<p>Predmet istraživanja u doktorskoj disertaciji je metoda za konstrukciju<br />lokalizacionih skupova za spektar i pseudospektar nelinearnih problema<br />karakterističnih korena bazirana na Geršgorinovoj teoremi i njenim<br />generalizacijama koja koristi osobine poznatih podklasa H-matrica.<br />Navedena tvrđenja i primeri rasvetljavaju odnose između navedenih<br />lokalizacionih skupova, što je posebno značajno za primenu u praksi.<br />Sadržaj ovog rada time predstavlja polaznu tačku za dublja istraživanja na<br />temu konstrukcije lokalizacionih skupova za spektar i pseudospektar<br />nelinearnih problema karakterističnih korena Geršgorinovog tipa.</p> / <p>The subject of research in the doctoral dissertation is a method for constructing<br />spectra and pseudospectra localization sets for nonlinear eigenvalue problems<br />based on Ger&scaron;gorin theorem and its generalizations, that uses the properties of<br />well-known subclasses of H-matrices. Theorems and examples given in this<br />paper are showing relations between stated localization sets, which is very<br />important for practical applications. Therefore, the content of this paper represent<br />the starting point for deeper explorations on the subject of constructing spectra<br />and pseudospectra localization sets for Ger&scaron;gorin type nonlinear eigenvalue<br />problems.</p>

[pt] Neste trabalho, são apresentadas algumas aplicações da otimização topológica para problemas de autovalor onde o principal objetivo é maximizar um determinado autovalor, como por exemplo uma frequência natural de vibração ou uma carga crítica linearizada, usando elementos finitos poligonais em domínios bidimensionais arbitrários. A otimização topológica tem sido comumente utilizada para minimizar a flexibilidade de estruturas sujeitas a restrições de volume. A ideia desta técnica é distribuir uma certa quantidade de material em uma estrutura, sujeita a carregamentos e condições de contorno, visando maximizar a sua rigidez. Neste trabalho, o objetivo é obter uma distribuição ótima de material de maneira a maximizar uma determinada frequência natural (para mantê-la afastada da frequência de excitação externa, por exemplo) ou maximizar a menor carga crítica linearizada (para garantir um nível mais elevado de estabilidade da estrutura). Malhas poligonais construídas usando diagramas de Voronoi são empregadas na solução do problema de otimização topológica. As variáveis de projeto, i.e. as densidades do material, utilizadas no processo de otimização, são associadas a cada elemento poligonal da malha. Vários exemplos de otimização topológica, tanto para problemas de frequências naturais de vibração quanto para cargas críticas linearizadas, são apresentados para demonstrar a funcionalidade e a aplicabilidade da metodologia proposta. / [en] In this work, we present some applications of topology optimization for eigenvalue problems where the main goal is to maximize a specified eigenvalue, such as a natural frequency or a linearized buckling load using polygonal finite elements in arbitrary two-dimensional domains. Topology optimization has commonly been used to minimize the compliance of structures subjected to volume constraints. The idea is to distribute a certain amount of material in a given design domain subjected to a set of loads and boundary conditions such that to maximize its stiffness. In this work, the objective is to obtain the optimal material distribution in order to maximize the fundamental natural frequency (e.g. to keep it away from an external excitation frequency) or to maximize the lowest critical buckling load (e.g. to ensure a higher level of stability of the structures). We employ unstructured polygonal meshes constructed using Voronoi tessellations for the solution of the structural topology optimization problems. The design variables, i.e. material densities, used in the optimization scheme, are associated with each polygonal element in the mesh. We present several topology optimization examples for both eigenfrequency and buckling problems in order to demonstrate the functionality and applicability of the proposed methodology.

This thesis presents a means of enhancing the iterative calculation techniques used in electronic structure calculations, particularly Kohn-Sham DFT. Based on the subspace iteration method of the FEAST eigenvalue solving algorithm, this nonlinear FEAST algorithm (NLFEAST) improves the convergence rate of traditional iterative methods and dramatically improves their robustness. A description of the algorithm is given, along with the results of numerical experiments that demonstrate its effectiveness and offer insight into the factors that determine how well it performs.

