Thesis (Ph. D.)--Oregon State University, 1994. / Typescript (photocopy). Includes bibliographical references (leaves 41-45). Also available on the World Wide Web.

The present work presents a number of contributions to the areas of numerical integration, singular integrals, and boundary element methods. The first contribution is an elemental distortion technique, based on the Duffy transformation, used to improve efficiency for the numerical integration of near hypersingular integrals. Results show that this method can reduce quadrature expense by up to 75 percent over the standard Duffy transformation. The second contribution is an improvement to integration of weakly singular integrals by using regularization to smooth weakly singular integrals. Errors show that the method may reduce errors by several orders of magnitude for the same quadrature order. The final work investigated the use of regularization applied to hypersingular integrals in the context of the boundary element method in three dimensions. This work showed that by using the simple solutions technique, the BEM is reduced to a weakly singular form which directly supports numerical integration. Results support that the method is more efficient than the state-of-the-art.

We provide an L² theory for the local double Hilbert transform along an analytic surface (s, t ,φ(s, t )) in the Heisenberg group H¹, that is operator f ↦ Hφ f (x) := p.v.∫∣s∣,∣t∣≤1 f (x ∙ (s, t ,φ(s, t ))-¹) ds/s dt/t, where ∙ denotes the group operation in H1. This operator combines several features: it is amulti-parameter singular integral, its kernel is supported along a submanifold, and convolution is with respect to a homogeneous group structure. We reprove Hφ is always L²(H¹)→L²(H¹) bounded (a result first obtained in [Str12]) to illustrate the method and then refine it to characterize the largest class of polynomials P of degree less than d such that the operator HP is uniformly bounded when P ranges in the class. Finally, we provide examples of surfaces that can be treated by our method but not by the theory of [Str12].

In this expository thesis we describe the Bochner integral for functions taking values in a separable Banach space, and we describe how a number of standard definitions and results in real analysis can be extended for these functions, with an emphasis on Hilbert-space-valued functions. We then present a partial vector-valued version of a classical theorem on singular integrals.

Cette thèse vise à développer des outils d'analyse harmonique non-commutative. Elle porte plus précisément sur les inégalités de Khintchine non-commutatives et les intégrales singulières à valeurs opérateur. La première partie est dédiée à des questions d'interpolation des espaces Lp classiques. On généralise et on énonce de nouvelles caractérisations des espaces interpolés entre espaces Lp. Dans une seconde partie, on démontre une forme des inégalités de Khintchine non-commutatives valides dans tous les espaces interpolés entre espace Lp. Celle-ci permet d’unifier les cas p < 2 et p > 2 ainsi que de traiter les espaces Lp faibles, même pour p = 1 ou 2. En s'appuyant sur la première partie, on caractérise les espaces dans lesquels les formules usuelles pour les inégalités de Khintchine sont valides. Dans une dernière partie, on donne une preuve simplifiée de l'inégalité de type (1,1) faible pour les intégrales singulières non-commutatives, un résultat précédemment obtenu par Parcet. Cette simplification nous permet de retrouver rapidement deux autres résultats connus : la pseudolocalisation Lp et l’inégalité de type faible pour les intégrales singulières non-commutatives dont le noyau est à valeurs dans un espace de Hilbert. / The purpose of this thesis is to develop tools of noncommutative harmonic analysis. More precisely, it deals with noncommutative Khintchine inequalities and operator-valued singular integrals. The first part is dedicated to questions of interpolation between classical Lp-spaces. We generalize and state new characterisations of interpolation spaces between Lp-spaces. In a second part, we introduce a form of the noncommutative Khintchine inequalities which holds in every interpolation space between two Lp-spaces. It enables us to unify the cases p < 2 and p > 2 and to deal with weak Lp-spaces even when p = 1 or 2. By relying on the first part, we characterize spaces in which the usual formulas for Khintchine inequalities hold. In a last part, we give a simplified proof of the weak boundedness of noncommutative singular integrals, a result previously obtained by Parcet. This simplification allows us to recover quickly two results: the Lp pseudolocalisation and the weak type inequality for noncommutative singular integrals associated to Hilbert-valued kernels.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

