε model and a moving boundary condition with two oscillating deflection profiles, parabolic and logarithmic, applied to the diaphragm. The actuators modeled are typical Bimorph and Thunder piezoelectric actuators. The initial conditions for the actuators are obtained experimentally resulting in 0.396mm and 0.07mm respectively when driven with a sinusoidal wave input at 1524 V/m and 4064 V/m. Although the velocity boundary numerical model gave overall better results than the current moving-boundary numerical model, the moving-boundary model is more accurate since it better approximates the movement of the diaphragm. From an optimizing viewpoint the moving boundary is more suitable to attempt to optimize the design because displacement magnitude of the diaphragm can be measured directly from experiments. For the higher displacement Bimorph actuator, a logarithmic profile matches the experimental results, whereas the parabolic profile provided better results for the relatively small displacement Thunder actuator. It is thus hypothesized that both tested actuators, Bimorph and Thunder, oscillate according to the specified logarithmic and parabolic profile respectively. Cavity height was briefly investigated for the Bimorph actuator. Results show that cavity height did not make a difference in the centerline velocity for the numerical model. The model fails to consider the important effect of the dynamic coupling of the actuator displacement and the pressure that develops inside the cavity. The pressure values obtained are comparable to the theoretical blocking pressure for the Bimorph in the cavity. The results of this study show that jet formation and development has unique characteristics for each actuator and cavity configuration. The smaller orifice cavity configuration produced a faster, longer, thinner jet with larger vortices than the bigger orifice. During max expulsion, t = 0.25T, and max ingestion, t = 0.75T, a low-pressure area localized at the corners of the orifice, inlet and exit respectively, were observed. All cavity configurations passed all three known jet formation criterions that include, Lo/Do>1, Re > 50, and Re/S2 > 0.16.

This thesis presents a finite element procedure for the solution of the Shallow-Water Equations dealing with tide generated flows in estuaries and coastal areas incorporating a dynamic land-water interface. The flow algorithm employed is an implementation of the explicit Two-Step Taylor-Galerkin finite element method. The 2-dimensional, depth averaged flow domain is discretised into linear triangular 3-noded elements. A background mesh storing the bathymetric information of the domain is generated using a procedure based on the Delaunay Triangulation technique but with the nodes optimally positioned to take into account the contours of the bed. The moving boundary component is separate from the flow computation. They are coupled at the end of each time-step. Nodes on the land-water interface are moved according to the tidal level and bed slope. Elements and nodes are deleted or created along the boundary so as to ensure good quality triangular elements at all stages of the tide. To limit computational time in the moving boundary algorithm, only the affected areas are meshed. The frequency of movement is monitored so that an optimum balance between accurate presentation of the land-water boundary and fast computational needs is achieved. The presence of structures in the intertidal zone is also dealt with.

A two-dimensional (2-D), mathematical model is adopted to investigate the development of circulation patterns for compressible, laminar, and shear driven flow inside a rectangular cavity. The bottom of the cavity is free to move at a specified speed and the aspect ratio of the cavity is changed from 1.0 to 1.5. The vertical sides and the bottom of the cavity are assumed insulated. The cavity is filled with a compressible fluid with Prandtl number, Pr =1. The governing equations are solved numerically using the commercial Computational Fluid Dynamics (CFD) package ANSYS FLUENT 2015 and compared with the results for the primitive variables of the problem obtained using in house CFD code based on Coupled Modified Strongly Implicit Procedure (CMSIP). The simulations are carried out for the unsteady, lid driven cavity flow problem with moving boundary (bottom) for different Reynolds number, Mach numbers, bottom velocities and high initial pressure and temperature.

