Magistro darbe pateikiami nauji rezultatai, gauti tiriant diskretųjį Šturmo ir Liuvilio uždavinį su viena klasikine (arba Noimano) ir antra nelokalia dvitaške kraštine sąlyga. Analitinėje dalyje pateikiama teorija, reikalinga nagrinėjamo uždavinio tyrimui ir trumpai pristatomi rezultatai, gauti tiriant panašų uždavinį. Projektinėje dalyje ištirta diferencialinio uždavinio ir baigtinių skirtumų schemų kompleksinės spektro dalies priklausomybė nuo nelokaliųjų kraštinių sąlygų parametrų $\gamma$ ir $\xi$. Dauguma tyrimo rezultatų pateikiama kompleksinės ir realiosios charakteristinių funkcijų grafikais. / In this Master thesis presented new results which are got investigated the Sturm--Liouville problem with one classical (or Neumann) and another two-point nonlocal boundary condition. In the analytical part are presented the theory, which is necessary to study the problem and presented the results of investigation a similar task. In the design part are investigated the spectrum in complex plane depends on the nonlocal boundary conditions parameters $\gamma$ and $\xi$ in differential problem and in the finite difference schemes. Simulation results are presented as graphs of complex-real characteristic functions.

While computed tomographic angiography (CTA) has emerged as a powerful noninvasive option that allows for direct visualization of arterial stenosis(AS), it cant assess the hemodynamic abnormality caused by an AS. Alternatively, trans-stenotic pressure gradient (TSPG) and fractional flow reserve (FFR) are well-validated hemodynamic indices to assess the ischemic severity of an AS. However, they have significant restriction in practice due to invasiveness and high cost. To fill the gap, a new computational modality, called <i>InVascular</i> has been developed for non-invasive quantification TSPG and/or FFR based on patient's CTA, aiming to quantify the hemodynamic abnormality of the stenosis and help to assess the therapeutic/surgical benefits of treatment for the patient. Such a new capability gives rise to a potential of computation aided diagnostics and therapeutics in a patient-specific environment for ASs, which is expected to contribute to precision planning for cardiovascular disease treatment. <i>InVascular</i> integrates a computational modeling of diseases arteries based on CTA and Doppler ultrasonography data, with cutting-edge Graphic Processing Unit (GPU) parallel-computing technology. Revolutionary fast computing speed enables noninvasive quantification of TSPG and/or FFR for an AS within a clinic permissible time frame. In this work, we focus on the implementation of inlet and outlet boundary condition (BC) based on physiological image date and and 3-element Windkessel model as well as lumped parameter network in volumetric lattice Boltzmann method. The application study in real human coronary and renal arterial system demonstrates the reliability of the in vivo pressure quantification through the comparisons of pressure waves between noninvasive computational and invasive measurement. In addition, parametrization of worsening renal arterial stenosis (RAS) and coronary arterial stenosis (CAS) characterized by volumetric lumen reduction (S) enables establishing the correlation between TSPG/FFR and S, from which the ischemic severity of the AS (mild, moderate, or severe) can be identified. In this study, we quantify TSPG and/or FFR for five patient cases with visualized stenosis in coronary and renal arteries and compare the non-invasive computational results with invasive measurement through catheterization. The ischemic severity of each AS is predicted. The results of this study demonstrate the reliability and clinical applicability of <i>InVascular</i>.

