Дисертації з теми "Numerical analysis methods"

Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.

This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is reduced, yet the convergence is recovered regardless of parity.

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed. In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure. An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods. A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics.
QC 20110503

Meshless methods have been developed to avoid the numerical burden imposed by meshing in the Finite Element Method. Such methods are especially attrac- tive in problems that require repeated updates to the mesh, such as problems with discontinuities or large geometrical deformations. Although meshing is not required for solving problems with meshless methods, the use of meshless methods gives rise to different challenges. One of the main challenges associated with meshless methods is imposition of essential boundary conditions. If exact interpolants are used as shape functions in a meshless method, imposing essen- tial boundary conditions can be done in the same way as the Finite Element Method. Another attractive feature of meshless methods is that their use involves compu- tations that are largely independent from one another. This makes them suitable for implementation to run on highly parallel computing systems. Highly par- allel computing has become widely available with the introduction of software development tools that enable developing general-purpose programs that run on Graphics Processing Units. In the current work, the Moving Regularized Interpolation method has been de- veloped, which is a novel method of constructing meshless shape functions that achieve exact interpolation. The method is demonstrated in data interpolation and in partial differential equations. In addition, an implementation of the Element-Free Galerkin method has been written to run on a Graphics Processing Unit. The implementation is described and its performance is compared to that of a similar implementation that does not make use of the Graphics Processing Unit.

Velmi přesná následná astrometrie je nezbytným předpokladem sledování blízkozemních objektů, které mohou představovat riziko srážky se Zemí. Tato práce přináší ucelený přehled přesné astrometrie, obsahuje potřebnou matematickou teorii, postup předzpracování snímků v astronomii, a nastiňuje použití filtrů. Navrhuje nové metody pro vyrovnání pozadí snímků před provedením astrometrického měření pro případ, kdy nejsou dostupné kalibrační snímky. Tyto metody jsou založeny na vytvoření syntetického flatfieldu pomocí aplikování filtru na snímek a následné užití tohoto flatfieldu pro odstranění pozadí snímku. Metody byly otestovány na vzorových snímcích a vzápětí použity k získání astrometrických pozic prvního mezihvězdného objektu 1I/2017 U1 ('Oumuamua).

Thesis (Ph.D.)--City University of Hong Kong, 2009.
"Submitted to the Department of Building and Construction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [170]-184)

