Magister Scientiae - MSc / Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

Cette thèse montre la possibilité d’une application rigoureuse de la méthode d’intégrationnumérique double-exponentielle introduite par Takahasi et Morien 1974, et sa pertinence pour lescalculs à grande précision en théorie des nombres. Elle contient en particulier une étude détailléede cette méthode, des critères simples sur son champ d’application, et des estimations rigoureusesdes termes d’erreur.Des paramètres explicités et précis permettent de l’employer aisément pour le calcul garantide fonctions définies par des intégrales.Cette méthode est également appliquée en détail au calcul de transformées de Mellin inversesde facteurs gamma intervenant dans les calculs numériques de fonctions L. Par une étude unifiée,ce travail démontre la complexité d’un algorithme de M. Rubinstein et permet de proposer desalgorithmes de calcul de valeurs de fonctions L quelconques dont le résultat est garanti et dont lacomplexité est meilleure en la précision. / This thesis contains a detailed study of the so-called double exponential integration formulasintroduced by Takahasi and Moriin 1974,and provides explicit bounds forarigorous applicationof the method in number theory.Accurate parameters are given, which makes it possible to use it as a blackbox for the rigorouscomputation of functions defined by integrals.It also deals with numerical computations of L functions. The complexity of analgorithm dueto M. Rubinstein is proven. In the context of double-exponential transformation, a new algorithmis provided whose complexity is low in terms of precision.

Orientadores: Simone Andrea Pozza, Varese Salvador Timóteo / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Tecnologia / Made available in DSpace on 2018-08-26T20:52:16Z (GMT). No. of bitstreams: 1 Pereira_MatheusFernando_M.pdf: 7089543 bytes, checksum: 5e3fe7245aa3630ebca8dda3ddd08eb3 (MD5) Previous issue date: 2014 / Resumo: Diversas técnicas vêm sendo apresentadas para resolução da equação da difusão, a qual é empregada para estimativas da concentração de poluentes em função do espaço e do tempo, levando-se conta fatores como fonte emissora, condições meteorológicas, características do meio e velocidade em que o poluente é carreado. Neste estudo, foi empregado um algoritmo de passo variável para a resolução da equação da difusão unidimensional e avaliação da influência do parâmetro de heterogeneidade do meio, da velocidade do fluxo e do coeficiente de dispersão na variação da concentração de poluentes em função do espaço e do tempo. As simulações foram realizadas utilizando as mesmas condições iniciais e de contorno adotadas em dois estudos abordados recentemente na literatura, e de acordo com os resultados, verificou-se que características como meios de menor heterogeneidade, baixa velocidade inicial do fluxo e baixo coeficiente de dispersão implicam em menores valores de concentração, facilitando a dispersão de poluentes. O método utilizado é caracterizado pela rápida convergência, simplicidade do código e baixo tempo computacional, podendo ser utilizado como base para resolução da equação da difusão bi e tridimensional / Abstract: Several techniques have been employed for solving advection-diffusion equation, which is used to estimate pollutants concentration as function of time and space, taking account factors such as emission source, meteorological conditions, medium characteristics and the velocity in which pollutant is adduced. In this study, we used an adaptive-step algorithm for solving one-dimensional advection-diffusion equation, and evaluating the influence of medium inhomogeneity parameter, flow velocity and dispersion coefficient in the pollutants concentration variation as function of space and time. Simulations were performed using the same initial and boundary conditions adopted by Kumar et al. (2010) and by Savovic and Djordjevich (2012), and according to the results, it was found that characteristics such as medium of less inhomogeneity, low initial flow velocity and low dispersion coefficient imply in lower concentration and facilitate pollutants dispersion. The method is characterized by rapid convergence, simplicity of the code and low computational time, and it can be used as a basis for solving the two and the three dimensional advection-diffusion equation / Mestrado / Tecnologia e Inovação / Mestre em Tecnologia

This thesis deals with continuous system simulation. The systems can be described by system of differential equations or block diagram. Differential equations are usually solved by numerical methods that are integrated into simulation software such as Matlab, Maple or TKSL. Taylor series method has been used for numerical solutions of differential equations. The presented method has been proved to be both very accurate and fast and also procesed in parallel systems. The aim of the thesis is to design, implement and compare a few versions of the parallel system.

