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Dissertations / Theses on the topic 'Partial differential equations, finite element method, Oseen equations'
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1
Höhne, Katharina. "Analysis and numerics of the singularly perturbed Oseen equations." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-188322.
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Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term.
In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square.
The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.
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Wärnegård, Johan. "A Cut Finite Element Method for Partial Differential Equations on Evolving Surfaces." Thesis, KTH, Numerisk analys, NA, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-190802.
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This thesis deals with cut finite element methods (CutFEM) for solving partial differential equations (PDEs) on evolving interfaces. Such PDEs arise for example in the study of insoluble surfactants in multiphase flow. In CutFEM, the interface is embedded in a larger mesh which need not respect the geometry of the interface. For example, the mesh of a two dimensional space containing a curve, may be used in order to solve a PDE on the curve. Consequently, in time-dependent problems, a fixed background mesh, in which the time-dependent domain is embedded, may be used. The cut finite element method requires a representation of the interface. Previous work on CutFEM has mostly been done using linear segments to represent the interfaces. Due to the linear interface representation the proposed methods have been of, at most, second order. Higher order methods require better than linear interface representation. In this thesis, a second order CutFEM is implemented using an explicit spline representation of the interface and the convection-diffusion equation for surfactant transport along a deforming interface is solved on a curve subject to a given velocity field. The markers, used to explicitly represent the interface, may due to the velocity field spread out alternately cluster. This may cause the interface representation to worsen. A method for keeping the interface markers evenly spread, proposed by Hou et al., is numerically investigated in the case of convection-diffusion. The method, as implemented, is shown to not be useful.Denna masteruppsats behandlar cut finite element methods (CutFEM) för att lösa partiella differentialekvationer (PDEs) på dynamiska gränsytor. Sådana ekvationer uppstår exempelvis i studiet av olösliga surfaktanter i flerfasflöde. I CutFEM innesluts gränsytan av ett större nät som ej behöver anpassas efter gränsytans geometri. Exempelvis kan ett tvådimensionellt nät användas för att lösa en PDE på en kurva som innesluts av nätet. Följaktligen kan ett fixt nät användas i tidberoende problem. CutFEM kräver en representation av gränsytan. I tidigare arbete har linjära segment använts för att representera gränsytan. På grund av den linjära representation av gränsytan har föreslagna metoder varit av högst andra ordningen. För att gå till högre ordningens metoder krävs en bättre representation av gränsytan. I denna uppsats implementeras CutFEM tillsammans med en explicit splinerepresentation av gränsytan för att lösa konvektions- och diffusionsekvationen för transport av surfaktanter längsmed en rörlig kurva. Metoden är av andra ordningens noggrannhet. Markörerna som används för att explicit representera ytan kan, på grund av hastighetsfältet, ömsom ansamlas ömsom spridas ut. Därvid kan approximationen av gränsytan försämras. En metod för att behålla markörerna jämt utspridda, framförd av Hou et al., undersöks numeriskt. Som implementerad i denna uppsats döms metoden ej vara användbar.
3
Prinja, Gaurav Kant. "Adaptive solvers for elliptic and parabolic partial differential equations." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/adaptive-solvers-for-elliptic-and-parabolic-partial-differential-equations(f0894eb2-9e06-41ff-82fd-a7bde36c816c).html.
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In this thesis our primary interest is in developing adaptive solution methods for parabolic and elliptic partial differential equations. The convection-diffusion equation is used as a representative test problem. Investigations are made into adaptive temporal solvers implementing only a few changes to existing software. This includes a comparison of commercial code against some more academic releases. A novel way to select step sizes for an adaptive BDF2 code is introduced. A chapter is included introducing some functional analysis that is required to understand aspects of the finite element method and error estimation. Two error estimators are derived and proofs of their error bounds are covered. A new finite element package is written, implementing a rather interesting error estimator in one dimension to drive a rather standard refinement/coarsening type of adaptivity. This is compared to a commercially available partial differential equation solver and an investigation into the properties of the two inspires the development of a new method designed to very quickly and directly equidistribute the errors between elements. This new method is not really a refinement technique but doesn't quite fit the traditional description of a moving mesh either. We show that this method is far more effective at equidistribution of errors than a simple moving mesh method and the original simple adaptive method. A simple extension of the new method is proposed that would be a mesh reconstruction method. Finally the new code is extended to solve steady-state problems in two dimensions. The mesh refinement method from one dimension does not offer a simple extension, so the error estimator is used to supply an impression of the local topology of the error on each element. This in turn allows us to develop a new anisotropic refinement algorithm, which is more in tune with the nature of the error on the parent element. Whilst the benefits observed in one dimension are not directly transferred into the two-dimensional case, the obtained meshes seem to better capture the topology of the solution.
4
Andrš, David. "Adaptive hp-FEM for elliptic problems in 3D on irregular meshes." To access this resource online via ProQuest Dissertations and Theses @ UTEP, 2008. http://0-proquest.umi.com.lib.utep.edu/login?COPT=REJTPTU0YmImSU5UPTAmVkVSPTI=&clientId=2515.
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Wells, B. V. "A moving mesh finite element method for the numerical solution of partial differential equations and systems." Thesis, University of Reading, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.414567.
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Massey, Thomas Christopher. "A Flexible Galerkin Finite Element Method with an A Posteriori Discontinuous Finite Element Error Estimation for Hyperbolic Problems." Diss., Virginia Tech, 2002. http://hdl.handle.net/10919/28245.
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A Flexible Galerkin Finite Element Method (FGM) is a hybrid class of finite element methods that combine the usual continuous Galerkin method with the now popular discontinuous Galerkin method (DGM). A detailed description of the formulation of the FGM on a hyperbolic partial differential equation, as well as the data structures used in the FGM algorithm is presented. Some hp-convergence results and computational cost are included. Additionally, an a posteriori error estimate for the DGM applied to a two-dimensional hyperbolic partial differential equation is constructed. Several examples, both linear and nonlinear, indicating the effectiveness of the error estimate are included.Ph. D.
7
Qiao, Zhonghua. "Numerical solution for nonlinear Poisson-Boltzmann equations and numerical simulations for spike dynamics." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/727.
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Haque, Md Z. "An adaptive finite element method for systems of second-order hyperbolic partial differential equations in one space dimension." Ann Arbor, Mich. : ProQuest, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3316356.
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Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U.Title from PDF title page (viewed Mar. 16, 2009). Source: Dissertation Abstracts International, Volume: 69-08, Section: B Adviser: Peter K. Moore. Includes bibliographical references.
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Kay, David. "The p- and hp- finite element method applied to a class of non-linear elliptic partial differential equations." Thesis, University of Leicester, 1997. http://hdl.handle.net/2381/30510.
