Thèses sur le sujet « Hamiltonian Boundary Value Method »
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1
Heredia, N. Fernando. « A new method for boundary value problems ». Thesis, Monterey, California. Naval Postgraduate School, 1985. http://hdl.handle.net/10945/21467.
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Rockstroh, Parousia. « Boundary value problems for the Laplace equation on convex domains with analytic boundary ». Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/273939.
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In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.
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Ugail, Hassan. « Method of boundary based smooth shape design ». ACTA Press, 2005. http://hdl.handle.net/10454/2685.
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The discussion in this paper focuses on how boundary
based smooth shape design can be carried out. For this we
treat surface generation as a mathematical boundary-value
problem. In particular, we utilize elliptic Partial Differential
Equations (PDEs) of arbitrary order. Using the methodology
outlined here a designer can therefore generate the
geometry of shapes satisfying an arbitrary set of boundary
conditions. The boundary conditions for the chosen PDE
can be specified as curves in 3-space defining the profile
geometry of the shape.
We show how a compact analytic solution for the chosen
arbitrary order PDE can be formulated enabling complex
shapes to be designed and manipulated in real time.
This solution scheme, although analytic, satisfies exactly,
even in the case of general boundary conditions, where the
resulting surface has a closed form representation allowing
real time shape manipulation. In order to enable users
to appreciate the powerful shape design and manipulation
capability of the method, we present a set of practical examples.
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Chan, Kwok Cheung. « Shooting method for singularly perturbed two-point boundary value problems ». HKBU Institutional Repository, 1998. http://repository.hkbu.edu.hk/etd_ra/274.
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Bozkaya, Canan. « Boundary Element Method Solution Of Initial And Boundary Value Problems In Fluid Dynamics And Magnetohydrodynamics ». Phd thesis, METU, 2008. http://etd.lib.metu.edu.tr/upload/12609552/index.pdf.
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In this thesis, the two-dimensional initial and boundary value problems invol-ving convection and diffusion terms are solved using the boundary element method (BEM). The fundamental solution of steady magnetohydrodynamic (MHD) flow equations in the original coupled form which are convection-diffusion type is established in order to apply the BEM directly to these coupled equations with the most general form of wall conductivities. Thus, the solutions of MHD flow in rectangular ducts and in infinite regions with mixed boundary conditions are obtained for high values of Hartmann number, M.
For the solution of transient convection-diffusion type equations the dual reciprocity boundary element method (DRBEM) in space is combined with the differential quadrature method (DQM) in time. The DRBEM is applied with the fundamental solution of Laplace equation treating all the other terms in the equation as nonhomogeneity. The use of DQM eliminates the need of iteration and very small time increments since it is unconditionally stable. Applications include unsteady MHD duct flow and elastodynamic problems. The transient Navier-Stokes equations which are nonlinear in nature are also solved with the DRBEM in space - DQM in time procedure iteratively in terms of stream function and vorticity. The procedure is applied to the lid-driven cavity flow for moderate values of Reynolds number. The natural convection cavity flow problem is also solved for high values of Rayleigh number when the energy equation is added.
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Alsaedy, Ammar, et Nikolai Tarkhanov. « The method of Fischer-Riesz equations for elliptic boundary value problems ». Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/6179/.
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We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
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Mokhtarzadeh, M. R. « A general global approximation method for the solution of boundary value problems ». Thesis, Loughborough University, 1998. https://dspace.lboro.ac.uk/2134/14478.
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A general global approximation scheme is developed and its generality is demonstrated by the derivation of classical Lagrange and Hermite interpolation and finite difference and finite element approximations as its special cases. It is also shown that previously reported general approximation techniques which use the idea of moving least square are also special cases of the present method. The combination of the developed general global approximation technique with the weighted residual methods provides a very powerful scheme for the solution of the boundary value problems formulated in terms of differential equations. Although this application is the main purpose of the this project, nevertheless, the power and flexibility of the developed approximation allows it to be used in many other areas. In this study the following applications of the described approximation are developed: 1- data fitting (including curve and surface fitting) 2- plane mapping (both in cases where a conformal mapping exists and for non-conformal mapping) 3- solution of eigenvalue problems with particular application to spectral expansions used in the modal representation of shallow water equations 4- solution of ordinary differential equations (including Sturm-Liouville equations, non-homogeneous equations with non-smooth right hand sides and 4th order equations) 5- elliptic partial differential equations (including Poisson equations with non-smooth right hand sides) A computer program which can handle the above applications is developed. This program utilises symbolic, numerical and graphical and the programming language provided by the Mathematica package.
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Chibi, Ahmed-Salah. « Defect correction and Galerkin's method for second-order elliptic boundary value problems ». Thesis, Imperial College London, 1989. http://hdl.handle.net/10044/1/47378.
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Yang, Zhiyun. « A Cartesian grid method for elliptic boundary value problems in irregular regions / ». Thesis, Connect to this title online ; UW restricted, 1996. http://hdl.handle.net/1773/6759.
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Poullikkas, Andreas. « The Method of Fundamental Solutions for the solution of elliptic boundary value problems ». Thesis, Loughborough University, 1997. https://dspace.lboro.ac.uk/2134/27141.
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We investigate the use of the Method of Fundamental Solutions (MFS) for the numerical solution of elliptic problems arising in engineering. In particular, we examine harmonic and biharmonic problems with boundary singularities, certain steady-state free boundary flow problems and inhomogeneous problems. The MFS can be viewed as an indirect boundary method with an auxiliary boundary. The solution is approximated by a linear combination of fundamental solutions of the governing equation which are expressed in terms of sources located outside the domain of the problem. The unknown coefficients in the linear combination of fundamental solutions and the location of the sources are determined so that the boundary conditions are satisfied in a least squares sense. The MFS shares the same advantages of the boundary methods over domain discretisation methods. Moreover, it is relatively easy to implement, it is adaptive in the sense that it takes into account sharp changes in the solution and/or in the geometry of the domain and it can easily incorporate complicated boundary conditions.
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Womble, David Eugene. « The convergence of the method of lines for time dependent free boundary problems ». Diss., Georgia Institute of Technology, 1986. http://hdl.handle.net/1853/29154.
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Jin, Kang Meir Amnon J. « On the Lattice Boltzmann method implementation and applications / ». Auburn, Ala, 2008. http://repo.lib.auburn.edu/EtdRoot/2008/FALL/Mathematics_and_Statistics/Dissertation/Jin_Kang_9.pdf.
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McFall, Kevin Stanley. « An artificial neural network method for solving boundary value problems with arbitrary irregular boundaries ». Diss., Available online, Georgia Institute of Technology, 2006, 2006. http://etd.gatech.edu/theses/available/etd-04052006-154934/.
