In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions.

The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. While such techniques have been known for a long time, most simulations do not make use of them because of the lack of a robust implementation. In this work, we present recent progresses that have been made to develop adaptive mesh refinement features in Nek5000, a code based on the spectral element method. These developments are driven by the algorithmic challenges posed by future exascale supercomputers. First, we perform the study of the strong scaling of Nek5000 on three petascale machines in order to assess the scalability of the code and identify the current bottlenecks. It is found that strong scaling limit ranges between 5, 000 and 220, 000 degrees of freedom per core depending on the machine and the case. The need for synchronized and low latency communication for efficient computational fluid dynamics simulation is also confirmed. Additionally, we present how Hypre, a library for linear algebra, is used to develop a new and efficient code for performing the setup step required prior to the use of an algebraic multigrid solver for preconditioning the pressure equation in Nek5000. Finally, the main objective of this work is to develop new methods for estimating the error on a numerical solution of the Navier–Stokes equations via the resolution of an adjoint problem. These new estimators are compared to existing ones, which are based on the decay of the spectral coefficients. Then, the estimators are combined with newly implemented capabilities in Nek5000 for automatic grid refinement and adaptive mesh adaptation is carried out. The applications considered so far are steady and two-dimensional, namely the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. The use of adaptive mesh refinement techniques makes mesh generation easier and it is shown that a similar accuracy as with a static mesh can be reached with a significant reduction in the number of degrees of freedom. / <p>QC 20171114</p>

Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above.

In this dissertation, Richardson extrapolation and other computational techniques are used to develop a series of high accuracy high efficiency solution techniques for solving partial differential equations (PDEs). A Richardson extrapolation-based sixth-order method with multiple coarse grid (MCG) updating strategy is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly obtain a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. The MCG updating strategy directly computes a sixth-order solution on the fine grid by using various combinations of multiple coarse grids. A multiscale multigrid (MSMG) method is used to solve the linear systems resulting from fourth-order compact (FOC) discretizations. Numerical investigations show that the proposed methods compute high accuracy solutions and have better computational efficiency and scalability than the existing Richardson extrapolation-based sixth order method with iterative operator based interpolation. Completed Richardson extrapolation is explored to compute sixth-order solutions on the entire fine grid. The correction between the fourth-order solution and the extrapolated sixth-order solution rather than the extrapolated sixth-order solution is involved in the interpolation process to compute sixth-order solutions for all fine grid points. The completed Richardson extrapolation does not involve significant computational cost, thus it can reach high accuracy and high efficiency goals at the same time. There are three different techniques worked with Richardson extrapolation for computing fine grid sixth-order solutions, which are the iterative operator based interpolation, the MCG updating strategy and the completed Richardson extrapolation. In order to compare the accuracy of these Richardson extrapolation-based sixth-order methods, truncation error analysis is conducted on solving a 2D Poisson equation. Numerical comparisons are also carried out to verify the theoretical analysis. Richardson extrapolation-based high accuracy high efficiency computation is extended to solve unsteady-state equations. A higher-order alternating direction implicit (ADI) method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution obtained from a high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the effects of Richardson extrapolation on stable numerical solutions from the underlying ADI method.

Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of an effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations as well as the optimal linear complexity of computing time. In Chapter 5, we offer a new implicit finite difference scheme in time for solving the first-order necessary optimality system arising in optimal control of wave equations. With a five-point central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy, which is not restricted by the classical Courant-Friedrichs-Lewy (CFL) stability condition on the spatial and temporal step sizes. Moreover, based on its advantageous developed structure, an efficient preconditioned Krylov subspace method is provided and analyzed for solving the discretized sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of proposed preconditioned iterative solver. Finally, brief summaries and future research perspectives are given in Chapter 6.

Cette thèse porte sur le développement d’une méthode multigrilles parallèle visant à réduire les temps de calculs des simulations éléments finis dans le domaine de la mise en forme de pièces forgées en 3D. Ces applications utilisent une méthode implicite, caractérisées par une formulation mixte en vitesse/pression et une gestion du contact par pénalisation. Elles impliquent de grandes déformations qui rendent nécessaires des remaillages fréquents sur les maillages tétraédriques non structurés utilisés. La méthode multigrilles développée suit une approche hybride, se basant sur une construction géométrique des niveaux grossiers par déraffinement de maillage non emboîtés et sur une construction algébrique des systèmes linéaires intermédiaires et grossiers. Un comportement asymptotique quasi-linéaire et une bonne efficacité parallèle sont attendus afin de permettre la réalisation de simulations à grand nombre de degrés de liberté dans des temps plus raisonnables qu’aujourd’hui. Pour cela, l’algorithme de déraffinement de maillages est compatible avec le calcul parallèle, ainsi que les opérateurs permettant les transferts de champs entre les différents niveaux de maillages partitionnés. Les spécificités des problèmes à traiter ont mené à la sélection d'un lisseur plus complexe que ceux utilisés plus fréquemment dans la littérature. Sur la grille la plus grossière, une méthode de résolution directe est utilisée, en séquentiel comme en calcul parallèle. La méthode multigrilles est utilisée en tant que préconditionneur d’une méthode de résidu conjugué et a été intégrée au logiciel FORGE NxT et montre un comportement asymptotique et une efficacité parallèle proches de l’optimal. Le déraffinement automatique de maillages permet une compatibilité avec les remaillages fréquents et permet à la méthode multigrilles de simuler un procédé du début à la fin. Les temps de calculs sont significativement réduits, même sur des simulations avec des écoulements particuliers, sur lesquelles la méthode multigrilles ne peut être utilisée de manière optimale. Cette robustesse permet, par exemple, de réduire de 4,5 à 2,5 jours le temps de simulation d’un procédé. / A parallel multigrid method is developed to reduce large computational costs involved by the finite element simulation of 3D metal forming applications. These applications are characterized by a mixed velocity/pressure implicit formulation with a penalty formulation to enforce contact and lead to large deformations, handled by frequent remeshings of unstructured meshes of tetrahedral. The developed multigrid method follows a hybrid approach where the different levels of non-nested meshes are geometrically constructed by mesh coarsening, while the linear systems of the intermediate and coarse levels result from the algebraic approach. A close to linear asymptotical behavior is expected along with parallel efficiency in order to allow simulations with large number of degrees of freedom under reasonable computation times. These objectives lead to a parallel mesh coarsening algorithm and parallel transfer operators allowing fields transfer between the different levels of partitioned meshes. Physical specificities of metal forming applications lead to select a more complex multigrid smoother than those classically used in literature. A direct resolution method is used on the coarsest mesh, in sequential and in parallel computing. The developed multigrid method is used as a preconditioner for a Conjugate Residual algorithm within FORGE NxT software and shows an asymptotical behavior and a parallel efficiency close to optimal. The automatic mesh coarsening algorithm enables compatibility with frequent remeshings and allows the simulation of a forging process from beginning to end with the multigrid method. Computation times are significantly reduced, even on simulations with particular material flows on which the multigrid method is not optimal. This robustness allows, for instance, reducing from 4.5 to 2.5 days the computation of a forging process.