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The analysis of the p- and hp-versions of the finite element methods has been studied in much detail for the Hilbert spaces W1,2 (omega). The following work extends the previous approximation theory to that of general Sobolev spaces W1,q(Q), q 1, oo . This extension is essential when considering the use of the p and hp methods to the non-linear a-Laplacian problem. Firstly, approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces W1,q(Q) are given. This analysis shows that the traditional view of avoiding the use of high order polynomial finite element methods is incorrect, and that the rate of convergence of the p version is always at least that of the h version (measured in terms of number of degrees of freedom). It is also shown that, if the solution has certain types of singularity, the rate of convergence of the p version is twice that of the h version. Numerical results are given, confirming the results given by the approximation theory. The p-version approximation theory is then used to obtain the hp approximation theory. The results obtained allow both non-uniform p refinements to be used, and the h refinements only have to be locally quasiuniform. It is then shown that even when the solution has singularities, exponential rates of convergence can be achieved when using the /ip-version, which would not be possible for the h- and p-versions.
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Salvatierra, Marcos Marreiro. "Modelagem matematica e simulação computacional da presença de materiais impactantes toxicos em casos de dinamica populacional com competição inter e intra-especifica." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307288.
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Orientador: João Frederico da Costa Azevedo MeyerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: A proposta deste trabalho é criar um modelo para descrever computacionalmente o convívio entre duas espécies competidoras com características de migração na presença de um material impactante tóxico. As equações a serem utilizadas deverão incluir os fenômenos de dispersão populacional, processos migratórios, dinâmicas populacionais densidade-dependentes e efeitos tóxicos de um material impactante evoluindo no meio, provocando um decaimento proporcional. Recorrendo a um instrumental consagrado, embora com desenvolvimento relativamente recente, será usado um sistema clássico do tipo Lotka-Volterra (conseqüentemente não-linear) combinado a Equações Diferenciais Parciais de Dispersão-Migração. O primeiro passo é a formulação variacional discretizada deste sistema visando o uso de Elementos Finitos combinados a um método de Crank-Nicolson. Em segundo lugar, virá a formulação de um algoritmo (conjuntamente com sua programação em ambiente MATLAB) que aproxima as soluções discretas relativas a cada população em cada ponto e ao longo do intervalo de tempo considerado nas simulações. Por fim, serão obtidas saídas gráficas úteis dos pontos de vista quantitativo e qualitativo para uso em conjunto com especialistas de áreas de ecologia e meio ambiente na avaliação e na calibração de modelos e programas, bem como no estudo de estratégias de preservação, impacto e recuperação de ambientes
Abstract: The purpose of this work is to create a model to computationally describe the coexistence of two competing species with migration features in the presence of a toxic impactant material. The equations must include the phenomena of populational dispersion, migratory processes, density-dependent populational dynamics and toxic effects of the evolutive presence of an impactant material developing in the environment, generating a proportional decrease in both populations. Resorting to well-established, although relatively recent, mathematical instruments a Lotka - Volterra type (and consequently nonlinear) system, including characteristics of a Migration-Dispersion PDE. The first step is the discrete variational formulation of this system aiming for the use of the Finite Element Method toghether with a Crank-Nicolson Method. Second, the formulation of an algorithm (together with a programme in MATLAB environment) that approximates the relative discrete solutions to each population in each point and along of the time interval considered in the simulations. Lastly, useful graphics will be obtained of the quantitative and qualitative viewpoints for use with specialists of the fields of ecology and environment and in the evaluation and calibration of models and programmes
Mestrado
Mestre em Matemática Aplicada
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Stewart, Dawn L. "Numerical Methods for Accurate Computation of Design Sensitivities." Diss., Virginia Tech, 1998. http://hdl.handle.net/10919/30561.
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This work is concerned with the development of computational methods for
approximating sensitivities of solutions to boundary value problems.
We focus on the continuous sensitivity equation method and
investigate the application of adaptive meshing and smoothing
projection techniques to enhance the basic scheme. The fundamental
ideas are first developed for a one dimensional problem and then
extended to 2-D flow problems governed by the incompressible
Navier-Stokes equations. Numerical experiments are conducted to test the
algorithms and to investigate the benefits of adaptivity and smoothing.Ph. D.
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Arthurs, Christopher J. "Efficient simulation of cardiac electrical propagation using adaptive high-order finite elements." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:ad31f06f-c4ed-4c48-b978-1ef3b12fe7a1.
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This thesis investigates the high-order hierarchical finite element method, also known as the finite element p-version, as a computationally-efficient technique for generating numerical solutions to the cardiac monodomain equation. We first present it as a uniform-order method, and through an a priori error bound we explain why the associated cardiac cell model must be thought of as a PDE and approximated to high-order in order to obtain the accuracy that the p-version is capable of. We perform simulations demonstrating that the achieved error agrees very well with the a priori error bound. Further, in terms of solution accuracy for time taken to solve the linear system that arises in the finite element discretisation, it is more efficient that the state-of-the-art piecewise linear finite element method. We show that piecewise linear FEM actually introduces quite significant amounts of error into the numerical approximations, particularly in the direction perpendicular to the cardiac fibres with physiological conductivity values, and that without resorting to extremely fine meshes with elements considerably smaller than 70 micrometres, we can not use it to obtain high-accuracy solutions. In contrast, the p-version can produce extremely high accuracy solutions on meshes with elements around 300 micrometres in diameter with these conductivities. Noting that most of the numerical error is due to under-resolving the wave-front in the transmembrane potential, we also construct an adaptive high-order scheme which controls the error locally in each element by adjusting the finite element polynomial basis degree using an analytically-derived a posteriori error estimation procedure. This naturally tracks the location of the wave-front, concentrating computational effort where it is needed most and increasing computational efficiency. The scheme can be controlled by a user-defined error tolerance parameter, which sets the target error within each element as a proportion of the local magnitude of the solution as measured in the H^1 norm. This numerical scheme is tested on a variety of problems in one, two and three dimensions, and is shown to provide excellent error control properties and to be likely capable of boosting efficiency in cardiac simulation by an order of magnitude. The thesis amounts to a proof-of-concept of the increased efficiency in solving the linear system using adaptive high-order finite elements when performing single-thread cardiac simulation, and indicates that the performance of the method should be investigated in parallel, where it can also be expected to provide considerable improvement. In general, the selection of a suitable preconditioner is key to ensuring efficiency; we make use of a variety of different possibilities, including one which can be expected to scale very well in parallel, meaning that this is an excellent candidate method for increasing the efficiency of cardiac simulation using high-performance computing facilities.
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Shepherd, David. "Numerical methods for dynamic micromagnetics." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/numerical-methods-for-dynamic-micromagnetics(e8c5549b-7cf7-44af-8191-5244a491d690).html.
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Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.
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Bringmann, Philipp. "Adaptive least-squares finite element method with optimal convergence rates." Doctoral thesis, Humboldt-Universität zu Berlin, 2021. http://dx.doi.org/10.18452/22350.
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Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert.
Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten.The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.
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Inforzato, Nelson Fernando. "Dispersão de poluentes num sistema ar-agua : modelagem matematica, aproximação numerica e simulação computacional." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307287.