Dans cette thèse, nous étudions la convergence de méthode de type éléments finis pour les équations de Maxwell en régime harmonique avec condition au bord d'impédance et l'équation de Helmholtz avec une couche parfaitement absorbante(PML). On étudie en premier, la formulation régularisée de l'équation de Maxwell en régime harmonique avec condition au bord d'impédance (qui consiste à ajouter le term ∇ div à l'équation originale pour avoir un problème elliptique) et on garde la condition d'impédance comme une condition au bord essentielle. Pour des domaines à bord régulier, le caractère bien posé de cette formulation est bien connu mais cela n'est pas le cas pour des domaines polyédraux convexes. On commence alors le premier chapitre par la preuve du caractère bien posé dans le cas du polyèdre convexe, qui est basé sur le fait que l'espace variationnel est inclus dans H¹. Dans le but d'avoir des estimations explicites en le nombre d'onde k de ce problème, il est obligatoire d'avoir des résultats de stabilité explicites en ce nombre d'onde. C'est aussi proposé, pour quelques situations particulières, dans ce chapitre. Dans le second chapitre on décrit les singularités d'arêtes et de coins pour notre problème. On peut alors déduire la régularité de la solution du problème original, ainsi que de son adjoint. On a tous les ingrédients pour proposer une analyse de convergence explicite en k pour une méthode d'éléments finis avec éléments de Lagrange. Dans le troisième chapitre, on considère une méthode d'éléments finis hp non conforme pour un domaine à bord régulier. Pour obtenir des estimations explicites en k, on introduit un résultat de décomposition, qui sépare la solution du problème original (ou de son adjoint) en une partie régulière mais fortement oscillante et une partie moins régulière mais peu oscillante. Ce résultat permet de montrer des estimations explicites en k. Le dernier chapitre est dédié à l'équation de Helmholtz avec une PML. L'équation de Helmholtz dans l'espace entier est souvent utilisée pour modéliser la diffraction d'onde acoustique (en régime harmonique), avec la condition de radiation à l'infini de Sommerfeld. L'ajout d'une PML est une façon pour passer d'un domaine infini à un domaine fini, elle correspond à l'ajout d'une couche autour du domaine de calcul qui absorbe très vite toutes les ondes sortantes. On propose en premier un résultat de stabilité explicite en k. On propose alors deux schémas numériques, une méthode d'éléments finis hp et une méthode multi- échelle basée sur un sous-espace local de correction. Le résultat de stabilité est utilisé pour mettre en relation de choix des paramètres des méthodes numériques considérées avec k. Nous montrons aussi des estimations d'erreur a priori. A la fin de ces chapitres, des tests numériques sont proposés pour confirmer nos résultats théoriques. / In this thesis, we propose wavenumber explicit convergence analyses of some finite element methods for time-harmonic Maxwell's equations with impedance boundary condition and for the Helmholtz equation with Perfectly Matched Layer (PML). We first study the regularized formulation of time-harmonic Maxwell's equations with impedance boundary conditions (where we add a ∇ div-term to the original equation to have an elliptic problem) and keep the impedance boundary condition as an essential boundary condition. For a smooth domain, the wellposedness of this formulation is well-known. But the well-posedness for convex polyhedral domain has been not yet investigated. Hence, we start the first chapter with the proof of the well-posedness in this case, which is based on the fact that the variational space is embedded in H¹. In order to perform a wavenumber explicit error analysis of our problem, a wavenumber explicit stability estimate is mandatory. We then prove such an estimate for some particular configurations. In the second chapter, we describe the corner and edge singularities for such problem. Then we deduce the regularity of the solution of the original and the adjoint problem, thus we have all ingredients to propose a explicit wavenumber convergence analysis for h-FEM with Lagrange element. In the third chapter, we consider a non conforming hp-finite element approximation for domains with a smooth boundary. To perform a wavenumber explicit error analysis, we split the solution of the original problem (or its adjoint) into a regular but oscillating part and a rough component that behaves nicely for large frequencies. This result allows to prove convergence analysis for our FEM, again explicit in the wavenumber. The last chapter is dedicated to the Helmholtz equation with PML. The Helmholtz equation in full space is often used to model time harmonic acoustic scattering problems, with Sommerfeld radiation condition at infinity. Adding a PML is a way to reduce the infinite domain to a finite one. It corresponds to add an artificial absorbing layer surrounding a computational domain, in which scattered wave will decrease very quickly. We first propose a wavenumber explicit stability result for such problem. Then, we propose two numerical discretizations: an hp-FEM and a multiscale method based on local subspace correction. The stability result is used to relate the choice of the parameters in the numerical methods to the wavenumber. A priori error estimates are shown. At the end of each chapter, we perform numerical tests to confirm our theoritical results.

