Laboratory test of reinforced and prestressed concrete structures have been used widely to explore the behavior of reinforced and prestressed concrete components and structures; Such tests are often time-consuming and costly. However, numerical models have been shown to compare favorably with experiments. Thus, computations are viewed nowadays as efficient alternatives to tests, time-wise and cost-wise. In the research reported in this thesis, finite-element model were used in a study of pretressed structural components in order to correlate levels of pretension with deflection and strain measurements. The two main objectives were to develop a suitable finite element model of prestressed concrete beams and to forecast beam prestension on the basis of deformations resulting from specified simple load, e.g., a uniformly distributed transverse load. A commercial finite-element analysis package (ANSYS 12) was used to set up, use and evaluate the computational model. Furthermore, a finite-difference model was employed in order to ascertain the validity of ANSYS results by comparison with engineering beam theory taking into account the applied pretension. This study demonstrates the potential usefulness of deflection and strain measurements as indicators of the pretension applied or remaining in prestressed concrete beams. / text

Optical whispering-gallery mode (WGM) cavities, which exhibit extraordinary spatial and temporal confinement of light, are one of the leading transducers for examining molecular recognition at low particle counts. With the advent of hybrid photonic-plasmonic and increasingly sophisticated forms of these resonators, the importance of supporting numerical methods has correspondingly become evident. In response, we adopt a full-vector finite difference approximation in order to solve for WGM's in terms of their field distributions, resonant wavelengths, and quality factors in the context of naturally discontinuous permittivity structure. A segmented Taylor series and alignment/rotation operator are utilized at such singularities in conjunction with arbitrarily spaced grid points. Simulations for microtoroids, with and without dielectric nanobeads, and plasmonic microdisks are demonstrated for short computation times and shown to be in agreement with data in the literature. Constricted surface plasmon polariton (SPP) WGM's are also featured within this document. The module of this thesis is devised as a keystone for composite WGM models that may guide experiments in the field. / Graduate

