On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domainMinisterio de Economía y Competitividad BIA2016-75042-C2-1-RFondos FEDER POCI-01-0247-FEDER-01775

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper, we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.Comment: 35 pages, 18 figure

Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers.

Partial Differential Equations (PDE) are extensively used in Applied Sciences to model real-world problems. The solution u of a PDE is normally not available in closed form, and so it is necessary to approximate it by means of some numerical method. Despite the differences among the various methods, the principle on which all of them are based is essentially the same: they first discretize the PDE by introducing a mesh, related to some discretization parameter n, and then they compute the corresponding numerical solution u_n, which will converge to u when n tends to infinity, i.e., when the mesh is progressively refined. Now, if both the PDE and the numerical method are linear, the computation of u_n reduces to solving a certain linear system A_n * u_n = f_n whose size d_n tends to infinity with n. In addition, the sequence of discretization matrices A_n often enjoys an asymptotic spectral distribution described by a certain matrix-valued function f, which takes values in the space of Hermitian matrices of a certain size s. This means that, for large n, the eigenvalues of A_n are approximately given by a uniform sampling of the eigenvalue functions lambda_i(f), i=1,...,s, over the domain of f. In this situation, f is called the (spectral) symbol of the sequence of matrices A_n. The identification and the study of the symbol are two interesting issues in themselves, because they provide an accurate information about the asymptotic global behavior of the eigenvalues of A_n. In particular, the number s coincides with the number of "branches" that compose the asymptotic spectrum of A_n. However, the knowledge of the symbol f and of its properties is not only interesting in itself, but can also be used for practical purposes. In particular, it can be used to design effective preconditioned Krylov and multigrid solvers for the linear systems associated with A_n. The reason is clear: the convergence properties of preconditioned Krylov and multigrid methods strongly depend on the spectral features of the matrix to which they are applied. Hence, the spectral information provided by the symbol can be conveniently used for designing fast solvers of this kind. The purpose of this thesis is to present some specific examples, of interest in practical applications, in which the above philosophical discussion comes to life. As our model PDE, we consider classical second-order elliptic differential equations. Concerning the numerical methods that we employ for their solution, we make three choices: the classical Qp Lagrangian Finite Element Method (FEM), the Galerkin B-spline Isogeometric Analysis (IgA) and the B-spline IgA Collocation Method. We first identify and study the symbol f that characterizes the asymptotic spectrum of the discretization matrices A_n arising from these approximation techniques. Then, by exploiting the properties of the symbol, we design fast iterative solvers for the matrices A_n associated with the two numerical methods based on IgA (the Galerkin B-spline IgA and the B-spline IgA Collocation Method)

Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers.

Partial Differential Equations (PDE) are extensively used in Applied Sciences to model real-world problems. The solution u of a PDE is normally not available in closed form, and so it is necessary to approximate it by means of some numerical method. Despite the differences among the various methods, the principle on which all of them are based is essentially the same: they first discretize the PDE by introducing a mesh, related to some discretization parameter n, and then they compute the corresponding numerical solution u_n, which will converge to u when n tends to infinity, i.e., when the mesh is progressively refined. Now, if both the PDE and the numerical method are linear, the computation of u_n reduces to solving a certain linear system A_n * u_n = f_n whose size d_n tends to infinity with n. In addition, the sequence of discretization matrices A_n often enjoys an asymptotic spectral distribution described by a certain matrix-valued function f, which takes values in the space of Hermitian matrices of a certain size s. This means that, for large n, the eigenvalues of A_n are approximately given by a uniform sampling of the eigenvalue functions lambda_i(f), i=1,...,s, over the domain of f. In this situation, f is called the (spectral) symbol of the sequence of matrices A_n. The identification and the study of the symbol are two interesting issues in themselves, because they provide an accurate information about the asymptotic global behavior of the eigenvalues of A_n. In particular, the number s coincides with the number of "branches" that compose the asymptotic spectrum of A_n. However, the knowledge of the symbol f and of its properties is not only interesting in itself, but can also be used for practical purposes. In particular, it can be used to design effective preconditioned Krylov and multigrid solvers for the linear systems associated with A_n. The reason is clear: the convergence properties of preconditioned Krylov and multigrid methods strongly depend on the spectral features of the matrix to which they are applied. Hence, the spectral information provided by the symbol can be conveniently used for designing fast solvers of this kind. The purpose of this thesis is to present some specific examples, of interest in practical applications, in which the above philosophical discussion comes to life. As our model PDE, we consider classical second-order elliptic differential equations. Concerning the numerical methods that we employ for their solution, we make three choices: the classical Qp Lagrangian Finite Element Method (FEM), the Galerkin B-spline Isogeometric Analysis (IgA) and the B-spline IgA Collocation Method. We first identify and study the symbol f that characterizes the asymptotic spectrum of the discretization matrices A_n arising from these approximation techniques. Then, by exploiting the properties of the symbol, we design fast iterative solvers for the matrices A_n associated with the two numerical methods based on IgA (the Galerkin B-spline IgA and the B-spline IgA Collocation Method)