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Orientador: João Frederico da Costa Azevedo MeyerTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Este trabalho estuda um problema evolutivo de difusão-advecção num sistema ar-água tri-dimensional. Apresenta-se um modelo e o correspondente sistema de EDPs que reúne as equações clássicas de Difusão-advecçãojreação em 3D e a equação de Stokes, junto com condições de contorno descritivas da dinâmica do poluente também na interface entre ar e água. Verificam-se existência e unicidade de solução na formulação variacional. São apresentadas discretizações espacial (elementos finitos de segunda ordem com SUPG) e temporal (CrankNicolson). São obtidas estimativas de erro a priori para Galerkin contínuo e GalerkinjCrank-Nicolson. Apresenta-se um programa computacional para simulações de diferentes cenários com resultados numéricos e saída gráfica para visualização de caráter qualitativo. Evidencia-se, assim, o potencial deste trabalho como suporte robusto na avaliação de estratégias de descarte de poluentes
Abstract: This work considers a three-dimensional air-water system pollution discharge problem, modelling it with a system of partial differential equations which includes both the diffusionadvection evolutionary and Stoke's equations. Appropriate boundary conditions are considered, including for the air-water interface, and special attention is dedicated to existence and uniqueness results. ln terms of the numerical approximation, space discretization is undertaken with three-dimensional second-order finite elements, and, in time, a Crank-Nicolson scheme is adopted. A priori estimates are given for the continuous Galerkin and for the GalerkinjCrankNicolson approximations. A numerical algorithm is presented and the qualitative visual output is used to emphasize the potential for simulating and discussing pollution discharge strategies
Doutorado
Doutor em Matemática Aplicada
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Wolmuth, Leidy Diane. "Modelagem e simulações do comportamento evolutivo de poluentes em corpos aquaticos de grande extensão : o caso da represa do rio Manso." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307277.
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Orientador: João Frederico da Costa Azevedo MeyerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Este trabalho modela o comportamento evolutivo de poluentes em corpos aquáticos de grande extensão. O tratamento é bidimensional na variável espacial, que descreve a superfície de um lago ou represa, por exemplo, e a concentração de poluentes é considerada em cada ponto, e a cada instante. O modelo apóia-se no uso de uma Equação Diferencial Parcial de Difusão-advecção com certas características especiais nas condições de contorno expressas, de modo genérico, numa expressão dita de Robin. Dada a geometria (em geral obtida via mapas ou imagens aéreas), a solução da EDP resultante desta modelagem é usada não em sua formulação clássica, mas num ambiente variacional (ou fraco), e é aproximada por métodos de comprovada confiabilidade, o método de Elementos Finitos via Galerkin para o espaço e o método de Crank-Nicolson para o tempo (ambos métodos de segunda ordem de aproximação). Um algoritmo para simulações computacionais é apresentado num ambiente Matlab, bem como alguns resultados numéricos usados para a produção de saídas gráficas qualitativamente adequadas para se avaliarem cenários possíveis de impacto.
Abstract: This work presents a model for the evolutive description of the movements of pollutant spills in large water bodies. It uses a bidimensional spatial approach in modelling a lake or a reservoir, for example and considers pollutant concentration on each planar point and at each moment in time. For this, the Diffusive-advective Partial Differential Equation is adopted with special border conditions, expressed, generically in the Robin formulation. Due to the geometry of the studied domain (in general obtained from aerial imagery), the solution of the resulting PDE is not used in ists strong formulation but rather in the weak or variational one, and it is approximated with reliable methods: Galerkin Method through the Finite element option for space, as well as Crank-Nicolson for approximation in time (both of these methods are of second order). An algorithm for this numerical scheme is presented in a Matlab environment and numerical results are graphically presented in order to enable a qualitative evaluation of possible impact cenarios.
Mestrado
Biomatematica
Mestre em Matemática Aplicada
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Neves, Odacir Almeida. "Simulação numerica de dispersão de poluentes pelo metodo de elementos finitos baseado em volumes de contole." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263431.
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Orientadores: Luiz Felipe Mendes de Moura, João Batista Campos SilvaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: A dispersão de poluentes no meio ambiente é um problema de grande interesse, por afetar diretamente a qualidade do ar, principalmente, nas grandes cidades. Ferramentas experimentais e numéricas têm sido utilizadas para prever o comportamento da dispersão de espécies poluentes na atmosfera. Códigos computacionais escritos na linguagem de programação fortran 90 foram desenvolvidos para obter simulações bidimensionais das equações de Navier-Stokes e de transporte de calor ou massa em regiões com obstáculos, variando a posição da fonte poluidora e simulações tridimensionais de equações de transporte arbitrando um campo de velocidade. Utilizaram-se, no primeiro caso, elementos finitos lagrangeanos quadrilaterais de quatro e de nove pontos nodais e no segundo, elementos lagrangeanos hexaedrais de oito e de vinte e sete pontos nodais. Os resultados numéricos de algumas aplicações foram obtidos e, quando possível, comparados com resultados da literatura apresentando concordância sastisfatória
Abstract: The dispersion of pollutant species in the environment is a problem of interest due to the bad quality of the air that this can originate, mainly, in big cities. Numerical and experimental tools have been developed and used to predict the behavior of the dispersion of pollutants in the atmosphere. In this work, computational codes have been developed in Fortran 90 language to simulate the flow with heat and mass transfer by solving the Navier-Stokes equations and the transport equations in two-dimensional domains with obstacles inserted in the media representing for example an urban canyon. Simulations of the three-dimensional transport equations for a given profile of velocity have also been done. In the two-dimensional simulations, it was utilized finite element quadrilateral Lagrangians of four and nine nodes; and in the three-dimensional simulations, it was utilized hexaedral finite elements Lagrangians of eight and twenty-seven nodes. The numerical results of some applications have been obtained and, when possible, compared to results from the literature. Both presented satisfactory concordance
Doutorado
Termica e Fluidos
Doutor em Engenharia Mecânica
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Gomes, Luciana Takata 1984. "Um estudo sobre o espalhamento da dengues usando equações diferenciais parciais e logica fuzzy." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307555.
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Orientador: Laecio Carvalho de BarrosDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: A doença a ser analisada é a dengue e, com este intuito, são criados alguns modelos matemáticos para simular sua evolução no distrito sul da cidade de Campinas. Divide-se a população humana local em três compartimentos, de acordo com o estado dos indivíduos - suscetível, infectante ou recuperado. A interação destas diferentes populações de humanos com a de mosquitos Aedes aegypti determina o comportamento da doença no domínio especificado. As variáveis de estado do modelo são as populações de humanos e a população de mosquitos, cuja divisão em compartimentos depende do modelo adotado. Seus valores são determinísticos e representam a densidade das populações em cada ponto do domínio. O trabalho contempla informações de especialistas a respeito do comportamento da doença e das condições para a proliferação e espalhamento do mosquito vetor. Tais condições, consideradas de natureza incerta, acabam por determinar o risco de contração da doença e, consequentemente, parâmetros dos modelos. A modelagem resulta em sistemas de Equações Diferencias Parciais, com alguns de seus parâmetros incertos. Para a obtenção de soluções (valores das variáveis em questão ao longo do tempo e sobre o domínio espacial citado), utilizam-se ferramentas de solução numérica (métodos dos Elementos Finitos e de Crank-Nicolson). Parâmetros relacionados ao comportamento da população do mosquito são avaliados por meio de Sistemas Baseados em Regras Fuzzy, aos quais são fornecidos, como entradas, as informações dos especialistas a respeito das condições do ambiente.