Το παρακάτω κείμενο αποτελεί διπλωματική εργασία που εκπονήθηκε στο Εργαστήριο Ασύρματης Επικοινωνίας του τμήματος Ηλεκτρολόγων Μηχανικών και Τεχνολογίας Υπολογιστών του Πανεπιστημίου Πατρών. Σκοπός μας, ήταν η επίλυση των εξισώσεων του Maxwell σε προβλήματα που αφορούν το σφαιρικό και το κυλινδρικό σύστημα συντεταγμένων. Στην προσπάθεια αυτή κάναμε χρήση της μεθόδου επίλυσης των πεπερασμένων διαφορών στο πεδίο του χρόνου (F.D.T.D) σε δύο ειδών εφαρμογές: η πρώτη αφορά έναν κυλινδρικό κυματοδηγό, τον οποίο μοντελοποιήσαμε με τη βοήθεια του κυλινδρικού συστήματος συντεταγμένων και η δεύτερη αφορά μια κωνική κεραία UWB, την οποία μοντελοποιήσαμε με τη βοήθεια του σφαιρικού συστήματος συντεταγμένων. Η προσομοίωση αυτών των δύο εφαρμογών γίνεται με τη βοήθεια του προγραμματιστικού περιβάλλοντος της Matlab. Επειδή η μέθοδος F.D.T.D επιλύει τις εξισώσεις του Maxwell στο χρόνο μας προσφέρει τη δυνατότητα της οπτικής απεικόνισης των πεδίων σε διάφορες χρονικές στιγμές, κάτι που μας επιτρέπει να παρατηρούμε τη χρονική εξέλιξη των φαινομένων. Η μέθοδος γίνεται ιδιαίτερα ελκυστική λόγω του επιπρόσθετου χαρακτηριστικού της απευθείας επίλυσης των εξισώσεων στροβιλισμού του Maxwell, καθιστώντας παράλληλα περιττή την επίλυση της κυματικής εξίσωσης. Στο πρώτο κεφάλαιο γίνεται μια εισαγωγή στις υπολογιστικές τεχνικές στον ηλεκτρομαγνητισμό. Επίσης γίνεται μια πρώτη αναφορά στη μέθοδο των πεπερασμένων διαφορών στο πεδίο του χρόνου (F.D.T.D), στις δυνατότητες της μεθόδου, στο πεδίο εφαρμογής της καθώς και στα πλεονεκτήματά της. Στο κεφάλαιο δύο παρουσιάζονται οι εξισώσεις του Maxwell. Συγκεκριμένα παρουσιάζονται οι εξισώσεις στροβιλισμού και οι βαθμωτές διαφορικές εξισώσεις που προκύπτουν από αυτές στις τρεις και δύο διαστάσεις. Στις δύο διαστάσεις γίνεται διάκριση σε εγκάρσιο ηλεκτρικό ρυθμό (Transverse Electric) και εγκάρσιο μαγνητικό ρυθμό (Transverse Magnetic). Τέλος παρουσιάζονται οι εξισώσεις του Maxwell που ισχύουν για τα σκεδαζόμενα πεδία. Στο κεφάλαιο τρία παρουσιάζονται τα βασικά στοιχεία της μεθόδου F.D.T.D, τα οποία πρέπει να γίνουν κατανοητά προκειμένου να αναδειχθούν τα πλεονεκτήματα και τα μειονεκτήματά της. Συγκεκριμένα παρουσιάζονται οι εξισώσεις πεπερασμένων διαφορών, που προκύπτουν από τις βαθμωτές διαφορικές εξισώσεις, που προκύπτουν από τις εξισώσεις στροβιλισμού του Maxwell. Στη συνέχεια παρουσιάζονται βασικά χαρακτηριστικά της μεθόδου, όπως η επιλογή του χωρικού και χρονικού βήματος και η δημιουργία πηγών. Μετά παρουσιάζεται η σημαντικότερη απορροφητική οριακή συνθήκη PML του Berenger και τέλος οι υπολογιστικές απαιτήσεις του αλγορίθμου F.D.T.D. Στο κεφάλαιο τέσσερα επιλύονται προβλήματα σε δύο διαστάσεις και συγκεκριμένα το πρόβλημα των ρυθμών διάδοσης TM και ΤΕ εντός κυλινδρικού κυματοδηγού με υλικό εντός του τον αέρα. Στο κεφάλαιο πέντε επιλύονται προβλήματα σε δύο διαστάσεις. Συγκεκριμένα παρουσιάζει την ανάπτυξη ενός σετ εργαλείων λογισμικού που είναι χρήσιμα στην ανάλυση κεραιών και δομών εξαιρετικά ευρείας ζώνης (UWB). Αυτά τα εργαλεία χρησιμοποιούνται στην εκτέλεση προσομοίωσης με τη μέθοδο των πεπερασμένων διαφορών στο πεδίο του χρόνου (FDTD) μίας κωνικής κεραίας με συνεχές κύμα (CW) και παλμικές διεγέρσεις UWB. Η κεραία αναλύεται με τη χρήση εξισώσεων σφαιρικών συντεταγμένων FDTD που προέρχονται από τις βασικές αρχές. Τα αποτελέσματα της προσομοίωσης για τη διέγερση τύπου συνεχούς κύματος (CW) συγκρίνονται με τα αποτελέσματα από προσομοιώσεις και μετρήσεις σε δημοσιευμένες πηγές· τα αποτελέσματα της διέγερσης UWB είναι νέα. Τα παραπάνω προβλήματα κάνουν σαφές το πόσο σημαντικό είναι να γνωρίζουμε σε βάθος τα χαρακτηριστικά της μεθόδου προκειμένου να φτάσουμε στην λύση τους. Σε περιπτώσεις όπου γνωρίζουμε τη λύση εκ των προτέρων (είτε ποιοτικά ή ποσοτικά) έχουμε τη δυνατότητα να επαληθεύσουμε την ορθότητα των αποτελεσμάτων της F.D.T.D. Οι λύσεις των προβλημάτων β ασίζονται στην εύρεση των ολικών πεδίων. Ο εναλλακτικός τρόπος της εύρεσης των σκεδαζόμενων πεδίων δεν χρησιμοποιείται. Βέβαια στα προβλήματα ακτινοβολίας κεραιών υποχρεωτικά εφαρμόζεται η διατύπωση των ολικών πεδίων. Μέσω αυτής της εργασίας έγινε σαφής η ικανότητα της F.D.T.D να εφαρμόζεται σε μεγάλη ποικιλία προβλημάτων. Κάτι που δεν έγινε σαφές είναι η δυνατότητα της μεθόδου να συνδυάζεται με άλλες μεθόδους, κάτι που μπορεί να επιφέρει σημαντική καταστολή ή και εξάλειψη των μειονεκτημάτων της. Με αυτό τον τρόπο δημιουργούνται νέες υβριδικές μέθοδοι. Με τις μεθόδους εύρεσης των ηλεκτρομαγνητικών πεδίων (όπως είναι η F.D.T.D) έχουμε τη δυνατότητα να δούμε τον ηλεκτρομαγνητισμό από νέα σκοπιά, κατανοώντας τον καλύτερα και προσαρμόζοντάς τον στις σύγχρονες ανάγκες της εποχής. / -