Evaluarea ordinului de aproximare prin operatori liniari si pozitivi

Auswertung der Approximationsgüte durch positive lineare Operatoren Abstract: The areas of research covered by this thesis can be roughly divided into three parts: aspects of quantitative approximation, studying of shape-preservation properties and over-iteration for some selected operators. Regarding the quantitative approximation we study traditional problems like: direct estimates, degree of simultaneous approximation or global smoothness preservation (Chapters 2, 3). On the other hand, we also present some non-classical issues like: estimates for the Peano remainder, a quantitative Voronovskaja theorem or estimates for differences of two positive linear operators (Chapter 5). Our object of study are different classes of operators: rational type operators, composite Beta type operators, but also other types that cannot be classified like: the BLaC-wavelet operator and the King operator

Multi-scale modelling describing thermal behaviour of polymeric materials. Scalable lattice-Boltzmann models based upon the theory of Grmela towards refined thermal performance prediction of polymeric materials at micro and nano scales.

Micrometer injection moulding is a type of moulding in which moulds have geometrical design features on a micrometer scale that must be transferred to the geometry of the produced part. The difficulties encountered due to very high shear and rapid heat transfer of these systems has motivated this investigation into the fundamental mathematics behind polymer heat transfer and associated processes. The aim is to derive models for polymer dynamics, especially heat dynamics, that are considerably less approximate than the ones used at present, and to translate this into simulation and optimisation algorithms and strategies, Thereby allowing for greater control of the various polymer processing methods at micrometer scales

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Robust multigrid methods for Isogeometric discretizations applied to poroelasticity problems

