Publicación: Transformation methods for the integration of singular and near-singular functions in XFEM
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2017-07-11
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Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias. Departamento de Estadística, Investigación Operativa y Cálculo Numérico
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Esta tesis doctoral aborda el problema de la integración numérica de funciones singulares y casi-singulares, en dos y tres dimensiones, usando métodos de transformación de variables. Se incluye el análisis de transformaciones con un propósito geométrico, tales que transforman el dominio físico en un dominio maestro estandarizado, y transformaciones de naturaleza algebraica, con el propósito de suavizar las casisingularidades del integrando. Las transformaciones del elemento físico en el dominio maestro se describen en el Capítulo 2. Se presenta el caso más general de una transformación isoparamétrica degenerada que es homogénea en una de sus variables, y se justifica su equivalencia con la transformación polar en el caso bidimensional. Estas transformaciones inducen una factorización de ciertos tipos de núcleo singular en una parte radial y otra angular, permitiendo un tratamiento separado y específico de cada factor. La integración singular en dos dimensiones se examina en el Capítulo 3. El núcleo radial se regulariza completamente por medio de un nuevo esquema que suprime su singularidad. Con respecto al núcleo angular, se muestra que tiene la misma forma que el núcleo casi-singular en una dimensión, de forma que el mismo conjunto de transformaciones se puede aplicar satisfactoriamente a ambos núcleos. El núcleo casi-singular en dos dimensiones es el objeto del Capítulo 4. Aunque el tratamiento del núcleo angular es idéntico al del Capítulo anterior, el núcleo radial admite un nuevo conjunto de transformaciones de regularización, aprovechando un factor lineal presente en el jacobiano de la transformación isoparamétrica. Se considera también la generalización de este problema a triángulos adyacentes, en los cuales el punto fuente está situado fuera del dominio de integración. La extensión de la integración singular a dominios tridimensionales se analiza en el Capítulo 5. El tratamiento del núcleo radial es muy similar al realizado en el Capítulo 3, mientras que en lo referido al núcleo angular en dos variables, su restricción a la frontera del dominio bidimensional se comporta de manera muy similar a la del núcleo casi-singular en una dimensión, por lo que el mismo conjunto de transformaciones de suavizado ya empleadas en los Capítulos 3 y 4 se puede reutilizar de forma satisfactoria en esta situación. Finalmente, el Capítulo 6 presenta una prueba de la forma óptima de la conocida transformación cúbica, usada como una de las alternativas más habituales para la regularización del núcleo angular descrito en los tres Capítulos anteriores. Todos los métodos propuestos se han sometido a ensayos numéricos exhaustivos, mostrando que son capaces de sobrepasar en rendimiento a los métodos existentes en una amplia variedad de situaciones.
This doctoral thesis addresses the problem of numerical integration of singular and near-singular functions, in two and three dimensions, using variable transformation methods. It includes the analysis of transformations with a geometric purpose, i.e., they map the physical domain onto a parent, standard domain, and transformations of an algebraic nature, with the purpose of softening the (near-)singularities in the integrand. Transformations used to map the physical element onto the parent domain are described in chapter 2. The most general case of a degenerate isoparametric map, such that it is homogeneous in one of its variables is presented, and its equivalence to the polar transformation is justified in the two-dimensional case. These maps induce a factorization of certain types of integral kernels into a radial and an angular part, allowing a separate, specific treatment of each factor. The two-dimensional singular integration problem is examined in chapter 3. The radial kernel is completely regularized by means of a new scheme that removes its singularity. Regarding the angular kernel, it is shown to have the same form as the one-dimensional near-singular kernel, and thus the same set of transformations can be successfully applied to both kernels. The two-dimensional near-singular kernel is the subject of chapter 4. Whilst the treatment of the angular kernel is exactly the same as in chapter 3, the radial kernel admits a whole new set of regularizing maps, taking advantage of the linear factor in the Jacobian of the degenerate isoparametric transformation. The generalization of the problem to adjacent triangles, in which the source point lies outside the integration domain is also considered. The extension of the singular integration to three-dimensional domains is covered in chapter 5. The treatment of the radial kernel is very similar as in chapter 3, whereas the bivariate angular kernel, restricted to the boundary of the bidimensional angular domain, behaves very similarly to the near-singular one dimensional kernel, and yet the same set of softening transformations as in chapter 3 and chapter 4 can be suitable re-utilized in this situation. Lastly, chapter 6 presents a proof of the optimal form of the well-known cubic transformation, employed as one of the most common alternatives to regularize the angular kernel in the three previous chapters. All proposed methods have been extensively tested from the numerical point of view, showing that they are able to outperform the existing methods for a broad variety of situations.
This doctoral thesis addresses the problem of numerical integration of singular and near-singular functions, in two and three dimensions, using variable transformation methods. It includes the analysis of transformations with a geometric purpose, i.e., they map the physical domain onto a parent, standard domain, and transformations of an algebraic nature, with the purpose of softening the (near-)singularities in the integrand. Transformations used to map the physical element onto the parent domain are described in chapter 2. The most general case of a degenerate isoparametric map, such that it is homogeneous in one of its variables is presented, and its equivalence to the polar transformation is justified in the two-dimensional case. These maps induce a factorization of certain types of integral kernels into a radial and an angular part, allowing a separate, specific treatment of each factor. The two-dimensional singular integration problem is examined in chapter 3. The radial kernel is completely regularized by means of a new scheme that removes its singularity. Regarding the angular kernel, it is shown to have the same form as the one-dimensional near-singular kernel, and thus the same set of transformations can be successfully applied to both kernels. The two-dimensional near-singular kernel is the subject of chapter 4. Whilst the treatment of the angular kernel is exactly the same as in chapter 3, the radial kernel admits a whole new set of regularizing maps, taking advantage of the linear factor in the Jacobian of the degenerate isoparametric transformation. The generalization of the problem to adjacent triangles, in which the source point lies outside the integration domain is also considered. The extension of the singular integration to three-dimensional domains is covered in chapter 5. The treatment of the radial kernel is very similar as in chapter 3, whereas the bivariate angular kernel, restricted to the boundary of the bidimensional angular domain, behaves very similarly to the near-singular one dimensional kernel, and yet the same set of softening transformations as in chapter 3 and chapter 4 can be suitable re-utilized in this situation. Lastly, chapter 6 presents a proof of the optimal form of the well-known cubic transformation, employed as one of the most common alternatives to regularize the angular kernel in the three previous chapters. All proposed methods have been extensively tested from the numerical point of view, showing that they are able to outperform the existing methods for a broad variety of situations.
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Facultades y escuelas::Facultad de Ciencias