Two new solutions to the heat conduction equation, describing transient heating of an evaporating droplet, are suggested. Both solutions take into account the effect of the reduction of the droplet radius due to evaporation, assuming that this radius is a linear function of time. It has been pointed out that the new approach predicts lower droplet surface temperatures and slower evaporation rates compared with the traditional approach. New solutions to the heat conduction equation, describing transient heating of an evaporating droplet, are suggested, assuming that the time evolution of droplet radius Rd(t) is known. The results of calculations are compared with the results obtained using the previously suggested approach, when the droplet radius was assumed to be a linear function of time during individual time steps, for typical Diesel engine-like conditions. Both solutions predict the same results which indicates that both models are likely to be correct. Two new solutions to the equation, describing the diffusion of species during multi-component droplet evaporation, are suggested. The first solution is the explicit analytical solution to this equation while the second one reduces the solution of the differential species diffusion equation to the solution of the Volterra integral equation of the second kind. Both solutions take into account the effect of the reduction of the droplet radius due to evaporation, assuming that this radius is a linear function of time. The analytical solution has been incorporated into a zero dimensional CFD code and applied to the analysis of bi-component (50% ethanol- 50% acetone mixture) droplet evaporation at atmospheric pressure. The transient heat conduction equation, describing heating of a body immersed into gas with inhomogeneous temperature distribution, is solved analytically, assu- ming that, at a certain distance from the body, gas temperature remains constant. The solution is applied to modelling of body heating in conditions close to those observed in Diesel engines. In the long time limit, the distribution of temperature in the body and gas practically does not depend on the initial distribution of gas temperature.

A major drawback in the operation of mechanical heart valve prostheses is thrombus formation in the near valve region potentially due to the high shear stresses present in the leakage jet flows through small gaps between leaflets and the valve housing. Detailed flow analysis in this region during the valve closure phase is of interest in understanding the relationship between shear stress and platelet activation. An efficient Cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three-dimensional immersed solid bodies as well as fluid-fluid interfaces. The embedded boundaries are represented using Levelsets and treated in a sharp manner without the use of source terms to represent boundary effects. The resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second-order accuracy. When dealing with problems of disparate length scales encountered in many applications, it is necessary to resolve the physically important length scales adequately to ensure accuracy of the solution. Fixed grid methods often have the disadvantage of heavy mesh requirement for well resolved calculations. A quadtree based adaptive local mesh refinement scheme is developed to complement the sharp interface Cartesian grid method scheme for efficient and optimized calculations. Detailed timing and accuracy data is presented for a variety of benchmark problems involving moving boundaries. The above method is then applied to modeling heart valve closure and predicting thrombus formation. Leaflet motion is calculated dynamically based on the fluid forces acting on it employing a fluid-structure interaction algorithm. Platelets are modeled and tracked as point particles by a Lagrangian particle tracking method which incorporates the hemodynamic forces on the particles. Leaflet closure dynamics including rebound is analyzed and validated against previous studies. Vortex shedding and formation of recirculation regions are observed downstream of the valve, particularly in the gap between the valve and the housing. Particle exposure to high shear and entrapment in recirculation regions with high residence time in the vicinity of the valve are observed corresponding to regions prone to thrombus formation.

In atherosclerosis, the arterial lining undergoes a specific sequence of inflammatory responses to an injury to the cells that line the blood vessel and to low density lipoprotein (LDL) particles from the blood stream that penetrate through this injury into the arterial wall. We model the events that take place inside the blood vessel wall that occur immediately after such an injury with a system of partial differential equations that involve the LDL particles, two proinflammatory cytokines, monocyte-derived macrophages and their lipid-filled counterparts, foam cells. The model includes the chemical and physical interactions with the endothelial cells that line the arterial wall. These interactions are formulated as boundary conditions. Through numerical simulations, we show that different LDL concentrations in the blood stream and different immune responses can qualitatively affect the development of a plaque. Numerical bifurcation analysis at the quasi-steady state through AUTO shows that there exists of a fold bifurcation when the flux of LDL into the plaque from the blood is high. An atherosclerotic plaque that develops within the intima, deforms the intima locally as macrophages and foam cells accumulate. We model the structure of the developing plaque by cell pressure and cell sorting models to account for the limited space within the intima. We do this by modelling cell movement in crowded tissue in a discrete space and extend this to a spatial domain where cells also moves due to cell pressure and chemotaxis. We model the mechanics of the physical interactions on the two bounding interfaces, (the lumen-intima boundary and the intima-media boundary) and of the tissue inside the domain and add advective terms to ensure that the mechanics of the cellular species is consistent with the underlying tissue deformation. Using a finite element solver, we produce numerical results in one dimension across the intima and in two dimensions as a cross section of an artery. With appropriate parameter values, this moving boundary problem produces results in agreement with the current theory on compensatory enlargement in atherosclerotic remodelling