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Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2006.George Vachtsevanos, Committee Member ; Nader Sadegh, Committee Member ; J. Robert Mahan, Committee Chair ; Ali Siadat, Committee Member ; Zhuomin Zhang, Committee Member.
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Chien, Chyou-Chi. « On the evaluation of hyper-singular integrals arising in the boundary element method for elasticity and acoustic problems ». Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/21264.
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Mudavanhu, Blessing. « A new renormalization method for the asymptotic solution of multiple scale singular perturbation problems / ». Thesis, Connect to this title online ; UW restricted, 2002. http://hdl.handle.net/1773/6794.
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Ritter, Baird S. « Solution strategies for second order, nonlinear, one dimensional, two point boundary value problems by FEM analysis ». Thesis, Monterey, California : Naval Postgraduate School, 1990. http://handle.dtic.mil/100.2/ADA246063.
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Thesis (M.S. in Mechanical Engineering)--Naval Postgraduate School, December 1990.Thesis Advisor: Salinas, D. "December 1990." Description based on title screen as viewed on April 1, 2010. DTIC Identifier(s): Boundary value problems, finite element analysis, differential equations, problem solving, theses, interpolation, iterations, one dimensional, computer programs, approximation/mathematics, linearity. Author(s) subject terms: Galerkin FEM, nonlinear, quasilinearization, linearization, interpolation, iteration, differential equation, convergence. Includes bibliographical references (p. 164). Also available in print.
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VARGAS, JOSE VIRIATO COELHO. « INITIAL VALUE METHOD FOR THE SOLUTION OF NON-SIMILAR BOUNDARY LAYERS APPLIED TO A WEDGE IN MEXED CONVECTION ». PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1992. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=19598@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROO presente trabalho apresenta o método de varredura de valores iniciais para a solução de problemas não-similares de camada limite, aplicado a uma cunha em convecção mista. É feita uma análise dos efeitos da força de empuxo nas características de transferência de calor e atrito, na superfície da cunha submetida a um escoamento laminar em convecção forçada. Os casos analisados referem-se a temperatura uniforme na superfície e a fluxo de calor uniforme através da superfície. Apresentam-se resultados numéricos para a placa vertical e a cunha em 120 graus celsius (problema similar), os quais foram comparados com os disponíveis na literatura, apresentando-se excelente concordância. Ainda são apresentados resultados para a cunha em 90 graus celsius e em 180 graus Celsius. Os resultados foram obtidos para o número de Prandtl 0,7 em uma larga faixa do parâmetro de não-similaridade (0 a 100 para temperatura uniforme e 0 a 5 para fluxo de calor uniforme). Em geral, verifica-se que para ambos os casos estudados de condições da superfície, o coeficiente de atrito local e o número de Nusselt local aumentam com o aumento da força de empuxo para o escoamento ajudado e diminuem com o aumento da força de empuxo para o escoamento ajudado e diminuem com o aumento da força de empuxo para o escoamento oposto. Verifica-se também que os efeitos de convecção natural diminuem com o aumento do ângulo da cunha. Compara-se, por fim os resultados de transferência de calor entre os dois casos estudados. O método possibilita a obtenção de resultados com o uso de um microcomputador PC AT-286 com o co-processador matemático, sem o uso de dupla precisão. As tolerâncias utilizadas para a convergência são as mesmas dos resultados disponíveis na literatura. Adicionalmente, o tempo computacional necessário para a obtenção das soluções foi bastante reduzido. Para toda a faixa de variação do parâmetro de não-similaridade, o equacionamento utilizado foi o mesmo, baseado em parâmetros de correlação para convecção forçada em uma superfície plana com injeção de massa, com ótima concordância com resultados disponíveis na literatura.
The present work introduces a method for searching initial values, to solve non-similar Boundary Layer problems. The new method has been applied to the problem of mixed convection on a Wedge. An analysis is performed to study the effects of buoyancy force on the heat transfer and friction characteristics of laminar forced convection flow which is either maintained at a uniform temperature or subjected to a uniform temperature or subjected to a uniform heat flux. Numerical results are presented for Prandtl number of 7,0 over a wide range of values of the buoyancy force parameters (0 to 100 for uniform temperature and 0 to 5 for uniform heat flux). The results for the vertical plate and for the similar wedge (120 graus celsius) Have been compared with solutions availabein the literature, showing an excellent agreement. In addition, solutions for the 90 degree celsius wedge and for the flat wall (180 degree celsius are also obtained. In general, it is found that for both surface heating conditions, the local friction factor and the local Nusselt number increase with increasing bouyancy force for assisting flow and decrease with inscreasing bouyancy force for opposing flow. Further, the effects of the buoyancy force on these two quantities are found to diminish as the angle of the wedge increases. A comparison is also made of the results beteween the case of uniform wall temperature and the case of uniform surface heat flux. The method turns possible to obtain results with the use of a microcomputer PC AT-286 with a Math co-processor, discarding the use of double precision. The tolerances for convergence are the same as the results available in the literature. The necessary computacional time to get the solution was greatly reduced. For the entire range of the buoyancy force parameter, the governing equations are the same, based upon forced convection parameters. To illustrate the flexibility of the method, the surface mass trasnfer problem of uniform injection (blowing) in a flat plate under forced cnvection, has been solved and the results compared with the available ones in the literature.
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Nyamayaro, Takura T. A. « On the design and implementation of a hybrid numerical method for singularly perturbed two-point boundary value problems ». University of the Western Cape, 2014. http://hdl.handle.net/11394/4326.
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>Magister Scientiae - MScWith the development of technology seen in the last few decades, numerous solvers have been developed to provide adequate solutions to the problems that model different aspects of science and engineering. Quite often, these solvers are tailor-made for specific classes of problems. Therefore, more of such must be developed to accompany the growing need for mathematical models that help in the understanding of the contemporary world. This thesis treats two point boundary value singularly perturbed problems. The solution to this type of problem undergoes steep changes in narrow regions (called boundary or internal layer regions) thus rendering the classical numerical procedures inappropriate. To this end, robust numerical methods such as finite difference methods, in particular fitted mesh and fitted operator methods have extensively been used. While the former consists of transforming the continuous problem into a discrete one on a non-uniform mesh, the latter involves a special discretisation of the problem on a uniform mesh and are known to be more accurate. Both classes of methods are suitably designed to accommodate the rapid change(s) in the solution. Quite often, finite difference methods on piece-wise uniform meshes (of Shishkin-type) are adopted. However, methods based on such non-uniform meshes, though layer-resolving, are not easily extendable to higher dimensions. This work aims at investigating the possibility of capitalising on the advantages of both fitted mesh and fitted operator methods. Theoretical results are confirmed by extensive numerical simulations.
