Poroelasticity problem: numerical difficulties and efficient multigrid solution

This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot’s consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of “Vanka”-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed

Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(NM\log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semialgebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with nonsmooth solutions and a nonlinear problem with applications in porous media, are presented

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

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 /

Efficient resolution of singularly perturbed coupled systems: Equations of reaction-diffusion

In this communication we consider a class of singularly perturbed linear system of reaction-diffusion type coupled in the reaction terms. To approximate its solution, in [3] J.L. Gracia, F. Lisbona, A uniformly convergent scheme for a system of reaction–diffusion equations, To appear in J. Comp. Appl. Math. the backward Euler method and the central difference scheme on a layer–adapted mesh of Shishkin type was used. We propose a new semi-implicit method which decouples the linear system to be solved at each time level and we prove that it is a uniformly convergent scheme (with respect to the diffusion parameters) in the discrete maximum norm. We display some numerical experiments illustrating in practice the theoretical results. From these examples we can see both the uniform convergence of the numerical method and also its efficiency to approximate the solution of the reaction–diffusion system

Estudio numérico de un modelo de propagación de enfermedades

Este trabajo de fin de grado consiste en un estudio de los sistemas de modelizacion de enfermedades SIR y SEIR, junto con la discretización de ambos modelos. Para ello se estudiará el metodo de ecuaciones en diferencias finitas y el imex

Estudio numérico de un modelo de propagación de enfermedades infecciosas.

En el trabajo se realiza el estudio de varios modelos (SIR y SEIR) empleados para intentar predecir y modelar enfermedades infecciosas entre la población. Además se da un ejemplo real sobre la aplicación de dichos modelos.<br /

Cálculo de variaciones para el procesamiento de imágenes

En este trabajo definiremos el cáculo varacional y utilizaremos desarrollos de este para el procesamiento de imágenes, en particular, para la eliminación de ruido en una imagen. Resolveremos un problema de eliminación de ruido con su correspondiente implemenatación en Matlab

Resolución analítica y numérica del modelo Black-Scholes

En el trabajo se comienza deduciendo el modelo Black-Scholes para la valoración de opciones europeas y explicando la base matemática que está detrás del modelo. En los dos siguientes capítulos se obtiene una solución analítica del modelo, en el primero de ellos, y una solución numérica, en el segundo. En cada uno de estos capítulos se explica el procedimiento utilizado para obtener dichas soluciones. Finalmente, ayudados del lenguaje de programación Matlab, se comparan los dos tipos de soluciones con un ejemplo concreto de opción europea Call.<br /

Simulación numérica en registro de imágenes

Dadas dos imágenes, el registro de imágenes consiste en hallar una transformación óptima de modo que al aplicarla sobre una de ellas se obtenga una imagen similar a la otra. Concretamente, para tratar el registro no paramétrico es necesaria la introducción del cálculo de variaciones, herramienta empleada para hallar extremos locales de funcionales