The aim of this work is to find a stable scheme which would solve the Stokes problem of the fluid flow, in which an elastic structure is immersed. Unlike most of the schemes solving fluid-structure interaction problems, in our scheme meshes of fluid and structure do not have to coincide. We have restricted ourselves to two-dimensional domain occupied by fluid with one-dimensional im- mersed structure. To describe a fluid-structure interaction, we have used an Immersed boundary method. At first we consider the strucure to be massless. We have modified an existing scheme and made it unconditionally stable, which was mathematically proven and numerically tested. Then we have proposed a modification where the structure is not massless and also proved the uncondi- tional stability in this case. The proposed schemes were implemented using the Freefem++ software and tested on aneurysm-like geometry. We have tested the behavior of our scheme in case when the qrowing aneurysm touches an obstacle, for example a bone (with no-slip condition on the bone boundary). Powered by TCPDF (www.tcpdf.org)

Droplet evaporation (and condensation) is one of the most common instancesof multiphase flow with phase change, encountered in nature as well as intechnical and industrial applications. Examples include falling rain drops, fogsand mists, aerosol applications like electronic cigarettes and inhalation drugdelivery, engineering applications like spray combustion, spray wet scrubbing orgas absorption, spray drying, flame spray pyrolysis.Multiphase flow with phase change is a challenging topic due to the inter-twined physical phenomena that govern its dynamics. Numerical simulation isan outstanding tool that enables us to gain insight in the details of the physics,often in cases when experimental studies would be too expensive, impracticalor limited.In the present work we focus on simulation of the evaporation of smalldroplets. We perform simulation of evaporation of a pure and two−componentdroplet, that includes detailed thermodynamics and variable physical andtransport properties. Some of the conclusions drawn include the importance ofenthalpy transport by species diffusion in the thermal budget of the system, andthe identification and characterization of evaporating regimes for an azeotropicdroplet.In the second part we develop a method based on the immersed boundaryconcept for interface resolved numerical simulation of laminar and turbulentflows with a large number of spherical droplets that undergo evaporation orcondensation. / Droppförångning (och kondensation) är en av de vanligaste fallen av flerfasflöde med fasförändring, både i naturen och i tekniska och industriella tillämpningar. Exempel är fallande regndroppar, dimma, aerosol-tillämpningar som elektroniskacigaretter och läkemedelsleverans via inandning, tekniska tillämpningar som sprayförbränning, våtskrubbning med sprayning, gasabsorption, spraytorkning samt flammsprayspyrolys. Flerfasflöde med fasförändring är ett utmanande ämne på grund av de sammanflätade fysikaliska fenomen som styr dess dynamik. Numerisk simulering är ett utmärkt verktyg som gör det möjligt för oss att få insikt i detaljerna i fysiken, ofta i fall då experimentella studier skulle vara för dyra, opraktiska eller begränsade. I det nuvarande arbetet fokuserar vi på simulering av förångning av små droppar. Vi utför simulering av förångning av en ren och två−komponentdroppe, som inkluderar detaljerad termodynamik samt varierande fysikaliska och transportegenskaper. Några av de slutsatser som dras inbegriper betydelsen av entalpitransport genom diffusion av olika ämnen i systemets termiska budget samt identifieringen och karakterisering av förångningsregimer för en azeotropiskdroppe. I den andra delen utvecklar vi en metod baserad på det nedsänkta rand konceptet för gränssnittskompletterad numerisk simulering av laminära och turbulenta flöden med ett stort antal sfäriska droppar som genomgår förångning eller kondensering. / <p>QC 20171117</p>

<p>Mathematical and computational modeling allow for the rationalization of complex phenomenon observed in our reality. Through the careful selection of assumptions, the intractable task of simulating reality can be reduced to the simulation of a practical system whose behavior can be replicated. The development of computational models allow for the full comprehension of the defined system, and the model itself can be used to evaluate the results of thousands of simulate experiments to aid in the rational design process.</p> <p>Biomedical engineering is the application of engineering principles to the field of medicine and biology. This discipline is composed of numerous diverse subdisciplines that span from genetic engineering to biomechanics. Each of these subdisciplines is concerned with its own complex and seemingly chaotic systems, whose behavior is difficult to characterize. The development and application of computational modeling to rationalize these systems is often necessary in this field and will be the focus of this thesis.</p> <p>This thesis is centered on the development and application of mathematical and computational modeling in three diverse systems in biomedical engineering. First, computational modeling is employed to investigate the behavior of key proteins in the post-synapse centered around learning and memory. Second, computational modeling is utilized to characterize the drug release rate from implantable drug delivery depots, and produce a tool to aid in the tailoring of the release rate. Finally, computational modeling is utilized to understand the motion of particles through an inertial focusing microfluidics chip and optimize the size selective capture efficiency.</p> <p> </p>