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Li, Boning. « Extending the scaled boundary finite-element method to wave diffraction problems ». University of Western Australia. School of Civil and Resource Engineering, 2007. http://theses.library.uwa.edu.au/adt-WU2007.0173.
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[Truncated abstract] The study reported in this thesis extends the scaled boundary finite-element method to firstorder and second-order wave diffraction problems. The scaled boundary finite-element method is a newly developed semi-analytical technique to solve systems of partial differential equations. It works by employing a special local coordinate system, called scaled boundary coordinate system, to define the computational field, and then weakening the partial differential equation in the circumferential direction with the standard finite elements whilst keeping the equation strong in the radial direction, finally analytically solving the resulting system of equations, termed the scaled boundary finite-element equation. This unique feature of the scaled boundary finite-element method enables it to combine many of advantages of the finite-element method and the boundaryelement method with the features of its own. ... In this thesis, both first-order and second-order solutions of wave diffraction problems are presented in the context of scaled boundary finite-element analysis. In the first-order wave diffraction analysis, the boundary-value problems governed by the Laplace equation or by the Helmholtz equation are considered. The solution methods for bounded domains and unbounded domains are described in detail. The solution process is implemented and validated by practical numerical examples. The numerical examples examined include well benchmarked problems such as wave reflection and transmission by a single horizontal structure and by two structures with a small gap, wave radiation induced by oscillating bodies in heave, sway and roll motions, wave diffraction by vertical structures with circular, elliptical, rectangular cross sections and harbour oscillation problems. The numerical results are compared with the available analytical solutions, numerical solutions with other conventional numerical methods and experimental results to demonstrate the accuracy and efficiency of the scaled boundary finite-element method. The computed results show that the scaled boundary finite-element method is able to accurately model the singularity of velocity field near sharp corners and to satisfy the radiation condition with ease. It is worth nothing that the scaled boundary finite-element method is completely free of irregular frequency problem that the Green's function methods often suffer from. For the second-order wave diffraction problem, this thesis develops solution schemes for both monochromatic wave and bichromatic wave cases, based on the analytical expression of first-order solution in the radial direction. It is found that the scaled boundary finiteelement method can produce accurate results of second-order wave loads, due to its high accuracy in calculating the first-order velocity field.
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Jin, Jicheng. « Finite element methods for some elliptic problems with singularity and problems on unbounded domains ». HKBU Institutional Repository, 2004. http://repository.hkbu.edu.hk/etd_ra/597.
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Kuhn, Wolfgang. « Rigorous and reasonable error bounds for the numerical solution of dynamical systems ». Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/28941.
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Starkloff, Hans-Jörg. « Stochastic finite element method with simple random elements ». Universitätsbibliothek Chemnitz, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200800596.
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We propose a variant of the stochastic finite element method, where the random
elements occuring in the problem formulation are approximated by simple random
elements, i.e. random elements with only a finite number of possible values.
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Rattana, Amornrat, et Christine Böckmann. « Matrix methods for computing Eigenvalues of Sturm-Liouville problems of order four ». Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/5927/.
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This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.
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Ugail, Hassan, M. I. G. Bloor et M. J. Wilson. « Implementing automatic design optimisation in an interactive environment ». American Institute of Aeronautics and Astronautics, 2000. http://hdl.handle.net/10454/2942.
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Aoyama, Taiki, Shota Fukumoto et Hideyuki Azegami. « Shape optimization of continua using NURBS as basis functions ». Springer, 2013. http://hdl.handle.net/2237/21124.
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Yoerger, Edward J. Jr. « Vertical Acoustic Propagation in the Non-Homogeneous Layered Atmosphere for a Time-Harmonic, Compact Source ». ScholarWorks@UNO, 2019. https://scholarworks.uno.edu/td/2709.
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In this work we study vertical, acoustic propagation in a non-homogeneous media for a spatially-compact, time-harmonic source. An analytical, 2-layer model is developed representing the acoustic pressure disturbance propagating in the atmosphere. The validity of the model spans the distance from the Earth's surface to 30,000 meters. This includes the troposphere (adiabatic), ozone layer (isothermal), and part of the stratosphere (isothermal). The results of the model derivation in the adiabatic region yield pressure solutions as Bessel functions of the First (J) and Second (Y) Kind of order $-\frac{7}{2}$ with an argument of $2 \Omega \tau$ (where $\Omega$ represents a dimensionless frequency and $\tau$ is a dimensionless vertical height in z (vertical coordinate)). For an added second layer (isothermal region), the pressure solution is a decaying sinusoidal, exponential function above the first layer.
In particular, the vertical, acoustic propagation is examined for various configurations. These are divided into 2 basic classes. The first class consists of examining the pressure response function when the source is located on boundary interfaces, while the second class consists of situations where the source is arbitrarily located within a finite layer. In all instances, a time-harmonic, compact source is implicitly understood. However, each class requires a different method of solution. The first class conforms to a general boundary value problem, while the second requires the use of Green's functions method.
In investigating problems of the first class, 3 different scenarios are examined. In the first case, we apply our model to a semi-infinite medium with a time-harmonic source ($e^{-i \omega t}$) located on the ground. In the next 2 cases, a semi-infinite medium is overlain on the previous medium at a height of z=13,000 meters. Thus, there exist two boundaries: the ground and the layer interface between the 2 media. Sources placed at these interfaces represent the 2nd and 3rd scenarios, respectively. The solutions to all 3 cases are of the form $A \frac{J_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}} + B \frac{Y_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}}$, where \textit{A} and \textit{B} are constants determined by the boundary conditions.
For the 2nd class, we examine the application to a time-harmonic, compact source placed arbitrarily within the 1st layer. The method of Green's functions is used to obtain a particular solution for the model equations. This result is compared with a Fast Field Program (FFP) which was developed to test these solutions. The results show that the response given by the Green's function compares favorably with that of the FFP.
Keywords: Linear Acoustics, Inhomogeneous Medium, Layered Atmosphere, Boundary Value Problem, Green's Function Method
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畔上, 秀幸, et Hideyuki Azegami. « 領域最適化問題の一解法 ». 日本機械学会, 1994. http://hdl.handle.net/2237/7238.
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Weichselbaum, Andreas. « Nanoscale Quantum Dynamics and Electrostatic Coupling ». Ohio University / OhioLINK, 2004. http://www.ohiolink.edu/etd/view.cgi?ohiou1091115085.
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Ozdemir, Nilufer A. « The method of moments solution of a nonconformal volume integral equation via the IE-FFT algorithm for electromagnetic scattering from penetrable objects ». Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1182258230.
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Desiderio, Luca. « H-matrix based Solver for 3D Elastodynamics Boundary Integral Equations ». Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLY002/document.