This master thesis is covering the modeling of the melting process in a tank filled with AdBlue. Due to AdBlue freezing at temperatures below -11 degree there is a need to add heat to be able to secure dosing in all situations. A rig for simulating freezing conditions is created with the possibility to store AdBlue in temperatures down to -40 degree. Temperatures are measured in and around the tank containing AdBlue and in the equipment used for adding heat. Two models are created from physical relations to estimate the mass of AdBlue melted, a static temperature model and a dynamic temperature model. The static model shows good results when calibrated at this specific setup and is very easy to use. The dynamic temperature model is more advanced but describes the real physical system better without external calibration.

Thesis (Ph. D.) -- University of Maryland, College Park, 2005.
Thesis research directed by: Mechanical Engineering. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

Many recent efforts in the field of microfluidics have been focused on reducing the size and the complexity of devices and on simplifying the methods of analysis performed with them. Gradient elution moving boundary electrophoresis (GEMBE) is a recently described counterflow electrophoresis method that was developed to simplify the analysis of ions in complex matrices. In this thesis, the improvement of the limit of detection of GEMBE and reduction of the GEMBE channel length is investigated.

Integration of simple and robust device components required for the successful adaptation of many analytical methods to multiplexed and field-portable devices often has negative effects on detection sensitivity, such as in the optical detection components in a capillary electrophoresis (CE) system. One of the simplest methods to improve sensitivity in the CE field is known as sample stacking. This method involves preparing the sample in a buffer with a different concentration (and conductivity) than that of the run buffer, such that when an electric field is applied the analyte concentration is increased at the boundary between the two different buffer concentrations. A method in which the sample is prepared in a buffer at a lower concentration than the run buffer has been implemented. This method achieves a significantly greater signal enhancement than expected for sample stacking. The concentration enhancement ability of this method is demonstrated utilizing GEMBE with channel current detection.

Current GEMBE device construction methods impose limitations on the minimum length of the separation channel. One technique well suited for minimizing the size of the GEMBE separation channel is multiphoton absorption polymerization (MAP). Because MAP is a non-linear optical fabrication method, polymerization is limited to a small region near the focal point of a laser beam. As a result, three-dimensional structures with small feature sizes can be easily created. The 3D capabilities of MAP have been exploited to create channels with circular cross sections and ∼300 &mgr;m lengths for GEMBE. The integration of device components fabricated with MAP and molded with PMDS allows visualization of the GEMBE separations, and provides insights into the effect of channel length on GEMBE step width.

