Optimizing Talbot's Contours for the Inversion of the Laplace Transform

Talbot's method for the numerical inversion of the Laplace Transform consists of numerically integrating the Bromwich integral on a special contour by means of the trapezoidal or midpoint rules. In this paper we address the issue of how to choose the parameters that define the contour, for the particular situation when parabolic PDEs are solved. In the process the well known subgeometric convergence rate O(e -c \sqrt N) of this method is improved to the geometric rate O(e -cN) with N the number of nodes in the integration rule. The value of the maximum decay rate c is explicitly determined. Numerical results involving two versions of the heat equation are presented. With the choice of parameters derived here, the rule-of-thumb is that to achieve an accuracy of 10 -l at any given time t, the associated elliptic problem has to be solved no more that l times.\ud \ud Supported by the National Research Foundation in South Africa under grant NRF528

Numerical methods for option pricing under jump-diffusion models.

Wu, Tao.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 56-61).Abstracts in English and Chinese.Chapter 1 --- Background and Organization --- p.7Chapter 2 --- Parallel Talbot method for solving partial integro- differential equations --- p.9Chapter 2.1 --- Introduction --- p.9Chapter 2.2 --- Initial-boundary value problem --- p.11Chapter 2.3 --- Spatial discretization and semidiscrete problem --- p.12Chapter 2.4 --- Parallel Talbot method --- p.15Chapter 2.4.1 --- Φ-functions and Talbot quadrature --- p.15Chapter 2.4.2 --- Control on nonnormality and feasibility con- straints --- p.18Chapter 2.4.3 --- Optimal parameterization of parabolic Talbot contour --- p.22Chapter 2.5 --- Numerical experiments --- p.26Chapter 2.6 --- Conclusion --- p.32Chapter 3 --- Memory-reduction Monte Carlo method for pricing American options --- p.37Chapter 3.1 --- Introduction --- p.37Chapter 3.2 --- Exponential Levy processes and the full-storage method --- p.39Chapter 3.3 --- Random number generators --- p.41Chapter 3.4 --- The memory-reduction method --- p.43Chapter 3.5 --- Numerical examples --- p.45Chapter 3.5.1 --- Black-Scholes model --- p.46Chapter 3.5.2 --- Merton's jump-diffusion model --- p.48Chapter 3.5.3 --- Variance gamma model --- p.50Chapter 3.5.4 --- Remarks on the efficiency of the memory-reduction method --- p.52Chapter 3.6 --- Conclusion --- p.53Chapter 3.7 --- Appendix --- p.5

Effects of Phase Transformations and Dynamic Material Strength on Hydrodynamic Instability Evolution in Metals

abstract: Hydrodynamic phenomena such as the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities can be described by exponential/linear growth of surface perturbations at a bimaterial interface when subjected to constant/impulsive acceleration. A challenge in designing systems to mitigate or exploit these effects is the lack of accurate material models at large dynamic strain rates and pressures. In particular, little stress-strain constitutive information at large strain rates and pressures is available for transient material phases formed at high pressures, and the continuum effect the phase transformation process has on the instability evolution. In this work, a phase-aware isotropic strength model is developed and partially validated with a novel RM-based instability experiment in addition to existing data from the literature. With the validated material model additional simulations are performed to provide insight into to the role that robust material constitutive behavior (e.g., pressure, temperature, rate dependence) has on RM instability and how RM instability experiments can be used to characterize and validated expected material behavior. For phase aware materials, particularly iron in this work, the simulations predict a strong dependence on the Atwood number that single phase materials do not have. At Atwood numbers close to unity, and pressures in the high pressure stability region, the high pressure phase dominates the RM evolution. However, at Atwood numbers close to negative one, the RM evolution is only weakly affected by the high-pressure phase even for shocks well above the phase transformation threshold. In addition to RM evolution this work looks at the closely related shock front perturbation evolution. Existing analytical models for isentropic processes in gases and liquids are modified for metal equation of states and plastic behavior for the first time. It is found that the presence of a volume collapsing phase transformation with increased pressure causes shock front perturbations to decay sooner, while plastic strength has the opposite effect which is significantly different from the effect viscosity has. These results suggest additional experimental setups to validate material models, or relevant material parameters that can be optimized for system design objectives, e.g., minimize feed through perturbations in inertial confinement fusion capsules.Dissertation/ThesisDoctoral Dissertation Mechanical Engineering 201

