Thesis--Ph. D., University of Hong Kong. / Mimeographed.

A systematic procedure is followed to develop singularity-reduced integral equations for modeling cracks in two-dimensional, linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations (IEs) for "generalized displacement" and "generalized stress"; these serve as the basis for the development of a Symmetric Galerkin Boundary Element Method (SGBEM). The implementation is carried out to allow treatment of general mixed boundary conditions, an arbitrary number of cracks, and multi-region domains (in which regions having different material properties are bonded together). Finally, a procedure for calculation of T-stress, the constant term in the asymptotic series expansion of crack-tip stress field, is developed for anisotropic elastic media. The pair of weak-form boundary IEs that is derived (one for generalized displacement and the other one for generalized stress) are completely regularized in the sense that all kernels that appear are (at most) weakly-singular. This feature allows standard Co elements to be utilized in the SGBEM, and such elements are employed everywhere except at the crack tip. A special crack-tip element is developed to properly model the asymptotic behavior of the relative crack-face displacements. This special element contains "extra" degrees of freedom that allow the generalized stress intensity factors to be directly obtained from the solution of the governing system of discretized equations. It should be noted that while the integral equations contain only weakly-singular kernels (and so are integrable in the usual sense) there remains a need to devise special integration techniques to accurately evaluate these integrals as part of the numerical implementation. Various examples for crack problems are treated to illustrate the accuracy and versatility of the proposed procedure for both unbounded and finite domains and for both single-region and multi-region problems. It is found that highly accurate fracture data can be obtained using relatively course meshes. Finally, this dissertation addresses the development of a numerical procedure to calculate T-stress for crack problems in general anisotropic elastic media. T-stress is obtained from the sum of crack-face displacements which are computed via a (regularized) integral equation of the boundary data. Two approaches for computing the derivative of the sum of crack-face displacements are proposed: one uses numerical differentiation, and the other one uses a weak-form integral equation. Various examples are examined to demonstrate that highly accurate results are obtained by means of both approaches. / text

(Uncorrected OCR) Abstract of thesis entitled liField analysis in power supply lines by integraJ. equation methodll subm1 tted by FOe, PIK YlJE for the degree of Ph.D at the University of Hong Kong in December, 1974. Abstract In this thesis, the integral equation (I.E.) method has been employed successfully to solve field problems in power supply lines. Though the I.E. method is mathematically quite involved, it is shown that it is possible to treat the integral equation as a system of linear equations. Hence the transformed simultaneous linear equations can be considered as the starting point for solving problems either in overhead lines or \Ulderground power cables. In overhead lines, especially in Extra-High-Voltage and Ultra- High-Voltage systems, an evaluation of the electric field near each conductor, especially the maximum electric field, is essential as corona and radio interference become important considerations in the design of such lines. The I.E. method has many advantages over the other existing methods in calculating the potential gradient at the surface of the overhead lines in that it yieldS reasonably accurate results with comparatively simple numerical computations. The difference between the present method and the existing methods is the basic assumption. In the I.E. method, subconductor surfaces are treated as equipotential lines whereas in other existing methods, the subconductor surfaces usually do not coincide with the simulated equipotential lines. The method can also be applied equally well to symmetrical or asymmetrical bundle conductors with or without ground wires. Other parameters such as capacitances, potential gradients at the earth surface etc. are also included in the computer program. In underground cable systems, the I. E. method proves to be very effective in tackling the thermal field problem, especially when the cables are buried shallow and thus the earth i s surface can no longer be treated as isothermal. .It has been found that the thermal resistance (external) obtained wi::;h a non-isothermal earth surface is considerably higher than that obtained under the assuumption of an isothermal earth surface. With non.-isothermal earth surface, the temperature difference on the earth surface between the spot vertically above the cable and the spot at a distance eClual to twice the depth of burial of the cable away could be as high as lOoe. The finite-difference or fini te-搪lement method could likewise be used to solve the problem of a non-isothermal ea>:>th surface, but the computer storage re'luired and the execution time would be much more than that using the I.E. method. / abstract / toc / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

Σκοπός της παρούσας διδακτορικής διατριβής είναι η ανάπτυξη αριθμητικής μεθόδου, η οποία επιλύει προβλήματα δισδιάστατης στατικής ελαστικότητας, καθώς και δυναμικής ελαστικότητας στο πεδίο των συχνοτήτων και στο πεδίο του χρόνου. Το κύριο χαρακτηριστικό της είναι ότι η προσέγγιση του άγνωστου πεδίου γίνεται με την τοποθέτηση σημείων και όχι με τη χρήση κάποιου πλέγματος όπως γίνεται στις μέχρι τώρα κλασικές μεθοδολογίες των πεπερασμένων ή συνοριακών στοιχείων. Μέρος της παρούσας διατριβής αποτελεί και η ανάπτυξη προγράμματος ηλεκτρονικού υπολογιστή, ο οποίος υποστηρίζει πλήρως τα όσα αναφέρονται στην παρούσα εργασία. Η παρούσα διατριβή αποτελείται από δύο ενότητες. Στην πρώτη ενότητα, η οποία περιλαμβάνει τα πρώτα τρία κεφάλαια, παρατίθεται το θεωρητικό υπόβαθρο της μεθοδολογίας. Στη δεύτερη ενότητα περιγράφονται διάφορες τεχνικές λεπτομέρειες, όπως ολοκληρώσεις και προσέγγιση πεδίου και δίνονται αρκετά παραδείγματα, τα οποία πιστοποιούν την ακρίβεια και την αξιοπιστία της. / -

In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations. / Thesis (Ph.D.)-University of Natal, 1995.

