Non-axisymmetric instabilities in self-gravitating tori around black holes, and solving Einstein constraints with superconvergent finite element methods

This thesis contains results on two related projects. In the first project, we explore non-axisymmetric instabilities in general relativistic accretion disks around black holes. Such disks are created as transient structures in several astrophysical scenarios, including mergers of compact objects and core collapse of massive stars. These disks are suggested for the role of cenral engines of gamma-ray bursts. We address the stability of these objects against the runaway and non-axisymmetric instabilities in the three-dimensional hydrodynamical fully general relativistic treatment. We explore three slender and moderately slender disk models with varying disk-to-black hole mass ratio. None of the models that we consider develop the runaway instability during the time span of the simulations, despite large radial axisymmetric oscillations, induced in the disks by the initial data construction procedure. All models develop unstable non-axisymmetric modes on a dynamical timescale. In simulations with dynamical general relativistic treatment, we observe two distinct types of instabilities: the Papaloizou-Pringle instability and the so-called Intermediate instability. The development of the nonaxisymmetric mode with azimuthal number m=1 is enhanced by the outspiraling motion of the black hole. The overall picture of the unstable modes in our disk models is similar to the Newtonian case. In the second project, we experiment with solving the Einstein constraint equations using finite elements on semistructured triangulations of multiblock grids. We illustrate our approach with a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $\psi$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, we evolve the constructed initial data using a high order finite-differencing multi-block approach and extract gravitational waves from the numerical solution

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

HDG methods and data-driven techniques for the van Roosbroeck model and its applications

Noninvasive estimation of doping inhomogeneities in semiconductors is relevant for many industrial applications. The goal is to estimate experimentally the unknown doping profile of a semiconductor by means of reproducible, indirect and non--destructive measurements. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the indirect detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions while measuring the resulting voltage drop or current at the contacts. These technologies lead to inverse problems for which we still do not have a complete theoretical framework. In this thesis, we present three different data-driven approaches based on least squares, multilayer perceptrons, and residual neural networks. We compare the three strategies after having optimized the relevant hyperparameters and we measure the robustness of our approaches with respect to noise. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by a numerical solution of the forward problem using a physics-preserving finite volume method stabilized using the Scharfetter--Gummel scheme. In view of the need of generating larger datasets for trainings, we study the possibility to apply high-order Discontinuous Galerkin methods to the forward problem, preserving the stability properties of the Scharfetter--Gummel scheme. We prove that the Hybridizable Discontinuous Galerkin methods (HDG), a family of high-order DG methods, are equivalent to the Scharfetter--Gummel scheme on uniform unidimensional grids for a specific choice of the HDG stabilization parameter. This result is generalized to two and three dimensions using an approach based on weighted scalar products, and on local Slotboom changes of variables. We show that the proposed numerical scheme is well-posed, and numerically validate that it has the same properties of classical HDG methods, including optimal convergence and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter-Gummel stabilized finite volume scheme (i.e., it produces the same system matrix)

An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 271-281).Improving the autonomy, efficiency, and reliability of partial differential equation (PDE) solvers has become increasingly important as powerful computers enable engineers to address modern computational challenges that require rapid characterization of the input-output relationship of complex PDE governed processes. This thesis presents work toward development of a versatile PDE solver that accurately predicts engineering quantities of interest to user-prescribed accuracy in a fully automated manner. We develop an anisotropic adaptation framework that works with any localizable error estimate, handles any discretization order, permits arbitrarily oriented anisotropic elements, robustly treats irregular features, and inherits the versatility of the underlying discretization and error estimate. Given a discretization and any localizable error estimate, the framework iterates toward a mesh that minimizes the error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model based on an affine-invariant metric interpolation framework; and optimization of the surrogate model to drive the mesh toward optimality. The combination of the framework with a discontinuous Galerkin discretization and an a posteriori output error estimate results in a versatile PDE solver for reliable output prediction. The versatility and effectiveness of the adaptive framework are demonstrated in a number of applications. First, the optimality of the method is verified against anisotropic polynomial approximation theory in the context of L2 projection. Second, the behavior of the method is studied in the context of output-based adaptation using advection-diffusion problems with manufactured primal and dual solutions. Third, the framework is applied to the steady-state Euler and Reynolds-averaged Navier-Stokes equations. The results highlight the importance of adaptation for high-order discretizations and demonstrate the robustness and effectiveness of the proposed method in solving complex aerodynamic flows exhibiting a wide range of scales. Fourth, fully-unstructured space-time adaptivity is realized, and its competitiveness is assessed for wave propagation problems. Finally, the framework is applied to enable spatial error control of parametrized PDEs, producing universal optimal meshes applicable for a wide range of parameters.by Masayuki Yano.Ph.D