Multitud de problemas en ciencia e ingeniería se plantean como ecuaciones en derivadas parciales (EDPs). Si la frontera del recinto donde esas ecuaciones han de satisfacerse se desconoce a priori, se habla de "Problemas de frontera libre", propios de sistemas estacionarios no dependientes del tiempo, o bien de "Problemas de frontera móvil", asociados a problemas de evolución temporal, donde la frontera cambia con el tiempo. La solución a dichos problemas viene dada por la expresión de la(s) variable(s) dependiente(s) de la(s) EDP(s) junto con la función que determina la posición de la frontera. Dado que este tipo de problemas carece en la mayoría de los casos de solución analítica conocida, se hace preciso recurrir a métodos numéricos que permitan obtener una solución lo suficientemente aproximada, y que además mantenga propiedades cualitativas de la solución del modelo continuo de EDP(s). En este trabajo se ha abordado el estudio numérico de algunos problemas de frontera móvil provenientes de diversas disciplinas. La metodología aplicada consta de dos pasos sucesivos: aplicación de la transformación de Landau o "Front-fixing transformation" al modelo en EDP(s) con el fin de mantener inmóvil la frontera del dominio, y posterior discretización a través de un esquema en diferencias finitas. De ahí se obtienen esquemas numéricos que se implementan por medio de la herramienta MATLAB. Mediante un exhaustivo análisis numérico, se estudian propiedades del esquema y de la solución numérica (positividad, estabilidad, consistencia, monotonía, etc.). En el primer capítulo de este trabajo se revisa el estado del arte del campo objeto de estudio, se justifica la necesidad de disponer de métodos numéricos adaptados a este tipo de problemas y se describe brevemente la metodología empleada en nuestro enfoque. El Capítulo 2 se dedica a un problema perteneciente a la Biología Matemática y que consiste en determinar la evolución de la población de una especie invasora que se propaga en un hábitat. Este modelo consiste en una ecuación de difusión-reacción unida a una condición tipo Stefan. Los resultados del análisis numérico confirman la existencia de una dicotomía propagación-extinción en la evolución a largo plazo de la densidad de población de la especie invasora. En particular, se ha podido precisar el valor del coeficiente de la condición de Stefan que separa el comportamiento de propagación del de extinción. Los Capítulos 3 y 4 se centran en un problema de Química del Hormigón con interés en Ingeniería Civil: el proceso de carbonatación del hormigón, fenómeno evolutivo que lleva consigo la degradación progresiva de la estructura afectada y finalmente su ruina, si no se toman medidas preventivas. En el Capítulo 3 se considera un sistema de dos EDPs de tipo parabólico con dos incógnitas. Para su resolución, hay que considerar además las condiciones iniciales, las de contorno y las de tipo Stefan en la frontera. Los resultados numéricos confirman la tendencia de la ley de evolución de la frontera móvil hacia una función del tipo "raíz cuadrada del tiempo". En el Capítulo 4 se considera un modelo más general que el anterior, en el que intervienen seis especies químicas que se encuentran tanto en la zona carbonatada como en la no carbonatada. En el Capítulo 5 se aborda un problema de transmisión de calor que aparece en diversos procesos industriales; en este caso, en el enfriamiento durante la colada de metal fundido, donde la fase sólida avanza y la líquida se va extinguiendo. La frontera móvil (frente de solidificación) separa ambas fases, siendo su posición en cada instante la variable a determinar, junto con las temperaturas en cada fase. Después de la adecuada transformación y discretización, se implementa un esquema en diferencias finitas, subdividiendo el proceso en tres estadios temporales, a fin de tratar las singularidades asociadas a posicione
Many problems in science and engineering are formulated as partial differential equations (PDEs). If the boundary of the domain where these equations are to be solved is not known a priori, we face "Free-boundary problems", which are characteristic of non-time dependent stationary systems; besides, we have "Moving-boundary problems" in temporal evolution processes, where the border changes over time. The solution to these problems is given by the expression of the dependent variable(s) of PDE(s), together with the function that determines the position of the boundary. Since the analytical solution of this type of problems is lacked in most cases, it is necessary to resort to numerical methods that allow an accurate enough solution to be obtained, and which also maintain the qualitative properties of the solution(s) of the continuous model. This work approaches the numerical study of some moving-boundary problems that arise in different disciplines. The applied methodology consists of two successive steps: firstly, the so-called Landau transformation, or "Front-fixing transformation", which is used in the PDE(s) model to maintain the boundary of the domain immobile; later, we proceed to its discretization with a finite difference scheme. Different numerical schemes are obtained and implemented through the MATLAB computational tool. Properties of the scheme and the numerical solution (positivity, stability, consistency, monotonicity, etc.) are studied by an exhaustive numerical analysis. The first chapter of this work reports the state of the art of the field under study, justifies the need to adapt numerical methods to this type of problem, and briefly describes the methodology used in our approach. Chapter 2 presents a problem in Mathematical Biology that consists in determining over time the evolution of an invasive species population that spreads in a habitat. This problem is modelled by a diffusion-reaction equation linked to a Stefan-type condition. The results of the numerical analysis confirm the existence of a spreading-vanishing dichotomy in the long-term evolution of the population density of the invasive species. In particular, it is possible to determine the value of the coefficient of the Stefan condition that separates the propagation behaviour from extinction. Chapters 3 and 4 focus on a problem of Concrete Chemistry with an interest in Civil Engineering: the carbonation of concrete, an evolutionary phenomenon that leads to the progressive degradation of the affected structure and its eventual ruin if preventive measures are not taken. Chapter 3 considers a system of two parabolic type PDEs with two unknowns. For its resolution, the initial and boundary conditions have to be considered together with the Stefan conditions on the carbonation front. The numerical analysis results agree with those obtained in a previous theoretical study. The dynamics of the concentrations and the moving boundary confirm the long-term behaviour of the evolution law for the moving boundary as a "square root of time". Chapter 4 considers a more general model than the previous one, which includes six chemical species, defined in both the carbonated and non-carbonated zones, whose concentrations have to be found. Chapter 5 addresses a heat transfer problem that appears in various industrial processes; in this case, the solidification of metals in casting processes, where the solid phase advances and liquid reduces until it is depleted. The moving boundary (the solidification front) separates both phases. Its position in each instant is the variable to be determined together with the temperature profiles in both phases. After suitable transformation, discretization is carried out to obtain a finite difference scheme to be implemented. The process was subdivided into three temporal stages to deal with the singularities associated with the moving boundary position in the initialisation and depletion stages.
Multitud de problemes en ciència i enginyeria es plantegen com a equacions en derivades parcials (EDPs). Si la frontera del recinte on eixes equacions han de satisfer-se es desconeix a priori, es parla de "Problemas de frontera lliure", propis de sistemes estacionaris no dependents del temps, o bé de "Problemas de frontera mòbil", associats a problemes d'evolució temporal, on la frontera canvia amb el temps. Atés que este tipus de problemes manca en la majoria dels casos de solució analítica coneguda, es fa precís recórrer a mètodes numèrics que permeten obtindre una solució prou aproximada a l'exacta, i que a més mantinga propietats qualitatives de la solució del model continu d'EDP(s). En aquest treball s'ha abordat l'estudi numèric d'alguns problemes de frontera mòbil provinents de diverses disciplines. La metodologia aplicada consta de dos passos successius: en primer lloc, s'aplica l'anomenada transformació de Landau o "Front-fixing transformation" al model en EDP(s) a fi de mantindre immòbil la frontera del domini; posteriorment, es procedix a la seva discretització a través d'un esquema en diferències finites. D'ací s'obtenen esquemes numèrics que s'implementen per mitjà de la ferramenta informàtica MATLAB. Per mitjà d'una exhaustiva anàlisi numèrica, s'estudien propietats de l'esquema i de la solució numèrica (positivitat, estabilitat, consistència, monotonia, etc.). En el primer capítol d'aquest treball es revisa l'estat de l'art del camp objecte d'estudi, es justifica la necessitat de disposar de mètodes numèrics adaptats a aquest tipus de problemes i es descriu breument la metodologia emprada en el nostre enfocament. El Capítol 2 es dedica a un problema pertanyent a la Biologia Matemàtica i que consistix a determinar l'evolució en el temps de la distribució de la població d'una espècie invasora que es propaga en un hàbitat. Este model consistix en una equació de difusió-reacció unida a una condició tipus Stefan, que relaciona les funcions solució i frontera mòbil a determinar. Els resultats de l'anàlisi numèrica confirmen l'existència d'una dicotomia propagació-extinció en l'evolució a llarg termini de la densitat de població de l'espècie invasora. En particular, s'ha pogut precisar el valor del coeficient de la condició de Stefan que separa el comportament de propagació del d'extinció. Els Capítols 3 i 4 se centren en un problema de Química del Formigó amb interés en Enginyeria Civil: el procés de carbonatació del formigó, fenomen evolutiu que comporta la degradació progressiva de l'estructura afectada i finalment la seua ruïna, si no es prenen mesures preventives. En el Capítol 3 es considera un sistema de dos EDPs de tipus parabòlic amb dos incògnites. Per a la seua resolució, cal considerar a més, les condicions inicials, les de contorn i les de tipus Stefan en la frontera. Els resultats de l'anàlisi numèrica s'ajusten als obtinguts en un estudi teòric previ. S'han dut a terme experiments numèrics, comprovant la tendència de la llei d'evolució de la frontera mòbil cap a una funció del tipus "arrel quadrada del temps". En el Capítol 4 es considera un model més general, en el que intervenen sis espècies químiques les concentracions de les quals cal trobar, i que es troben tant en la zona carbonatada com en la no carbonatada. En el Capítol 5 s'aborda un problema de transmissió de calor que apareix en diversos processos industrials; en aquest cas, en el refredament durant la bugada de metall fos, on la fase sòlida avança i la líquida es va extingint. La frontera mòbil (front de solidificació) separa ambdues fases, sent la seua posició en cada instant la variable a determinar, junt amb les temperatures en cada una de les dos fases. Després de l'adequada transformació i discretització, s'implementa un esquema en diferències finites, subdividint el procés en tres estadis temporals, per tal de tractar les singularitats asso
Piqueras García, MÁ. (2018). Numerical Methods for Multidisciplinary Free Boundary Problems: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/107948
TESIS

This thesis reports onreview and research work carried out on the numerical analysis of elastomers. The two numerical techniques investigated for this purpose are the finite and boundary element methods. The finite element method is studied so that existing theory is used to develop a finite element code both to review the finite element method as applied to the stress analysis of elastomers and to provide a comparison of results and numerical approach with the boundary element method. The research work supported on in this thesis covers the application of the boundary element method to the stress analysis of elastomers. To this end a simplified regularization approach is discussed for the removal of strong and hypersingularities generated in the system on non-linear boundary integral equations. The necessary programming details for the implementation of the boundary element method are discussed based on the code developed for this research. Both the finite and boundary element codes developed for this research use the Mooney-Rivlin material model as the strain energy based constitutive stress strain function. For validation purposes four test cases are investigated. These are the uni-axial patch test, pressurized thick wall cylinder, centrifugal loading of a rotating disk and the J-Integral evaluation for a centrally cracked plate. For the patch test and pressurized cylinder, both plane stress and strain have been investigated. For the centrifugal loading and centrally cracked plate test cases only plane stress has been investigated. For each test case the equivalent results for an equivalent FEM program mesh have been presented. The test results included in this thesis prove that the FE and BE derivations detailed in this work are correct. Specifically the simplified domain integral singular and hyper-singular regularization approach was shown to lead to accurate results for the test cases detailed. Various algorithm findings specific to the BEM implementation of the theory are also discussed.