Abstract: The aim of this work is to study dengue and, with this purpose, some mathematical models were created to simulate its evolution in the southern district of the city of Campinas. The human population was subdivided into three compartments, according to the state of the individuals { susceptible, infectious or recovered. The interaction between these different populations and the Aedes aegypti mosquito population establishes the behaviour of the disease in the specified domain. The state variables of the models are the human populations and the mosquito population, whose compartmental division depends on the adopted model. Its values are deterministic and represent population densities in each point of the domain. This work takes into account specialists' information concerning the behaviour of the disease and the conditions of the proliferation and spread of the mosquito vector. These conditions, whose nature is considered uncertain, determine the risk of contraction of the disease and, consequently, the model parameters. The modelling results in systems of partial differential equations with some of its parameters being uncertain. To obtain the solutions (variable values according to time and the cited domain), numerical solution tools are used (Finite Elements and Crank-Nicolson methods). Parameters related to the behaviour of mosquito populations are evaluated through the Fuzzy Rules Based Systems, to which are provided, as entries, the specialists' information with respect to the environmental conditions.
Mestrado
Mestre em Matemática Aplicada
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Paillere, Henri J. "Multidimensional upwind residual distribution schemes for the Euler and Navier-Stokes equations on unstructured grids." Doctoral thesis, Universite Libre de Bruxelles, 1995. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212553.
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Une approche multidimensionelle pour la résolution numérique des équations d'Euler et de Navier-Stokes sur maillages non-structurés est proposée. Dans une première partie, un exposé complet des schémas de distribution, dits de "fluctuation-splitting" ,est décrit, comprenant une étude comparative des schémas décentrés, positifs et de 2ème ordre, pour résoudre l'équation de convection à coefficients constants, ainsi qu'une étude théorique et numérique de la précision des schémas sur maillages réguliers et distordus. L'extension à des lois de conservation non-linéaires est aussi abordée, et une attention particulière est portée au problème de la linéarisation conservative. Dans une deuxième partie, diverses discrétisations des termes visqueux pour l'équation de convection-diffusion sont développées, avec pour but de déterminer l'approche qui offre le meilleur compromis entre précision et coût. L'extension de la méthode aux systèmes des lois de conservation, et en particulier à celui des équations d'Euler de la dynamique des gaz, représente le noyau principal de la thèse, et est abordée dans la troisième partie. Contrairement aux schémas de distribution classiques, qui reposent sur une extension formelle du cas scalaire, l'approche développée ici repose sur une décomposition du résidu par élément en équations scalaires, modélisant le transport de variables caracteristiques. La difficulté vient du fait que les équations d'Euler instationnaires ne se diagonalisent pas, et admettent une infinité de solutions élémentaires (ondes simples) se propageant dans toutes les directions d'espace. En régime stationnaire, en revanche, les équations se diagonalisent complètement dans le cas des écoulements supersoniques, et partiellement dans le cas des écoulements subsoniques. Ainsi, les équations sous forme conservative peuvent être remplacées par un système équivalent comprenant deux équations totalement découplées, exprimant l'invariance de l'entropie et de l'enthalpie totale le long des lignes de courant, et deux autres équations, modélisant les effets purement acoustiques. En régime supersonique, celles-ci se découplent aussi, et expriment la convection le long des lignes de Mach d'invariants de Riemann généralisés. La discrétisation de ces équations par des schémas scalaires décentrés permet de simuler des écoulements continus et discontinus avec une grande précision et sans oscillations. Finalement, dans une dernière partie, l'extension aux équations de Navier-Stokes est abordée, et la discrétisation des termes visqueux par une approche éléments finis est proposée. Les résultats numériques confirment la précision et la robustesse de la méthode.
Doctorat en sciences appliquées
info:eu-repo/semantics/nonPublished
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Miyaoka, Tiago Yuzo 1990. "Impacto ambiental e populações que interagem : uma modelagem inovadora, aproximação e simulações computacionais." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307267.
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Orientador: João Frederico da Costa Azevedo MeyerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Este trabalho trata da modelagem matemática e da simulação computacional de um problema de dinâmica populacional, mais precisamente a interação de um poluente tóxico a duas espécies que competem entre si por espaço e alimento. A modelagem é feita a partir de dispersão e advecção populacional juntamente com o modelo clássico de Lotka-Volterra e reprodução do tipo de Verhulst, mas com um termo inovador para a interação entre poluente e população. Este termo inovador visa a melhoria do modelo a médio e longo prazos, pois tem comportamento assintótico em relação ao tempo. Temos assim um sistema de equações diferenciais parciais não-linear, cuja solução analítica é impossível de ser obtida. Recorremos então a métodos numéricos e simulações computacionais para obter soluções aproximadas. Para isso, utilizamos os métodos de Elementos Finitos (com elementos triangulares de primeira ordem) nas variáveis espaciais e de Diferenças Finitas (mais especificamente, o método de Crank-Nicolson) na temporal, além do método preditor-corretor de Douglas e Dupont para tratar não linearidades, detalhando o procedimento de se obter um software capaz de gerar cenários qualitativamente realistas (os parâmetros utilizados foram estimados). Com o software obtido apresentamos gráficos das soluções aproximadas em cenários hipotéticos distintos, de forma a poder analisar possíveis impactos ambientais causados pela poluição despejada no meio ambiente
Abstract: This work treats the mathematical modeling and computational simulation of a populational dynamics problem, more precisely the interaction of a toxic pollutant in two species which compete with each other for space and food. The modeling is done from populational dispersion and advection together with the classical model of Lotka-Volterra and Verhulst type reproduction, but with a innovative term for the interaction of pollutant and population. This innovative term aims the improvement of the model in the medium and long time, because it has asymptotic behaviour in relation to time. Therefore we have a system of non linear partial differential equations, whose analytical solution is impossible to be obtained. We then appeal to numerical methods and computational simulations to obtain approximated solutions. For this, we use the Finite Elements method (with first order triangular elements) in spatial variables and Finite Differences method (more specifically the Crank-Nicolson method), in addition to the Douglas and Dupont predictor-corrector method to treat non linearities, detailing the process of obtaining a software capable of generating qualitatively realistic scenarios (the parameters used were estimated). With the obtained software we present plots of approximate solutions in different hypothetical scenarios, in order to analyze possible enviromental impacts caused by pollution released into the environment
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada
21
Romão, Estaner Claro 1979. "Estudo numérico da aplicação do método dos elementos finitos de Galerkin e dos mínimos quadrados na solução da equação da convecção-difusão-reação tridimensional." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263390.