Classically, researchers in Computational Physics and specifically in Computational Electromagnetics have sought to find numerical solutions to complex physical problems. Several techniques have been developed to accomplish such tasks, each of which having advantages over their counterparts. Typically, each solution method has been developed separately despite having numerous commonalities with other methods. This fact motivates a unified software tool to house each solution method to avoid duplicating previous efforts. Subsequently, these solution methods can be used alone or in conjunction with one another in a straightforward manner. The aforementioned goals can be accomplished by using an Object Oriented software approach. Thus, the goal of the presented research was to incorporate a specific solution technique, an Integral Equation Nystrm method, into a general, Object Oriented software framework.

Integral equations arise in many scienti c and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any eld to give rise to integral equations. Integral equations also has signi cant application in mathematical physics models, such as di rac- tion problems, scattering in quantum mechanics, conformal mapping and water waves. The Volterra's population growth model, biological species living together, propagation of stocked sh in a new lake, the heat transfer and the heat radiation are among many areas that are described by integral equations. For limited applicability of analytical techniques, the numer- ical solvers often are the only viable alternative. General computational techniques of solving integral equation involve discretization and generates equivalent system of linear equations. In most of the cases the discretization produces dense matrix. Multigrid methods are widely used to solve partial di erential equation. We discuss the multigrid algorithms to solve integral equations and propose usages of distributive relaxation and the Kaczmarz method. / Department of Mathematical Sciences

Accurate modeling of current flow and network parameter extraction in 2-D and 3-D conductors has an important application in signal integrity of high-speed interconnects. In this thesis, we propose a new rigorous single-source Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) for magnetostatic analysis of 2-D transmission lines and broadband resistance and inductance extraction in 3-D interconnects. Furthermore, the novel integral equation can be used for the solution of full-wave scattering problems on penetrable 2-D cylinders of arbitrary cross-section under transverse magnetic polarization. The new integral equation is derived from the classical Volume Electric Field Integral Equation (V-EFIE) by representing the electric field inside a conductor or a scatterer as a superposition of the cylindrical waves emanating from the conductor’s surface. This converts the V-EFIE into a surface integral equation involving only a single unknown function on the surface. The novel equation features a product of integral operators mapping the field from the conductor surface to its volume and back to its surface terming the new equation the Surface-Volume-Surface EFIE. The number of unknowns in the proposed SVS-EFIE is approximately the square root of the number of degrees of freedom in the traditional V-EFIE; therefore, it allows for substantially faster network parameter extraction and solutions to 2-D scattering problems without compromising the accuracy. The validation and benchmark of the numerical implementation of the Method of Moment discretization of the novel SVS-EFIE has been done via comparisons against numerical results obtained by using alternative integral equations, data found in literature, simulation results acquired from the CAD software, and analytic formulas.

The purpose of this thesis is to look into the difficulty of valuing American options, put as well as call, on an asset that pays continuous dividends. The authors are willing to demonstrate how mentioned above securities can be priced using a simple approximation of the Kim integral equations by quadrature formulas. This approach is compared with closed form American Option price formula proposed by Bjerksund-Stenslands in 2002. The results obtained by Bjerksund-Stenslands method are numerically compared by authors to the Kim’s. In Joon Kim’s approximation seems to be more accurate and closer to the chosen “true” value of an American option, however, Bjerksund-Stenslands model is demonstrating a higher speed in calculations.

Today, nonhomogeneous materials are used in many technological applications. Nonhomogeneity can be introduced intentionally in order to improve the thermomechanical performance of material systems. The concept of functionally graded materials (FGMs) is an example of such an application. Nonhomogeneity can also be an intrinsic property of some of the natural materials such as natural soil. The main interest in this study is on the contact mechanics of nonhomogeneous surfaces. There is an extensive volume of literature on the contact mechanics of nonhomogeneous materials. In most of these studies, the elastic gradation is assumed to exist in depth direction. But, it is known that elastic gradation may also exist laterally. This may either occur naturally as in the case of natural soil or may be induced as a result of the applied processing technique as in the case of FGMs. The main objective in this study is therefore to examine the effect of the lateral nonhomogeneities on the contact stress distribution at the surface of an elastically graded material. In the model developed to examine this problem, a laterally graded surface is assumed to be in sliding contact with a rigid stamp of arbitrary profile. The problem is formulated using the theory of elasticity and reduced to a singular integral equation. The integral equation is solved numerically using a collocation approach. By carrying out parametric studies, the effects of the nonhomogeneity constants, coefficient of friction and stamp location on the contact stress distribution and on the required contact forces are studied.