This thesis is concerned with a class of explicit numerical methods for multiscale differential equations, including the Heterogeneous Multiscale Methods (HMM) and Projective Integration (PI) methods. These techniques have been developed within the last decade and successfully applied to a wide range of multiscale problems. We examine the HMM and PI methods when applied to multiscale systems for which the dynamics converges rapidly to a lower dimensional manifold defined in terms of the slow degrees of freedom, and provide rigorous convergence results for the methods under these conditions. The analysis allows us to establish the differences between several formulations of HMM, as well as develop a PI method with slightly better accuracy and stability properties than existing PI formulations. We corroborate our results by numerical simulations. We then compare the HMM and PI methods, with insight on how to select numerical parameters, the difficulties faced by each method, and how one might bypass these difficulties.

This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8.

Thesis (Ph.D.)--George Mason University, 2009.
Vita: p. 108. Thesis co-directors: John Wallin, Thomas Wanner. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computational Science and Informatics. Title from PDF t.p. (viewed Oct. 12, 2009). Includes bibliographical references (p. 102-107). Also issued in print.

Research on interfacial phenomena has a long history, which has attracted tremendous interest in recent years. One of the most successful tools is the phase-field approach. As phase-field models usually involve very complex dynamics and it is nontrivial to obtain analytical solutions, numerical methods have played an important role in various simulations. This thesis is mainly devoted to developing accurate, efficient and robust numerical methods and the related numerical analysis for three representative phase-field models, namely the Allen-Cahn equation, the Cahn-Hilliard equation and the thin film models. The first part of this thesis is mainly concentrated on the stability analysis for these three models, with particular attention to the Allen-Chan equation. We have established three stability criterion, i.e., nonlinear energy stability, L∞-stability and L2-stability. As shared by most phase-field models, one of the intrinsic properties of the Allen- Cahn and the Cahn-Hilliard equations is that they satisfy a nonlinear stability re- lationship, usually expressed as a time-decreasing free energy functional. We have studied several stabilized temporal discretization for both the Allen-Cahn and the Cahn-Hilliard equations so that the relevant nonlinear energy stability can be pre- served. The corresponding temporal discretization schemes are linear and are of second-order accuracy. We also apply multi-step implicit-explicit methods to ap- proximate the Allen-Cahn equation. We demonstrate that by suitably choosing the parameters in multi-step implicit-explicit methods the nonlinear energy stability can be preserved. Apart from studying the energy stability for the Allen-Cahn equation, we also establish the numerical maximum principle for some fully discretized schemes. We further extend our analysis technique to the fractional-in-space Allen-Cahn equation. A more general Allen-Cahn-type equation with a nonlinear degenerate mobility and a logarithmic free energy is also considered. The third stability under investigation is the L2-stability. We prove that the con- tinuum Allen-Cahn equation satisfies a uniform Lp-stability. Furthermore, we show that both semi-discretized Fourier Galerkin and Fourier collocation methods can in- herit this stability for p = 2, i.e., L2-stability. Based on the established L2-stability, we accomplish the spectral convergence estimate for the Fourier Galerkin methods. We adopt the second-order Strang splitting schemes in the temporal direction with Fourier collocation methods to demonstrate the uniform L2-stability in the fully dis- cretized scheme. Another contribution of this thesis is to propose a p-adaptive spectral deferred correction methods for the long time simulations for all three models. We develop a high-order accurate and energy stable scheme to simulate the phase-field models by combining the semi-implicit spectral deferred correction method and first-order stabilized semi-implicit schemes. It is found that the accuracy improvement may affect the overall energy stability. To compromise the accuracy and stability, a local p- adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. Numerical results demonstrate the high effectiveness of our proposed numerical strategy. Keywords: Phase-field models, Allen-Cahn equations, Cahn-Hilliard equations, thin film models, nonlinear energy stability, maximum principle, L2-stability, adaptive simulations, stabilized semi-implicit schemes, finite difference, Fourier spectral meth- ods, spectral deferred correction methods, convex splitting

Thesis (Ph. D.)--Michigan State University. Electrical and Computer Engineering, 2008.
Title from PDF t.p. (viewed on Apr. 8, 2009) Includes bibliographical references (p. 148-153). Also issued in print.

During the epoch when the first collapsed structures formed (6<z<50) our Universe went through an extended period of changes. Some of the radiation from the first stars and accreting black holes in those structures escaped and changed the state of the Intergalactic Medium (IGM). The era of this global phase change in which the state of the IGM was transformed from cold and neutral to warm and ionized, is called the Epoch of Reionization.In this thesis we focus on numerical methods to calculate the effects of this escaping radiation. We start by considering the performance of the cosmological radiative transfer code C2-Ray. We find that although this code efficiently and accurately solves for the changes in the ionized fractions, it can yield inaccurate results for the temperature changes. We introduce two new elements to improve the code. The first element, an adaptive time step algorithm, quickly determines an optimal time step by only considering the computational cells relevant for this determination. The second element, asynchronous evolution, allows different cells to evolve with different time steps. An important constituent of methods to calculate the effects of ionizing radiation is the transport of photons through the computational domain or ``ray-tracing''. We devise a novel ray tracing method called PYRAMID which uses a new geometry - the pyramidal geometry. This geometry shares properties with both the standard Cartesian and spherical geometries. This makes it on the one hand easy to use in conjunction with a Cartesian grid and on the other hand ideally suited to trace radiation from a radially emitting source. A time-dependent photoionization calculation not only requires tracing the path of photons but also solving the coupled set of photoionization and thermal equations. Several different solvers for these equations are in use in cosmological radiative transfer codes. We conduct a detailed and quantitative comparison of four different standard solvers in which we evaluate how their accuracy depends on the choice of the time step. This comparison shows that their performance can be characterized by two simple parameters and that the C2-Ray generally performs best.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Submitted. Paper 3: Submitted.