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Orientador: Luiz Felipe Mendes de MouraTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Este trabalho trata da aplicação do Método dos Elementos Finitos nas variantes Galerkin e Mínimos Quadrados com equações auxiliares para a solução numérica da equação diferencial parcial que modela a convecção-difusão-reação definida sobre um domínio tridimensional em regime permanente. Na discretização espacial foram utilizados elementos hexaedrais com oito (elemento linear) e vinte e sete (elemento quadrático) nós, no qual foram adotadas funções de interpolação de Lagrange nas coordenadas locais. Transformando toda a formulação do problema das coordenadas globais para as coordenadas locais, o Método da Quadratura de Gauss-Legendre foi utilizado para integração numérica dos coeficientes das matrizes dos elementos. Adicionalmente, à formulação pelos dois métodos, um código computacional foi implementado para simular o fenômeno proposto. Dispondo de soluções analíticas, várias análises de erro numérico foram realizadas a partir das normas L2 (erro médio no domínio) e L? (maior erro cometido no domínio), validando assim os resultados numéricos. Um caso real é proposto e analisado
Abstract: This paper the application of the Finite Element Method in variants Galerkin and Least Squares with auxiliary equations for the numerical solution of partial differential equation that models the convection-diffusion-reaction defined over a three-dimensional domain in steady state. In the spatial discretization were used hexahedrons elements with eight (linear element) and twenty-seven (quadratic element) nodes, which were adopted Lagrange interpolation functions in local coordinates. Transforming the problem of global coordinates to local coordinates, the method of Gauss-Legendre quadrature was used for numerical integration of the coefficients of the matrices of the elements. Additionally, the formulation by the two methods, a computer code was implemented to simulate the phenomenon proposed. Offering analytical solutions, several numerical error analysis were performed from L2 norms (average error in the domain) and L? (higher error in the domain), thus validating the numerical results. A real case is proposed and analyzed
Doutorado
Termica e Fluidos
Doutor em Engenharia Mecânica
22
Wang, Hao. "Incremental sheet forming process : control and modelling." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a80370f5-2287-4c6b-b7a4-44f06211564f.
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Incremental Sheet Forming (ISF) is a progressive metal forming process, where the deformation occurs locally around the point of contact between a tool and the metal sheet. The final work-piece is formed cumulatively by the movements of the tool, which is usually attached to a CNC milling machine. The ISF process is dieless in nature and capable of producing different parts of geometries with a universal tool. The tooling cost of ISF can be as low as 5–10% compared to the conventional sheet metal forming processes. On the laboratory scale, the accuracy of the parts created by ISF is between ±1.5 mm and ±3mm. However, in order for ISF to be competitive with a stamping process, an accuracy of below ±1.0 mm and more realistically below ±0.2 mm would be needed. In this work, we first studied the ISF deformation process by a simplified phenomenal linear model and employed a predictive controller to obtain an optimised tool trajectory in the sense of minimising the geometrical deviations between the targeted shape and the shape made by the ISF process. The algorithm is implemented at a rig in Cambridge University and the experimental results demonstrate the ability of the model predictive controller (MPC) strategy. We can achieve the deviation errors around ±0.2 mm for a number of simple geometrical shapes with our controller. The limitations of the underlying linear model for a highly nonlinear problem lead us to study the ISF process by a physics based model. We use the elastoplastic constitutive relation to model the material law and the contact mechanics with Signorini’s type of boundary conditions to model the process, resulting in an infinite dimensional system described by a partial differential equation. We further developed the computational method to solve the proposed mathematical model by using an augmented Lagrangian method in function space and discretising by finite element method. The preliminary results demonstrate the possibility of using this model for optimal controller design.
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Karlsson, Christian. "A comparison of two multilevel Schur preconditioners for adaptive FEM." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-219939.
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There are several algorithms for solving the linear system of equations that arise from the finite element method with linear or near-linear computational complexity. One way is to find an approximation of the stiffness matrix that is such that it can be used in a preconditioned conjugate residual method, that is, a preconditioner to the stiffness matrix. We have studied two preconditioners for the conjugate residual method, both based on writing the stiffness matrix in block form, factorising it and then approximating the Schur complement block to get a preconditioner. We have studied the stationary reaction-diffusion-advection equation in two dimensions. The mesh is refined adaptively, giving a hierarchy of meshes. In the first method the Schur complement is approximated by the stiffness matrix at one coarser level of the mesh, in the second method it is approximated as the assembly of local Schur complements corresponding to macro triangles. For two levels the theoretical bound of the condition number is 1/(1-C²) for either method, where C is the Cauchy-Bunyakovsky-Schwarz constant. For multiple levels there is less theory. For the first method it is known that the condition number of the preconditioned stiffness matrix is O(l²), where l is the number of levels of the preconditioner, or, equivalently, the number mesh refinements. For the second method the asymptotic behaviour is not known theoretically. In neither case is the dependency of the condition number of C known. We have tested both methods on several problems and found the first method to always give a better condition number, except for very few levels. For all tested problems, using the first method it seems that the condition number is O(l), in fact it is typically not larger than Cl. For the second method the growth seems to be superlinear.
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Bernabeu, Llinares Miguel Oscar. "An open source HPC-enabled model of cardiac defibrillation of the human heart." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:9ca44896-8873-4c91-9358-96744e28d187.
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Sudden cardiac death following cardiac arrest is a major killer in the industrialised world. The leading cause of sudden cardiac death are disturbances in the normal electrical activation of cardiac tissue, known as cardiac arrhythmia, which severely compromise the ability of the heart to fulfill the body's demand of oxygen. Ventricular fibrillation (VF) is the most deadly form of cardiac arrhythmia. Furthermore, electrical defibrillation through the application of strong electric shocks to the heart is the only effective therapy against VF. Over the past decades, a large body of research has dealt with the study of the mechanisms underpinning the success or failure of defibrillation shocks. The main mechanism of shock failure involves shocks terminating VF but leaving the appropriate electrical substrate for new VF episodes to rapidly follow (i.e. shock-induced arrhythmogenesis). A large number of models have been developed for the in silico study of shock-induced arrhythmogenesis, ranging from single cell models to three-dimensional ventricular models of small mammalian species. However, no extrapolation of the results obtained in the aforementioned studies has been done in human models of ventricular electrophysiology. The main reason is the large computational requirements associated with the solution of the bidomain equations of cardiac electrophysiology over large anatomically-accurate geometrical models including representation of fibre orientation and transmembrane kinetics. In this Thesis we develop simulation technology for the study of cardiac defibrillation in the human heart in the framework of the open source simulation environment Chaste. The advances include the development of novel computational and numerical techniques for the solution of the bidomain equations in large-scale high performance computing resources. More specifically, we have considered the implementation of effective domain decomposition, the development of new numerical techniques for the reduction of communication in Chaste's finite element method (FEM) solver, and the development of mesh-independent preconditioners for the solution of the linear system arising from the FEM discretisation of the bidomain equations. The developments presented in this Thesis have brought Chaste to the level of performance and functionality required to perform bidomain simulations with large three-dimensional cardiac geometries made of tens of millions of nodes and including accurate representation of fibre orientation and membrane kinetics. This advances have enabled the in silico study of shock-induced arrhythmogenesis for the first time in the human heart, therefore bridging an important gap in the field of cardiac defibrillation research.
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Barkman, Patrik. "Grey-box modelling of distributed parameter systems." Thesis, KTH, Beräkningsvetenskap och beräkningsteknik (CST), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-240677.