Multitud de problemas en ciencia e ingeniería se plantean como ecuaciones en derivadas parciales (EDPs). Si la frontera del recinto donde esas ecuaciones han de satisfacerse se desconoce a priori, se habla de "Problemas de frontera libre", propios de sistemas estacionarios no dependientes del tiempo, o bien de "Problemas de frontera móvil", asociados a problemas de evolución temporal, donde la frontera cambia con el tiempo. La solución a dichos problemas viene dada por la expresión de la(s) variable(s) dependiente(s) de la(s) EDP(s) junto con la función que determina la posición de la frontera. Dado que este tipo de problemas carece en la mayoría de los casos de solución analítica conocida, se hace preciso recurrir a métodos numéricos que permitan obtener una solución lo suficientemente aproximada, y que además mantenga propiedades cualitativas de la solución del modelo continuo de EDP(s). En este trabajo se ha abordado el estudio numérico de algunos problemas de frontera móvil provenientes de diversas disciplinas. La metodología aplicada consta de dos pasos sucesivos: aplicación de la transformación de Landau o "Front-fixing transformation" al modelo en EDP(s) con el fin de mantener inmóvil la frontera del dominio, y posterior discretización a través de un esquema en diferencias finitas. De ahí se obtienen esquemas numéricos que se implementan por medio de la herramienta MATLAB. Mediante un exhaustivo análisis numérico, se estudian propiedades del esquema y de la solución numérica (positividad, estabilidad, consistencia, monotonía, etc.). En el primer capítulo de este trabajo se revisa el estado del arte del campo objeto de estudio, se justifica la necesidad de disponer de métodos numéricos adaptados a este tipo de problemas y se describe brevemente la metodología empleada en nuestro enfoque. El Capítulo 2 se dedica a un problema perteneciente a la Biología Matemática y que consiste en determinar la evolución de la población de una especie invasora que se propaga en un hábitat. Este modelo consiste en una ecuación de difusión-reacción unida a una condición tipo Stefan. Los resultados del análisis numérico confirman la existencia de una dicotomía propagación-extinción en la evolución a largo plazo de la densidad de población de la especie invasora. En particular, se ha podido precisar el valor del coeficiente de la condición de Stefan que separa el comportamiento de propagación del de extinción. Los Capítulos 3 y 4 se centran en un problema de Química del Hormigón con interés en Ingeniería Civil: el proceso de carbonatación del hormigón, fenómeno evolutivo que lleva consigo la degradación progresiva de la estructura afectada y finalmente su ruina, si no se toman medidas preventivas. En el Capítulo 3 se considera un sistema de dos EDPs de tipo parabólico con dos incógnitas. Para su resolución, hay que considerar además las condiciones iniciales, las de contorno y las de tipo Stefan en la frontera. Los resultados numéricos confirman la tendencia de la ley de evolución de la frontera móvil hacia una función del tipo "raíz cuadrada del tiempo". En el Capítulo 4 se considera un modelo más general que el anterior, en el que intervienen seis especies químicas que se encuentran tanto en la zona carbonatada como en la no carbonatada. En el Capítulo 5 se aborda un problema de transmisión de calor que aparece en diversos procesos industriales; en este caso, en el enfriamiento durante la colada de metal fundido, donde la fase sólida avanza y la líquida se va extinguiendo. La frontera móvil (frente de solidificación) separa ambas fases, siendo su posición en cada instante la variable a determinar, junto con las temperaturas en cada fase. Después de la adecuada transformación y discretización, se implementa un esquema en diferencias finitas, subdividiendo el proceso en tres estadios temporales, a fin de tratar las singularidades asociadas a posicione
Many problems in science and engineering are formulated as partial differential equations (PDEs). If the boundary of the domain where these equations are to be solved is not known a priori, we face "Free-boundary problems", which are characteristic of non-time dependent stationary systems; besides, we have "Moving-boundary problems" in temporal evolution processes, where the border changes over time. The solution to these problems is given by the expression of the dependent variable(s) of PDE(s), together with the function that determines the position of the boundary. Since the analytical solution of this type of problems is lacked in most cases, it is necessary to resort to numerical methods that allow an accurate enough solution to be obtained, and which also maintain the qualitative properties of the solution(s) of the continuous model. This work approaches the numerical study of some moving-boundary problems that arise in different disciplines. The applied methodology consists of two successive steps: firstly, the so-called Landau transformation, or "Front-fixing transformation", which is used in the PDE(s) model to maintain the boundary of the domain immobile; later, we proceed to its discretization with a finite difference scheme. Different numerical schemes are obtained and implemented through the MATLAB computational tool. Properties of the scheme and the numerical solution (positivity, stability, consistency, monotonicity, etc.) are studied by an exhaustive numerical analysis. The first chapter of this work reports the state of the art of the field under study, justifies the need to adapt numerical methods to this type of problem, and briefly describes the methodology used in our approach. Chapter 2 presents a problem in Mathematical Biology that consists in determining over time the evolution of an invasive species population that spreads in a habitat. This problem is modelled by a diffusion-reaction equation linked to a Stefan-type condition. The results of the numerical analysis confirm the existence of a