The approximate greatest descent (AGD) method and a two-phase AGD method (AGDN) are proposed as new methods for a nonlinear least squares problem. Numerical experiments show that these methods outperform existing methods including the Levenberg-Marquardt method. However, the AGDN method outperforms the AGD method with a faster convergence. If the AGDN method fails due to singularity of the Hessian matrix, the AGD method should be used.

The use of guided ultrasonic waves (GUW) has increased considerably in the fields of non-destructive (NDE) testing and structural health monitoring (SHM) due to their ability to perform long range inspections, to probe hidden areas as well as to provide a complete monitoring of the entire waveguide. Guided waves can be fully exploited only once their dispersive properties are known for the given waveguide. In this context, well stated analytical and numerical methods are represented by the Matrix family methods and the Semi Analytical Finite Element (SAFE) methods. However, while the former are limited to simple geometries of finite or infinite extent, the latter can model arbitrary cross-section waveguides of finite domain only. This thesis is aimed at developing three different numerical methods for modelling wave propagation in complex translational invariant systems. First, a classical SAFE formulation for viscoelastic waveguides is extended to account for a three dimensional translational invariant static prestress state. The effect of prestress, residual stress and applied loads on the dispersion properties of the guided waves is shown. Next, a two-and-a-half Boundary Element Method (2.5D BEM) for the dispersion analysis of damped guided waves in waveguides and cavities of arbitrary cross-section is proposed. The attenuation dispersive spectrum due to material damping and geometrical spreading of cavities with arbitrary shape is shown for the first time. Finally, a coupled SAFE-2.5D BEM framework is developed to study the dispersion characteristics of waves in viscoelastic waveguides of arbitrary geometry embedded in infinite solid or liquid media. Dispersion of leaky and non-leaky guided waves in terms of speed and attenuation, as well as the radiated wavefields, can be computed. The results obtained in this thesis can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup.

The use of guided ultrasonic waves (GUW) has increased considerably in the fields of non-destructive (NDE) testing and structural health monitoring (SHM) due to their ability to perform long range inspections, to probe hidden areas as well as to provide a complete monitoring of the entire waveguide. Guided waves can be fully exploited only once their dispersive properties are known for the given waveguide. In this context, well stated analytical and numerical methods are represented by the Matrix family methods and the Semi Analytical Finite Element (SAFE) methods. However, while the former are limited to simple geometries of finite or infinite extent, the latter can model arbitrary cross-section waveguides of finite domain only. This thesis is aimed at developing three different numerical methods for modelling wave propagation in complex translational invariant systems. First, a classical SAFE formulation for viscoelastic waveguides is extended to account for a three dimensional translational invariant static prestress state. The effect of prestress, residual stress and applied loads on the dispersion properties of the guided waves is shown. Next, a two-and-a-half Boundary Element Method (2.5D BEM) for the dispersion analysis of damped guided waves in waveguides and cavities of arbitrary cross-section is proposed. The attenuation dispersive spectrum due to material damping and geometrical spreading of cavities with arbitrary shape is shown for the first time. Finally, a coupled SAFE-2.5D BEM framework is developed to study the dispersion characteristics of waves in viscoelastic waveguides of arbitrary geometry embedded in infinite solid or liquid media. Dispersion of leaky and non-leaky guided waves in terms of speed and attenuation, as well as the radiated wavefields, can be computed. The results obtained in this thesis can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup.

Dalla sua concezione a metà degli anni novanta, il legno lamellare a strati incrociati, anche noto come CLT o X-Lam, ha raggiunto grande popolarità tra i materiali da costruzione grazie alle numerose innate qualità, gli sforzi a livello mondiale per costruire strutture affidabili in zone a rischio sismico e la necessità di costruire un ambiente più eco-sostenibile. Molti test sono stati fatti negli ultimi 15 anni, volti a comprendere meglio il comportamento delle connessioni in edifici in CLT, di parti strutturali o di intere strutture in CLT, in modo da fornire regole affidabili per i progettisti per progettare strutture in CLT sotto ogni condizione di carico. Sulla base di questi test, molti sono stati i modelli numerici che sono stati suggeriti negli anni. Questi rappresentano uno strumento fondamentale per la progettazione di strutture in CLT quando insorgono specifiche problematiche ed un approccio analitico da solo non è sufficiente. Nonostante i molti anni di sforzi, non esistono ancora affidabili metodologie di progetto nella quasi totalità dei codici a livello mondiale e ancora molte sono le incognite relative al comportamento delle strutture in CLT a molti livelli (connessioni, parti strutturali, strutture). Questa tesi riassume tre anni di ricerche numeriche, le quali hanno affrontato diversi problemi relativi al comportamento di elementi strutturali e strutture in CLT sotto azioni dinamiche. Durante la prima parte di questo percorso l’attenzione è stata posta sulla continuazione di un precedente studio, portato avanti durante la tesi di laurea magistrale, il quale era incentrato sulla formulazione di un metodo semplificato per la costruzione di un dominio resistente sforzo normale-momento flettente per pannelli in CLT connessi alla base da connessioni tipo hold-down e angle bracket. In mancanza di risultati di test di interesse, la concentrazione è stata rivolta ancora alla formulazione di metodi semplificati per la progettazione di elementi strutturali in CLT. È stato analizzato il problema delle connessioni pannello-pannello all’interno di una stessa parete. In particolare, è stata studiata la rigidezza di queste connessioni in relazione al comportamento ribaltante di pareti a due pannelli attraverso l’analisi di test a scala reale indipendenti e analisi numeriche agli elementi finiti. Una formula per il calcolo di queste connessioni è stata dapprima proposta e poi, dopo ulteriori analisi, rivista e corretta. Per estendere l’analisi e considerare elementi strutturali più complessi, è stata investigata, a livello di analisi numerica, l’influenza del solaio e delle connessioni parete-solaio superiore sul comportamento ribaltante delle pareti, prendendo in considerazione configurazioni con e senza solaio, variando diversi parametri di modo da ottenere risultati statisticamente significativi. Nell’estate del 2017 il candidato ha partecipato attivamente al NHERI TallWood Project, una ricerca statunitense intesa a testare strutture in CLT per fornire regole di progettazione per tali strutture nei futuri codici nazionali. Sponsorizzato dalla Colorado State University, nella persona del Prof. John W. van de Lindt, il candidato ha collaborato alla preparazione di un edificio con due orizzontamenti fuori terra testato sulla tavola vibrante della UCSD a San Diego (California) Per valutare il più corretto valore di smorzamento per strutture in CLT sotto l’azione di eventi sismici di bassa intensità, sono stati riprodotti numericamente ed analizzati i test su tavola vibrante del progetto SOFIE a 0,15 g. Ulteriori considerazioni sono state fatte sul ruolo dell’attrito su questo tipo di strutture e sul problema delle analisi lineari per strutture in CLT (risposta non simmetrica di connessioni caricate in tensione-compressione).
Since its conception in the mid 90’s, cross-laminated timber, known also as CLT or X-Lam, has achieved a great popularity as construction material thanks to its numerous intrinsic qualities, worldwide effort to build reliable structures in seismic-prone areas and necessity to build a more eco-friendly environment. Many tests have been carried out in the last 15 years, aimed to better understand the behavior of connections in CLT buildings, CLT assemblies and CLT structures in order to provide reliable rules for designers to design structures made of CLT in any loading condition. Based on these tests, many numerical models have been suggested through the years. They represent a fundamental tool for the design of CLT structures when specific design problems arise. Despite many years of efforts, reliable design rules are still missing in almost every code worldwide and many are still the unknown related to CLT structures behavior at many levels (connections, assemblies, structures). This thesis summarizes three years of numerical investigations, which have faced different problems related to the comprehension of CLT assemblies and structures behavior under dynamic loading conditions. The first part of this path focused on the continuation of a previous study made within the Master Degree thesis, which was the formulation of a simplified method to obtain an axial-load/bending moment limit domain for a CLT panel connected to the supporting surface through hold-down and angle bracket connections. Without test results of interest, the focus of the study returned to be the formulation of simple methods for CLT assemblies design. The problem of panel-to-panel connections was investigated. In particular, the stiffness of such connections related to the rocking behavior of 2-panel wall assemblies was studied through full-scale tests and FE numerical analyses. A formula for the design of these connections was firstly suggested and then, after further analyses, revised and corrected. In order to extend the analyses and consider more complex assemblies, the influence of diaphragm and wall-to-diaphragm connections stiffness on the rocking behavior of wall assemblies was numerically investigated, taking into account configuration with and without diaphragm, varying several parameters to obtain statistically significant results. In the summer of 2017 the candidate actively participated to the NHERI TallWood Project, an American research project intended to test CLT structures in order to provide design rules for these structures in the future US codes. Sponsored by the Colorado State University, in the person of Professor John W. van de Lindt, the candidate collaborated to the setup of a 2-story CLT building that was tested on the UCSD shaking table located in San Diego (California). In order to assess the most proper value of damping for CLT structures under low-intensity seismic events and to better investigate the potential of the component approach for the modelling of CLT structures, the 0,15 g shaking table tests of the 3-story building within the SOFIE Project were reproduced and analyzed. Further considerations on the role of friction for this type of structure have been made together with the problem of linear analyses for CLT structures (non-symmetric response for tension-compression loaded connections).