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Cette thèse porte sur l'étude théorique et numérique des méthodes rapides pour résoudre les équations de l'élastodynamique 3D en domaine fréquentiel, et se place dans le cadre d'une collaboration avec la société Shell en vue d'optimiser la convergence des problèmes d'inversion sismique. La méthode repose sur l'utilisation des éléments finis de frontière (BEM) pour la discrétisation et sur les techniques de matrices hiérarchiques (H-matrices) pour l'accélération de la résolution du système linéaire. Dans le cadre de cette thèse on a développé un solveur direct pour les BEMs en utilisant une factorisation LU et un stockage hiérarchique. Si le concept des H-matrices est simple à comprendre, sa mise en oeuvre requiert des développements algorithmiques importants tels que la gestion de la multiplication de matrices représentées par des structures différentes (compressées ou non) qui ne comprend pas mois de 27 sous-cas. Un autre point délicat est l'utilisation des méthodes d'approximations par matrices compressées (de rang faible) dans le cadre des problèmes vectoriels. Une étude algorithmique a donc été faite pour mettre en oeuvre la méthode des H-matrices. Nous avons par ailleurs estimé théoriquement le rang faible attendu pour les noyaux oscillants, ce qui constitue une nouveauté, et montré que la méthode est utilisable en élastodynamique. En outre on a étudié l'influence des divers paramètres de la méthode en acoustique et en élastodynamique 3D, à fin de calibrer leur valeurs numériques optimales. Dans le cadre de la collaboration avec Shell, un cas test spécifique a été étudié. Il s'agit d'un problème de propagation d'une onde sismique dans un demi-espace élastique soumis à une force ponctuelle en surface. Enfin le solveur direct développé a été intégré au code COFFEE développé a POEMS (environ 25000 lignes en Fortran 90)This thesis focuses on the theoretical and numerical study of fast methods to solve the equations of 3D elastodynamics in frequency-domain. We use the Boundary Element Method (BEM) as discretization technique, in association with the hierarchical matrices (H-matrices) technique for the fast solution of the resulting linear system. The BEM is based on a boundary integral formulation which requires the discretization of the only domain boundaries. Thus, this method is well suited to treat seismic wave propagation problems. A major drawback of classical BEM is that it results in dense matrices, which leads to high memory requirement (O (N 2 ), if N is the number of degrees of freedom) and computational costs.Therefore, the simulation of realistic problems is limited by the number of degrees of freedom. Several fast BEMs have been developed to improve the computational efficiency. We propose a fast H-matrix based direct BEM solver
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Kulkarni, Mandar S. « Multi-coefficient Dirichlet Neumann type elliptic inverse problems with application to reflection seismology ». Birmingham, Ala. : University of Alabama at Birmingham, 2009. https://www.mhsl.uab.edu/dt/2010r/kulkarni.pdf.
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Thesis (Ph. D.)--University of Alabama at Birmingham, 2009.Title from PDF t.p. (viewed July 21, 2010). Additional advisors: Thomas Jannett, Tsun-Zee Mai, S. S. Ravindran, Günter Stolz, Gilbert Weinstein. Includes bibliographical references (p. 59-64).
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El, Bouzid Hassan. « Méthodes d'éléments finis raffinés pour quelques problèmes aux limites dans des domaines non-réguliers ». Valenciennes, 1996. https://ged.uphf.fr/nuxeo/site/esupversions/78c75dce-f930-4497-9077-c57356a9fac0.
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Dans ce travail, nous proposons différentes méthodes d'éléments finis raffines pour deux types de problèmes: le premier concerne des problèmes aux limites dans des polygones et le second le système de Stokes dans des polygones ou des polyèdres. Les solutions de ces problèmes présentent des singularités, en conséquence, l'utilisation d'une méthode d'éléments finis classique n'aboutit pas à un ordre optimal de convergence. En dimension 2, la forme explicite de ces singularités permet d'adopter certaines stratégies de façon à restaurer l'ordre optimal de convergence ; parmi celles-ci, citons la méthode de raffinement de maillage, la méthode d'adjonction de singularités et la méthode des fonctions singulières duales. En dimension 3, la structure des singularités étant plus complexe, seule la méthode de raffinement de maillage est actuellement applicable pour des domaines polyèdraux généraux. Pour le premier type de problèmes, les méthodes utilisées jusqu'à présent sont en général conformes, ce qui, dans le cas du problème des plaques, induit un cout de résolution assez important. Pour minimiser ce cout, nous proposons ici d'utiliser la méthode d'éléments finis non conformes raffinés. Sont étudiés notamment dans cette thèse: le problème de Dirichlet pour l'operateur de Laplace et l'équation des plaques encastrées. Pour le second, nous analysons une méthode d'éléments finis mixtes raffinés (conformes et non conformes) afin d'approcher la solution du système de Stokes. Les majorations d'erreurs optimales du cas des domaines reguliers sont démontrées pour les deux types de problèmes considérés.
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Ячменьов, Володимир Олександрович, Владимир Александрович Ячменев, Volodymyr Oleksandrovych Yachmenov et С. А. Терновский. « Решение начально-краевых задач для уравнений с дробными производными ». Thesis, Сумский государственный университет, 2014. http://essuir.sumdu.edu.ua/handle/123456789/39366.
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Исследование диффузионных процессов аномальной природы, отклоняющихся от классической гауссовской диффузии, приводит к необходимости решения начально-краевых задач для дифференциальных уравнений с дробными производными.
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Ladecký, Martin. « Isogeometrická analýza a její použití v mechanice kontinua ». Master's thesis, Vysoké učení technické v Brně. Fakulta stavební, 2018. http://www.nusl.cz/ntk/nusl-371938.
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Thesis deals with solving the problems of continuum mechanics by method of Isogeometric analysis. This relatively young method combines the advantages of precise NURBS geometry and robustness of the classical finite element method. The method is described on procedure of solving a plane Poissons boundary value problem. Solver is implemented in MatLab and algorithms are attached to the text.
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Joubert, Dominique. « Numerical methods for pricing American put options under stochastic volatility / Dominique Joubert ». Thesis, North-West University, 2013. http://hdl.handle.net/10394/10202.
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The Black-Scholes model and its assumptions has endured its fair share of criticism.
One problematic issue is the model’s assumption that market volatility is constant.