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Grey-box models are constructed by combining model components that are derived from first principles with components that are identified empirically from data. In this thesis a grey-box modelling method for describing distributed parameter systems is presented. The method combines partial differential equations with a multi-layer perceptron network in order to incorporate prior knowledge about the system while identifying unknown dynamics from data. A gradient-based optimization scheme which relies on the reverse mode of automatic differentiation is used to train the network. The method is presented in the context of modelling the dynamics of a chemical reaction in a fluid. Lastly, the grey-box modelling method is evaluated on a one-dimensional and two-dimensional instance of the reaction system. The results indicate that the grey-box model was able to accurately capture the dynamics of the reaction system and identify the underlying reaction.Hybridmodeller konstrueras genom att kombinera modellkomponenter som härleds från grundläggande principer med modelkomponenter som bestäms empiriskt från data. I den här uppsatsen presenteras en metod för att beskriva distribuerade parametersystem genom hybridmodellering. Metoden kombinerar partiella differentialekvationer med ett neuronnätverk för att inkorporera tidigare känd kunskap om systemet samt identifiera okänd dynamik från data. Neuronnätverket tränas genom en gradientbaserad optimeringsmetod som använder sig av bakåt-läget av automatisk differentiering. För att demonstrera metoden används den för att modellera kemiska reaktioner i en fluid. Metoden appliceras slutligen på ett en-dimensionellt och ett två-dimensionellt exempel av reaktions-systemet. Resultaten indikerar att hybridmodellen lyckades återskapa beteendet hos systemet med god precision samt identifiera den underliggande reaktionen.
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Comte, Eloïse. "Pollution agricole des ressources en eau : approches couplées hydrogéologique et économique." Thesis, La Rochelle, 2017. http://www.theses.fr/2017LAROS029/document.
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Ce travail s’inscrit dans un contexte de contrôle de la pollution des ressources en eau. On s’intéresse plus particulièrement à l’impact des engrais d’origine agricole sur la qualité de l’eau, en alliant modélisation économique et hydrogéologique. Pour cela, nous définissons d’une part un objectif économique spatio-temporel prenant en compte le compromis entre l’utilisation d’engrais et les coûts de dépollution. D’autre part, nous décrivons le transport du polluant dans le sous-sol (3D en espace) par un système non linéaire d’équations aux dérivées partielles couplées de type parabolique (réaction-convection-dispersion) et elliptique dans un domaine borné. Nous prouvons l’existence globale d’une solution au problème de contrôle optimal. L’unicité est quant à elle démontrée par analyse asymptotique pour le problème effectif tenant compte de la faible concentration d’engrais en sous-sol. Nous établissons les conditions nécessaires d’optimalité et le problème adjoint associé à notre modèle. Quelques exemples analytiques sont donnés et illustrés. Nous élargissons ces résultats au cadre de la théorie des jeux, où plusieurs joueurs interviennent, et prouvons notamment l’existence d’un équilibre de Nash. Enfin, ce travail est illustré par des résultats numériques (2D en espace), obtenus en couplant un schéma de type Éléments Finis Mixtes avec un algorithme de gradient conjugué non linéaireThis work is devoted to water ressources pollution control. We especially focus on the impact of agricultural fertilizer on water quality, by combining economical and hydrogeological modeling. We define, on one hand, the spatio-temporal objective, taking into account the trade off between fertilizer use and the cleaning costs. On an other hand, we describe the pollutant transport in the underground (3D in space) by a nonlinear system coupling a parabolic partial differential equation (reaction-advection-dispersion) with an elliptic one in a bounded domain. We prove the global existence of the solution of the optimal control problem. The uniqueness is proved by asymptotic analysis for the effective problem taking into account the low concentration fertilizer. We define the optimal necessary conditions and the adjoint problem associated to the model. Some analytical results are provided and illustrated. We extend these results within the framework of game theory, where several players are involved, and we prove the existence of a Nash equilibrium. Finally, this work is illustrated by numerical results (2D in space), produced by coupling a Mixed Finite Element scheme with a nonlinear conjugate gradient algorithm
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Benoit, David. "Divers problèmes théoriques et numériques liés à la simulation de fluides non newtoniens." Thesis, Paris Est, 2014. http://www.theses.fr/2014PEST1004/document.
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Le chapitre 1 introduit les modèles et donne les principaux résultats obtenus. Dans le chapitre 2, on présente des simulations numériques d'un modèle macroscopique en deux dimensions. La méthode de discrétisation par éléments finis utilisée est décrite. Pour le cas test de l'écoulement autour d'un cylindre, les phénomènes en jeu dans les fluides vieillissants sont observés. Le chapitre 3 concerne l'étude mathématique de la version unidimensionnelle du système d'équations aux dérivées partielles utilisé pour les simulations. On montre que le problème est bien posé et on examine le comportement en temps long de la solution. Dans le dernier chapitre, des équations macroscopiques sont dérivées à partir d'une équation mésoscopique. L'analyse mathématique de cette équation mésoscopique est également menéeThis thesis is devoted to the modelling, the mathematical analysis and the simulation of non-Newtonian fluids. Some fluids in an intermediate liquid-solid phase are particularly considered: aging fluids. Modelling scales are macroscopic and mesoscopic. In Chapter 1, we introduce the models and give the main results obtained. In Chapter 2, we present numerical simulations of a macroscopic two-dimensional model. The finite element method used for discretization is described. For the flow past a cylinder test-case, phenomena at play in aging fluids are observed. The Chapter 3 contains a mathematical analysis of the one-dimensional version of the system of partial differential equations used for the simulations. We show well-posedness and investigate the longtime behaviour of the solution. In the last chapter, macroscopic equations are derived from a mesoscopic equation. The mathematical analysis of this mesoscopic equation is also carried out
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Pereira, Weslley da Silva. "Validação numérica de estimativas analíticas aplicadas à combustão em meios porosos." Universidade Federal de Juiz de Fora, 2015. https://repositorio.ufjf.br/jspui/handle/ufjf/402.
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É crescente o interesse na utilização de métodos térmicos para recuperação de óleo de média e alta viscosidade. Um desses métodos é a combustão in situ, que consiste na liberação de calor no interior do reservatório através da combustão do ar injetado. As componentes mais pesadas do óleo atuam como combustível para as reações exotérmicas e o calor gerado reduz a viscosidade do óleo, estimulando o fluxo em direção aos poços de produção. Os modelos matemáticos para este método de recuperação em geral são complexos. Portanto, a obtenção de soluções analíticas para tais modelos é inviável, sendo necessária a utilização de simulações computacionais. Diversos trabalhos apresentam estudos analíticos e numéricos de modelos unidimensionais para a combustão em meios porosos. Em trabalhos anteriores, estimativas analíticas para modelos unidimensionais foram obtidas. Neste trabalho, tais estimativas são ligeiramente generalizadas através da inclusão da pressão prevalecente. É proposto um modelo bidimensional para o processo de combustão in situ em meios porosos heterogêneos que considera pressão variável. Soluções numéricas são obtidas utilizando o método de elementos finitos para a discretização espacial, o esquema de diferenças finitas de Crank-Nicolson para discretização no tempo e o método de Newton para resolução das equações não lineares resultantes. Estimativas analíticas para a temperatura e velocidade da onda de combustão são obtidas através de um modelo unidimensional simplificado. Tais estimativas são validadas com sucesso para o modelo geral através das simulações. Uma outra simplificação unidimensional do modelo geral é simulada numericamente através de duas abordagens: a primeira é similar à utilizada para a solução do modelo geral; e a segunda é escrita como um problema de complementaridade. Os problemas de complementaridade não-linear são resolvidos pelo algoritmo FDA-NCP. As duas abordagens numéricas utilizadas são comparadas com uma estimativa analítica para a onda térmica e mostram bons resultados.