This work is concerned with the numerical solution of Friedrichs systems by discontinuous Galerkin finite element methods (DGFEMs). Friedrichs systems are boundary value problems with symmetric, positive, linear first-order partial differential operators and allow the unified treatment of a wide range of elliptic, parabolic, hyperbolic and mixed-type equations. We do not assume that the exact solution of a Friedrichs system belongs to a Sobolev space, but only require that it is contained in the associated graph space, which amounts to differentiability in the characteristic direction. We show that the numerical approximations to the solution of a Friedrichs system by the DGFEM converge in the energy norm under hierarchical h- and p- refinement. We introduce a new compatibility condition for the boundary data, from which we can deduce, for instance, the validity of the integration-by-parts formula. Consequently, we can admit domains with corners and allow changes of the inertial type of the boundary, which corresponds in special cases to the componentwise transition from in- to outflow boundaries. To establish the convergence result we consider in equal parts the theory of graph spaces, Friedrichs systems and DGFEMs. Based on the density of smooth functions in graph spaces over Lipschitz domains, we study trace and extension operators and also investigate the eigensystem associated with the differential operator. We pay particular attention to regularity properties of the traces, that limit the applicability of energy integral methods, which are the theoretical underpinning of Friedrichs systems. We provide a general framework for Friedrichs systems which incorporates a wide range of singular boundary conditions. Assuming the aforementioned compatibility condition we deduce well-posedness of admissible Friedrichs systems and the stability of the DGFEM. In a separate study we prove hp-optimality of least-squares stabilised DGFEMs.

Asian options are exotic financial derivative products which price must be calculated by numerical evaluation. In this thesis, we study certain ways of solving partial differential equations, which are associated with these derivatives. Since standard numerical techniques for Asian options are often incorrect and impractical, we discuss their variations, which are efficiently applicable for handling frequent numerical instabilities reflected in form of oscillatory solutions. We will show that this crucial problem can be treated and eliminated by adopting flux limiting techniques, which are total variation dimishing.

Understanding the mechanics of turbulent fluid flow is of key importance for industry and society as for example in aerodynamics and aero-acoustics. The massive computational cost for resolving all turbulent scales in a realistic problem makes direct numerical simulation of the underlying Navier-Stokes equations impossible. Recent advances in adaptive finite element methods offer a new powerful tool in Computational Fluid Dynamics (CFD). The computational cost for simulating turbulent flow can be minimized where the mesh is adaptively resolved, based on a posteriori error control. These adaptive methods have been implemented for efficient serial computations, but the extension to an efficient parallel solver is a challenging task. This work concerns the development of an adaptive finite element method for modern parallel computer architectures. We present efficient data structures and data decomposition methods for distributed unstructured tetrahedral meshes. Our work also concerns an efficient parallellization of local mesh refinement methods such as recursive longest edge bisection. We also address the load balance problem with the development of an a priori predictive dynamic load balancing method. Current results are encouraging with almost linear strong scaling to thousands of cores on several modern architectures.
QC 20110223