spreading-vanishing dichotomy in the long-term evolution of the population density of the invasive species. In particular, it is possible to determine the value of the coefficient of the Stefan condition that separates the propagation behaviour from extinction. Chapters 3 and 4 focus on a problem of Concrete Chemistry with an interest in Civil Engineering: the carbonation of concrete, an evolutionary phenomenon that leads to the progressive degradation of the affected structure and its eventual ruin if preventive measures are not taken. Chapter 3 considers a system of two parabolic type PDEs with two unknowns. For its resolution, the initial and boundary conditions have to be considered together with the Stefan conditions on the carbonation front. The numerical analysis results agree with those obtained in a previous theoretical study. The dynamics of the concentrations and the moving boundary confirm the long-term behaviour of the evolution law for the moving boundary as a "square root of time". Chapter 4 considers a more general model than the previous one, which includes six chemical species, defined in both the carbonated and non-carbonated zones, whose concentrations have to be found. Chapter 5 addresses a heat transfer problem that appears in various industrial processes; in this case, the solidification of metals in casting processes, where the solid phase advances and liquid reduces until it is depleted. The moving boundary (the solidification front) separates both phases. Its position in each instant is the variable to be determined together with the temperature profiles in both phases. After suitable transformation, discretization is carried out to obtain a finite difference scheme to be implemented. The process was subdivided into three temporal stages to deal with the singularities associated with the moving boundary position in the initialisation and depletion stages.
Multitud de problemes en ciència i enginyeria es plantegen com a equacions en derivades parcials (EDPs). Si la frontera del recinte on eixes equacions han de satisfer-se es desconeix a priori, es parla de "Problemas de frontera lliure", propis de sistemes estacionaris no dependents del temps, o bé de "Problemas de frontera mòbil", associats a problemes d'evolució temporal, on la frontera canvia amb el temps. Atés que este tipus de problemes manca en la majoria dels casos de solució analítica coneguda, es fa precís recórrer a mètodes numèrics que permeten obtindre una solució prou aproximada a l'exacta, i que a més mantinga propietats qualitatives de la solució del model continu d'EDP(s). En aquest treball s'ha abordat l'estudi numèric d'alguns problemes de frontera mòbil provinents de diverses disciplines. La metodologia aplicada consta de dos passos successius: en primer lloc, s'aplica l'anomenada transformació de Landau o "Front-fixing transformation" al model en EDP(s) a fi de mantindre immòbil la frontera del domini; posteriorment, es procedix a la seva discretització a través d'un esquema en diferències finites. D'ací s'obtenen esquemes numèrics que s'implementen per mitjà de la ferramenta informàtica MATLAB. Per mitjà d'una exhaustiva anàlisi numèrica, s'estudien propietats de l'esquema i de la solució numèrica (positivitat, estabilitat, consistència, monotonia, etc.). En el primer capítol d'aquest treball es revisa l'estat de l'art del camp objecte d'estudi, es justifica la necessitat de disposar de mètodes numèrics adaptats a aquest tipus de problemes i es descriu breument la metodologia emprada en el nostre enfocament. El Capítol 2 es dedica a un problema pertanyent a la Biologia Matemàtica i que consistix a determinar l'evolució en el temps de la distribució de la població d'una espècie invasora que es propaga en un hàbitat. Este model consistix en una equació de difusió-reacció unida a una condició tipus Stefan, que relaciona les funcions solució i frontera mòbil a determinar. Els resultats de l'anàlisi numèrica confirmen l'existència d'una dicotomia propagació-extinció en l'evolució a llarg termini de la densitat de població de l'espècie invasora. En particular, s'ha pogut precisar el valor del coeficient de la condició de Stefan que separa el comportament de propagació del d'extinció. Els Capítols 3 i 4 se centren en un problema de Química del Formigó amb interés en Enginyeria Civil: el procés de carbonatació del formigó, fenomen evolutiu que comporta la degradació progressiva de l'estructura afectada i finalment la seua ruïna, si no es prenen mesures preventives. En el Capítol 3 es considera un sistema de dos EDPs de tipus parabòlic amb dos incògnites. Per a la seua resolució, cal considerar a més, les condicions inicials, les de contorn i les de tipus Stefan en la frontera. Els resultats de l'anàlisi numèrica s'ajusten als obtinguts en un estudi teòric previ. S'han dut a terme experiments numèrics, comprovant la tendència de la llei d'evolució de la frontera mòbil cap a una funció del tipus "arrel quadrada del temps". En el Capítol 4 es considera un model més general, en el que intervenen sis espècies químiques les concentracions de les quals cal trobar, i que es troben tant en la zona carbonatada com en la no carbonatada. En el Capítol 5 s'aborda un problema de transmissió de calor que apareix en diversos processos industrials; en aquest cas, en el refredament durant la bugada de metall fos, on la fase sòlida avança i la líquida es va extingint. La frontera mòbil (front de solidificació) separa ambdues fases, sent la seua posició en cada instant la variable a determinar, junt amb les temperatures en cada una de les dos fases. Després de l'adequada transformació i discretització, s'implementa un esquema en diferències finites, subdividint el procés en tres estadis temporals, per tal de tractar les singularitats asso
Piqueras García, MÁ. (2018). Numerical Methods for Multidisciplinary Free Boundary Problems: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/107948
TESIS