Immersed boundary (IB) methods are attractive due to their ability to simulate flow over complex geometries on a simple Cartesian mesh. Unlike conformal grid formulation, the mesh does not need to conform to the shape and orientation of the boundary. This eliminates the need for complex mesh and/or re-meshing in simulations with moving/morphing boundaries, which can be cumbersome and computationally expensive. However, the imposition of boundary conditions in IB methods is not straightforward and numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In a nutshell, IB methods in the literature often suffer from numerical oscillations, implementation complexity, time-step restriction, burred interface, and lack of generality. This limits their ability to mimic conformal grid results and enforce Neumann boundary conditions. In addition, there is no generic IB capable of solving flow with multiple potentials, closely/loosely packed structures as well as IBs of infinitesimal thickness. This dissertation describes a novel 2$ ^{\text{nd}} $ order direct forcing immersed boundary method designed for simulation of two- and three-dimensional incompressible flow problems with complex immersed boundaries. In this formulation, each cell cut by the IB is reshaped to conform to the shape of the IB. IBs are modeled as a series of 2D planes in 3D space that connect seamlessly at the edges of the cut cells, in a way that mimics conformal grid. IBs are represented in a continuous and consistent fashion from one cell to another, thus eliminating spatial pressure oscillations originating from inconsistent description of the IB as well as the traditional stair-step problem, leading to a more accurate resolution of the boundary layer. Boundary conditions are enforced at the exact location of the IB devoid of interpolation, which guarantees sound simulations even on grids with high aspect ratio, and enables simulations of flow packed with multiple IBs in close proximity. Boundary conditions for each phase across the IB are enforced independently, yielding a unique capability to solve flows with zero-thickness IBs. Simulations of a large number of 2D and 3D test cases confirm the prowess of the devised immersed boundary method in solving flows over multiple loosely/closely-packed IBs; stationary, moving and highly morphing IBs; as well as IBs with zero-thickness. Extension of the proposed scheme to solve flow with multiple potentials is demonstrated by simulating transfer and transport of a passive scalar from an array of side-by-side and tandem cylinders in cross-flow. Aquatic vegetation represented by a colony of circular cylinders with low to high solid fraction is simulated to showcase the prowess of the current numerical technique in solving flow with closely packed structures. Aquatic vegetation studies are extended to a colony of flat plates with different orientations to show the capability of the developed method in modeling zero-thickness structures.

Flush-mounted cavity hot-wire probes have been around since two decades, but have typically not been applied as often compared to the traditional wall hot-wires mounted several wire diameters above the surface. While the latter suffer from heat conduction from the hot wire to the substrate in particular when used in air flows, the former is belived to significantly enhance the frequency response of the sensor. The recent work using a cavity hotwire by Gubian et al. (2019) came to the surprising conclusion that the magnitute of the fluctuating wall-shear stress τ+w,rms reaches an asymptotic value of 0.44 beyond the friction Reynolds number Re τ ∼ 600. In an effort to explain this result, which is at odds with the majority of the literature, the present work combines direct numerical simulations (DNS) of a turbulent channel flow with a cavity modelled using the immersed boundary method, as well as an experimental replication of the study of Gubian et al. in a turbulent boundary layer to explain how the contradicting results could have been obtained. It is shown that the measurements of the mentioned study can be replicated qualitatively as a result of measurement problems. We will present why cavity hot-wire probes should neither be used for quantitative nor qualitative measurements of wall-bounded flows, and that several experimental short-comings can interact to sometimes falsely yield seemingly correct results.

