A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners

Developments in the Extended Finite Element Method and Algebraic Multigrid for Solid Mechanics Problems Involving Discontinuities

In this dissertation, some contributions related to computational modeling and solution of solid mechanics problems involving discontinuities are discussed. The main tool employed for discrete modeling of discontinuities is the extended finite element method and the primary solution method discussed is the algebraic multigrid. The extended finite element method has been shown to be effective for both weak and strong discontinuities. With respect to weak discontinuities, a new approach that couples the extended finite element method with Monte Carlo simulations with the goal of quantifying uncertainty in homogenization of material properties of random microstructures is presented. For accelerated solution of linear systems arising from problems involving cracks, several new methods involving the algebraic multigrid are presented. In the first approach, the Schur complement of the linear system arising from XFEM is used to develop a Hybrid-AMG method such that crack-conforming aggregates are formed. Another alternative approach involves transforming the original linear system into a modified system that is amenable for a direct application of algebraic multigrid. It is shown that if only Heaviside-enrichments are present, a simple transformation based on the phantom-node approach is available, which decouples the linear system along the discontinuities such that crack conforming aggregates are automatically generated via smoother aggregation algebraic multigrid. Various numerical examples are presented to verify the accuracy of the resulting solutions and the convergence properties of the proposed algorithms. The parallel scalability performance of the implementation are also discussed

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposatsPostprint (published version

HAZniCS -- Software Components for Multiphysics Problems

We introduce the software toolbox HAZniCS for solving interface-coupled multiphysics problems. HAZniCS is a suite of modules that combines the well-known FEniCS framework for finite element discretization with solver and graph library HAZmath. The focus of the paper is on the design and implementation of a pool of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics. The robustness and efficiency of the numerical algorithms and methods is shown in several numerical examples, namely the Darcy-Stokes equations that model flow of cerebrospinal fluid in the human brain and the mixed-dimensional model of electrodiffusion in the brain tissue

Parallel Element-Based Algebraic Multigrid for H (Curl) And H (Div) Problems Using the Parelag Library

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers (ParELAG) library, to produce multilevel preconditioners and solvers for H (curl) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and the Auxiliary-Space Maxwell Solver or the Auxiliary-Space Divergence Solver on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library