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Durante a vida produtiva de um poço de petróleo, problemas devido à produção de sólidos podem ocasionar gastos excessivos por danos nos equipamentos ou redução de produtividade do poço. Por causa destes problemas, a instalação de sistemas de contenção de sólidos na etapa de completação é uma das mais complexas e fundamentais fases na construção do poço. A alteração no estado de tensões atuante sobre a formação é uma das principais fontes de carregamento dos sistemas de contenção mecânica de sólidos. Este trabalho visa simular as tensões atuantes no sistema de contenção de sólidos (gravel packing e stand alone) instalados em uma formação com potencial de produção de sólidos, permitindo a otimização de projetos para este tipo de sistemas. Para isso foi utilizado o modelo de Mohr Coulomb solucionado numericamente no software comercial de elementos finitos Abaqus que foi escolhido devido a sua enorme capacidade de resolver problemas não lineares. Os resultados obtidos foram então comparados com ensaios experimentais que apresentaram comportamento bastante semelhante com os obtidos numericamente. Além disso, foi observada a capacidade do gravel packing de suportar as tensões até determinado estado de tensões.
During the production steps of a petroleum well, issues regarding sand production may have hight costs due to damages in the equipment or reduction of the well’s productivity. Such problems make the application of sand control systems in the completion phase one of the most complex and essential parts in the construction of the well. This work aims to simulate the behavior of different sand control methods (gravel packing and stand alone) taking into account mechanical interaction between the formation and sand control screens. For the development of the present study, elastoplastic (Mohr Coulomb) models are used to represent granular materials with the commercial FEM software Abaqus, chosen due to its versality in the solution of non-linear problems named out previously. Numerical simulations were compared to experimental tests which presented similar behavior regarding the numerical analysis. In addition, it was observed the capability of the gravel packing to withstand the stresses up to a certain state of stress.

Pipe junctions arc a regular feature of piping and pressure vessel systems and are often the subject of multiple loads. acting simultaneously and at irregular intervals. Due to the nature and complexity of the loading. the subject has received a significant amount of study from designers and stress analysts to resolve some of the difficulties in stressing pressure structures. An extensive finite element (FE) analysis was carried out on 92 reinforced buttwelded pipe junctions manufactured by the collaborating company. Spromak Ltd. After comparing the resulting effective stress factor (ESF) data with ESFs for un-reinforced fahricated tee (UFT) it was concluded that, for the majority of loads, reinforced branch outlets appear better able to contain stresses than their un-reinforced counterparts. The linear FE study was followed by the inelastic analysis of three reinforced branch junctions. The purpose of the research was to investigate the potential use of such analysis as a tool for estimating the bursting pressure of pipe junctions and satisfying customer requirement for proof of a products performance under internal pressure. Results obtained showed that small displacement analysis is unsuitable for estimating the bursting pressure of a pipe junction, whilst the large displacement results were similar to those obtained using a hand-calculation. Ultimately, the study concluded that inelastic analysis was too expensive, offering little by way of insight into the problem than could be found by using classical stress analysis techniques. Following on from the study of reinforced branch outlets, this thesis described work undertaken with British Energy Ltd. to extend their current capability of stress prediction in UFT junctions using a FE based neural network approach. Upon completion of training new neural networks, the PIPET program was tested against new, previously unseen, FE data generated for this study with good results. The program was further evaluated by comparing the output from PIPET with FE data obtained from reviewed literature. For the pressure load case, a significant proportion of the data obtained from said literature was within the PIPET predicted stress ranges. with the new version of PIPET tending to calculate slightly lower stresses than the original program. However, whilst the pressure load case comparisons proved useful, the branch bending cases showed less concordance with PIPET's predicted stress ranges.