In the present study, a computational method was developed, validated and applied for the determination of parameters of a constitutive model for a ceramic material. An optimization algorithm, based on a direct search method, was applied to the determination of the load-displacement response of the specimen, and for the identification of the parameters of the constitutive model.A one-dimensional nonlinear initial-boundary value problem of wave propagation in a composite bar made of dissimilar materials was formulated and solved numerically. Convergence of the numerical scheme was studied, and range of convergence was established. Numerical scheme was validated for a number of benchmark problems with known analytical solutions, and for the problems solved using finite element method. Investigation of the accuracy of the displacement and strain responses was conducted; known limitations of the Kolsky's method for split Hopkinson pressure bar were revealed. For numerical examples considered in the present study, comparison of performance of the optimized finite-difference solver and of the finite element code LS-DYNA showed that the finite-difference code is about 10 times faster.Developed method and solutions were applied for the identification of the parameters of the Johnson-Holmquist constitutive model for five sets of experimental data for aluminum oxide AD995. Results of analysis revealed significant sensitivity of stress response to variation of fractured strength model parameters and damage model parameters.For the determined values of parameters, detailed parametric study of stress field, damage accumulation, and velocity field, was conducted with the help of the finite element method.It was found that the accuracy of the simulation using the JH-2 constitutive model changes with the rate of damage accumulation in the ceramic material.The damage patterns and history of damage development, obtained numerically, agreed qualitatively with the fracture history and its patterns, observed in the recovered Macor ceramics available in the literature.A method for image analysis of the photographic images of the lateral sides of the recovered specimen was proposed. It was used to quantify density of the damage in the specimen and to establish a better integral approach to predict amount of damage inside the specimen.

For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada

This paper presents modifications to two widely used numerical groundwater flow models in an effort to improve upon the interaction between a well of finite length and conductivity with the surrounding formation. The first objective is to discard the common assumptions about flux- or head-based boundary conditions along the well screen by coupling pipe flow hydraulics and groundwater flow. The second objective is to avoid restricting the wellbore hydraulics to a single flow regime. Five flow regimes (laminar through rough-turbulent), based on Reynolds number and pipe roughness, are considered. The modifications are integrated into the highly versatile, well-documented and well-tested models HydroGeoSphere (finite-element/finite-difference) and USGS MODFLOW (finite-difference). Verification of the algorithm and code and is performed by comparing results to: 1) the idealized, analytical Theis solution; 2) the original, unmodified code; and 3) the results of a third party numerical solution that also accounts for variable frictional wellbore losses. Results highlight the inadequacy of either a uniform flux or a uniform head assumption along the wellbore. The solution also tends to produce much steeper hydraulic gradients in those portions of the aquifer nearest the pump intake than have previously been predicted. Systems most affected by in-well hydraulic losses include those for which well screen is long, pumping rate is large, pipe diameter is small, pipe roughness is large (either through design or aging) and aquifer conductivity is high. Improved modeling of the non-linear hydraulic conditions within the well screen can particularly influence the interpretation of wellbore flowmeter and tracer tests, leading to more precise knowledge of the variation of local aquifer hydraulic conductivity along well screens. Aquifer drawdown curves, solute transport and inflow velocities will also be influenced, which can impact capture zones and remediation costs. Given that the solution is incorporated within the HydroGeoSphere and MODFLOW models, it presents the additional advantage over existing approaches of offering a wide range of modeling capabilities, such as three-dimensional flow, arbitrary well inclination and surface-subsurface flow integration. / Thesis (Master, Civil Engineering) -- Queen's University, 2008-01-04 17:27:50.629

This thesis deals with the numerical solution of ordinary differential equations (ODEs) using finite difference (FD) methods. In particular, boundary summation equation (BSE) preconditioning for FD approximations for ODEs with constant coefficients on locally refined meshes is studied. Firstly, the BSE for FD approximations of ODEs with constant coefficients is derived on a locally refined mesh. Secondly, the obtained linear system of equations are solved by the iterative method GMRES. Then, the arithmetic complexity and convergence rate of the iterative solution of the BSE formulation are discussed. Finally, numerical experiments are performed to compare the new approach with the FD approach. The results show that the BSE formulation has low arithmetic complexity and the convergence rate of the iterative solvers is fast and independent of the number of grid points.

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions.

Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013

Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013