The theme of the research is on the use of the hypersingular boundary element method for the modelling of surface waves. Surface waves in solids are known to be partially reflected & transmitted and mode converted into body waves at stress discontinuities, which suggests that a formulation continuous in stress and strain might prove beneficial for modelling purposes. Such continuity can be achieved with a subparametric approach where the geometry is approximated using linear elements and the field variables, displacement and traction, are approximated using cubic Hermitian and linear shape functions respectively. The higher order polynomial for approximating displacement is intended to be a more accurate representation of the physics relating to surface wave phenomena, especially at corners, and thus, is expected to capture this behaviour with greater accuracy than the standard isoparametric approach. The subparametric approach affords itself to continuity in stress and strain by imposing a smoothness in the elements, which is not available to the isoparametric approach. As the attention is focused primarily on the modelling of surface waves on the boundary of a medium rather than the interior, the boundary element method lends itself appropriately to this end.A 2D semi analytical integration scheme is employed to evaluate the integrals appearing in the hypersingular boundary integral formulation. The integration scheme is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to be evaluated analytically. The scheme makes use of the aforementioned subparametric approach and is applied to linear elements for the use in steady state elastodynamic boundary element method problems. The steady state problem is used as it is a simplified problem and is sufficient to permit the investigation of surface vibration at a constant motion. The 2D semi analytical integration scheme presented can be naturally extended to 3D.A particular focus and novelty of the work is the application of different limiting approaches to determine the free terms common to boundary integral methods. The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a circular/spherical shaped surface centred at a singularity and allowing the radius of the circle/sphere to tend to zero. This approach is revisited in order to ascertain whether incorrect results are possible as a consequence of shape dependency, which is a recognised issue for hypersingular integrals.Two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. The first approach, the point limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach, shrinkage rates and path dependency are examined. A related and second approach, the boundary limiting method, involves moving an invariant, but shrinking, boundary toward the singularity, again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, that is, by comparison with the direct method. Unlike the circular/spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not an issue. A particular highlight of the point limiting approach is the ability to obtain free terms in mixed formulation, which is not available to the circular/spherical approach.There are three numerical problems considered in this research. The first problem considers the longitudinal vibration of a square plate. This is a problem for which a known analytical solution exists and is used to verify the equation formulation and integration scheme adopted for the isoparametric and subparametric formulations. Both formulations are as accurate as each other and produce results that are in keeping with the analytical solution, thus instilling confidence in their predictions.The second problem considers the simulation of surface waves on a square plate. Various boundaries of a square plate have displacement conditions imposed on them as a result of surface wave propagation. The results indicate that the surface wave behaviour is not captured. However, the analytical solution does not make any consideration for the effects from corners; the analytical solution is for a Rayleigh wave propagating upon a planar surface. It does not take into account the wave phenomena encountered at corners. Therefore, these results cannot be used to validate the predictions obtained on the boundary of the problem considered. The purpose of this problem is to illustrate the impact of corners on the surface wave propagation. Sensitivity studies are conducted to illustrate the effect of corners on the computed solution at the boundary.The final problem considers the simulation of surface waves on a circular plate. Various portions of the boundary of the circular plate have displacement conditions imposed on them as a result of surface wave propagation on curved surfaces. The results indicate that the isoparametric and subparametric predictions are similar to one another. However, both displacement profiles predict the presence of other waves. Given the multi faceted nature of the mesh, the computed solution is picking up mode conversion and partial reflection & transmission of surface waves. In reality, this is not expected as the surface of the boundary is smooth. However, due to the discretisation there are many corners in this problem.