Epithelial tissues are known to deform in response to chemical signals and mechanical forces or due to trauma, such as tumours and wounds. In this thesis, a novel free boundary mathematical model of epithelial tissues with mechanobiological coupling is developed to study how the deformation of epithelial tissues impacts tumour growth and wound healing. Mechanobiological coupling is introduced in a discrete modelling framework and new reaction-diffusion equations are derived. Case studies involving the Rac-Rho pathway and activator-inhibitor patterning demonstrate that the reaction-diffusion equations accurately reflect the biological processes included in the discrete model.

In this study, an attempt is made to obtain convergent and stable solutions of the K-E turbulence model equations for non-reacting turbulent flows over an isothermal solid surface in regression. A physics based mathematical model is used to describe the flow and temperature field over the moving surface. The flow is assumed to be two-dimensional, unsteady, incompressible with boundary layer approximations. Parabolized form of the standard K-E equations is adopted to simulate turbulence in the flow. Regression of the solid surface causes the bounds of the solution domain to change with time, therefore a coordinate transformation is used in the vertical direction. The computational domain with fixed boundaries is discretized using an orthogonal grid system where a coordinate stretching is used in the vertical direction. A second order accurate, explicit finite difference technique is used for discretization of the governing equations. The final set of discretized equations is then solved using a solution algorithm specifically developed for this study. The verification of the solution algorithm includes a grid independence study, time increment study, and a comparison of the steady state results for the laminar and the turbulent flow cases. Finally, a parametric study is conducted using the proposed solution algorithm to test the stability of the numerical results for different Reynolds numbers, regression rates, and surface temperatures. It is concluded that the proposed numerical solution algorithm is capable of providing convergent and stable solutions of the two-equation turbulence model.

Using the Navier-Stokes equations along with a continuity equation, a one-dimensional model is developed to describe the spin coating process of thin polymeric films. The resulting model is a system of a parabolic partial differential equation coupled with an integral equation as well as with an ordinary differential equation describing the motion of a moving boundary. Viscosity and diffusivity are allowed to be varied in the model. To be able to perform the finite element approximation of the model equations, the moving boundary is fixed. Then the finite element method is applied along with the so called Method of Lines resulting in a semi-discrete problem, a large system of ordinary differential equations which is then solved with MATLAB. We present an existence and uniqueness result what concerns the semi-discrete solutions. Finally, we illustrate numerically the behavior of the solutions to our model.

This work deals with the one-dimensional Stefan problem with a general time- dependent boundary condition at the fixed boundary. The solution will be obtained using a discrete random walk method and the results will be compared qualitatively with analytical- and finite difference method solutions. A critical part has been to model the moving boundary with the random walk method. The results show that the random walk method is competitive in relation to the finite difference method and has its advantages in generality and low effort to implement. The finite difference method has, on the other hand, higher accuracy for the same computational time with the here chosen step lengths. For the random walk method to increase the accuracy, longer execution times are required, but since the method is generally easily adapted for parallel computing, it is possible to speed up. Regarding applications for the Stefan problem, there are a large range of examples such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for oil extraction using steam assisted gravity drainage.