Acoustic plane wave intearctions with a rough seabed with a continuously varying density and sound speed in a fluid-like sediment layer overlying an elastic basement is considered in this thesis. The acoustic properties in the sediment layer possess an exponential type of variation in density and one of the three classes of sound speed profiles, which are constant, k^2-linear, or inverse-square variations. Analytical solutions for the Helmholtz equation in the sediment layer, combined with a formulation based upon boundary perturbation theory, facilitate numerical implementation for the solution of coherent field. The coherent reflection coefficients corresponding to the aformentioned density and sound speed profiles for various frequencies, roughness parameters, basement stiffness, are numerically generated and analyzed. Physical interpretations are provided for various results. This simple model characterizes three important features of an realistic sea floor, including seabed roughness, sediment inhomogenieties, and basement shear property,%Two dimensions is considered in the seafloor environment and the random roughness is belong to one dimension space.% , therefore, provides a canonical model for the study of seabed acoustics. The variation of the acoustic properties takes such a form that it is not only geologically realistic, but also renders analytical solutions for the Helmholtz equation, thus facilitating the formulation of the problem. The computational algorithm for the spatial spectrum of the scattered field due to random seabed has been developed based upon a boundary perturbation method. %About scattering field, only one time reflection from the sediment is taked account of, because the higher numerical order is, the lower scattering energy exist.% The results have shown that, while the coherent field mainly depends upon the gross structure of the rough seabed represented by the RMS roughness, the scattered field heavily depends upon the details of the roughness structure specialized by the roughness power spectrum and the spatial correlation length of the rough surface. The dependence of the spatial spectrum on the sediment stratification is also carefully examined.

Exact analytical solutions for the quasistationary electromagnetic fields in the presence of conducting objects require the field solutions both internal and external to the conductors. Such solutions are limited for certain canonically shaped objects but are useful in testing the accuracy of various approximate models and numerical methods developed to solve complex problems related to real world conducting objects and in calibrating instruments designed to measure various field quantities. Theoretical investigations of quasistationary electromagnetic fields also aid in improving the understanding of the physical phenomena of electromagnetic induction. This thesis presents rigorous analytical expressions derived as benchmark solutions for the quasistationary field quantities both inside and outside, Joule losses and the electromagnetic forces acting upon a conducting spheroid placed in the proximity of a non-uniform field produced by current-carrying turns. These expressions are used to generate numerous numerical results of specified accuracy and selected results are presented in a normalized form for extended ranges of the spheroid axial ratio, the ratio of the depth of penetration to the semi-minor axis and the position of the inducing turns relative to the spheroids. They are intended to constitute reference data to be employed for comprehensive comparisons of results from approximate numerical methods or from boundary impedance models used for real world conductors. Approximate boundary conditions such as the simpler perfect electric conductor model or the Leontovich surface impedance boundary condition model can be used to obtain approximate solutions by only analyzing the field external to the conducting object. The range of validity of these impedance boundary condition models for the analysis of axisymmetric eddy-current problems is thoroughly investigated. While the simpler PEC model can be employed only when the electromagnetic depth of penetration is much smaller than the smallest local radius of curvature, the results obtained using the surface impedance boundary condition model for conducting prolate and oblate spheroids of various axial ratios are in good agreement with the exact results for skin depths of about 1/5 of the semi-minor axis when calculating electromagnetic forces and for skin depths less than 1/20 of the semi-minor axis when calculating Joule losses.

Disertacijoje nagrinėjami nestacionarių matematinių modelių nestandartinėse srityse skaitiniai sprendimo algoritmai. Uždavinio formulavimo sritis yra šakotosios struktūros (ang. branching structures), kurių išsišakojimo taškuose apibrėžiami tvermės dėsniai. Tvermės dėsnių skaitinė analizė ir nestandartinių kraštinių sąlygų analizė skiria nagrinėjamus uždavinius nuo klasikinių aprašytų literatūroje matematinės fizikos uždavinių. Disertacijoje suformuluoti uždaviniai apima skaitinių algoritmų šakotose struktūrose su skirtingais srautų tvermės dėsniais stabilumo ir konvergavimo tyrimą, lygiagrečiųjų algoritmų sudarymą ir taikymą, skaitinių schemų uždaviniams su nelokaliomis integralinėmis sąlygomis tyrimą. Disertacijoje sprendžiami taikomieji neurono sužadinimo ir impulso relaksacijos lazerio apšviestame puslaidininkyje uždaviniai, netiesinio modelio identifikavimo uždavinys. Disertaciją sudaro įvadas, penki skyriai, rezultatų apibendrinimas, literatūros šaltinių sąrašas ir autorės publikacijų disertacijos tema sąrašas. Įvadiniame skyriuje formuluojama problema, aprašytas tyrimų objektas, darbo aktualumas, formuluojami darbo tikslai ir uždaviniai, aprašoma tyrimų metodika, darbo mokslinis naujumas, darbo rezultatų praktinė reikšmė, pateikti ginamieji teiginiai ir disertacijos struktūra. Pabaigoje pristatomi pranešimai konferencijose disertacijos tema. Pirmajame skyriuje pateikta matematinių modelių nestandartinėse srityse arba su nestandartinėmis sąlygomis apžvalga. Antrajame... [toliau žr. visą tekstą] / The numerical algorithms for non-stationary mathematical models in non-standard domains are investigated in the dissertation. The problem definition domain is represented by branching structures with conjugation equations considered at the branching points. The numerical analysis of the conjugation equations and non-classical boundary conditions distinguish considered problems among the classical problems of mathematical physics presented in the literature. The scope of the dissertation covers the investigation of stability and convergence of the numerical algorithms on branching structures with different conjugation equations, the construction and implementation of parallel algorithms, the investigation of the numerical schemes for the problems with nonlocal integral conditions. The modeling of the excitation of neuron and photo-excited carrier decay in a semiconductor, also the problem of the identification of nonlinear model are considered in the dissertation. The dissertation consists of an introduction, five chapters, main conclusions, bibliography and the list of the author's publications on the topic of dissertation. Introductory chapter covers the problem formulation and the object of research, the topicality of the thesis, the aims and objectives of the dissertation, the methodology of research, scientific novelty and the practical value of the achieved results. The defended thesis and structure of the dissertation are given in this chapter. The first chapter... [to full text]