The famous Moore's law states: Since the invention of the integrated circuit, the number of transistors that can be placed on an integrated circuit has increased exponentially, doubling approximately every two years. As a result of the downscaling of the size of the transistor, quantum effects have become increasingly important while affecting significantly the device performances. Nowadays, at the nanometer scale, inter-atomic interactions and quantum mechanical properties need to be studied extensively. Device and material simulations are important to achieve these goals because they are flexible and less expensive than experiments. They are also important for designing and characterizing new generation of electronic device such as silicon nanowire or carbon nanotube (CNT) transistors. Several modeling methods have been developed and applied to electronic structure calculations, such as: Hartree-Fock, density functional theory (DFT), empirical tight-binding, etc. For transport simulations, most of the device community focuses on studying the stationary problem for obtaining characteristics such as I-V curves. The non-equilibrium transport problem is then often addressed by solving a multitude of time-independent Schrodinger-type equation for all possible energies. On the other hand, for many other electronic applications including high-frequency electronics response (e.g. when a time-dependent potential is applied to the system), the description of the system behavior necessitate insights on the time dependent electron dynamics. To address this problem, it is then necessary to solve a time-dependent Schrodinger-type equation. In this thesis, we will focus on solving time-dependent problems with application to CNTs. We will be identifying all the numerical difficulties and propose new effective modeling and numerical schemes to address the current limitations in time-dependent quantum simulations. we will point out that two numerical errors may occur: an integration error and the anti-commutation issue error; the direct computation above being mathematically equivalent to performing the integration of the time dependent Hamiltonian using a rectangle numerical quadrature formula along the total simulation times. After careful study and many numerical experiments, we found that the Gaussian quadrature scheme provides a good trade off between computational consumption and numerically accuracy, meanwhile unitary, stability and time reversal properties are well preserved. The new Gaussian quadrature integration scheme uses (i) much fewer points in time to approximate the integral of the Hamiltonian, (ii) ordered exponential to factorize the time evolution operator, (iii) FEM discretize techniques (iv) and at last, the FEAST eigenvalue solver to diagonalize and solve each exponential.

Max-plus代数は，実数全体に無限小元を付加した集合に，加法として最大値をとる演算，乗法として通常の加法を考えた代数系である．本論文では，max-plus線形方程式に対するCramerの公式の類似物を用いて，線形方程式の解空間の基底が構成できることを示した．さらに固有値問題に関連して，max-plus行列の固有ベクトルの概念を2通りの観点から拡張した． / The max-plus algebra is the semiring with addition "max" and multiplication "+". In the present thesis, the author gives a combinatorial characterization of solutions of linear systems in terms of the max-plus Cramer's rule. Further, the author extends the concept of eigenvectors of max-plus matrices from two different perspectives. / 博士(理学) / Doctor of Philosophy in Science / 同志社大学 / Doshisha University