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown's force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy's theorem in continuum mechanics to a magnetoelastic material. The proof of Brown's formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown's force formula. One obtains an additional nonlinear surface term which allows one to regard Brown's assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure. / Das Thema dieser Arbeit ist eine mathematisch strenge Herleitung von Formeln für die magnetische Kraft, die auf einen Teil eines beschränkten, magnetischen Körpers durch seine Umgebung ausgeübt wird. Zunächst wird die magnetische Kraft in einem kontinuierlichen System auf Grundlage der makroskopischen Magnetostatik betrachtet. Mit Bezug auf W. F. Brown, der eine vor allem physikalisch motivierte Herleitung der Kraftformel gegeben hat, wird diese auch Brownsche Kraftformel genannt. Das Oberflächenintegral in dieser Formel zeigt eine nichtlineare Abhängigkeit von der Normalen. Um Cauchys Theorem aus der Kontinuumsmechanik in einem magnetoelastischen Material anwenden zu können, nimmt Brown an, dass die Oberflächenkraft einen zusäatzlichen Term enthält, der den nichtlinearen Ausdruck aufhebt. Der Beweis der Brownschen Kraftformel in dieser Arbeit beruht auf einer geeigneten Regularisierung eines hypersingulären Kerns und benutzt Methoden für singuläre Integrale. Danach gehen wir von einem diskreten, periodischen System von magnetischen Dipolen aus und betrachten die Kraft zwischen einem Teil einer beschränkten Menge und der Umgebung. Um zum Kontinuumslimes überzugehen, starten wir von der üblichen Kraftformel für wechselwirkende magnetische Dipole. Es zeigt sich, dass sich der Limes der diskreten Kraft von der Brownschen Kraftformel unterscheidet. Man erhält einen zusätzlichen nichtlinearen Oberflächenterm, der es ermöglicht, Browns Annahme als Konsequenz des atomistischen Zugangs zu sehen. Kurzreichweitige Effekte führen zudem zu einem linearen Oberflächenterm im Kontinuumlimes der diskreten Kraft. Dieser Zusatzterm enthält eine gewisse Gittersumme, die von einem hypersingulären Kern und der Struktur des zugrundeliegenden Gitters abhängt.

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown's force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy's theorem in continuum mechanics to a magnetoelastic material. The proof of Brown's formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown's force formula. One obtains an additional nonlinear surface term which allows one to regard Brown's assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure.

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown''s force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy''s theorem in continuum mechanics to a magnetoelastic material. The proof of Brown''s formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown''s force formula. One obtains an additional nonlinear surface term which allows one to regard Brown''s assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure. / Das Thema dieser Arbeit ist eine mathematisch strenge Herleitung von Formeln für die magnetische Kraft, die auf einen Teil eines beschränkten, magnetischen Körpers durch seine Umgebung ausgeübt wird. Zunächst wird die magnetische Kraft in einem kontinuierlichen System auf Grundlage der makroskopischen Magnetostatik betrachtet. Mit Bezug auf W. F. Brown, der eine vor allem physikalisch motivierte Herleitung der Kraftformel gegeben hat, wird diese auch Brownsche Kraftformel genannt. Das Oberflächenintegral in dieser Formel zeigt eine nichtlineare Abhängigkeit von der Normalen. Um Cauchys Theorem aus der Kontinuumsmechanik in einem magnetoelastischen Material anwenden zu können, nimmt Brown an, dass die Oberflächenkraft einen zusäatzlichen Term enthält, der den nichtlinearen Ausdruck aufhebt. Der Beweis der Brownschen Kraftformel in dieser Arbeit beruht auf einer geeigneten Regularisierung eines hypersingulären Kerns und benutzt Methoden für singuläre Integrale. Danach gehen wir von einem diskreten, periodischen System von magnetischen Dipolen aus und betrachten die Kraft zwischen einem Teil einer beschränkten Menge und der Umgebung. Um zum Kontinuumslimes überzugehen, starten wir von der üblichen Kraftformel für wechselwirkende magnetische Dipole. Es zeigt sich, dass sich der Limes der diskreten Kraft von der Brownschen Kraftformel unterscheidet. Man erhält einen zusätzlichen nichtlinearen Oberflächenterm, der es ermöglicht, Browns Annahme als Konsequenz des atomistischen Zugangs zu sehen. Kurzreichweitige Effekte führen zudem zu einem linearen Oberflächenterm im Kontinuumlimes der diskreten Kraft. Dieser Zusatzterm enthält eine gewisse Gittersumme, die von einem hypersingulären Kern und der Struktur des zugrundeliegenden Gitters abhängt.

