Ce document traite premièrement des diverses tentatives de modélisation et de simulation de la nage anguilliforme puis élabore une nouvelle technique, basée sur la méthode de la frontière immergée généralisée et la théorie des poutres de Reissner-Simo. Cette dernière, comme les équations des fluides polaires, est dérivée de la mécanique des milieux continus puis les équations obtenues sont discrétisées afin de les amener à une résolution numérique. Pour la première fois, la théorie des schémas de Runge-Kutta additifs est combinée à celle des schémas de Runge-Kutta-Munthe-Kaas pour engendrer une méthode d’ordre de convergence formel arbitraire. De plus, les opérations d’interpolation et d’étalement sont traitées d’un nouveau point de vue qui suggère l’usage des splines interpolatoires nodales en lieu et place des fonctions d’étalement traditionnelles. Enfin, de nombreuses vérifications numériques sont faites avant de considérer les simulations de la nage. / This paper first discusses various attempts at modeling and simulating anguilliform swimming, then we develop a new technique, based on a method of generalized immersed boundary and the beam theory of Reissner-Simo. Subsequent to the derivation of the equations of polar fluids, the beam theory is derived from continuum mechanics and the resulting equations are then discretized, allowing a numerical solution. For the first time, the theory of additive Runge-Kutta schemes are combined with the Runge-Kutta-Munthe-Kaas method to generate schemes of arbitrarily high formal order of convergence. Moreover, the interpolation and spreading operations are handled from a new point of view that suggests the use of interpolatory nodal splines instead of spreading traditional functions. Finally, many numerical verifications are done before considering simulations of swimming.

Tato disertační práce se zabývá analýzou stability a konvergence klasických numerických metod pro řešení obyčejných diferenciálních rovnic. Jsou představeny klasické jednokrokové metody, jako je Eulerova metoda, Runge-Kuttovy metody a nepříliš známá, ale rychlá a přesná metoda Taylorovy řady. V práci uvažujeme zobecnění jednokrokových metod do vícekrokových metod, jako jsou Adamsovy metody, a jejich implementaci ve dvojicích prediktor-korektor. Dále uvádíme generalizaci do vícekrokových metod vyšších derivací, jako jsou např. Obreshkovovy metody. Dvojice prediktor-korektor jsou často implementovány v kombinacích modů, v práci uvažujeme tzv. módy PEC a PECE. Hlavním cílem a přínosem této práce je nová metoda čtvrtého řádu, která se skládá z dvoukrokového prediktoru a jednokrokového korektoru, jejichž formule využívají druhých derivací. V práci je diskutována Nordsieckova reprezentace, algoritmus pro výběr proměnlivého integračního kroku nebo odhad lokálních a globálních chyb. Navržený přístup je vhodně upraven pro použití proměnlivého integračního kroku s přístupe vyšších derivací. Uvádíme srovnání s klasickými metodami a provedené experimenty pro lineární a nelineární problémy.

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.

La simulation numérique de systèmes d’équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L’objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l’intégrateur numérique, et au niveau de la décomposition de l’intervalle de temps. Nous montrons l’efficacité limitée au nombre d’étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l’introduction d’un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d’abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l’erreur globale, nous permettent de construire une extension parallèle de l’extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résidu. / This PhD Thesis deals with the development of parallel numerical methods for solving Ordinary and Algebraic Differential Equations. ODE and DAE are commonly arising when modeling complex dynamical phenomena. We first show that the parallelization across the method is limited by the number of stages of the RK method or DIMSIM. We introduce the Schur complement into the linearised linear system of time integrators. An automatic framework is given to build a mask defining the relationships between the variables. Then the Schur complement is coupled with Jacobian Free Newton-Krylov methods. As time decomposition, global time steps resolutions can be solved by parallel nonlinear solvers (such as fixed point, Newton and Steffensen acceleration). Two steps time decomposition (Parareal, Pita,...) are developed with a new definition of their grids to solved stiff problems. Global error estimates, especially the Richardson extrapolation, are used to compute a good approximation for the second grid. Finally we propose a parallel deferred correction

Made available in DSpace on 2014-10-09T12:51:51Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:06:32Z (GMT). No. of bitstreams: 0 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Tese (Doutoramento) / IPEN/T / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP / FAPESP:00/11446-0