A video sequence often contains a number of objects. For each object, the motion of its projection on the video frames is affected by its movement in 3-D space, as well as the movement of the camera. Video object segmentation refers to the task of delineating and distinguishing different objects that exist in a series of video frames. Segmentation of moving objects from a two-dimensional video is difficult due to the lack of depth information at the boundaries between different objects. As the motion incoherency of a region is intrinsically linked to the presence of such boundaries and vice versa, a failure to recognise a discontinuity in the motion field, or the use of an incorrect motion, often leads directly to errors in the segmentation result. In addition, many defects in a segmentation mask are also located in the vicinity of moving object boundaries, due to the unreliability of motion estimation in these regions. The approach to segmentation in this work comprises of three stages. In the first part, a phase-based method is devised for detection of moving object boundaries. This detection scheme is based on the characteristics of a phase-matched difference image, and is shown to be sensitive to even small disruptions to a coherent motion field. In the second part, a spatio-temporal approach for object segmentation is introduced, which involves a spatial segmentation in the detected boundary region, followed by a motion-based region-merging operation using three temporally adjacent video frames. In the third stage, a multiple-frame approach for stabilisation of object masks is introduced to alleviate the defects which may have existed earlier in a local segmentation, and to improve upon the temporal consistency of object boundaries in the segmentation masks along a sequence. The feasibility of the proposed work is demonstrated at each stage through examples carried out on a number of real video sequences. In the presence of another object motion, the phase-based boundary detection method is shown to be much more sensitive than direct measures such as sum-of-squared error on a motion-compensated difference image. The three-frame segmentation scheme also compares favourably with a recently proposed method initiated from a non-selective spatial segmentation. In addition, improvements in the quality of the object masks after the stabilisation stage are also observed both quantitatively and visually. The final segmentation result is then used in an experimental object-based video compression framework, which also shows improvements in efficiency over a contemporary video coding method.

Doctor of Philosophy
Department of Mechanical and Nuclear Engineering
Mingjun Wei
In this study, a global Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order model (ROM) is proposed. It is extended from usual fixed-domain problems to more general fluid-solid systems with moving boundaries/interfaces. The idea of the extension is similar to the immersed boundary method in numerical simulations which uses embedded forcing terms to represent boundary motions and domain changes. This immersed boundary method allows a globally defined fixed domain including both fluid and solid, where POD-Galerkin projection can be directly applied. However, such a modified approach cannot get away with the unsteadiness of boundary terms which appear as time-dependent coefficients in the new Galerkin model. These coefficients need to be pre-computed for prescribed periodic motion, or worse, to be computed at each time step for non-prescribed (e.g. with fluid-structure interaction) or non-periodic situations. Though computational time for each unsteady coefficient is smaller than the coefficients in a typical Galerkin model, because the associated integration is only in the close neighborhood of moving boundaries. The time cost is still much higher than a typical Galerkin model with constant coefficients. This extra expense for moving-boundary treatment eventually undermines the value of using ROMs. An aggressive approach is to decompose the moving boundary/domain to orthogonal modes and derive another low-order model with fixed coefficients for boundary motion. With this domain decomposition, an approach including two coupled low-order models both with fixed coefficients is proposed. Therefore, the new global ROM with decomposed approach is more efficient. Though the model with the domain decomposition is less accurate at the boundary, it is a fair trade-off for the benefit on saving computational cost. The study further shows, however, that the most time-consuming integration in both approaches, which come from the unsteady motion, has almost negligible impact on the overall dynamics. Dropping these time-consuming terms reduces the computation cost by at least one order while having no obvious effect on model accuracy. Based on this global POD-Galerkin based ROM with forcing term, an improved ROM which can handle the parametric variation of body motions in a certain range is also presented. This study shows that these forcing terms not only represent the moving of the boundary, but also decouple the moving parameters from the computation of model coefficients. The decoupling of control parameters provides the convenience to adapt the model for the prediction on states under variation of control parameters. An improved ROM including a shit mode seems promising in model adaptation for typical problems in a fixed domain. However, the benefit from adding a shit mode to model diminishes when the method is applied to moving-boundary problems. Instead, a combined model, which integrates data from a different set of parameters to generate the POD modes, provides a stable and accurate ROM in a certain range of parametric space for moving-boundary problems. By introducing more data from a different set of parameters, the error of the new model can be further reduced. This shows that the combined model can be trained by introducing more and more information. With the idea of the combined model, the improved global ROM with forcing terms shows impressive capability to predict problems with different unknown moving parameters, and can be used in future parametric control and optimization problems.