El análisis isogeométrico (IGA) elimina la barrera existente entre elementos finitos (FEA) y el diseño geométrico asistido por ordenador (CAD). Debido a esto, IGA es un método novedoso que está recibiendo una creciente atención en la literatura y recientemente se ha convertido en tendencia. Muchos esfuerzos están siendo puestos en el diseño de solvers eficientes y robustos para este tipo de discretizaciones. Dada la optimalidad de los métodos multimalla para elementos finitos, la aplicación de estosmétodos a discretizaciones isogeométricas no ha pasado desapercibida. Nosotros pensamos firmemente que los métodos multimalla son unos candidatos muy prometedores a ser solvers eficientes y robustos para IGA y por lo tanto en esta tesis apostamos por su aplicación. Para contar con un análisis teórico para el diseño de nuestros métodos multimalla, el análisis local de Fourier es propuesto como principal análisis cuantitativo. En esta tesis, a parte de considerar varios problemas escalares, prestamos especial atención al problema de poroelasticidad, concretamente al modelo cuasiestático de Biot para el proceso de consolidación del suelo. Actualmente, el diseño de métodos multimalla robustos para problemas poroelásticos respecto a parámetros físicos o el tamaño de la malla es un gran reto. Por ello, la principal contribución de esta tesis es la propuesta de métodos multimalla robustos para discretizaciones isogeométricas aplicadas al problema de poroelasticidad.La primera parte de esta tesis se centra en la construcción paramétrica de curvas y superficies dado que estas técnicas son la base de IGA. Así, la definición de los polinomios de Bernstein y curvas de Bézier se presenta como punto de partida. Después, introducimos los llamados B-splines y B-splines racionales no uniformes (NURBS) puesto que éstas serán las funciones base consideradas en nuestro estudio.La segunda parte trata sobre el análisis isogeométrico propiamente dicho. En esta parte, el método isoparamétrico es explicado al lector y se presenta el análisis isogeométrico de algunos problemas. Además, introducimos la formulación fuerte y débil de los problemas anteriores mediante el método de Galerkin y los espacios de aproximación isogeométricos. El siguiente punto de esta tesis se centra en los métodos multimalla. Se tratan las bases de los métodos multimalla y, además de introducir algunos métodos iterativos clásicos como suavizadores, también se introducen suavizadores por bloques como los métodos de Schwarz multiplicativos y aditivos. Llegados a esta parte, nos centramos en el LFA para el diseño de métodos multimalla robustos y eficientes. Además, se explican en detalle el análisis estándar y el análisis basado en ventanas junto al análisis de suavizadores por bloques y el análisis para sistemas de ecuaciones en derivadas parciales.Tras introducir las discretizaciones isogeométricas, los métodos multimalla y el LFA como análisis teórico, nuestro propósito es diseñar métodos multimalla eficientes y robustos respecto al grado polinomial de los splines para discretizaciones isogeométricas de algunos problemas escalares. Así, mostramos que el uso de métodos multimalla basados en suavizadores de tipo Schwarz multiplicativo o aditivo produce buenos resultados y factores de convergencia asintóticos robustos. La última parte de esta tesis está dedicada al análisis isogeométrico del problema de poroelasticidad. Para esta tarea, se introducen el modelo de Biot y su discretización isogeométrica. Además, presentamos una novedosa estabilización de masa para la formulación de dos campos de las ecuaciones de Biot que elimina todas las oscilaciones no físicas en la aproximación numérica de la presión. Después, nos centramos en dos tipos de solvers para estas ecuaciones poroelásticas: Solvers desacoplados y solvers monolíticos. En el primer grupo, le dedicamos una especial atención al método fixed-stress y a un método iterativo propuesto por nosotros que puede ser aplicado de forma automática a partir de la estabilización de masa ya mencionada.Por otro lado, realizamos un análisis de von Neumann para este método iterativo aplicado al problema de Terzaghi y demostramos su estabilidad y convergencia para los pares de elementos Q1 Q1, Q2 Q1 y Q3 Q2 (con suavidad global C1). Respecto al grupo de solvers monolíticos, nosotros proponemos métodos multimalla basados en suavizadores acoplados y desacoplados. En esta parte, métodosIsogeometric analysis (IGA) eliminates the gap between finite element analysis (FEA) and computer aided design (CAD). Due to this, IGA is an innovative approach that is receiving an increasing attention in the literature and it has recently become a trending topic. Many research efforts are being devoted to the design of efficient and robust solvers for this type of discretization. Given the optimality of multigrid methods for FEA, the application of these methods to IGA discretizations has not been unnoticed. We firmly think that they are a very promising approach as efficient and robust solvers for IGA and therefore in this thesis we are concerned about their application. In order to give a theoretical support to the design of multigrid solvers, local Fourier analysis (LFA) is proposed as the main quantitative analysis. Although different scalar problems are also considered along this thesis, we make a special focus on poroelasticity problems. More concretely, we focus on the quasi-static Biot's equations for the soil consolidation process. Nowadays, it is a very challenging task to achieve robust multigrid solvers for poroelasticity problems with respect physical parameters and/or the mesh size. Thus, the main contribution of this thesis is to propose robust multigrid methods for isogeometric discretizations applied to poroelasticity problems. The first part of this thesis is devoted to the introduction of the parametric construction of curves and surfaces since these techniques are the basis of IGA. Hence, with the definition of Bernstein polynomials and B\'ezier curves as a starting point, we introduce B-splines and non-uniform rational B-splines (NURBS) since these will be the basis functions considered for our numerical experiments. The second part deals with the isogeometric analysis. In this part, the isoparametric approach is explained to the reader and the isogeometric analysis of some scalar problems is presented. Hence, the strong and weak formulations by means of Galerkin's method are introduced and the isogeometric approximation spaces as well. The next point of this thesis consists of multigrid methods. The basics of multigrid methods are explained and, besides the presentation of some classical iterative methods as smoothers, block-wise smoothers such as multiplicative and additive Schwarz methods are also introduced. At this point, we introduce LFA for the design of efficient and robust multigrid methods. Furthermore, both standard and infinite subgrids local Fourier analysis are explained in detail together with the analysis for block-wise smoothers and the analysis for systems of partial differential equations. After the introduction of isogeometric discretizations, multigrid methods as our choice of solvers and LFA as theoretical analysis, our goal is to design efficient and robust multigrid methods with respect to the spline degree for IGA discretizations of some scalar problems. Hence, we show that the use of multigrid methods based on multiplicative or additive Schwarz methods provide a good performance and robust asymptotic convergence rates. The last part of this thesis is devoted to the isogeometric analysis of poroelasticity. For this task, Biot's model and its isogeometric discretization are introduced. Moreover, we present an innovative mass stabilization of the two-field formulation of Biot's equations that eliminates all the spurious oscillations in the numerical approximation of the pressure. Then, we deal with two types of solvers for these poroelastic equations: Decoupled and monolithic solvers. In the first group we devote special attention to the fixed-stress split method and a mass stabilized iterative scheme proposed by us that can be automatically applied from the mass stabilization formulation mentioned before. In addition, we perform a von Neumann analysis for this iterative decoupled solver applied to Terzaghi's problem and demonstrate that it is stable and convergent for pairs Q1-Q1, Q2-Q1 and Q3-Q2 (with global smoothness C1). Regarding the group of monolithic solvers, we propose multigrid methods based on coupled and decoupled smoothers. Coupled additive Schwarz methods are proposed as coupled smoothers for isogeometric Taylor-Hood elements. More concretely, we propose a 51-point additive Schwarz method for the pair Q2-Q1. In the last part, we also propose to use an inexact version of the fixed-stress split algorithm as decoupled smoother by applying iterations of different additive Schwarz methods for each variable. For the latter approach, we consider the pairs of elements Q2-Q1 and Q3-Q2 (with global smoothness C1). Finally, thanks to LFA we manage to design efficient and robust multigrid solvers for the Biot's equations and some numerical results are shown.<br /