Evaluating the industrial application of non-destructive inspection of composites using transient thermography

Thesis (MEng)--Stellenbosch University, 2016.ENGLISH ABSTRACT: Transient thermography is a non-destructive testing method used in the detection and visualization of sub-surface flaws. Transient thermography could use one of two heating methods: step and square-pulse heating. Both these methods rely on observing the temperature rise of a surface that is subjected to a constant heat flux, while square pulse thermography also observes the subsequent thermal decay after the heat has been removed. The transient methods have not been thoroughly explored in literature with respect to the more popular methods, such as pulsed and lock-in thermography. Particular interest has been placed on investigating transient thermography on fiber-reinforced polymer (FRP) materials and its application in industry. Composites are prone to flaws such as delaminations, voids and inclusions that do not accurately represent flat-bottom holes, which are commonly evaluated in experimental work. Therefore, the inspection of thin artificial air-gaps and Teflon® delaminations were investigated. These artificial flaws can be considered to represent either a fully-separated or contacting delamination. A significant reduction in defect contrast and definition was observed for the thin delaminations, which is ascribed to the lower thermal resistance than that for flat-bottom holes. Further studies investigated the qualitative and quantitative performance of thermographic inspection on defective samples provided by an industrial partner. Experimental results demonstrated that variability in core geometry, ply arrangement, surface and sub-surface anomalies could be identiffied. The smallest detectable anomaly was found to be 1 mm wide, which was a spatial resolution limitation of the infrared camera. The investigated samples exhibited small radius and low resistance defects. It was found that current techniques to quantify defect depth are inadequate, especially if an accurate reference depth cannot be found. Thermography data is typically associated with subtle defect signatures that are strongly affected by non-uniform heating and surface variability. Advanced processing methods have been shown to help mitigate these effects. Various processing methods are reviewed from literature. Several methods were tested here for the first time, such as: multiscale retinex, matched filters, Markov error contrast and modified differential absolute contrast (IDAC) for step thermography. Transient thermography has shown to be a strong competitor amongst other thermographic methods for its simple application, relatively fast inspection times, and high thermal contrast for low defect resistance cases. It further enables the use of an entry-level infrared camera. The ndings of the artificial samples reported a maximum defect depth up to 7 mm was observed for clear Plexiglas®. The clear Plexiglas® can be considered to be the least optimal case of heating with optical excitation and has a low thermal emissivity. For the carbon and glass fibre reinforced polymers, a maximum detectable defect depth of 5 mm was observed, which is considered to be comparative or even better than pulsed thermography. The method was particularly better for low diffusivity materials, such as glass fibre composites.AFRIKAANSE OPSOMMING: Oorgangstermografie is 'n nie-destruktiewe tegniek om defekte onder die oppervlak waar te neem en te visualiseer. Oorgangstermografie kan een van twee verhittingsmetodes gebruik: stap en vierkant puls verhitting. Beide tegnieke is gebaseer op die waarneming van die temperatuur styging van 'n oppervlak onderwerp aan 'n konstante warmtelas, terwyl vierkant puls verhitting ook die temperatuur daling waarneem nadat die warmtelas verwyder is. In vergelyking met meer populêre metodes, soos gepulseerde en geslote termografie, is die oorgangsmetodes nog nie ewe deeglik beskryf in die literatuur nie. Daar is veral belangstelling in ondersoeke na oorgangstermografie vir veselversterkte polimere en die toepassing daarvan in industrie. Saamgesteldemateriale is geneig om defekte soos delaminasie, leemtes en inklusies te hê wat nie goed voorgestel word deur plat bodem gate nie, soos algemeen gebruik in eksperimentele werk. Hier is die gebruik van dun, kunsmatige, luggapings en Te on® delaminasies ondersoek. 'n Beduidende verlaging in kontras en definisie is waargeneem vir dun delaminasies wat toegeskryf kan word aan die feit dat dit 'n laer termiese weerstand het as plat bodem gate. Verdere ondersoeke na die kwalitatiewe en kwantitatiewe vermoë van die termografiese inspeksie van defektiewe onderdele voorsien deur 'n industriële vennoot is gedoen. Eksperimentele resultate het getoon dat variasies in die kern geometrie, laag oriëntasie, oppervlak en sub-oppervlak afwykings geïdenti fiseer kan word. Die kleinste, waarneembare afwyking was 1 mm wyd, wat toegeskryf word aan die beperkte ruimtelike resolusie van die infrarooikamera. Die ondersoekte voorbeelde het klein radius en lae weerstand defekte getoon. Dit is gevind dat bestaande tegnieke om defek diepte te vind deur die gebruik van inversie metodes ontoereikend is, veral wanneer 'n verwysingsdiepte nie akkuraat bepaal kan word nie. Termografiese data word dikwels geassosieer met fyn defek kenmerke wat sterk beïnvloed word deur oneweredige verhitting en oppervlakte variasies. Dit is al gevind dat gevorderde verwerkingsmetodes die effek hiervan kan verminder. Verskeie van hierdie tegnieke, soos gevind in die literatuur, is oorweeg. Nuwe metodes, soos multiskaal retinex, bypassende lters, Markov fout kontras en aangepaste differensiële absolute kontras, word ook beskryf en ge-evalueer. Die prosesseringsmetodes is geïmplimenteer in 'n oopbron sagteware pakket en is getoets met voorbeelde uit die industrie. Dit is getoon dat oorgangstermografie 'n sterk mededinger is in die versameling termografiese tegnieke vernaamlik as gevolg van die eenvoudige toepassing daarvan, relatief vinnige inspeksie tye en hoë termiese kontras vir gevalle waar die termiese weerstand van die defek laag is. Verder is dit moontlik om intreevlak infrarooikameras te gebruik met hierdie tegnieke. Gebaseer op toetse met kunsmatige defekte kon foute so diep as 7 mm onder die oppervlak gevind word in helder Plexiglas®. Helder Plexiglas® is nie 'n ideale materiaal vir hierdie tegnieke nie as gevolg van die materiaal se lae termiese emmisiwiteit. Defekte so diep as 5 mm kon gevind word in koolstof- en glasvesel versterkte polimere. Dit is vergelykbaar met en selfs beter as gepulseerde termografie. Die tegniek het veral beter resultate gelewer met materiale met lae diffusiwiteit, soos saamgeselde veselglas materiale

