Isogeometric preconditioners based on fast solvers for the Sylvester equation

We consider large linear systems arising from the isogeometric discretization of the Poisson problem on a single-patch domain. The numerical solution of such systems is considered a challenging task, particularly when the degree of the splines employed as basis functions is high. We consider a preconditioning strategy which is based on the solution of a Sylvester-like equation at each step of an iterative solver. We show that this strategy, which fully exploits the tensor structure that underlies isogeometric problems, is robust with respect to both mesh size and spline degree, although it may suffer from the presence of complicated geometry or coefficients. We consider two popular solvers for the Sylvester equation, a direct one and an iterative one, and we discuss in detail their implementation and efficiency for 2D and 3D problems on single-patch or conforming multi-patch NURBS geometries. Numerical experiments for problems with different domain geometries are presented, which demonstrate the potential of this approach

A parallel multigrid solver for multi-patch Isogeometric Analysis

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF) for the financial support through the DK W1214-04, while the second author was supported by the FWF grant NFN S117-0

Reduced Order Modeling based Inexact FETI-DP solver for lattice structures

This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness-to-weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine-scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so-called inexact FETI-DP method where the local, cell-wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell-wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro-scale data, thereby making our method also multiscale and matrix-free. The solver is tested against various 2D and 3D analyses. It shows major gains with respect to black-box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes.Comment: 30 pages, 12 figures, 2 table

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 /