This dissertation presents a detailed account of the study undertaken on the subject of heat and mass transfer phenomena in porous media. The current work specifically targets the general reaction-diffusion systems arising in separation processes using porous sorbent particles. These particles are comprised of pore channels spanning length scales over almost three orders of magnitude while involving a variety of physical processes such as mass diffusion, heat transfer and surface adsorption-desorption. A novel methodology is proposed in this work that combines models that account for the multi-scale and multi-physics phenomena involved. Pore-resolving DNS calculations using an immersed boundary method (IBM) framework are used to simulate the macro-scale physics while the phenomena at smaller scales are modeled using a sub-pore modeling technique. The IBM scheme developed as part of this work is applicable to complex geometries on curvilinear grids, while also being very efficient, consuming less than 1% of the total simulation time per time-step. A new method of implementing the conjugate heat transfer (CHT) boundary condition is proposed which is a direct extension of the method used for other boundary conditions and does not involve any complex interpolations like previous CHT implementations using IBM. Detailed code verification and validation studies are carried out to demonstrate the accuracy of the developed method. The developed IBM scheme is used in conjunction with a stochastic reconstruction procedure based on simulated annealing. The developed framework is tested in a two-dimensional channel with two types of porous sections - one created using a random assembly of square blocks and another using the stochastic reconstruction procedure. Numerous simulations are performed to demonstrate the capability of the developed framework. The computed pressure drops across the porous section are compared with predictions from the Darcy-Forchheimer equation for media composed of different structure sizes. The developed methodology is also applied to CO2 diffusion studies in porous spherical particles of varying porosities. For the pore channels that are unresolved by the IBM framework, a sub-pore modeling methodology developed as part of this work which solves a one-dimensional unsteady diffusion equation in a hierarchy of scales represented by a fractal-type geometry. The model includes surface adsorption-desorption, and heat generation and absorption. It is established that the current framework is useful and necessary for reaction-diffusion problems in which the adsorption time scales are very small (diffusion-limited) or comparable to the diffusion time scales. Lastly, parametric studies are conducted for a set of diffusion-limited problems to showcase the powerful capability of the developed methodology. / Ph. D.

This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111]. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the time-harmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. {In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical time-dependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fully-discrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models time-Harmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.

O escoamento em torno de corpos tem sido objeto de estudo de muitos pesquisadores e é muito explorado experimental e computacionalmente, devido a sua grande aplicabilidade na engenharia. No entanto, simular computacionalmente este tipo de escoamento requer uma atenção especial ao escolher o tipo malha a ser utilizado. Em muitos casos faz-se necessário o uso de uma malha que se adapte ao contorno do obstáculo, o que pode ocasionar um aumento no esforço computacional. Um maneira de contornar este problema é a utilização do Método das Fronteiras Imersas, que possibilita o uso de malha cartesiana na simulação computacional do escoamento em torno de obstáculos. Isso é possível através da adição de um termo forçante nas equações que modelam o escoamento, e assim as forças que agem sobre o contorno do corpo são transferidas diretamente para a malha. O objetivo deste trabalho de mestrado foi implementar o método das Fronteiras Imersas e simular o escoamento em torno de um cilindro circular em repouso, movimentando-se na mesma direção do escoamento, na direção perpendicular ao escoamento, ou rotacionando em torno do próprio eixo. As simulações computacionais possibilitaram a captura do fenômeno de Atrelagem Síncrona, caracterizado pela sincronia entre a frequência de desprendimento natural de vórtices e a frequência de oscilação do mesmo. O Método das Fronteiras Imersas mostrou um ótimo desempenho quando comparado a resultados experimentais e numéricos encontrados na literatura / The flow around bodies have been studied by many researchers. Both experimental and computational approaches have been extensively explored in researches on flow around bodies and have been applied in many engeneering problems. However, to choose an appropriate type of mesh to perform computational simulations of this type of problem requires special attention. In many cases, it is necessary to use a mesh that is able to conform to the boundary if a given obstacle. The need to perform this adaptation may increase the computational effort. The Immersed Boundary Method enables the use of cartesian meshes to perform computational simulations of flows around obstacles. The idea of this method is to add a forcing term in the equations that model the flow. Thus, the forces applied on the body boundaries are directly transfered to the mesh. The aim of this work was to perform a computational implementation of the Immersed Boundary Method to simulate the flow over a oscilating circular cylinder. This oscilation may be inline with the flow, cross-flow, or rotating. The computational simulations enabled the capture of the lock-in phenomena, which consists of the syncronization between the vortex shedding frequency and the cylinder oscilation frequency. The results obtained from the computational simulations using the Immersed Boundary Method were in good agreement with the numerical and experimental results found in the literature