Optical testing in adverse environments, ophthalmology and applications where characterization by curvature is leveraged all have a common goal: accurately estimate wavefront shape. This dissertation investigates wavefront sensing techniques as applied to optical testing based on gradient and curvature measurements. Wavefront sensing involves the ability to accurately estimate shape over any aperture geometry, which requires establishing a sampling grid and estimation scheme, quantifying estimation errors caused by measurement noise propagation, and designing an instrument with sufficient accuracy and sensitivity for the application. Starting with gradient-based wavefront sensing, a zonal least-squares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a pre-defined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the pre-defined matrix across discontinuous boundary. The results show that the RMS deviation error of the estimated wavefront from the original wavefront can be less than λ/130~ λ/150 (for λ equals 632.8nm) after about twelve iterations and less than λ/100 after as few as four iterations. The presented approach to handling irregular pupil shapes applies equally well to wavefront estimation from curvature data. A defining characteristic for a wavefront estimation algorithm is its error propagation behavior. The error propagation coefficient can be formulated as a function of the eigenvalues of the wavefront estimation-related matrices, and such functions are established for each of the basic estimation geometries (i.e. Fried, Hudgin and Southwell) with a serial numbering scheme, where a square sampling grid array is sequentially indexed row by row. The results show that with the wavefront piston-value fixed, the odd-number grid sizes yield lower error propagation than the even-number grid sizes for all geometries. The Fried geometry either allows sub-sized wavefront estimations within the testing domain or yields a two-rank deficient estimation matrix over the full aperture; but the latter usually suffers from high error propagation and the waffle mode problem. Hudgin geometry offers an error propagator between those of the Southwell and the Fried geometries. For both wavefront gradient-based and wavefront difference-based estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimum-norm least-squares solution. Noll’s theoretical result, which was extensively used as a reference in the previous literature for error propagation estimate, corresponds to the Southwell geometry with an odd-number grid size. For curvature-based wavefront sensing, a concept for a differential Shack-Hartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic Shack-Hartmann sensor with the collimated beam split into three output channels, along each of which a lenslet array is located. Three Hartmann grid arrays are generated by three lenslet arrays. Two of the lenslets shear in two perpendicular directions relative to the third one. By quantitatively comparing the Shack-Hartmann grid coordinates of the three channels, the differentials of the wavefront slope at each Shack-Hartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmann-based sensor allows us to uniquely determine the principal curvatures and directions more accurately than prior methods. Measurement of local curvatures as opposed to slopes is unique because curvature is intrinsic to the wavefront under test, and it is an absolute as opposed to a relative measurement. A zonal least-squares-based wavefront estimation algorithm was developed to estimate the wavefront shape from the Laplacian curvature data, and validated. An implementation of the DSH curvature sensor is proposed and an experimental system for this implementation was initiated. The DSH curvature sensor shares the important features of both the Shack-Hartmann slope sensor and Roddier’s curvature sensor. It is a two-dimensional parallel curvature sensor. Because it is a curvature sensor, it provides absolute measurements which are thus insensitive to vibrations, tip/tilts, and whole body movements. Because it is a two-dimensional sensor, it does not suffer from other sources of errors, such as scanning noise. Combined with sufficient sampling and a zonal wavefront estimation algorithm, both low and mid frequencies of the wavefront may be recovered. Notice that the DSH curvature sensor operates at the pupil of the system under test, therefore the difficulty associated with operation close to the caustic zone is avoided. Finally, the DSH-curvature-sensor-based wavefront estimation does not suffer from the 2π-ambiguity problem, so potentially both small and large aberrations may be measured.

No description available.

One of the problems with the modern radio communication is the lack of availableradio frequencies. Recent studies have shown that, while the available licensed radiospectrum becomes more occupied, the assigned spectrum is significantly underutilized.To alleviate the situation, cognitive radio (CR) technology has been proposedto provide an opportunistic access to the licensed spectrum areas. Secondary CRsystems need to cyclically detect the presence of a primary user by continuouslysensing the spectrum area of interest. Radiowave propagation effects like fading andshadowing often complicate sensing of spectrum holes. When spectrum sensing isperformed in a cooperative manner, then the resulting sensing performance can beimproved and stabilized. In this thesis, two fully distributed and adaptive cooperative Primary User (PU)detection solutions for CR networks are studied. In the first part of this thesis we study a distributed energy detection schemewithout using any fusion center. Due to reduced communication such a topologyis more energy efficient. We propose the usage of distributed, diffusion least meansquare (LMS) type of power estimation algorithms with different network topologies.We analyze the resulting energy detection performance by using a commonframework and verify the theoretical findings through simulations. In the second part of this thesis we propose a fully distributed detection scheme,based on the largest eigenvalue of adaptively estimated correlation matrices, assumingthat the primary user signal is temporally correlated. Different forms of diffusionLMS algorithms are used for estimating and averaging the correlation matrices overthe CR network. The resulting detection performance is analyzed using a commonframework. In order to obtain analytic results on the detection performance, theadaptive correlation matrix estimates are approximated by a Wishart distribution.The theoretical findings are verified through simulations. / <p>QC 20170908</p>