In meshfree methods, partial differential equations are solved on an unstructured cloud of points distributed throughout the computational domain. In collocated meshfree methods, the differential operators are directly approximated at each grid point based on a local cloud of neighboring points. The set of neighboring nodes used to construct the local approximation is determined using a variable search radius. The variable search radius establishes an implicit nodal connectivity and hence a mesh is not required. As a result, meshfree methods have the potential flexibility to handle problem sets where the computational grid may undergo large deformations as well as where the grid may need to undergo adaptive refinement. In this work we develop the sharp interface formulation of the immersed boundary method for collocated meshfree approximations. We use the framework to implement three meshfree methods: General Finite Differences (GFD), Smoothed Particle Hydrodynamics (SPH), and Moving Least Squares (MLS). We evaluate the numerical accuracy and convergence rate of these methods by solving the 2D Poisson equation. We demonstrate that GFD is computationally more efficient than MLS and show that its accuracy is superior to a popular corrected form of SPH and comparable to MLS. We then use GFD to solve several canonic steady state fluid flow problems on meshfree grids generated using uniform and variable radii Poisson disk algorithm.

La caractérisation et la simulation des écoulements internes au sein des moteurs àpropergol solide, en considérant des mécanismes physiques fortement couplés, constituentl’objectif principal de ce mémoire de thèse. Dans cette optique, la conjonction entrefluide/régression de surface/couplage fluide structure a imposé de déployer une stratégiepropre lors du développement de la modélisation numérique. En effet, le modèle intègre untraitement de frontière immergée couplé avec un suivi de frontière mobile afin de pouvoirrendre compte de la formidable variation géométrique interne subie au cours d’un tir. Côtéfluide, un maillage automatique est nécessaire et la gestion de ce dernier s’appuie sur undéveloppement récursif avec structure hiérarchisée de type 2n tree. Une attention particulièrea été portée sur le solveur lui-même avec une approche explicite en temps et un schémanumérique basé sur l’approche de Roe avec limiteur de flux au second ordre. Des cas testsont été réalisés afin de valider le solveur et les différentes conditions aux limites introduites,notamment des conditions spécifiques développées pour les besoins de simulation. Lespremiers résultats soulignent tout l’intérêt du modèle proposé et sauf erreur de notre part,pour la première fois, l’analyse des sources tourbillonnaires responsables des instabilités ausein de ces moteurs a été étudiée en intégrant les effets du changement continu de géométrie.Finalement, la faisabilité d’une interaction forte entre solveur fluide et solveur solide a étéréalisée sur un modèle simplifié d’un moteur segmenté.L’ensemble des développements permet un accès aux mécanismes couplés complexeset aux fortes interactions au sein des moteurs à propergol solide et offre de nouvellesperspectives dans la caractérisation des mécanismes fortement couplés
Characterization and simulation of internaI flow within the solid rocket motors, considering the physicalmechanisms strongly coupled, are the main focus of this thesis objective. In this context, the conjunctionbetween fluid/regression surface/fluid coupling structure imposed deploy c1ean during the development ofnumerical modeling strategy. Indeed, the model incorporates treatment coupled with an immersed boundarytracking moving boundary in order to realize the tremendous internai geometric variation experienced during ashot. Fluid side, an automatic mesh is required and the management of the latter is based on a recursivehierarchical structure development with type 2" tree. Particular attention was paid to the solver itself with anexplicit approach to time and a numerical scheme based on the approach of Roe with flow limiter in the secondorder. Tests cases were conducted to validate the sol ver and different boundary conditions introduced, inc1udingspecific conditions developed for the purpose of simulation. The first results emphasize the interest of theproposed and unless our error model, for the first time, the analysis of the sources responsible vortex instabilitiesin these engines has been studied by incorporating the effects of continuous change in geometry. Finally, thefeasibility of a strong interaction between fluid and solid solver was conducted on a simplified model of a multiengine.AlI the developments allows access to complex mechanisms coupled and strong interactions in solidrocket motors and off ers new insights into the characterization of strongly coupled mechanisms