Este trabalho pode ser dividido em três etapas principais. Inicialmente é proposta uma formulação estabilizada do método dos elementos finitos (MEF) para solução de problemas de escoamento incompressível governado pela equação de Navier-Stokes. Esta formulação foi implementada em um código computacional e testada através de diversos exemplos numéricos. Alguns elementos finitos com diferentes pares de função de interpolação da velocidade e pressão, consagrados na literatura, e também elementos finitos menos populares, foram investigados e seus resultados e performance comparados. A segunda etapa consiste na formulação do problema estrutural. Buscou-se por uma formulação dinâmica, não linear, capaz de simular movimentos complexos de estruturas sujeitas a grandes deslocamentos e grandes deformações durante longos intervalos de tempo. A etapa final deste trabalho é a proposição de um método para solução de problemas de Interação Fluido Estrutura (IFE) que utiliza o conceito de fronteiras imersas como alternativa a abordagens ALE (Arbitrary Lagrangian Eulerian) clássicas. Elementos Finitos Generalizados, juntamente com Multiplicadores de Lagrange, são utilizados para prover descontinuidade nos campos de velocidade e pressão do fluido ao longo da interface com a estrutura. O acoplamento dos dois problemas é realizado utilizando um método implícito e alternado (staggered scheme), que possui a vantagem de permitir, facilmente, a implementação de códigos computacionais desenvolvidos para resolver isoladamente o problema fluido e/ou estrutural. / This work is divided in three parts. Initially, it is presented a stabilized Finite Element Method formulation to solve fluid flow problems governed by the incompressible Navier-Stokes Equations. This formulation was implemented in a computer code and validated throughout several numeric simulations. Some well-known finite elements with different pairs of velocity/pressure approximations, as well as some other less popular elements, were investigated and their performance compared. The second part describes the Structural Problem formulation. This formulation is able to simulate nonlinear dynamic problems involving large displacements and finite strains during long period of time. In the final part of this work, it is proposed a Fluid-Structure Interaction method based on an immersed interface approach in opposition to classical ALE (Arbitrary Lagrangian Eulerian) approaches. Generalized Finite Elements, together with Lagrange Multipliers, are used to provide velocity and pressure discontinuities on the fluid domain across the immersed interface. To couple both fluid and structural problems, an implicit staggered scheme is adopted, which allows the easy implementation of already developed black box computer codes.

No desenvolvimento de códigos paralelos, a biblioteca PETSc se destaca como uma ferramenta prática e útil. Com o uso desta ferramenta, este trabalho apresenta um estudo sobre resolvedores de sistemas lineares aplicados a escoamentos incompressíveis de fluidos em microescala, além de uma análise de seu comportamento em paralelo. Após um estudo dos diversos aspectos dos métodos de fronteira imersa, é apresentado um método de fronteira imersa paralelo de primeira ordem. Na sequência, é apresentada uma proposta de melhoria na precisão do método, baseada na minimização da distância entre a condição de contorno exata e aproximada, no sentido de mínimos quadrados. O desenvolvimento de uma ferramenta paralela eficiente é demonstrado na solução numérica de problemas envolvendo escoamentos incompressíveis de fluidos viscosos com fronteiras imersas / In the development of parallel codes, PETSc library has an important position as a practical and useful tool. With this tool, this work presents a study about linear system solvers applied to incompressible flow in microscale problems, furthermore an analysis of the parallel behavior of these methods is presented. After a study of several aspects of immersed boundary methods, and taking advantage of the flexibility of PETSc, a parallel first order immersed boundary method is presented. Thereafter, an improvement in the accuracy of the method is presented, based on the minimization of the distance between exact and approximated boundary conditions, in the least square sense. The development of a parallel and efficient tool is demonstrated in the numerical solution of incompressible viscous flow problems with immersed boundary