Made available in DSpace on 2014-10-09T12:51:51Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:06:32Z (GMT). No. of bitstreams: 0 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Foram estudados os processos de transferência de energia (TE) dos níveis 4I13/2 ; 4I11/2 do Er3+ e 5I7 ; 5I6 do Ho3+ no LiYF4 (YLF) e no ZBLAN, para os íons desativadores Nd3+, Tb3+ ou Eu3+. Os microparâmetros de interação dessas TE foram obtidos utilizandose o método da integral de sobreposição e os resultados indicaram que o íon de Eu3+ é o melhor desativador do primeiro estado excitado do Ho3+ no YLF e o Nd3+ é o mais eficiente desativador do Er3+ no YLF e ZBLAN. A dependência temporal das fluorescências do Er3+ em 1,5 ; 2,7 m e do Ho3+ em 2,1 ; 2,9 m foram medidas utilizandose excitações laser pulsadas provenientes do Nd-YAG+2 +OPO sintonizável. Foi proposto um critério geral para a discriminação dos processos de TE assistidos pela migração da excitação entre doadores (difusão ou saltos). Verificou-se que o modelo de migração por difusão ajusta melhor os processos de TE do segundo estado excitado do doador (Er3+ or Ho3+) independentemente da razão CD-D / CD-A, enquanto que o modelo de migração por saltos aplica-se ao primeiro estado excitado do doador. Foi proposta uma modificação no modelo de saltos que descreve os resultados experimentais para sistemas com CD-D / CD-A> 10. Utilizando-se os parâmetros de TE verificamos que os melhores sistemas para a ação laser em 3µm são: Ho:Eu:YLF, Ho:Nd:YLF e Er:Nd:YLF. Por outro lado, verifica-se que os sistemas com baixas concentrações (1,5mol%) de Er:Nd; Er:Tb e Er:Eu no ZBLAN não apresentaram potencial de inversão de população para a ação laser em 2,8 m. Os processos de conversão ascendente no sistema Er:YLF foram estudados em função da concentração de Er3+, sendo que os processos de absorção de estado excitado (ESA) e de conversão ascendente por transferência de energia (ETU) foram discriminados utilizando as curvas de decaimento resolvidas no tempo. Observou-se que o comprimento de onda de excitação em 980nm é o mais adequado para o bombeamento do sistema Er:YLF para a emissão laser quase contínua (cw) em 2,8 m. A técnica de pump-probe foi utilizada a fim de serem investigados os efeitos nos tempos de vida do sistema Er:YLF, sendo verificada uma diminuição da contribuição da migração da excitação nos tempos de vida do primeiro e segundo estados excitados do Er3+ no YLF com o aumento da potência de bombeamento cw ( pump ). As taxas experimentais de TE, determinadas para os melhores sistemas por meio do parâmetro RN, foram utilizadas no sistema de equações de taxa e, resolvendo-as pelo método numérico de Runge-Kutta de 4a ordem, pudemos avaliar a densidade de população invertida para as emissões em 2,8 m do Er3+ e 2,9 m do Ho3+ em função das taxas de bombeamento e das concentrações dos íons ativador e desativador. A melhor concentração do sistema Er:YLF foi de 20mol% e no sistema Er:Nd:YLF foi de 4mol% de Er3+ e 1,5mol% de Nd3+. Verificou-se que o sistema Er(4mol%):Nd(1.5mol%):YLF favorece o aumento da freqüência máxima de operação de 14Hz no Er:YLF para 391Hz baseando-se na medida do tempo de vida do nível laser inferior (4I13/2) do Er3+. O sistema Ho3+ no YLF é otimizado utilizando-se 0,6mol% de Ho3+ e 1,5mol% de Nd3+ ou pelo sistema Ho(3mol%):Eu(1,2mol%):YLF, sendo que a densidade de população invertida observada no sistema Ho(3mol%):Eu(1,2mol%) é 2,4 vezes maior que no sistema Ho(0,6mol%):Nd(1,5mol%). / Tese (Doutoramento) / IPEN/T / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP / FAPESP:00/11446-0

In this thesis the Ramberg-Osgood nonlinear model for describing the behavior of many different materials is investigated. A brief overview of the model as it is currently used in the literature is undertaken and several misunderstandings and possible pitfalls in its application is pointed out, especially as it pertains to more recent approaches to finding solutions involving the model. There is an investigation of the displacement of a cantilever beam under a combined loading consisting of a distributed load across the entire length of the beam and a point load at its end and new solutions to this problem are provided with a mixture of numerical techniques, which suggest strong mathematical consistency within the model for all theoretical assumptions made. A physical experiment was undertaken and the results prove to be inaccurate when using parameters derived from tensile tests, but when back calculating parameters from the beam test the model has a 14.40% error at its extreme against the experimental data suggesting the necessity for further testing.

Thesis is dedicated to evaluation of roundoff impact on numerical integration methods accuracy and effectivity. Contains theoretical expectations taken from existing literature, implementation of chosen methods, experimental measurement of attained accuracy under different circumstances and their comparison with regard to time complexity. Library contains Runge-Kutta methods to order 7 and Adams-Bashforth methods to order 20 implemented using C++ templates which allow optional arbitrary-precision arithmetic. Small models with known analytic solution were used for experiments.

This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)

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