Réponse élastodynamique d'une plaque stratifiée anisotrope : approches comparées. : Vers le développement de méthodes hybrides.

This work adresses the direct problem of the propagation of an elastodynamic field radiated by a source in an anisotropic layered medium. Applications concern non destructive evaluation of composite plates by ultrasonic guided waves. In the lower frequencies, these materials can be modeled as homogeneous, anisotropic and dissipative media. Two causal approaches are studied and developped to solve the wave equation, and their interest is discussed regarding to the widely used harmonic modal method. One of these methods is modal, and is formulated directly in the time domain. It allows to deal easily with anisotropy, even in 3D ; however it also suffers classical shortcomings such as the high cost of the unestablished regime or the difficulty to deal with open waveguides. The other method is a formulation of the so-called partial-waves method in the Laplace domain. Its attractiveness relies in its versatility and in the fact that computational costs can be very acceptable. In a second time, we consider using both methods to solve problems of diffraction by defects. A boundary element method based on the partial-waves approach is developped and leads to solve very efficiently the case of a planar defect. The possibility of treating more general defects is briefly discussed.Cette thèse traite de la résolution du problème direct de propagation d'un champ élastodynamique rayonné par une source dans un milieu stratifié anisotrope. Le contexte applicatif visé est le contrôle non destructif par ondes ultrasonores guidées de plaques de matériaux composites. Aux basses fréquences, ces matériaux sont assimilables à des milieux homogènes, anisotropes et dissipatifs. Deux approches causales sont étudiées et mises en oeuvre pour résoudre l'équation d'onde, et leur intérêt vis-à-vis de la méthode modale harmonique - la plus couramment employée dans ce domaine applicatif - est discuté. L'une des méthodes est modale et est formulée directement dans le domaine temporel. Elle permet de traiter facilement l'anisotropie, y compris en 3D, mais souffre des écueils classiques concernant le régime non-établi ou le cas du guide ouvert. L'autre approche est une formulation dans le domaine de Laplace de la méthode dite par ondes partielles. Elle présente l'intérêt d'être extrêmement polyvalente tout en conduisant à des coûts numériques tout à fait raisonnables. Dans un second temps, la possibilité d'exploiter ces deux méthodes pour résoudre des problèmes de diffraction par des défauts est étudiée. Une approche par éléments finis de frontière basée sur la méthode par ondes partielles est considérée. Elle permet de traiter efficacement le cas de défauts plans. L'extension à des défauts plus généraux est brièvement discutée