In this two-part study, developments in finite element technology and the application thereof to topology optimization are investigated. Ultimately, the developed finite elements and corresponding topology optimization procedures are aimed at, but not restricted to, aiding the design of piezoelectrically driven compliant mechanisms for micropositioning applications. The objective is to identify and exploit existing, or to develop new, finite element technologies to alleviate the numerical instabilities encountered in topology optimization. Checkerboarding and one-node connected hinges are two commonly encountered examples which can directly be attributed to inadequacies or deficiencies in the finite element solution of structural problems using 4-node bilinear isoparametric finite elements (denoted Q4). The numerical behaviour leading to checkerboard layouts stems from an over-stiff estimation of a checkerboard patch of Q4 elements. The numerical model of a one-node connected hinge using Q4 elements, on the other hand, possesses no (or very little) stiffness in rotation about the common node. In the first part of the study, planar finite elements with in-plane rotational (drilling) degrees of freedom are investigated. It is shown that the skew-symmetric part of the stress tensor can directly be used to quantitatively assess the validity of the penalty parameter ã, which relates the in-plane translations to the rotations. Thereafter, the variational formulations used to develop these planar finite elements with drilling degrees of freedom are extended to account for the piezoelectric effect. Several new piezoelectric elements that include in-plane rotational degrees of freedom (with and without assumed stress and electric flux density) are implemented, evaluated and shown to be accurate and stable. Furthermore, the application of alternative reduced order integration schemes to quadratic serendipity (Q8) and Lagrangian (Q9) elements is investigated. Reduced or selective reduced integration schemes are often used to enhance element accuracy by `softening' higher order deformation modes. However, application of reduced integration schemes to Q8 and Q9 elements is usually accompanied by element rank deficiencies. It is shown how the application of five and eight point modified integration schemes preserve the accuracy benefits of reduced integration, while preventing element rank deficiencies. In the second part of the investigation, the salient features of elements with drilling degrees are utilized in two schemes to prevent, or improve the modelling of, one-node connected hinges. In principle, the first scheme uses the rotations computed at interior nodes to detect excessive rotations at suspect nodes. The second scheme essentially replaces planar elements forming a one-node hinge, where appropriate, with a more realistic beam model of the material layout while other elements in the mesh are modelled using planar elements as usual. Next, the dependence of optimal topologies on element formulation is demonstrated. Attention is especially paid to plate and shell applications. It is shown that Mindlin-Reissner based elements, which employ selective reduced integration on shear terms, are not reliable in topology optimization problems. Conversely, elements based on an assumed natural strain formulation are shown to be stable and capable of reproducing thin plate topology results computed using shear-rigid elements. Furthermore, it is shown that an ad hoc treatment of rotational degrees of freedom in shell problems is sensitive to the related adjustable parameter, whereas optimal topologies, using a proper treatment of drilling degrees of freedom are not. Finally, the use of reduced order integration schemes as a strategy to reduce the stiffness of a checkerboard patch of elements is considered. It is demonstrated that employing the five and eight point integration schemes, used to enhance the accuracy of Q8 and Q9 elements, also significantly reduce the stiffness of a checkerboard patch of elements, thereby reducing the probability of observing checkerboard layouts in optimal topologies. / Thesis (PhD (Mechanical Engineering))--University of Pretoria, 2007. / Mechanical and Aeronautical Engineering / PhD / unrestricted

The Flex Representation Method (FRM) leverages unique computational advantages of splines to address limitations in the process of building CAE simulation models from CAD geometric models. Central to the approach is the envelope CAD domain that encapsulates a CAD model. An envelope CAD domain can be of arbitrary topological and geometric complexity. Envelope domains are constructed from spline representations, like U-splines, that are analysis-suitable. The envelope CAD domain can be used to approximate none, some, or all of the features in a CAD model. This yields additional simulation modeling options that simplify the model-building process while leveraging the properties of splines to control the accuracy and robustness of computed solutions. Modern integration techniques are adapted to envelope domains to maintain accurate solutions regardless of the CAD envelope chosen. The potential of the method is illustrated through several carefully selected benchmark problems.