The numerical algorithms for non-stationary mathematical models in non-standard domains are investigated in the dissertation. The problem definition domain is represented by branching structures with conjugation equations considered at the branching points. The numerical analysis of the conjugation equations and non-classical boundary conditions distinguish considered problems among the classical problems of mathematical physics presented in the literature. The scope of the dissertation covers the investigation of stability and convergence of the numerical algorithms on branching structures with different conjugation equations, the construction and implementation of parallel algorithms, the investigation of the numerical schemes for the problems with nonlocal integral conditions. The modeling of the excitation of neuron and photoexcited carrier decay in a semiconductor, also the problem of the identification of nonlinear model are considered in the dissertation. / Disertacijoje nagrinėjami nestacionarių matematinių modelių nestandartinėse srityse skaitiniai sprendimo algoritmai. Uždavinio formulavimo sritis yra šakotosios strukturos (ang. branching structures), kurių išsišakojimo taškuose apibrežiami tvermės dėsniai. Tvermės dėsnių skaitinė analizė ir nestandartinių kraštinių sąlygų analizė skiria nagrinėjamus uždavinius nuo klasikinių aprašytų literatūroje matematinės fizikos uždaviniu. Disertacijoje suformuluoti uždaviniai apima skaitinių algoritmų šakotose struktūrose su skirtingais srautų tvermės dėsniais stabilumo ir konvergavimo tyrimą, lygiagrečiųjų algoritmų sudarymą ir taikymą, skaitinių schemų uždaviniams su nelokaliomis integralinėmis sąlygomis tyrimą. Disertacijoje sprendžiami taikomieji neurono sužadinimo ir impulso relaksacijos lazerio apšviestame puslaidininkyje uždaviniai, netiesinio modelio identifikavimo uždavinys.

Exact analytical solutions for the quasistationary electromagnetic fields in the presence of conducting objects require the field solutions both internal and external to the conductors. Such solutions are limited for certain canonically shaped objects but are useful in testing the accuracy of various approximate models and numerical methods developed to solve complex problems related to real world conducting objects and in calibrating instruments designed to measure various field quantities. Theoretical investigations of quasistationary electromagnetic fields also aid in improving the understanding of the physical phenomena of electromagnetic induction. This thesis presents rigorous analytical expressions derived as benchmark solutions for the quasistationary field quantities both inside and outside, Joule losses and the electromagnetic forces acting upon a conducting spheroid placed in the proximity of a non-uniform field produced by current-carrying turns. These expressions are used to generate numerous numerical results of specified accuracy and selected results are presented in a normalized form for extended ranges of the spheroid axial ratio, the ratio of the depth of penetration to the semi-minor axis and the position of the inducing turns relative to the spheroids. They are intended to constitute reference data to be employed for comprehensive comparisons of results from approximate numerical methods or from boundary impedance models used for real world conductors. Approximate boundary conditions such as the simpler perfect electric conductor model or the Leontovich surface impedance boundary condition model can be used to obtain approximate solutions by only analyzing the field external to the conducting object. The range of validity of these impedance boundary condition models for the analysis of axisymmetric eddy-current problems is thoroughly investigated. While the simpler PEC model can be employed only when the electromagnetic depth of penetration is much smaller than the smallest local radius of curvature, the results obtained using the surface impedance boundary condition model for conducting prolate and oblate spheroids of various axial ratios are in good agreement with the exact results for skin depths of about 1/5 of the semi-minor axis when calculating electromagnetic forces and for skin depths less than 1/20 of the semi-minor axis when calculating Joule losses.