There is a growing interest in using thermal methods for the recovery of medium and high viscosity oil. One of these methods is the in-situ combustion, which consists in release heat within the reservoir through combustion of the injected air. The heavier oil components are used as fuel for exothermic reactions and the generated heat reduces the oil viscosity, stimulating the flow towards the production well. In general, the mathematical models for this recovery method are complex. Therefore, the analytical solutions for such models are impossible, requiring numerical simulations. Several works present analytical and numerical studies of one-dimensional models for combustion in porous media. In previous works analytical estimates for one dimensional models were obtained. Here these estimates are slightly generalized by including the prevailing pressure. We propose a two-dimensional model for the in-situ combustion process in heterogeneous porous media, considering variable pressure. Numerical results are obtained using the finite element method for spatial discretization, Crank-Nicolson finite difference scheme for time discretization and Newton’s method for the arising nonlinear equations. Analytical estimates for combustion wave speed and combustion wave temperature are obtained using one-dimensional simplified model. These estimates are successfully validated in the general model through the simulation results. Another one-dimensional simplification of the general model is numerically simulated by two approaches: the first is similar to the one previously described; and the second one is written as a complementarity problem. The arising nonlinear complementarity problems are solved by the FDA-NCP algorithm. Both numerical approaches are compared to the analytical estimate for the thermal wave, showing good agreement.
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Vieira, Jardel 1991. "Um estudo computacional de equações pseudo-parabólicas para mecânica dos fluidos e fenômenos de transporte em meios porosos." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307021.
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Orientador: Eduardo Cardoso de AbreuDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O foco desta dissertação de mestrado consiste em um estudo computacional de equações pseudo-parabólicas em mecânica dos fluidos e fenômenos de transporte de fluidos em meios porosos. Serão considerados problemas de valor de contorno e inicial associados a duas classes de modelos de equações de evolução pseudo-parabólicas: um modelo de advecção-difusão com termo pseudo-parabólico que exibe um certo caráter dispersivo e um outro modelo pseudo-parabólico "puro", i.e., sem a presença do termo de advecção. O primeiro modelo se relaciona com a modelagem física do fluxo de duas fases incompressíveis em dinâmica de fluidos em meios porosos, onde são considerados modelos de pressão capilar dinâmica, ou seja, em que os efeitos dinâmicos são também incluídos na diferença de pressão entre as fases fluidas. Uma discussão sobre a relevância física em aplicações e da importância matemática do sistema governante de equações para pressão capilar dinâmica em fenômenos de transporte de fluidos em meios porosos é também feita de modo a indicar algum suporte à escolha dos métodos estudados para aproximação numérica dos modelos consideradores. Além disso, um conjunto de experimentos numéricos é apresentado e discutido para avaliar a qualidade das soluções obtidas do estudo proposto, bem como para justificar variações dos métodos numéricos estudados. Especificamente, para o modelo pseudo-parabólico puro, os resultados são comparados com soluções analíticas para o caso linear. Para o modelo pseudo-parabólico com o termo de advecção, é avaliado se os resultados dos métodos numéricos empregados concordam qualitativamente com resultados da literatura
Abstract: The focus of this work consists of a computational study of pseudo-parabolic equations in fluid mechanics and transport phenomena in porous media. For concreteness, we consider initial-boundary value problems related to two classes of systems of evolution pseudo-parabolic equations: a advection-diffusion model, which in turn the pseudo-parabolic term exhibits a certain dispersive character, and a second of "purely" pseudo-parabolic nature, i.e., without the presence of advection term. The first model relates to the modeling of incompressible two-phase flow in porous media, which in turn takes into account the nonlinear dynamic capillary pressure effects, where the dynamic effects are also included into the pressure difference between the fluid phases. Further, a discussion of the physical and mathematical relevance of the governing system of equations for dynamic capillary pressure in porous media fluid transport phenomena is also made in order to drive the choice of the numerical approximations for the differential models under investigation. Moreover, a set of numerical experiments are presented and discussed to address the quality of the obtained solutions proposed study, as well as to justify variations of the numerical methods studied. Specifically, to the purely pseudo-parabolic model, the results are compared along with analytical solutions with respect to a linear case. On the other hand, to the nonlinear pseudo-parabolic model with advection term, it is performed numerical experiments in order to account the correct qualitative behavior of the computed solutions against the available results discussed in the recent literature
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada
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Li, Xiaodong. "Observation et commande de quelques systèmes à paramètres distribués." Phd thesis, Université Claude Bernard - Lyon I, 2009. http://tel.archives-ouvertes.fr/tel-00456850.
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L'objectif principal de cette thèse consiste à étudier plusieurs thématiques : l'étude de l'observation et la commande d'un système de structure flexible et l'étude de la stabilité asymptotique d'un système d'échangeurs thermiques. Ce travail s'inscrit dans le domaine du contrôle des systèmes décrits par des équations aux dérivées partielles (EDP). On s'intéresse au système du corps-poutre en rotation dont la dynamique est physiquement non mesurable. On présente un observateur du type Luenberger de dimension infinie exponentiellement convergent afin d'estimer les variables d'état. L'observateur est valable pour une vitesse angulaire en temps variant autour d'une constante. La vitesse de convergence de l'observateur peut être accélérée en tenant compte d'une seconde étape de conception. La contribution principale de ce travail consiste à construire un simulateur fiable basé sur la méthode des éléments finis. Une étude numérique est effectuée pour le système avec la vitesse angulaire constante ou variante en fonction du temps. L'influence du choix de gain est examinée sur la vitesse de convergence de l'observateur. La robustesse de l'observateur est testée face à la mesure corrompue par du bruit. En mettant en cascade notre observateur et une loi de commande stabilisante par retour d'état, on souhaite obtenir une stabilisation globale du système. Des résultats numériques pertinents permettent de conjecturer la stabilité asymptotique du système en boucle fermée. Dans la seconde partie, l'étude est effectuée sur la stabilité exponentielle des systèmes d'échangeurs thermiques avec diffusion et sans diffusion. On établit la stabilité exponentielle du modèle avec diffusion dans un espace de Banach. Le taux de décroissance optimal du système est calculé pour le modèle avec diffusion. On prouve la stabilité exponentielle dans l'espace Lp pour le modèle sans diffusion. Le taux de décroissance n'est pas encore explicité dans ce dernier cas.
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Klimanis, Nils. "Generic Programming and Algebraic Multigrid for Stabilized Finite Element Methods." Doctoral thesis, 2006. http://hdl.handle.net/11858/00-1735-0000-0006-B38C-5.
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Gillette, Andrew Kruse. "Stability of dual discretization methods for partial differential equations." Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-05-3467.
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This thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra.
Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results.
Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties.
A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis.text
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Dilip, Jagtap Ameya. "Kinetic Streamlined-Upwind Petrov Galerkin Methods for Hyperbolic Partial Differential Equations." Thesis, 2016. http://etd.iisc.ernet.in/handle/2005/2658.