The past decade has seen numerous publications addressing this issue by adapting the
Black-Scholes model to incorporate stochastic volatility. In this dissertation, American
put options are priced under the Heston stochastic volatility model using the Crank-
Nicolson finite difference method in combination with the Projected Over-Relaxation
method (PSOR). Due to the early exercise facility, the pricing of American put options
is a challenging task, even under constant volatility. Therefore the pricing problem under
constant volatility is also included in this dissertation. It involves transforming the
Black-Scholes partial differential equation into the heat equation and re-writing the pricing
problem as a linear complementary problem. This linear complimentary problem is
solved using the Crank-Nicolson finite difference method in combination with the Projected
Over-Relaxation method (PSOR). The basic principles to develop the methods
necessary to price American put options are covered and the necessary numerical methods
are derived. Detailed algorithms for both the constant and the stochastic volatility
models, of which no real evidence could be found in literature, are also included in this
dissertation.MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013
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Acikgoz, Nazmiye. « Adaptive and Dynamic Meshing Methods for Numerical Simulations ». Diss., Georgia Institute of Technology, 2007. http://hdl.handle.net/1853/14521.
Texte intégralRésumé :
For the numerical simulation of many problems of engineering interest, it is desirable to have an automated mesh adaption tool. This is important especially for problems characterized by anisotropic features and require mesh clustering in the direction of high gradients. Another significant issue in meshing emerges in unsteady simulations with moving boundaries, where the boundary motion has to be accommodated by deforming the computational grid. Similarly, there exist problems where current mesh needs to be adapted to get more accurate solutions. To solve these problems, we propose three novel procedures.
In the first part of this work, we present an optimization procedure for three-dimensional anisotropic tetrahedral grids based on metric-driven h-adaptation. Through the use of topological and geometrical operators, the mesh is iteratively adapted until the final mesh minimizes a given objective function. We propose an optimization process based on an ad-hoc application of the simulated annealing technique, which improves the likelihood of removing poor elements from the grid. Moreover, a local implementation of the simulated annealing is proposed to reduce the computational cost.
Many challenging unsteady multi-physics problems are characterized by moving boundaries and/or interfaces. When the boundary displacements are large, degenerate elements are easily formed in the grid such that frequent remeshing is required. We propose a new r-adaptation technique that is valid for all types of elements (e.g., triangle, tet, quad, hex, hybrid) and deforms grids that undergo large imposed displacements at their boundaries. A grid is deformed using a network of linear springs composed of edge springs and a set of virtual springs. The virtual springs are constructed in such a way as to oppose element collapsing.
Both frequent remeshing, and exact-pinpointing of clustering locations are great challenges of numerical simulations, which can be overcome by adaptive meshing algorithms. Therefore, we conclude this work by defining a novel mesh adaptation technique where the entire mesh is adapted upon application of a force field in order to comply with the target mesh or to get more accurate solutions. The method has been tested for two-dimensional problems of a-priori metric definitions as well as for oblique shock clusterings.
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Brideson, Michael. « Electromagnetic induction tomography : a feasibility study ». Thesis, Queensland University of Technology, 2000.
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ROSSI, ELENA. « Balance Laws : Non Local Mixed Systems and IBVPs ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2016. http://hdl.handle.net/10281/103090.
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Scalar hyperbolic balance laws in several space dimensions play a central role in this thesis. First, we deal with a new class of mixed parabolic-hyperbolic systems on all R^n: we obtain the basic well-posedness theorems, devise an ad hoc numerical algorithm, prove its convergence and investigate the qualitative properties of the solutions. The extension of these results to bounded domains requires a deep understanding of the initial boundary value problem (IBVP) for hyperbolic balance laws. The last part of the thesis provides rigorous estimates on the solution to this IBVP, under precise regularity assumptions.
In Chapter 1 we introduce a predator-prey model. A non local and non linear balance law is coupled with a parabolic equation: the former describes the evolution of the predator density, the latter that of prey. The two equations are coupled both through the convective part of the balance law and the source terms. The drift term is a non local function of the prey density. This allows the movement of predators to be directed towards the regions where the concentration of prey is higher. We prove the well-posedness of the system, hence the existence and uniqueness of solution, the continuous dependence from the initial data and various stability estimates.
In Chapter 2 we devise an algorithm to compute approximate solutions to the mixed system introduced above. The balance law is solved numerically by a Lax-Friedrichs type method via dimensional splitting, while the parabolic equation is approximated through explicit finite-differences. Both source terms are integrated by means of a second order Runge-Kutta scheme. The key result in Chapter 2 is the convergence of this algorithm. The proof relies on a careful tuning between the parabolic and the hyperbolic methods and exploits the non local nature of the convective part in the balance law.
This algorithm has been implemented in a series of Python scripts. Using them, we obtain information about the possible order of convergence and we investigate the qualitative properties of the solutions. Moreover, we observe the formation of a striking pattern: while prey diffuse, predators accumulate on the vertices of a regular lattice.
The analytic study of the system above is on all R^n. However, both possible biological applications and numerical integrations suggest that the boundary plays a relevant role. With the aim of studying the mixed hyperbolic-parabolic system in a bounded domain, we noticed that for balance laws known results lack some of the estimates necessary to deal with the coupling. In Chapter 3 we then focus on the IBVP for a general balance law in a bounded domain. We prove the well-posedness of this problem, first with homogeneous boundary condition, exploiting the vanishing viscosity technique and the doubling of variables method, then for the non homogeneous case, mainly thanks to elliptic techniques. We pay particular attention to the regularity assumptions and provide rigorous estimates on the solution.
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Axelsson, Andreas, et kax74@yahoo se. « Transmission problems for Dirac's and Maxwell's equations with Lipschitz interfaces ». The Australian National University. School of Mathematical Sciences, 2002. http://thesis.anu.edu.au./public/adt-ANU20050106.093019.
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The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwells equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwells equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.
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Vogt, Andreas. « Analytische und numerische Untersuchung von direkten und inversen Randwertproblemen in Gebieten mit Ecken mittels Integralgleichungsmethoden ». [S.l.] : [s.n.], 2001. http://webdoc.sub.gwdg.de/diss/2002/vogt/vogt.pdf.
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Nouisri, Amine. « Identification paramétrique en dynamique transitoire : traitement d’un problème couplé aux deux bouts ». Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLN005/document.