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At many American colleges and universities, efforts to enhance the retention of a diverse group of students have become a priority. This study represents part of this effort at the University of Central Florida, a large public suburban state university in the South. Specifically, this investigation evaluated Pegasus '95 and the Academic Mentoring Program offered in the Summer and Fall Semesters of 1995 to specially-admitted students who fell short of regular admissions requirements. During the summer, Pegasus '95 provided testing, orientation, guided course work, study skills workshops, and mentoring, both individually and in the context of cohesive socialization groups of approximately 15 students each. In the Fall 1995 Semester, students were highly encouraged to participate in one-on-one mentoring in the Academic Mentoring Program (AMP) available through the Student Academic Resource Center (SARC), a university-based office which provides a variety of academic assistance services. A multiple regression analysis was conducted using the following independent predictor variables: gender, SAT/ACT scores, Pegasus participation, use of the AMP in the Fall 1995 semester, four summary scores from the College Student Inventory (CSI), and eight scaled scores from the Noncognitive Questionnaire (NCQ). Dependent variables were individual student GPA in the Summer and Fall 1995 semesters, cumulative GPA after two semesters, and enrolled credit hours into the Spring 1996 academic term. Overall, it was expected that a combination of predictor variables, including both traditional cognitive factors (SAT/ACT scores and high school GPA) and noncognitive factors (NCQ scores and CSI scores, Pegasus participation, and mentoring by the SARC) would significantly predict GP A and retention. The study found that a regression equation including gender, high school GPA, overall SAT scores and the eight NCQ scale scores significantly predicted Fall 1995 and cumulative GPA after two semesters but not Summer 1995 GPA or credit hours enrolled in Spring 1996. Attendance at Pegasus meetings was also shown to be significantly and positively associated with Fall 1995 GPA and cumulative GPA after two semesters but not of Summer 1995 GPA or credit hours enrolled in Spring 1996. Gender, high school GP A, the ACT score and the CSI Dropout Proneness scale significantly predicted credit hours enrolled in Spring 1996, as did use of the AMP program provided by the SARC. Of particular interest was the finding that including noncognitive factors in significant equations led to a greater explanation of the variance than could be obtained with any of the traditional cognitive measurements alone, suggesting that with academically disadvantaged students noncognitive measures must be considered in predicting who can succeed and persist in college.

Les nouveaux enjeux environnementaux motivent la recherche par les acteurs de l'industrie aéronautique de méthodes de calcul de trajectoires optimales. Les contributions de cette thèse se déclinent selon trois axes. Dans un premier temps, plusieurs techniques d'optimisation de trajectoire avion sont comparées sur un cas simple traité dans la bibliographie universitaire. Puis, un modèle réduit pour prendre en compte le niveau des nuisances sonores dans un algorithme d'optimisation de trajectoire est proposé.Enfin, un problème d'optimisation de trajectoire de montée d'un avion civil est résolu par une approche directe. Les spécificités du problème consistent en la présence de plusieurs phases au sein de la trajectoire, la formulation de contraintes égalités à vérifier par des composantes du vecteur d'état sur des intervalles de temps et enfin la difficulté d'intégration numérique du modèle de bruit. / Forthcoming environmental challenges stimulate the development of trajectory optimization methods by aeronautical actors. This contribution consists in three parts. First, several trajectory optimization techniques are compared. The comparison is based on a simple academic problem. After that, a model is proposed for considering noise nuisance level in the framework of trajectory optimization. Finally, the optimization problem of an ascent phase of a civil aircraft is solved using a direct approach. The specific issues of the problem are tackled with a general formulation. They consist in the presence of several phases along the trajectory, running state equality constraints and tough numerical integration of the noise model.