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Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-09-24T20:35:00Z No. of bitstreams: 2 Santos, Claudia Gomes de Oliveira - Dissertação - 2013.pdf: 1950939 bytes, checksum: 050c57553672656134c6b1264cb562a6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2014-09-24T20:42:50Z (GMT) No. of bitstreams: 2 Santos, Claudia Gomes de Oliveira - Dissertação - 2013.pdf: 1950939 bytes, checksum: 050c57553672656134c6b1264cb562a6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-09-24T20:42:50Z (GMT). No. of bitstreams: 2 Santos, Claudia Gomes de Oliveira - Dissertação - 2013.pdf: 1950939 bytes, checksum: 050c57553672656134c6b1264cb562a6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-07-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The Boundary Element Method (BEM) has been successfully employed in the analysis of various engineering problems. The BEM consists in a mathematical modeling, for a numerical solution of a system of integral equations, and in their cores may appear singularities. This paper presents the Classical and Hypersingular formulation of the Boundary Element Method for dimensional elastostatic problems with smooth boundary geometry. The improper integrals arising from the singularities of the core in the hypersingular formulation are treated by Hadamard finite parts. In the discretization process two types of interpolation are used, one traditional and the other special. Traditional interpolation is used in all bondary elements that have no point , special interpolation ensures the continuity of the tangential derivative of displacements on the element that contains the point . To accomplish this, a theoretical mathematics study of related topics was performed. The hypersingular formulation developed in this work was implemented through the Intel Visual Fortran compiler. Some problems were analyzed and the obtained results were compared with those of analytical solution or through the Finite Element Method. The results achieved were satisfactory validating the proposed formulation / O Método dos Elementos de Contorno (MEC) vem sendo empregado com sucesso na análise de diversos problemas de engenharia. O MEC consisti em uma modelagem matemática, para resolução numérica de um sistema de equações integrais, e que em seus núcleos podem aparecer singularidades. Nesse trabalho apresenta a formulação Clássica e Hipersingular do Método dos Elementos de Contorno para problemas de elastostática bidimensional com geometria de contornos não suaves. As integrais impróprias que surgem da singularidade do núcleo na formulação hipersingular são tratados por partes finitas de Hadamard. No processo de discretização utiliza-se de dois tipos de interpolação, uma tradicional e outra especial. A interpolação tradicional é utilizada em todos os elementos de contorno que não tem o ponto , a interpolação especial garante a continuidade da derivada tangencial dos deslocamentos no elemento que contém o ponto . Para a realização deste, foi realizado um estudo teórico-matemático dos tópicos afins. Implementou-se a formulação hipersingular desenvolvidas no trabalho através do compilador Intel Visual FORTRAN. Foram analisados alguns problemas e os resultados obtidos comparados àqueles de solução analítica ou através do Método dos Elementos Finitos. Os resultados alcançados mostraram-se satisfatórios validando a formulação proposta.

The presented master’s thesis deals with the application of the gradient elasticity in fracture mechanics problems. Specifically, the displacement and stress field around the crack tip is a matter of interest. The influence of a material microstructure is considered. Introductory chapters are devoted to a brief historical overview of gradient models and definition of basic equations of dipolar gradient elasticity derived from Mindlin gradient theory form II. For comparison, relations of classical elasticity are introduced. Then a derivation of asymptotic displacement field using the Williams asymptotic technique follows. In the case of gradient elasticity, also the calculation of the J-integral is included. The mathematical formulation is reduced due to the singular nature of the problem to singular integral equations. The methods for solving integral equations in Cauchy principal value and Hadamard finite part sense are briefly introduced. For the evaluation of regular kernel, a Gauss-Chebyshev quadrature is used. There also mentioned approximate methods for solving systems of integral equations such as the weighted residual method, especially the least square method with collocation points. In the main part of the thesis the system of integral equations is derived using the Fourier transform for straight crack in an infinite body. This system is then solved numerically in the software Mathematica and the results are compared with the finite element model of ceramic foam.