In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

[EN] The present PhD thesis is focused on numerical analysis and computing of finite difference schemes for several relevant option pricing models that generalize the Black-Scholes model. A careful analysis of desirable properties for the numerical solutions of option pricing models as the positivity, stability and consistency, is provided. In order to handle the free boundary that arises in American option pricing problems, various transformation techniques based on front-fixing method are applied and studied. Special attention is paid to multi-asset option pricing, such as exchange or spread option. Appropriate transformation allows eliminating of the cross derivative term. Transformation techniques of partial differential equations to remove convection and reaction terms are studied in order to simplify the models and avoid possible troubles of stability. This thesis consists of six chapters. The first chapter is an introduction containing definitions of option and related terms and derivation of the Black-Scholes equation as well as general aspects of theory of finite difference schemes, including preliminaries on numerical analysis. Chapter 2 is devoted to solve linear Black-Scholes model for American put and call options. A Landau transformation and a new front-fixing transformation are applied to the free boundary value problem. It leads to non-linear partial differential equation (PDE) in a fixed domain. Stable and consistent explicit numerical schemes are proposed preserving positivity and monotonicity of the solution in accordance with the behaviour of the exact solution. Efficiency of the front-fixing method demonstrated in Chapter 2 has motivated us to apply the method to some more complicated nonlinear models. A new change of variables resulting in a time dependent boundary instead of fixed one, is applied to nonlinear Black-Scholes model for American options, such as Barles and Soner and Risk Adjusted Pricing models. Chapter 4 provides a new alternative approach for solving American option pricing problem based on rationality of investor. There exists an intensity function that can be reduced in the simplest case to penalty approach. Chapter 5 deals with multi-asset option pricing. Appropriate transformation allows eliminating of the cross derivative term avoiding computational drawbacks and possible troubles of stability. Concluding remarks are given in Chapter 6. All the considered models and numerical methods are accompanied by several examples and simulations. The convergence rate is computed confirming the theoretical study of consistency. Stability conditions are tested by numerical examples. Results are compared with known relevant methods in the literature showing efficiency of the proposed methods.
[ES] La presente tesis doctoral se centra en la construcción de esquemas en diferencias finitas y el análisis numérico de relevantes modelos de valoración de opciones que generalizan el modelo de Black-Scholes. Se proporciona un análisis cuidadoso de las propiedades de las soluciones numéricas tales como la positividad, la estabilidad y la consistencia. Con el fin de manejar la frontera libre que surge en los problemas de valoración de opciones Americanas, se aplican y se estudian diversas técnicas de transformación basadas en el método de fijación de las fronteras (front-fixing). Se presta especial atención a la valoración de opciones de múltiples activos, como son las opciones ''exchange'' y ''spread''. Esta tesis se compone de seis capítulos. El primer capítulo es una introducción que contiene las definiciones de opción y términos relacionados y la derivación de la ecuación de Black-Scholes, así como aspectos generales de la teoría de los esquemas en diferencias finitas, incluyendo preliminares de análisis numérico. El capítulo 2 está dedicado a resolver el modelo lineal de Black-Scholes para opciones Americanas put y call. Para fijar las fronteras del problema de frontera libre se aplican transformaciones como la de Landau y un nuevo cambio de variable propuesto. La eficiencia del método front-fixing mostrada en el capítulo 2 ha motivado el estudio de su aplicación a algunos modelos no lineales más complicados. En particular, se propone un cambio de variables que lleva a una nueva frontera dependiente del tiempo en lugar de una fija. Este cambio se aplica a modelos no lineales de Black-Scholes para opciones Americanas, como son el de Barles y Soner y el modelo RAPM (Risk Adjusted Pricing Methodology). El capítulo 4 ofrece una nueva técnica para la resolución de problemas de valoración de opciones Americanas basada en la racionalidad de los inversores. Aparece una función de la intensidad que se puede reducir en el caso más simple a la técnica de penalización (penalty method). Este enfoque tiene en cuenta el posible comportamiento irracional de los inversores. En la sección 4.2 se aplica esta técnica al modelo de cambio de regímenes lo que lleva a un nuevo modelo que tiene en cuenta el posible ejercicio irracional, así como varios estados del mercado. El enfoque del parámetro de racionalidad junto con una transformación logarítmica permiten construir un esquema numérico eficiente sin aplicar el método front-fixing o la conocida formulación de LCP (Linear Complementarity Problem). El capítulo 5 se dedica a la valoración de opciones de activos múltiples. Una transformación apropiada permite la eliminación del término de derivadas cruzadas evitando inconvenientes computacionales y posibles problemas de estabilidad. Las conclusiones se muestran en el capítulo 6. Se pone en relieve varios aspectos de la presente tesis. Todos los modelos considerados y los métodos numéricos van acompañados de varios ejemplos y simulaciones. Se estudia la convergencia numérica que confirma el estudio teórico de la consistencia. Las condiciones de estabilidad son corroboradas con ejemplos numéricos. Los resultados se comparan con métodos relevantes de la bibliografía mostrando la eficiencia de los métodos propuestos.
[CAT] La present tesi doctoral se centra en la construcció d'esquemes en diferències finites i l'anàlisi numèrica de rellevants models de valoració d'opcions que generalitzen el model de Black-Scholes. Es proporciona una anàlisi cuidadosa de les propietats de les solucions numèri-ques com ara la positivitat, l'estabilitat i la consistència. A fi de manejar la frontera lliure que sorgix en els problemes de valoració d'opcions Americanes, s'apliquen i s'estudien diverses tècniques de transformació basades en el mètode de fixació de les fronteres (front-fixing). Es presta especial atenció a la valoració d'opcions de múltiples actius, com són les opcions ''exchange'' i ''spread''. Esta tesi es compon de sis capítols. El primer capítol és una introducció que conté les definicions d'opció i termes relacionats i la derivació de l'equació de Black-Scholes, així com aspectes generals de la teoria dels esquemes en diferències finites, incloent aspectes preliminars d'anàlisi numèrica. El 2n capítol està dedicat a resoldre el model lineal de Black-Scholes per a opcions Americanes ''put'' i ''call''. Per a fixar les fronteres del problema de frontera lliure s'apliquen transformacions com la de Landau i s'ha proposat un nou canvi de variable proposat. Açò porta a una equació diferencial en derivades parcials no lineal en un domini fix. L'eficiència del mètode front-fixing mostrada en el 2n capítol ha motivat l'estudi de la seua aplicació a alguns models no lineals més complicats. En particular, es proposa un canvi de variables que porta a una nova frontera dependent del temps en compte d'una fixa. Este canvi s'aplica a models no lineals de Black-Scholes per a opcions Americanes, com són el de Barles i Soner i el model RAPM (Risk Adjusted Pricing Methodology). El 4t capítol oferix una nova tècnica per a la resolució de problemes de valoració d'opcions Americanes basada en la racionalitat dels inversors. Apareix una funció de la intensitat que es pot reduir en el cas més simple a la tècnica de penalització (penal method) . Este enfocament té en compte el possible comportament irracional dels inversors. En la secció 4.2 s'aplica esta tècnica al model de canvi de règims el que porta a un nou model que té en compte el possible exercici irracional, així com diversos estats del mercat. L'enfocament del paràmetre de racionalitat junt amb una transformació logarítmica permeten construir un esquema numèric eficient sense aplicar el mètode front-fixing o la coneguda formulació de LCP (Linear Complementarity Problem). El 5é capítol es dedica a la valoració d'opcions d'actius múltiples. Una transformació apropiada permet l'eliminació del terme de derivades mixtes evitant inconvenients computacionals i possibles problemes d' estabilitat. Les conclusions es mostren al 6é capítol. Es posa en relleu diversos aspectes de la present tesi. Tots els models considerats i els mètodes numèrics van acompanyats de diversos exemples i simulacions. S'estu-dia la convergència numèrica que confirma l'estudi teòric de la consistència. Les condicions d'estabilitat són corroborades amb exemples numèrics. Els resultats es comparen amb mètodes rellevants de la bibliografia mostrant l'eficiència dels mètodes proposats.
Egorova, V. (2016). Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/68501
TESIS
Premiado