This dissertation investigates uncertainty quantification in dynamic problems. The Advanced Mean Value (AMV) method is used to calculate probabilistic sound power and the sensitivity of elastically supported panels with small uncertainty (coefficient of variation). Sound power calculations are done using Finite Element Method (FEM) and Boundary Element Method (BEM). The sensitivities of the sound power are calculated through direct differentiation of the FEM/BEM/AMV equations. The results are compared with Monte Carlo simulation (MCS). An improved method is developed using AMV, metamodel, and MCS. This new technique is applied to calculate sound power of a composite panel using FEM and Rayleigh Integral. The proposed methodology shows considerable improvement both in terms of accuracy and computational efficiency. In systems with large uncertainties, the above approach does not work. Two Spectral Stochastic Finite Element Method (SSFEM) algorithms are developed to solve stochastic eigenvalue problems using Polynomial chaos. Presently, the approaches are restricted to problems with real and distinct eigenvalues. In both the approaches, the system uncertainties are modeled by Wiener-Askey orthogonal polynomial functions. Galerkin projection is applied in the probability space to minimize the weighted residual of the error of the governing equation. First algorithm is based on inverse iteration method. A modification is suggested to calculate higher eigenvalues and eigenvectors. The above algorithm is applied to both discrete and continuous systems. In continuous systems, the uncertainties are modeled as Gaussian processes using Karhunen-Loeve (KL) expansion. Second algorithm is based on implicit polynomial iteration method. This algorithm is found to be more efficient when applied to discrete systems. However, the application of the algorithm to continuous systems results in ill-conditioned system matrices, which seriously limit its application. Lastly, an algorithm to find the basis random variables of KL expansion for non-Gaussian processes, is developed. The basis random variables are obtained via nonlinear transformation of marginal cumulative distribution function using standard deviation. Results are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential. In all the cases, it is found that the proposed algorithm matches very well with the known solutions and can be applied to solve non-Gaussian process using SSFEM. / Ph. D.