A computer code is developed using finite volume technique for solving steady twodimensional and axisymmetric compressible Euler and Navier-Stokes equations for internal flows by &ldquo / multi-block&rdquo / technique. For viscous flows, both laminar and turbulent flow properties can be used. Explicit one step second order accurate Lax-Wendroff scheme is used for time integration. Inviscid solutions are verified by comparing the results of test cases of a support project which was supported by ONERA/France for Turkey T-108, named &ldquo / 2-D Internal Flow Applications for Solid Propellant Rocket Motors&rdquo / . For laminar solutions, analytical flat plate solution is used for planar case and theoretical pipe flow solution is used for axisymmetric case for verification. Prandtl turbulent flow analogy is used in a flat plate solution to verify the turbulent viscosity calculation. The test cases solved with single-block code are compared with the ones solved with multi-block technique to verify the multi-block algorithm and good similarity is observed between single-block solutions and multi-block solutions. For the burning simulation of propellant of Solid Propellant Rocket Motors, injecting boundary is used. Finally, a segmented solid propellant rocket motor case is solved to show the multi-block algorithm&rsquo / s flexibility in solving complex geometries.

La réduction des nuisances sonores induites par les aéronefs civils autour des grands aéroports est devenue un enjeu sociétal important. Pour réduire le bruit propulsif de soufflante, devenu prépondérant au cours des dernières années avec l'avènement de turboréacteurs à double flux, les constructeurs sont amenés à généraliser l'utilisation de matériaux absorbants acoustiques (également appelés « liners »). Ce sujet de thèse s'inscrit dans le cadre de l'amélioration des outils CAA relativement à la prévision numérique de ces matériaux absorbants. Cette modélisation soulève plusieurs interrogations, liées à divers aspects tels que le type d'écoulement mis en jeu (inhomogénéités, couche limite, etc.), les niveaux acoustiques en présence (effets de non linéarité), les effets de diffraction induits par les ruptures d'impédance, etc. Cette étude consiste donc à valider et à améliorer la condition limite d'impédance, implémentée dans le solveur CAA sAbrinA.v0, développé par l'Onera. Des développements théoriques sont d'abord consacrés à la modélisation de l'impédance dans le domaine temporel, et conduisent à une discussion sur la généralisation de cette modélisation. Le travail consiste ensuite à simuler plusieurs cas tests canoniques de l'absorption du bruit par un revêtement acoustique, lesquels sont validés par confrontation avec d'autres résultats analytiques et/ou expérimentaux. Ce travail fournit de nouvelles connaissances sur la façon dont les matériaux absorbants acoustiques peuvent être modélisés et simulés de manière précise dans le cadre d'une approche CAA dans le domaine temporel. / The reduction of acoustic emission induced by civil aircraft around major airports has become an important societal issue. To reduce the fan noise, induced by the engines, which has become preponderant over the past years with the advent of turbofan engines, manufacturers are led to generalize the employment of acoustic absorbing materials (or acoustic “liners”). The present thesis is related to the numerical prediction of such absorbing materials, in the context of time-domain CAA (Computational AeroAcoustics) methods. Such modeling raises several key questions, which are related to various aspects such as the type of flow involved (boundary layers effects, etc.), the sound levels considered (non-linear phenomena), the diffraction effects induced by ruptures of impedance, etc. The present study then consists in validating and improving the time-domain impedance boundary condition implemented in Onera’s structured CAA solver (named sAbrinA.v0). Theoretical developments are first devoted to the modeling of impedance in the time-domain, and lead to a discussion on the generalization of this modeling. The work then consists in CAA-simulating several canonical tests of noise absorption by acoustic liners. Outputs are compared against experimental and/or analytical results, delivering new insight in the way noise absorption materials can be accurately modeled and simulated using time-domain CAA-approaches.