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Les travaux de thèse portent sur l'identification paramétrique en dynamique transitoire à partir des mesures fortement bruitées, l'un des objectifs à long terme étant de proposer une méthode d’identification peu intrusive afin de pouvoir être implémentée dans des codes de calcul éléments finis commerciaux. Dans ce travail, le concept de l'erreur en relation de comportement modifiée a été retenu pour traiter le problème d’identification des paramètres matériau. La minimisation de la fonctionnelle coût sous contraintes débouche, dans le cas de la dynamique transitoire, sur un problème dit « aux deux bouts » dans lequel il s’agit de résoudre un problème différentiel spatio-temporel avec des conditions à la fois initiales et finales en temps. Il en résulte un problème couplé entre les champs direct et adjoint dont le traitement est délicat. Dans un premier temps, des méthodes précédemment développées telles que la « méthode de Riccati » et la « méthode de tirs » ont été étudiées. Il est montré que l’identification par ces méthodes est robuste même pour des mesures fortement corrompues, mais qu’elles sont limitées par la complexité d’implémentation dans un code industriel, des problèmes de conditionnement ou de coût de calcul. Dans un second temps, une approche itérative basée sur une méthode de sur-relaxation a été développée et comparée à celles précédemment mentionnées sur des exemples académiques, validant l’intérêt de cette nouvelle approche. Enfin, des comparaisons ont été menées entre cette technique et une variante « discrétisée » de la formulation introduite par Bonnet et Aquino [Inverse Problems, vol. 31, 2015]This thesis deals with parameters identification in transient dynamic in case of highly noisy experimental data. One long-term goal is the derivation of a non-intrusive method dedicated to the implementation in a commercial finite element code.In this work, the modified error in the constitutive relation framework is used to treat the identification of material parameters. The minimization of the cost function under constraints leads, in the case of transient dynamics, to a « two points boundary value problem » in which the differential space-time problem involves both initial and final time conditions. This results in a problem coupling the direct and adjoint fields, whose treatment is difficult.In the first part, methods such as those based on the « Riccati equations » and the « shooting methods » have been studied. It is shown that the identification is robust even in the case of highly corrupted measures, but these methods are limited either by the implementation intrusiveness, conditioning problems or the numerical cost.In the second part, an iterative over-relaxation approach is developed and compared to the aforementioned approaches on academic problems in order to validate the interest of the method. Finally, comparisons are carried out between this approach and a « discretized » variation of the formulation introduced by Bonnet and Aquino [Inverse Problems, vol. 31, 2015]
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Silva, Clovis Antonio da. « Um novo algoritmo, naturalmente paralelizável, para o cálculo de permeabilidades equivalentes em reservatórios ». Universidade do Estado do Rio de Janeiro, 2008. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=1252.
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Conselho Nacional de Desenvolvimento Científico e TecnológicoNeste trabalho é apresentado um novo procedimento numérico para o upscaling de permeabilidade utilizando condições de contorno periódicas. Este procedimento combina decomposição de domínio com elementos finitos mistos na discretização do problema local de pressão-velocidade necessário para se encontrar as permeabilidades equivalentes.
A new numerical method is proposed for the permeabilities upscaling take into consideration periodic boundary conditions. This method combines domain decomposition with mixed finite elements in discretization of the local problem of pressure-velocity necessary to meet the equivalent permeabilities.
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Docherty, Paul J. « Developing literary Glasgow : towards a strategy for a reading, writing and publishing city ». Thesis, University of Stirling, 2018. http://hdl.handle.net/1893/28083.
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Since the 1990s, urban cultural policy in the UK has been bound to the cause of urban regeneration. Much has been written in examination and critique of this relationship, but what happens when the direction of strategic attention is reversed and civic leadership seeks to regenerate culture itself? The city of Glasgow, having made capital of culture over many decades, has moved towards a strategy for the development of literary Glasgow. This thesis documents a search for those factors crucial to that strategy. The research focuses on literary Glasgow as one aspect of the city’s cultural sector; identifies and examines gaps in the relationship between the civic cultural organisation and literary communities; and highlights those elements vital to the formation of a strategy for development of the literary in Glasgow. An extended period of participatory ethnographic research within the Aye Write! book festival and Sunny Govan Community Radio, is supplemented with data from interviews conducted across the literary sector and analysis of organisational documentation. Through these a gap has been identified between the policies and operations of a civic cultural organisation, and the desires of those engaged within the literary community. This gap is caused, in part, by the lack of a mechanism with which to reconcile contrasting narratives about the cultural essence of the city, or to negotiate the variations in definitions of value in relation to cultural engagement. The interdisciplinary approach builds upon insights from existing work within publishing studies, cultural policy, complexity theory and organisational studies to construct an understanding of the dynamics of Glasgow’s literary sector. This reveals the need for a framework in support of a landscape of practice, a desire for the placement of boundary objects to facilitate engagement, and the significance of value in relation to participation in literary activity. This work informs a strategy for literary Glasgow and contributes to conversations on strategies for cultural development in other cities.
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Nguyen, Thi Nhu Thao. « Modélisation mathématique et simulation de la dynamique spatiale de populations de campagnols dans l’est de la France ». Thesis, Bourgogne Franche-Comté, 2020. http://www.theses.fr/2020UBFCD031.
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L'objectif principal de la thèse est de proposer et d'analyser des modèles mathématiques basés sur des équations aux dérivées partielles (EDP) afin de décrire la dynamique spatiale de deux espèces de campagnols (Microtus arvalis et Arvicola terrestris), qui sont particulièrement surveillés dans l'est de la France. Les modèles que nous avons proposés reposent sur des EDP qui décrivent l'évolution de la densité de la population de campagnols en fonction du temps, de l'âge et de la position dans l'espace. Nous avons suivi deux approches complémentaires pour représenter la dynamique. Dans la première approche, nous avons proposé un premier modèle qui consiste en une EDP scalaire en structurée en temps, en âge, et en espace. Elle est complétée par une condition au bord non locale. Le flux est linéaire à coefficient constant dans la direction de l'âge mais contient un terme non local dans les directions de l'espace. De plus, l'équation contient un terme de second ordre par rapport aux variables spatiales. Nous avons démontré l'existence et la stabilité de solutions faibles entropiques pour le modèle en utilisant, la compacité par compensation établie par Panov et un argument du type doublement de variables. Dans la deuxième approche, nous nous sommes inspirés du modèle Multi Agents introduit par Marilleau-Lang-Giraudoux, où la dynamique spatiale des juvéniles est découplée de l'évolution locale dans chaque parcelle. Pour mettre en place ce deuxième modèle, nous avons introduit un graphe orienté dont les