Proceedings of the 12th UK Conference on Boundary Integral Methods (UKBIM12)

Boundary integral methods have become established for solving a wide variety of problems in science and engineering. UK based researchers have been active and made substantial contributions in the theory and development of boundary integral formulations, as well as their analysis, discretisation and numerical solution. The UKBIM conference series aims to provide a forum where recent developments in boundary integral methods can be discussed in an informal atmosphere. The first UK conference on boundary integral methods (UKBIM) was held at the University of Leeds in 1997. Subsequent UKBIM conferences have taken place in Brunel (1999), Brighton (2001), Salford (2003), Liverpool (2005), Durham (2007), Nottingham (2009), Leeds (2011), Aberdeen (2013), Brighton (2015) and Nottingham-Trent (2017). The success of these events has made the conference a regular event for researchers based in the UK, and elsewhere, who are working on all aspects of boundary integral methods. This book contains the abstracts and papers presented at the Twelfth UK Conference on Boundary Integral Methods (UKBIM 12), held at Oxford Brookes University in July 2019. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield. I am grateful to the members of the scientific committee for their advice and support during the past year, and to all the authors and reviewers for their hard work in producing the high quality peer-reviewed papers for this book

Adjoint-based optimization for inkjet printing

In this thesis the flow inside inkjet printhead microchannels is analysed using a two- parameter low Mach number expansion of the compressible Navier–Stokes equations and a reduced order model for the free surface flow inside the inkjet nozzle. The channel flow is separated into equations for an incompressible flow with no acoustic oscillations and equations for thermoviscous acoustic oscillations with no mean flow. This thesis concerns two types of optimal control problems. The optimal control problem of the first type is finding a velocity profile of the piezo-electric actuator that eliminates residual oscillations after a droplet is ejected. The cost function is the sum of the acoustic energy in the channel and the surface energy of the spherical cap of ink at the end of the nozzle at a given time. This problem is approached by obtaining the sensitivity of the total energy inside an inkjet microchannel with respect to boundary forcing using the adjoint method. Using gradient-based optimization algorithms, optimal waveforms are found that minimize the objective value at various final times and for geometries with increasing complexity. Physical interpretation to the optimal waveforms profiles is provided, and the exploited mechanisms are revealed. The optimal control problem of the second type is finding a shape of the inkjet printhead channel that maximises dissipation of the acoustic oscillations, without increasing the pressure drop required to drive the steady flow. Similarly, the adjoint approach is used to obtain the sensitivity of the acoustic flow eigenvalues with respect to boundary deformations in Hadamard form. Knowing the shape sensitivity of the incompressible flow viscous dissipation, the constrained optimization problem is solved to find a design that has the same viscous dissipation for the steady flow but a 40% larger decay rate for the oscillating flow. The final shape is not straightforward and would have been difficult to achieve through physical insight or trial and error. It could be improved further by adapting the parameters that describe the shape, but in this case the improvement would be small. The method is general and could be applied to many different applications in microfluidics. In summary, the methods in this thesis are promising techniques in the design and optimization of inkjet printheads. The discussed numerical techniques and the gained physical understanding can be used to automatically find the optimal design parameters, or, at a minimum, accelerate the experimental trial and error processes.This project has received funding from the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. H2020-MSCA-ITN-2015

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Higher order numerical methods for singular perturbation problems

Philosophiae Doctor - PhDIn recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.South Afric