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 /

Shape analysis of the corpus callosum of autistic and normal subjects in neuroimaging.

Early detection of human disease in today’s society can have an enormous impact on the severity of the disease that is manifested. Disease such as Autism and Dyslexia, which have no current cure or proven mechanism as to how they develop, can often have an adverse physical and physiological impact on the lifestyle of a human being. Although these disease are not fully curable, the severity handicaps that accompany them can be significantly reduced with the proper therapy, and thus the earlier that the disease is detected the faster therapy can be administered. The research in this thesis is an attempt at studying discriminatory shape measures of some brain structures that are known to carry changes from autistics to normal individuals. The focus will be on the corpus callosum. There has been considerable research done on the brain scans (MRI, CT) of autistic individuals vs. control (normal) individuals to observe any noticeable discrepancies through statistical analysis. The most common and powerful tool to analyze structures of the brain, once a specific region has been segmented, is using Registration to match like structures and record their error. The ICP algorithm (Iterative Closest Point) is commonly used to accomplish this task. Many techniques such as level sets and statistical methods can be used for segmentation. The Corpus Callosum (CC) and the cortical surface of the brain are currently where most Autism analysis is performed. It has been observed that the gyrification of the cortical surface is different in the two groups, and size as well as shape of the CC. An analysis approach for autism MRI is quite extensive and involves many steps. This thesis is limited to examination of shape measures of the CC that lend discrimination ability to distinguish between normal and autistic individuals from T1-weigheted MRI scans. We will examine two approaches for shape analysis, based on the traditional Fourier Descriptors (FD) method and shape registration (SR) using the procrustes technique. MRI scans of 22 autistic and 16 normal individuals are used to test the approaches developed in this thesis. We show that both FD and SR may be used to extract features to discriminate between the two populations with accuracy levels over 80% up to 100% depending on the technique

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

The Construction of Optimized High-Order Surface Meshes by Energy-Minimization

Despite the increasing popularity of high-order methods in computational fluid dynamics, their application to practical problems still remains challenging. In order to exploit the advantages of high-order methods with geometrically complex computational domains, coarse curved meshes are necessary, i.e. high-order representations of the geometry. This dissertation presents a strategy for the generation of curved high-order surface meshes. The mesh generation method combines least-squares fitting with energy functionals, which approximate physical bending and stretching energies, in an incremental energy-minimizing fitting strategy. Since the energy weighting is reduced in each increment, the resulting surface representation features high accuracy. Nevertheless, the beneficial influence of the energy-minimization is retained. The presented method aims at enabling the utilization of the superior convergence properties of high-order methods by facilitating the construction of coarser meshes, while ensuring accuracy by allowing an arbitrary choice of geometric approximation order. Results show surface meshes of remarkable quality, even for very coarse meshes representing complex domains, e.g. blood vessels

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Discontinuities in numerical radiative transfer

Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem

Raffinement de maillage adaptatif pour la simulation numérique des instabilités MHD dans les tokamaks : le code JOREK

The purpose of this paper is to illustrate both validity and advantages of the implementation of the adaptive mesh raffinement strategy in the recent version of the 3D non-linear MHD code JOREK which uses a technique based on the bicubic Bezier surfaces developed in the paper of Czarny-Huijsmans. We describe the physcal model and establish a refinement criteria. Then, we also present the numerical results of adaptive mesh raffinement simulation for the a tearing instability test case and to the test case of injection mechanism of a small pellet of frozen hydrogen into a tokamak

A novel parallel algorithm for surface editing and its FPGA implementation

A thesis submitted to the University of Bedfordshire in partial fulfilment of the requirements for the degree of Doctor of PhilosophySurface modelling and editing is one of important subjects in computer graphics. Decades of research in computer graphics has been carried out on both low-level, hardware-related algorithms and high-level, abstract software. Success of computer graphics has been seen in many application areas, such as multimedia, visualisation, virtual reality and the Internet. However, the hardware realisation of OpenGL architecture based on FPGA (field programmable gate array) is beyond the scope of most of computer graphics researches. It is an uncultivated research area where the OpenGL pipeline, from hardware through the whole embedded system (ES) up to applications, is implemented in an FPGA chip. This research proposes a hybrid approach to investigating both software and hardware methods. It aims at bridging the gap between methods of software and hardware, and enhancing the overall performance for computer graphics. It consists of four parts, the construction of an FPGA-based ES, Mesa-OpenGL implementation for FPGA-based ESs, parallel processing, and a novel algorithm for surface modelling and editing. The FPGA-based ES is built up. In addition to the Nios II soft processor and DDR SDRAM memory, it consists of the LCD display device, frame buffers, video pipeline, and algorithm-specified module to support the graphics processing. Since there is no implementation of OpenGL ES available for FPGA-based ESs, a specific OpenGL implementation based on Mesa is carried out. Because of the limited FPGA resources, the implementation adopts the fixed-point arithmetic, which can offer faster computing and lower storage than the floating point arithmetic, and the accuracy satisfying the needs of 3D rendering. Moreover, the implementation includes Bézier-spline curve and surface algorithms to support surface modelling and editing. The pipelined parallelism and co-processors are used to accelerate graphics processing in this research. These two parallelism methods extend the traditional computation parallelism in fine-grained parallel tasks in the FPGA-base ESs. The novel algorithm for surface modelling and editing, called Progressive and Mixing Algorithm (PAMA), is proposed and implemented on FPGA-based ES’s. Compared with two main surface editing methods, subdivision and deformation, the PAMA can eliminate the large storage requirement and computing cost of intermediated processes. With four independent shape parameters, the PAMA can be used to model and edit freely the shape of an open or closed surface that keeps globally the zero-order geometric continuity. The PAMA can be applied independently not only FPGA-based ESs but also other platforms. With the parallel processing, small size, and low costs of computing, storage and power, the FPGA-based ES provides an effective hybrid solution to surface modelling and editing