nœuds sont les parcelles (ou colonies). Dans chaque nœud, l'évolution de la colonie est décrite par une équation de transport structurée en temps et en âge, et les mouvements de dispersion dans l'espace sont représentés par les passages d'un nœud à un autre. Nous avons proposé une discrétisation du modèle, par des schéma volumes finis, et, grâce à des simulations numériques, nous avons pu illustrer le fait que le modèle est capable de reproduire certaines caractéristiques qualitatives de la dynamique spatiale observée dans la nature. Nous avons ensuite proposé un troisième modèle qui est un système proie-prédateur constitué d'une équation hyperbolique pour les prédateurs et d'une équation parabolique-hyperbolique pour les proies analogue à celle proposée dans le premier modèle. Le terme de force dans l'équation des prédateurs dépend de manière non localement de la densité des proies et les deux équations sont également couplées via des termes sources classiques de type Lotka-Volterra. Nous avons établi l'existence de solutions en appliquant la méthode de la viscosité évanescente, et nous avons établi un résultat de stabilité par un argument de type doublement de variables. Enfin nous avons proposé et validé un schéma de type volumes finis pour le premier modèle.La dernière partie de mes travaux de recherche est dédiée à un projet auquel j'ai participé lors d'une école d'été CEMRACS. Il concerne un sujet de biomathématiques différent du thème principal de la thèse et porte sur un modèle épidémiologique pour la salmonellose. Nous avons proposé un nouveau cadre de modélisation générique multi-échelles de la transmission hétérogène d'agents pathogènes dans une population animale. Au niveau intra-hôte, le modèle décrit l'interaction entre le microbiote commensal, le pathogène et la réponse inflammatoire. Des fluctuations aléatoires de la dynamique écologique du microbiote individuel et de la transmission à l'échelle inter-hôte sont ajoutées pour obtenir un modèle EDP de la distribution des agents pathogènes au niveau de la population. Une extension du modèle a, par ailleurs, été développé pour représenter la transmission entre plusieurs populations. Le comportement asymptotique ainsi que l'impact des stratégies de contrôle, y compris le nettoyage et l'administration d'antimicrobiens, sont étudiés par des simulations numériquesThe main objective of the thesis is to propose and analyze mathematical models based on partial differential equations (PDE) to describe the spatial dynamics of two species of voles (Microtus arvalis and Arvicola terrestris), which are particularly monitored in Eastern France. The models that we have proposed are based on PDE which describe the evolution of the density of the population of voles as a function of time, age and position in space. We have two complementary approaches to represent the dynamics. In the first approach, we propose a first model that consists of a scalar PDE depending on time, age, and space supplemented with a non-local boundary condition. The flux is linear with constant coefficient in the direction of age but contains a non-local term in the directions of space. Moreover, the equation contains a second order term in the spatial variables only. We have demonstrated the existence and stability of weak entropy solutions for the model by using, respectively, the Panov's theorem of the multidimensional compensated and a doubling of the variables type argument. In the second approach we were inspired by a Multi Agent model proposed by Marilleau-Lang-Giraudoux, where the spatial dynamics of juveniles is decoupled from local evolution in each plot. To apply this model, we have introduced a directed graph whose nodes are the plots. In each node, the evolution of the colony is described by a transport equation with two variables, time and age, and the movements of dispersion, in space, are represented by the passages from one node to the other. We have proposed a discretization of the model, by finite volume methods, and noticed that this approach manages to reproduce the qualitative characteristics of the spatial dynamics observed in nature. We also proposed to consider a predator-prey system consisting of a hyperbolic equation for predators and a parabolic-hyperbolic equation for preys, where the prey's equation is analogous to the first model of the vole populations. The drift term in the predators' equation depends nonlocally on the density of prey and the two equations are also coupled via classical source terms of Lotka-Volterra type. We establish existence of solutions by applying the vanishing viscosity method, and we prove stability by a doubling of variables type argument. Moreover, concerning the numerical simulation of the first model in one-dimensional space, we obtain a finite volume discretization by using the upwind scheme and then validate the numerical scheme.The last part of my thesis work is a project in which I participated during a Summer school CEMRACS. The project was on a subject of biomathematics different from that of the thesis (an epidemiological model for salmonellosis). A new generic multi-scale modeling framework for heterogeneous transmission of pathogens in an animal population is suggested. At the intra-host level, the model describes the interaction between the commensal microbiota, the pathogen and the inflammatory response. Random fluctuations in the ecological dynamics of the individual microbiota and transmission at the inter-host scale are added to obtain a PDE model of drift-diffusion of pathogen distribution at the population level. The model is also extended to represent transmission between several populations. Asymptotic behavior as well as the impact of control strategies, including cleaning and administration of antimicrobials, are studied by numerical simulation
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Hartmann, Marcelo. « Métodos de Monte Carlo Hamiltoniano na inferência Bayesiana não-paramétrica de valores extremos ». Universidade Federal de São Carlos, 2015. https://repositorio.ufscar.br/handle/ufscar/4601.
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Previous issue date: 2015-03-09In this work we propose a Bayesian nonparametric approach for modeling extreme value data. We treat the location parameter _ of the generalized extreme value distribution as a random function following a Gaussian process model (Rasmussem & Williams 2006). This configuration leads to no closed-form expressions for the highdimensional posterior distribution. To tackle this problem we use the Riemannian Manifold Hamiltonian Monte Carlo algorithm which allows samples from the posterior distribution with complex form and non-usual correlation structure (Calderhead & Girolami 2011). Moreover, we propose an autoregressive time series model assuming the generalized extreme value distribution for the noise and obtained its Fisher information matrix. Throughout this work we employ some computational simulation studies to assess the performance of the algorithm in its variants and show many examples with simulated and real data-sets.
Neste trabalho propomos uma abordagem Bayesiana não-paramétrica para a modelagem de dados com comportamento extremo. Tratamos o parâmetro de locação _ da distribuição generalizada de valor extremo como uma função aleatória e assumimos um processo Gaussiano para tal função (Rasmussem & Williams 2006). Esta situação leva à intratabilidade analítica da distribuição a posteriori de alta dimensão. Para lidar com este problema fazemos uso do método Hamiltoniano de Monte Carlo em variedade Riemanniana que permite a simulação de valores da distribuição a posteriori com forma complexa e estrutura de correlação incomum (Calderhead & Girolami 2011). Além disso, propomos um modelo de série temporal autoregressivo de ordem p, assumindo a distribuição generalizada de valor extremo para o ruído e determinamos a respectiva matriz de informação de Fisher. No decorrer de todo o trabalho, estudamos a qualidade do algoritmo em suas variantes através de simulações computacionais e apresentamos vários exemplos com dados reais e simulados.
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Franceschi, Sandro. « Approche analytique pour le mouvement brownien réfléchi dans des cônes ». Thesis, Tours, 2017. http://www.theses.fr/2017TOUR4046/document.