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In the last half a century, Computational Fluid Dynamics (CFD) has been established as an important complementary part and some times a significant alternative to Experimental and Theoretical Fluid Dynamics. Development of efficient computational algorithms for digital simulation of fluid flows has been an ongoing research effort in CFD.
An accurate numerical simulation of compressible Euler equations, which are the gov-erning equations of high speed flows, is important in many engineering applications like designing of aerospace vehicles and their components. Due to nonlinear nature of governing equations, such flows admit solutions involving discontinuities like shock waves and contact discontinuities. Hence, it is nontrivial to capture all these essential features of the flows numerically. There are various numerical methods available in the literature, the popular ones among them being the Finite Volume Method (FVM), Finite Difference Method (FDM), Finite Element Method (FEM) and Spectral method. Kinetic theory based algorithms for solving Euler equations are quite popular in finite volume framework due to their ability to connect Boltzmann equation with Euler equations. In kinetic framework, instead of dealing directly with nonlinear partial differential equations one needs to deal with a simple linear partial differential equation. Recently, FEM has emerged as a significant alternative to FVM because it can handle complex geometries with ease and unlike in FVM, achieving higher order accuracy is easier. High speed flows governed by compressible Euler equations are hyperbolic partial differential equations which are characterized by preferred directions for information propagation. Such flows can not be solved using traditional FEM methods and hence, stabilized methods are typically introduced. Various stabilized finite element methods are available in the literature like Streamlined-Upwind Petrov Galerkin (SUPG) method, Galerkin-Least Squares (GLS) method, Taylor-Galerkin method, Characteristic Galerkin method and Discontinuous Galerkin Method.
In this thesis a novel stabilized finite element method called as Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) method is formulated. Both explicit and implicit versions of KSUPG scheme are presented. Spectral stability analysis is done for explicit KSUPG scheme to obtain the stable time step. The advantage of proposed scheme is, unlike in SUPG scheme, diffusion vectors are obtained directly from weak KSUPG formulation.
The expression for intrinsic time scale is directly obtained in KSUPG framework. The accuracy and robustness of the proposed scheme is demonstrated by solving various test cases for hyperbolic partial differential equations like Euler equations and inviscid Burgers equation. In the KSUPG scheme, diffusion terms involve computationally expensive error and exponential functions. To decrease the computational cost, two variants of KSUPG scheme, namely, Peculiar Velocity based KSUPG (PV-KSUPG) scheme and Circular distribution based KSUPG (C-KSUPG) scheme are formulated. The PV-KSUPG scheme is based on peculiar velocity based splitting which, upon taking moments, recovers a convection-pressure splitting type algorithm at the macroscopic level. Both explicit and implicit versions of PV-KSUPG scheme are presented. Unlike KSUPG and PV-KUPG schemes where Maxwellian distribution function is used, the C-KUSPG scheme uses a simpler circular distribution function instead of a Maxwellian distribution function. Apart from being computationally less expensive it is less diffusive than KSUPG scheme.
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Michoski, Craig E. "Evolution equations in physical chemistry." Thesis, 2009. http://hdl.handle.net/2152/ETD-UT-2009-05-54.
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We analyze a number of systems of evolution equations that arise in the study of physical chemistry. First we discuss the well-posedness of a system of mixing compressible barotropic multicomponent flows. We discuss the regularity of these variational solutions, their existence and uniqueness, and we analyze the emergence of a novel type of entropy that is derived for the system of equations.
Next we present a numerical scheme, in the form of a discontinuous Galerkin (DG) finite element method, to model this compressible barotropic multifluid. We find that the DG method provides stable and accurate solutions to our system, and that further, these solutions are energy consistent; which is to say that they satisfy the classical entropy of the system in addition to an additional integral inequality. We discuss the initial-boundary problem and the existence of weak entropy at the boundaries. Next we extend these results to include more complicated transport properties (i.e. mass diffusion), where exotic acoustic and chemical inlets are explicitly shown.
We continue by developing a mixed method discontinuous Galerkin finite element method to model quantum hydrodynamic fluids, which emerge in the study of chemical and molecular dynamics. These solutions are solved in the conservation form, or Eulerian frame, and show a notable scale invariance which makes them particularly attractive for high dimensional calculations.
Finally we implement a wide class of chemical reactors using an adapted discontinuous Galerkin finite element scheme, where reaction terms are analytically integrated locally in time. We show that these solutions, both in stationary and in flow reactors, show remarkable stability, accuracy and consistency.text
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(9733025), Li Xue. "Rapid Modeling and Simulation Methods for Large-Scale and Circuit-Intuitive Electromagnetic Analysis of Integrated Circuits and Systems." Thesis, 2020.
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Accurate, fast, large-scale, and circuit-intuitive electromagnetic analysis is of critical importance to the design of integrated circuits (IC) and systems. Existing methods for the analysis of integrated circuits and systems have not satisfactorily achieved these performance goals. In this work, rapid modeling and simulation methods are developed for large-scale and circuit-intuitive electromagnetic analysis of integrated circuits and systems. The derived model is correct from zero to high frequencies where Maxwell's equations are valid. In addition, in the proposed model, we are able to analytically decompose the layout response into static and full-wave components with neither numerical computation nor approximation. This decomposed yet rigorous model greatly helps circuit diagnoses since now designers are able to analyze each component one by one, and identify which component is the root cause for the design failure. Such a decomposition also facilitates efficient layout modeling and simulation, since if an IC is dominated by RC effects, then we do not have to compute the full-wave component; and vice versa. Meanwhile, it makes parallelization straightforward. In addition, we develop fast algorithms to obtain each component of the inverse rapidly. These algorithms are also applicable for solving general partial differential equations for fast electromagnetic analysis.
The fast algorithms developed in this work are as follows. First, an analytical method is developed for finding the nullspace of the curl-curl operator in an arbitrary mesh for an arbitrary order of curl-conforming vector basis function. This method has been applied successfully to both a finite-difference and a finite-element based analysis of general 3-D structures. It can be used to obtain the static component of the inverse efficiently. An analytical method for finding the complementary space of the nullspace is also developed. Second, using the analytically found nullspace and its complementary space, a rigorous method is developed to overcome the low-frequency breakdown problem in the full-wave analysis of general lossy problems, where both dielectrics and conductors can be lossy and arbitrarily inhomogeneous. The method is equally valid at high frequencies without any need for changing the formulation. Third, with the static component part solved, the full-wave component is also ready to obtain. There are two ways. In the first way, the full-wave component is efficiently represented by a small number of high-frequency modes, and a fast method is created to find these modes. These modes constitute a significantly reduced order model of the complementary space of the nullspace. The second way is to utilize the relationship between the curl-curl matrix and the Laplacian matrix. An analytical method to decompose the curl-curl operator to a gradient-divergence operator and a Laplacian operator is developed. The derived Laplacian matrix is nothing but the curl-curl matrix's Laplacian counterpart. They share the same set of non-zero eigenvalues and eigenvectors. Therefore, this Laplacian matrix can be used to replace the original curl-curl matrix when operating on the full-wave component without any computational cost, and an iterative solution can converge this modified problem much faster irrespective of the matrix size. The proposed work has been applied to large-scale layout extraction and analysis. Its performance in accuracy, efficiency, and capacity has been demonstrated.