In this work we address the problem of shape partitioning which enables the decomposition of an arbitrary topology object into smaller and more manageable pieces called partitions. Several applications in Computer Aided Design (CAD), Computer Aided Manufactury (CAM) and Finite Element Analysis (FEA) rely on object partitioning that provides a high level insight of the data useful for further processing. In particular, we are interested in 2-manifold partitioning, since the boundaries of tangible physical objects can be mathematically defined by two-dimensional manifolds embedded into three-dimensional Euclidean space. To that aim, a preliminary shape analysis is performed based on shape characterizing scalar/vector functions defined on a closed Riemannian 2-manifold. The detected shape features are used to drive the partitioning process into two directions – a human-based partitioning and a thickness-based partitioning. In particular, we focus on the Shape Diameter Function that recovers volumetric information from the surface thus providing a natural link between the object’s volume and its boundary, we consider the spectral decomposition of suitably-defined affinity matrices which provides multi-dimensional spectral coordinates of the object’s vertices, and we introduce a novel basis of sparse and localized quasi-eigenfunctions of the Laplace-Beltrami operator called Lp Compressed Manifold Modes. The partitioning problem, which can be considered as a particular inverse problem, is formulated as a variational regularization problem whose solution provides the so-called piecewise constant/smooth partitioning function. The functional to be minimized consists of a fidelity term to a given data set and a regularization term which promotes sparsity, such as for example, Lp norm with p ∈ (0, 1) and other parameterized, non-convex penalty functions with positive parameter, which controls the degree of non-convexity. The proposed partitioning variational models, inspired on the well-known Mumford Shah models for recovering piecewise smooth/constant functions, incorporate a non-convex regularizer for minimizing the boundary lengths. The derived non-convex non-smooth optimization problems are solved by efficient numerical algorithms based on Proximal Forward-Backward Splitting and Alternating Directions Method of Multipliers strategies, also employing Convex Non-Convex approaches. Finally, we investigate the application of surface partitioning to patch-based surface quadrangulation. To that aim the 2-manifold is first partitioned into zero-genus patches that capture the object’s arbitrary topology, then for each patch a quad-based minimal surface is created and evolved by a Lagrangian-based PDE evolution model to the original shape to obtain the final semi-regular quad mesh. The evolution is supervised by asymptotically area-uniform tangential redistribution for the quads.
In questo lavoro affrontiamo il problema della partizione delle forme il cui scopo è la decomposizione di un oggetto di topologia arbitraria in parti più piccole e meglio gestibili chiamate partizioni. Svariate applicazioni in Computer Aided Design (CAD), Computer Aided Manufactury (CAM) e Finite Element Analysis (FEA) sfruttano tali decomposizioni in quanto forniscono un’informazione globale sulla forma. In particolare, siamo interessati al partizionamento di varietà topologiche di dimensioni 2, in quanto il bordo di oggetti fisici tangibili può essere definito matematicamente da varietà bidimensionali immerse nello spazio euclideo tridimensionale. A tale scopo, viene eseguita un’analisi preliminare sulla forma che fa uso di diverse funzioni scalari/vettoriali definite sulla varietà. Il processo di partizionamento si può affrontare da due punti di vista: uno basato sulla percezione visiva umana e un altro basato sullo spessore delle componenti della forma in esame. In particolare, ci concentriamo sulla funzione ’Diametro di forma’ che recupera informazioni volumetriche dalla superficie, fornendo così un naturale legame tra il volume dell’oggetto e il suo bordo; inoltre studiamo la decomposizione spettrale di opportune matrici di affinità che fornisce coordinate spettrali multidimensionali caratterizzanti la forma dell’oggetto; infine introduciamo una nuova base, denominata Lp Compressed Manifold Modes, di quasi-autofunzioni sparse e localizzate dell’operatore Laplace-Beltrami. Il problema di partizionamento può essere considerato un particolare problema inverso, pertanto è fomulato come un problema di regolarizzazione variazionale che ha come soluzione la cosiddetta funzione di partizionamento. Il funzionale da minimizzare è somma di un termine di fedeltà a un determinato set di dati e di un termine di regolarizzazione che promuove la sparsità, come ad esempio la norma Lp con p ∈ (0, 1) o altre funzioni di penalizzazione non convesse e parametrizzate, con parametro positivo, che controlla il grado di non convessità. I metodi proposti per ottenere la funzione di partizione, ispirati ai modelli variazionali di Mumford-Shah di funzionali costanti o smooth a tratti, incorporano un regolarizzatore non convesso per ridurre al minimo le lunghezze del contorno delle partizioni. Per la soluzione dei problemi di ottimizzazione non convessi e non smooth si propongono metodi numerici basati su Proximal Forward-Backward Splitting, Alternating Directions Method of Multipliers e strategie Convex Non-Convex. Inoltre, studiamo un’applicazione del partizionamento di forma nell’ambito della patchbased surface quadrangulation. A questo scopo la varietà viene prima suddivisa in patch di genere zero che catturano la topologia arbitraria dell’oggetto, quindi per ogni patch viene creata una superficie minima ad elementi quadrilateri che si evolve secondo un modello differenziale alle derivate parziali, seguendo un approccio Lagrangiano per ottenere una rappresentazione a griglie quadrilatere semi-regolari. L’evoluzione è supervisionata da una ridistribuzione tangenziale uniforme dell’area-asintotica dei quadrilateri.

In this thesis, two algorithms are introduced for use in ultrasound elastography. Ultrasound elastography is a technique developed in the last 20 years by which anomalous regions in soft tissue are located and diagnosed without the need for biopsy. Due to this, the relativity cheap cost of ultrasound imaging and the high level of accuracy in the methods, ultrasound elastography methods have shown great potential for the diagnosis of cancer in soft tissues. The algorithms introduced in this thesis represent an advance in this field. The first algorithm is a two-step iteration procedure consisting of two minimization problems - displacement estimation and elastic parameter calculation that allow for diagnosis of any anomalous regions within soft tissue. The algorithm represents an improvement on existing methods in several ways. A weighting factor is introduced for each different point in the tissue dependent on the confidence in the accuracy of the data at that point, an exponential substitution is made for the elasticity modulus, an adjoint method is used for efficient calculation of the gradient vector and a total variation regularization technique is used. Most importantly, an adaptive mesh refinement strategy is introduced that allows highly efficient calculation of the elasticity distribution of the tissue though using a number of degrees of freedom several orders lower than methods that use a uniform mesh refinement strategy. Results are presented that show the algorithm is robust even in the presence of significant noise and that it can locate a tumour of 4mm in diameter within a 5cm square region of tissue. Also, the algorithm is extended into 3 dimensions and results are presented that show that it can calculate a 3 dimensional elasticity distribution efficiently. This extension into 3-d is a significant advance in the field. The second algorithm is a one-step algorithm that seeks to combine the two problems of elasticity distribution and displacement calculation into one. As in the two-step algorithm, a weighting factor, exponential substitution for the elasticity parameter, adjoint method for calculation of the gradient vector, total variation regularization and adaptive mesh refinement strategy are incorporated. Results are presented that show that this original approach can locate tumours of varying sizes and shapes in the presence of varying levels of added artificial noise and that it can determine the presence of a tumour in images taken from breast tissue in vivo.

Better understanding of the dynamic behavior of structures is as important as using reliable methods to determine the dynamic behaviour. This study started by describing the importance of dynamic analysis in structures, and further exploring what is meant by “Modal Analysis” and the different methods used in obtaining the Modal properties of a simple structure. First, using “Mathematical Methods”, a “Simple Beam Element” was analyzed with Analytical analysis and Numerical modelling. In the Numerical model, the beam was modelled twice with different boundary conditions. The results of the mathematical methods were compared. Second, using “Experimental Methods”, the same simple beam element was analyzed using “Impact Testing” methods of experimental modal analysis. The results gotten with experimental methods were compared with both the analytical and numerical methods, and the comparison between these results shows that the errors are within reasonable range, and it will help future engineers to better design and develop engineering components.

This work is concerned with the development of computational methods for approximating sensitivities of solutions to boundary value problems. We focus on the continuous sensitivity equation method and investigate the application of adaptive meshing and smoothing projection techniques to enhance the basic scheme. The fundamental ideas are first developed for a one dimensional problem and then extended to 2-D flow problems governed by the incompressible Navier-Stokes equations. Numerical experiments are conducted to test the algorithms and to investigate the benefits of adaptivity and smoothing.
Ph. D.