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Le mouvement Brownien réfléchi de manière oblique dans le quadrant, introduit par Harrison, Reiman, Varadhan et Williams dans les années 80, est un objet largement analysé dans la littérature probabiliste. Cette thèse, qui présente l’étude complète de la mesure invariante de ce processus dans tous les cônes du plan, a pour objectif plus global d’étendre au cadre continu une méthode analytique développée initialement pour les marches aléatoires dans le quart de plan par Fayolle, Iasnogorodski et Malyshev dans les années 70. Cette approche est basée sur des équations fonctionnelles, reliant des fonctions génératrices dans le cas discret et des transformées de Laplace dans le cas continu. Ces équations permettent de déterminer et de résoudre des problèmes frontière satisfaits par ces fonctions génératrices. Dans le cas récurrent, cela permet de calculer explicitement la mesure invariante du processus avec rebonds orthogonaux, dans le chapitre 2, et avec rebonds quelconques, dans le chapitre 3. Les transformées de Laplace des mesures invariantes sont prolongées analytiquement sur une surface de Riemann induite par le noyau de l’équation fonctionnelle. L’étude des singularités et l’application de méthodes du point col sur cette surface permettent de déterminer l’asymptotique complète de la mesure invariante selon toutes les directions dans le chapitre 4Obliquely reflected Brownian motion in the quadrant, introduced by Harrison, Reiman, Varadhan and Williams in the eighties, has been studied a lot in the probabilistic literature. This thesis, which presents the complete study of the invariant measure of this process in all the cones of the plan, has for overall aim to extend to the continuous framework an analytic method initially developped for random walks in the quarter plane by Fayolle, Iasnogorodski and Malyshev in the seventies. This approach is based on functional equations which link generating functions in the discrete case and Laplace transform in the continuous case. These equations allow to determine and to solve boundary value problems satisfied by these generating functions. In the recurrent case, it permits to compute explicitly the invariant measure of the process with orthogonal reflexions, in the chapter 2, and with any reflexions, in the chapter 3. The Laplace transform of the invariant measure is analytically extended to a Riemann surface induced by the kernel of the functional equation. The study of singularities and the use of saddle point methods on this surface allows to determine the full asymptotics of the invariant measure along every directions in the chapter 4
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FRASCA, CACCIA GIANLUCA. « A new efficient implementation for HBVMs and their application to the semilinear wave equation ». Doctoral thesis, 2015. http://hdl.handle.net/2158/992629.
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In this thesis we have provided a detailed description of the low-rank Runge-Kutta family of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of Hamiltonian problems. In particular, we have studied in detail their main property: the conservation of polynomial Hamiltonians, which results into a practical conservation for generic suitably regular Hamiltonians. This property turns out to play a fundamental role in some problems where the error on the Hamiltonian, usually obtained even when using a symplectic method, would be not negligible to the point of affecting the dynamics of the numerical solution.
The research developed in this thesis has addressed two main topics. The first one is a new procedure, based on a particular splitting of the matrix defining the method, which turns out to be more effective of the well-known blended-implementation, as well as of a classical fixed-point iteration when the problem at hand is stiff. This procedure has been applied also to second order problems with separable Hamiltonian function, resulting in a cheaper computational cost.
The second topic addressed is the application of HBVMs for the full discretization of a method of lines approach to numerically solve Hamiltonian PDEs. In particular, we have considered the semilinear wave equation coupled with either periodic, Dirichlet or Neumann boundary conditions, and the application of a (practically) energy conserving HBVM method to the semi-discrete problem obtained by means of a second order finite-difference approximation in space. When the problem is coupled with periodic boundary conditions we have also considered the case of higher-order finite-difference spatial discretizations and the case when a Fourier-Galerkin method is used for the spatial semi-discretization. The proposed methods are able to provide a numerical solution such that the energy (which can be conserved or not, depending on the assigned boundary conditions) practically satisfies its prescribed variation in time. A few numerical tests for the sine-Gordon equation have given evidence that, for some problems, there is an effective advantage in using an energy-conserving method for the time integration, with respect to the use of a symplectic one. Moreover, even though HBVMs are implicit method, their computational cost for the considered problem turns out to be competitive even with respect to that of explicit solvers of the same order, which, furthermore, may suffer from stepsize restrictions due to stability reasons, whereas HBVMs are A-stable.
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(5930024), Kshitij Mall. « Advancing Optimal Control Theory Using Trigonometry For Solving Complex Aerospace Problems ». Thesis, 2019.
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Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.
In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.
This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.
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Chen, Chien-Chou, et 陳建州. « Semi-Analytic Method for Boundary Value Problems of ODEs ». Thesis, 2005. http://ndltd.ncl.edu.tw/handle/97394865236013850652.
Texte intégralRésumé :
碩士國立中山大學
應用數學系研究所
93
In this thesis, we demonstrate the capability of power series, combined with numerical methods, to solve boundary value problems and Sturm-Liouville eigenvalue problems of ordinary differential equations. This kind of schemes is usually called the numerical-symbolic, numerical-analytic or semi-analytic method. In the first chapter, we develop an adaptive algorithm, which automatically decides the terms of power series to reach desired accuracy. The expansion point of power series can be chosen freely. It is also possible to combine several power series piecewisely. We test it on several models, including the second and higher order linear or nonlinear differential equations. For nonlinear problems, the same procedure works similarly to linear problems. The only differences are the nonlinear recurrence of the coefficients and a nonlinear equation, instead of linear, to be solved. In the second chapter, we use our semi-analytic method to solve singularly perturbed problems. These problems arise frequently in fluid mechanics and other branches of applied mathematics. Due to the existence of boundary or interior layers, its solution is very steep at certain point. So the terms of series need to be large in order to reach the desired accuracy. To improve its efficiency, we have a strategy to select only a few required basis from the whole polynomial family. Our method is shown to be a parameter diminishing method. A specific type of boundary value problem, called the Sturm-Liouville eigenvalue problem, is very important in science and engineering. They can also be solved by our semi-analytic method. This is our focus in the third chapter. Our adaptive method works very well to compute its eigenvalues and eigenfunctions with desired accuracy. The numerical results are very satisfactory.
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Kao, Jeng-Hung, et 高政宏. « Regularized meshless method for boundary value problems with multiply-connected domain ». Thesis, 2006. http://ndltd.ncl.edu.tw/handle/48092373281516078268.
Texte intégralRésumé :
碩士國立臺灣海洋大學
河海工程學系
93
Abstract In this thesis, the regularized meshless method is adopted to solve Laplace problems and eigenproblems with multiply-connected domain, respectively. Here, the solution is represented by using double-layer potential. The subtracting and adding-back technique is used to regularize the singularity and hypersingularity of the kernel functions. Only boundary nodes on the real boundary are required by using the proposed technique in a different way of conventional MFS by putty singularities on fictition boundaries. A linear algebraic equation is obtained free of mesh generation. After matching boundary conditions, the unknown densities in the algebraic system can be easily determined. Test of convergence and sensitivity study of the proposed method are also done. Finally, several engineering problems including multiple inclusions problem under antiplane shear, piezoelectricity problems with multiple inclusions and multiply-connected acoustic eigenproblem, were given to demonstrate the validity of the proposed method. Numerical results agree well after comparing with the available exact solution and those of boundary element method, point-matching method and finite element method. A general-purpose program for multiple cavities and inclusions of various shapes and arbitrary positions was developed. Keywords: Laplace problem, multiple holes, multiply-connected domain, eigenproblem, method of fundamental solution, Regularized meshless method, inclusion, piezoelectricity, eigenproblem, antiplane shear