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) method must be employed. In this work an h-p Finite Element Method (FEM) is used to obtain the dominant Lambda mode associated with a configuration of a reactor core using the SPN approximation. The performance of the SPN (N= 1, 3, 5) approximations has been tested for different reactor benchmarks. / [ES] La ecuación de la difusión neutrónica es una aproximación de la ecuación del transporte de neutrones que describe la población de neutrones en el núcleo de un reactor nuclear. En particular, consideraremos reactores de tipo VVER y para simular su comportamiento se utilizará la ecuación de la difusión neutrónica para cuya discretización se hace uso de mallas hexagonales. La mayoría de los códigos de simulación de reactores nucleares utilizan aproximación multigrupo de energía de la ecuación de la difusión neutrónica para describir la distribución de neutrones en el interior del núcleo del reactor. Para estudiar el estado estacionario del reactor, es posible forzar la criticidad del reactor de forma artificial modificando las secciones eficaces de forma que se obtiene un problema de valores propios diferencial, conocido como el problema de los Modos Lambda, que se resuelve para obtener los valores propios dominantes del reactor y sus correspondientes funciones propias. Para discretizar este modelo se ha hecho uso de un método de elementos finitos con adaptabilidad h-p. Este método permite el uso de mallas heterogéneas, y de diferentes refinamientos como el uso mallas h-adaptativas, reduciendo el tamaño de los distintos nodos, y el p-refinado, aumentando el grado del polinomio de las funciones básicas utilizado en los desarrollos de la solución en los diferentes nodos. Se ha desarrollado un código basado en un método de elementos finitos de alto orden para resolver el problema de los Modos Lambda en un reactor con geometría hexagonal y se han obtenido los Modos dominantes para distintos problemas de referencia. Una vez que se ha obtenido la solución para la distribución de neutrones en estado estacionario, ésta se utiliza como condición inicial para la integración de la ecuación de difusión neutrónica dependiente del tiempo. Para simular el comportamiento de un reactor nuclear para un determinado transitorio, es necesario ser capaz de integrar la ecuación de la difusión neutrónica dependiente del tiempo en el interior del núcleo del reactor. La discretización espacial de esta ecuación se hace usando un método de elementos finitos de alto orden que permite refinados de tipo h-p para distintas geometrías. Los transitorios que implican el movimiento de los bancos de las barras de control tienen el problema conocido como el efecto 'rod-cusping'. Estudios anteriores, por lo general, han abordado este problema utilizando una malla fija y definiendo propiedades promedio para los materiales correspondientes a las celdas donde se tiene la barra de control parcialmente insertada. En el presente trabajo se propone el uso de un esquema de malla móvil, de forma que en mallado espacial va cambiando con el movimiento de la barra de control, evitando la necesidad de utilizar secciones eficaces equivalentes para las celdas parcialmente insertadas. El funcionamiento de este esquema de malla móvil propuesto se estudia resolviendo distintos problemas tipo. La precisión obtenida mediante de la teoría de la difusión en los cálculos de reactores es limitada cuando se tienen elementos de combustible complejos o se pretenden realizar cálculos en malla fina. Para mejorar estos resultados, es necesario disponer de un método que incorpore aproximaciones de orden superior de la ecuación del transporte de neutrones. Una posibilidad es hacer uso de las ecuaciones PN simplificadas (SPN ). En este trabajo se utiliza un método de elementos finitos h-p para obtener los modos dominantes asociados con una configuración dada del núcleo de un reactor nuclear con geometría hexagonal usando la aproximación SPN . El funcionamiento de las aproximaciones SPN (N = 1, 3, 5) se ha estudiado para distintos problemas de referencia. / [CA] L'equació de la difusió neutrònica és una aproximació de l'equació del transport de neutrons que descriu la població de neutrons en el nucli de un reactor nuclear. En particular, considerarem reactors de tipus VVER i per a simular el seu comportament s'utilitzarà l'equació de la difusió neutrónica que es discretitza fent ús de malles hexagonals. La majoria dels codis de simulació de reactors nuclears utilitzen l'aproximació multigrup d'energia de l'equació de la difusió neutrónica per a descriure la distribució de neutrons a l'interior del nucli del reactor. Per a estudiar l'estat estacionari del reactor, és possible forçar la seua criticitat de forma artificial modificant les seccions eficaces de manera que s'obté un problema de valors propis diferencial, conegut com el problema dels Modes Lambda, que es resol per a obtenir els valors propis dominants del reactor i les seues corresponents funcions pròpies. Per a discretitzar aquest model s'ha fet ús d'un mètode d'elements finits amb adaptabilitat h-p. Aquest mètode permet l'ús de malles heterogènies, i de diferents refinaments com l'ús malles h-adaptatives, reduint la grandària dels diferents nodes, i el p-refinat, augmentant el grau del polinomi de les funcions bàsiques utilitzat en els desenvolupaments de la solució en els diferents nodes. S'ha desenvolupat un codi basat en un mètode d'elements finits d'alt ordre per a resoldre el problema dels Modes Lambda en un reactor amb geometria hexagonal i s'han obtingut els Modes dominants per a diferents problemes de referència. Una vegada que s'ha obtingut la solució per a la distribució de neutrons en estat estacionari, aquesta s'utilitza com a condició inicial per a la integració de l'equació de difusió neutrònica depenent del temps. Per a simular el comportament d'un reactor nuclear per a un determinat transitori, és necessari ser capaç d'integrar l'equació de la difusió neutrónica depenent del temps a l'interior del nucli del reactor. La discretitzación espacial d'aquesta equació es fa usant un mètode d'elements finits d'alt ordre que permet refinats de tipus h-p per a diferents geometries. Els transitoris que impliquen el moviment dels bancs de les barres de control tenen el problema conegut com l'efecte 'rod-cusping'. Estudis anteriors, en general, han abordat aquest problema utilitzant una malla fixa i definint propietats equivalents per als materials corresponents a les cel·les on es té la barra de control parcialment inserida. En el present treball es proposa l'ús d'un esquema de malla mòbil, de manera que en mallat espacial va canviant amb el moviment de la barra de control, evitant la necessitat d'utilitzar seccions eficaces equivalents per a les cel·les parcialment inserides. El funcionament de aquest esquema de malla mòbil s'estudia resolent diferents problemes tipus. La precisió obtinguda mitjançant de la teoria de la difusió en els càlculs de reactors és limitada quan es tenen elements de combustible complexos o es pretenen realitzar càlculs en malla fina. Per a millorar aquests resultats, és necessari disposar d'un mètode que incorpore aproximacions d'ordre superior de l'equació del transport de neutrons. Una possibilitat és fer ús de les equacions PN simplificades (SPN ). En aquest treball s'utilitza un mètode d'elements finits h- p per a obtenir els modes dominants associats amb una configuració donada del nucli de un reactor amb geometria hexagonal usant l'aproximació SPN . El funcionament de les aproximacions SPN (N = 1, 3, 5) s'ha estudiat per a diferents problemes de referència. / Fayez Moustafa Moawad, R. (2016). Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/65353