Avec le développement des techniques d’imagerie telles que la tomographie par rayons X au cours des dernières années, il est maintenant possible de prendre en compte la microstructure réelle dans les simulations des matériaux composites. Cependant, la complexité des composites tels que des fibres inclinées et brisées, les vides, exige un grand nombre des données à l’échelle microscopique pour décrire ces détails et amène ainsi des problèmes difficiles en termes de temps de calcul et de mémoire lors de l’utilisation de méthodes de simulation traditionnelles comme la méthode Eléments Finis. Ces problèmes deviennent encore plus sérieux dans la simulation de l’endommagement, comme la propagation des fissures. Par conséquent, il est nécessaire d’étudier des méthodes numériques plus efficaces pour ce genre de problèmes à grande échelle. La méthode Multigrille (MG) est une méthode qui peut être efficace parce que son coût de calcul est proportionnel au nombre d’inconnues. Dans cette thèse, un solveur de MG efficace pour ces problèmes est développé. La méthode MG est appliquée pour résoudre le problème d’élasticité statique basé sur l’équation de Lamé et aussi le problème de la propagation de fissures basé sur une méthode de champ de phase. La précision des solutions MG est validée par une solution analytique classique d’Eshelby. Ensuite, le solveur MG est développé pour étudier le processus d’homogénéisation des composites et ses solutions sont comparées avec des solutions existantes de la littérature. Après cela, le programme de calcul MG est appliqué pour simuler l’effet de bord libre dans les matériaux composites stratifiés. Une structure stratifiée réelle donnée par tomographie X est d’abord simulé. Enfin, le solveur MG est encore développé, combinant une méthode de champ de phase, pour simuler la rupture quasi-fragile. La méthode MG présente l’efficacité à la fois en temps de calcul et en mémoire pour résoudre les problèmes ci-dessus. / With the development of imaging techniques like X-Ray tomography in recent years, it is now possible to take into account the microscopic details in composite material simulations. However, the composites' complex nature such as inclined and broken fibers, voids, requires rich data to describe these details and thus brings challenging problems in terms of computational time and memory when using traditional simulation methods like the Finite Element Method. These problems become even more severe in simulating failure processes like crack propagation. Hence, it is necessary to investigate more efficient numerical methods for this kind of large scale problems. The MultiGrid (MG) method is such an efficient method, as its computational cost is proportional to the number of unknowns. In this thesis, an efficient MG solver is developed for these problems. The MG method is applied to solve the static elasticity problem based on the Lame's equation and the crack propagation problem based on a phase field method. The accuracy of the MG solutions is validated with Eshelby's classic analytic solution. Then the MG solver is developed to investigate the composite homogenization process and its solutions are compared with existing solutions in the literature. After that, the MG solver is applied to simulate the free-edge effect in laminated composites. A real laminated structure using X-Ray tomography is first simulated. At last, the MG solver is further developed, combined with a phase field method, to simulate the brittle crack propagation. The MG method demonstrates its efficiency both in time and memory dimensions for solving the above problems.

Cartesian grids offer a valuable option to simulate aerodynamic flows around complex geometries such as multi-element airfoils, aircrafts, and rockets. Therefore, an adaptively-refined Cartesian grid generator and Euler solver are developed. For the mesh generation part of the algorithm, dynamic data structures are used to determine connectivity information between cells and uniform mesh is created in the domain. Marching squares and cubes algorithms are used to form interfaces of cut and split cells. Geometry-based cell adaptation is applied in the mesh generation. After obtaining appropriate mesh around input geometry, the solution is obtained using either flux vector splitting method or Roe&rsquo / s approximate Riemann solver with cell-centered approach. Least squares reconstruction of flow variables within the cell is used to determine high gradient regions of flow. Solution based adaptation method is then applied to current mesh in order to refine these regions and also coarsened regions where unnecessary small cells exist. Multistage time stepping is used with local time steps to increase the convergence rate. Also FAS multigrid technique is used in order to increase the convergence rate. It is obvious that implementation of geometry and solution based adaptations are easier for Cartesian meshes than other types of meshes. Besides, presented numerical results show the accuracy and efficiency of the algorithm by especially using geometry and solution based adaptation. Finally, Euler solutions of Cartesian grids around airfoils, projectiles and wings are compared with the experimental and numerical data available in the literature and accuracy and efficiency of the solver are verified.