Algebro-Differentialgleichungen (engl. differential-algebraic equations - DAEs) sind implizite singuläre gewöhnliche Differentialgleichungen, die restringierte dynamische Prozesse beschreiben. Sie unterscheiden sich von expliziten gewöhnlichen Differentialgleichungen dahingehend, dass Anfangswerte nicht beliebig vorgegeben werden können. Weiterhin sind in einer DAE neben Integrations- auch Differentiationsaufgaben involviert. Der Differentiationsindex gibt an, wieviele Differentiationen zur Lösung notwendig sind. Seit den 1980er Jahren wird vorwiegend an der Charakterisierung und Klassifizierung regulärer DAEs und der Konstruktion nebst Fundierung von Integrationsmethoden gearbeitet. I. Higueras, R. März und C. Tischendorf haben gezeigt, dass man lineare DAEs mit properem Hauptterm, A(t)(D(t)x(t))'' + B(t)x(t) = q(t), die regulär mit Traktabilitätsindex 2 sind, zuverlässig numerisch integrieren kann - im Unterschied zu linearen DAEs in Standardform. In Publikationen von R. Riaza und R. März wird die Klassifizierungen kritischer Punkten von linearen DAEs an die Verletzung bestimmter Rangbedingungen von Matrixfunktionen im Rahmen des Traktabilitätsindexes geknüpft. Im wesentlichen heißt ein kritischer Punkt harmlos, wenn der durch die inhärente Differentialgleichung beschriebene Fluß nicht tangiert ist. Gegenstand der vorliegenden Arbeit sind lineare quasi-proper formulierte DAEs. Es werden Index 2 DAEs mit harmlosen kritischen Punkten charakterisiert. Unter Verwendung von quasi-zulässigen Projektorfunktionen können neben DAEs, die fast überall gleiche charakteristische Werte haben, nun erstmalig auch solche mit Indexwechseln behandelt werden. Der Hauptteil der Arbeit besteht im Nachweis von Durchführbarkeit, Konvergenz und nur schwacher Instabilität von numerischen Integrationsmethoden (BDF, IRK(DAE)) für lineare Index 2 DAEs mit harmlosen kritischen Punkten, sowie in der Entwicklung von Fehlerschätzern und Schrittweitensteuerung. / Differential-algebraic equations (DAEs) are implicit singular ordinary differential equations, which describe dynamical processes that are restricted by some constraints. In contrast to explicit regular ordinary differential equations, for a DAE not any value can be imposed as an initial condition. Furthermore, DAEs involve not only integration problems but also differentiation problems. The differentiation index of a DAE indicates the number of differentiations required in order to solve a DAE. Since the 1980th, research focuses primarily on the characterization and classification of regular problem classes and the construction and foundation of integration methods for simulation software. I. Higueras, R. Maerz, and C. Tischendorf have shown that one can reliably integrate a general linear DAE with a properly stated leading term, A(t)(D(t)x(t))'' + B(t)x(t) = q(t), which is regular with tractability index 2 - in contrast to linear standard form DAEs. The first classification of critical points of linear DAEs has been published by R. Riaza and R. Maerz. Based on the tractability index, critical points are classified according to failures of certain rank conditions of matrix functions. Essentially, a critical point is said to be harmless, if the flow described by the inherent differential equation is not affected. The subject of this work are quasi-proper linear DAEs. Index-2 DAEs with harmless critical points are characterized. Under the application of quasi-admissible projector functions. Besides DAEs which have almost everywhere the same characteristic values, DAEs with index changes can now be discussed for the first time. The main part of the work is to provide a proof of feasibility, convergence, and only weak instability of numerical integration methods (BDF, IRK (DAE)) for linear index-2 DAEs with harmless critical points, as well as the development and testing of error estimators and stepsize control.