Este trabalho apresenta a formulação e implementação numérica do método dos elementos de contorno (MEC) para elasticidade linear tri-dimensional, com avaliação direta das integrais fracamente e fortemente singulares. A implementação segue a formulação tradicional do MEC direto, e a discretização do contorno das variáveis do problema é realizada com elementos descontínuos, permitindo o uso de malhas desconectadas ao longo das superfícies. O cálculo das integrais singulares é realizado através do uso de expansões assintóticas calculadas em torno de um ponto singular genérico. As expressões analíticas destas expansões são apresentadas no trabalho. Estas expansões serão subtraídas do núcleo original regularizando-o e a parte singular é integrada analiticamente, restando apenas uma integral regular, tornando ambas as integrais possíveis de serem calculadas com quadraturas de Gauss. É concluído que o presente método requer menos pontos de integração para o mesmo nível de erro quando comparado com outras técnicas. Alguns casos de elasticidade são resolvidos para ilustrar a eficiência e precisão do método. / This work presents the formulation and implementation of the boundary element method (BEM) to three dimensional linear elastostatics, with the direct evaluation of the strongly singular integral equations. The implementation follows the traditional direct BEM formulation, and the discretization of the boundary is carried out with discontinuous elements, enabling the use of disconnected meshes along the surfaces. The computation of the singular integral equations is accomplished by using the asymptotic expansions derived around a generic singular point. The analytical expressions for these expansions are presented in this work. The expansions are subtracted from the kernel to regularize it. This subtracted part is then added by computing a regular line integral along the boundary of the element. Both the integrals can be calculated with Gauss-type quadratures. It's observed that the present method needs less integration points for the same level of error when compared with other techniques. Several elasticity benchmarks are solved to demonstrate the eficiency and the accuracy of the present method.

Este trabalho apresenta a formulação e implementação numérica do método dos elementos de contorno (MEC) para elasticidade linear tri-dimensional, com avaliação direta das integrais fracamente e fortemente singulares. A implementação segue a formulação tradicional do MEC direto, e a discretização do contorno das variáveis do problema é realizada com elementos descontínuos, permitindo o uso de malhas desconectadas ao longo das superfícies. O cálculo das integrais singulares é realizado através do uso de expansões assintóticas calculadas em torno de um ponto singular genérico. As expressões analíticas destas expansões são apresentadas no trabalho. Estas expansões serão subtraídas do núcleo original regularizando-o e a parte singular é integrada analiticamente, restando apenas uma integral regular, tornando ambas as integrais possíveis de serem calculadas com quadraturas de Gauss. É concluído que o presente método requer menos pontos de integração para o mesmo nível de erro quando comparado com outras técnicas. Alguns casos de elasticidade são resolvidos para ilustrar a eficiência e precisão do método. / This work presents the formulation and implementation of the boundary element method (BEM) to three dimensional linear elastostatics, with the direct evaluation of the strongly singular integral equations. The implementation follows the traditional direct BEM formulation, and the discretization of the boundary is carried out with discontinuous elements, enabling the use of disconnected meshes along the surfaces. The computation of the singular integral equations is accomplished by using the asymptotic expansions derived around a generic singular point. The analytical expressions for these expansions are presented in this work. The expansions are subtracted from the kernel to regularize it. This subtracted part is then added by computing a regular line integral along the boundary of the element. Both the integrals can be calculated with Gauss-type quadratures. It's observed that the present method needs less integration points for the same level of error when compared with other techniques. Several elasticity benchmarks are solved to demonstrate the eficiency and the accuracy of the present method.

Este trabalho apresenta a formulação e implementação numérica do método dos elementos de contorno (MEC) para elasticidade linear tri-dimensional, com avaliação direta das integrais fracamente e fortemente singulares. A implementação segue a formulação tradicional do MEC direto, e a discretização do contorno das variáveis do problema é realizada com elementos descontínuos, permitindo o uso de malhas desconectadas ao longo das superfícies. O cálculo das integrais singulares é realizado através do uso de expansões assintóticas calculadas em torno de um ponto singular genérico. As expressões analíticas destas expansões são apresentadas no trabalho. Estas expansões serão subtraídas do núcleo original regularizando-o e a parte singular é integrada analiticamente, restando apenas uma integral regular, tornando ambas as integrais possíveis de serem calculadas com quadraturas de Gauss. É concluído que o presente método requer menos pontos de integração para o mesmo nível de erro quando comparado com outras técnicas. Alguns casos de elasticidade são resolvidos para ilustrar a eficiência e precisão do método. / This work presents the formulation and implementation of the boundary element method (BEM) to three dimensional linear elastostatics, with the direct evaluation of the strongly singular integral equations. The implementation follows the traditional direct BEM formulation, and the discretization of the boundary is carried out with discontinuous elements, enabling the use of disconnected meshes along the surfaces. The computation of the singular integral equations is accomplished by using the asymptotic expansions derived around a generic singular point. The analytical expressions for these expansions are presented in this work. The expansions are subtracted from the kernel to regularize it. This subtracted part is then added by computing a regular line integral along the boundary of the element. Both the integrals can be calculated with Gauss-type quadratures. It's observed that the present method needs less integration points for the same level of error when compared with other techniques. Several elasticity benchmarks are solved to demonstrate the eficiency and the accuracy of the present method.