57 research outputs found
A Two-Level Finite Element Discretization of the Streamfunction Formulation of the Stationary Quasi-Geostrophic Equations of the Ocean
In this paper we proposed a two-level finite element discretization of the
nonlinear stationary quasi-geostrophic equations, which model the wind driven
large scale ocean circulation. Optimal error estimates for the two-level finite
element discretization were derived. Numerical experiments for the two-level
algorithm with the Argyris finite element were also carried out. The numerical
results verified the theoretical error estimates and showed that, for the
appropriate scaling between the coarse and fine mesh sizes, the two-level
algorithm significantly decreases the computational time of the standard
one-level algorithm.Comment: Computers and Mathematics with Applications 66 201
Institute for Computer Applications in Science and Engineering (ICASE)
Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science during the period April 1, 1983 through September 30, 1983 is summarized
High Order Explicit Semi-Lagrangian Method for the Solution of Lagrangian Transport and Stochastic Differential Equations
A semi-Lagrangian method is developed for the solution of Lagrangian transport equations and stochastic differential equations that is consistent with Discontinuous Spectral Element Method (DSEM) approximations of Eulerian conservation laws. The method extends the favorable properties of DSEM that include its high-order accuracy, its local and boundary fitted properties and its high performance on parallel platforms for the concurrent semi-Lagrangian and Eulerian solution of a class of time-dependent problems that can be described by coupled Eulerian-Lagrangian formulations. Such formulations include the probabilistic models used for the simulation of chemically reacting turbulent flows or particle-laden flows. Motivated by the high-fidelity simulation of chemically, reacting, turbulent shear layers in propulsion systems, this thesis also reports on the stability of the interaction of two cold flow shear layers. Consistent with an explicit, DSEM discretization, the semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time according to a governing deterministic and/or stochastic differential equation and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. Stochastic samples are averaged on the quadrature nodes. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles from leaving the element\u27s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for advection and (stochastic) diffusion problems show that the method converges exponentially for constant and non-constant advection and diffusion velocities. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator. The linear and non-linear growth of instabilities in a two shear layer cold flow configuration is studied using a combination of Linear Stability Analysis (LSA) and Direct Numerical Simulation (DNS) based on a DSEM approximation of the first-principle Navier-Stokes equations. The linear growth of spatial and temporal LSA modes for a single shear layer is verified using DNS. Then DNS is used to study the transition from linear growth to non-linear instabilities. In DNS of a non-parallel flow with a spatially growing viscous LSA mode an absolute instability is observed. In that case, a match between DNS and LSA is difficult because of spurious modes introduced by the reflections at the boundaries. In a temporal analysis, it is found that the growth of the temporal mode is greater with an increasing difference in the strength of the two shear layers for both inviscid and viscous flows. Longer DNS runs show that presence of a stronger shear layer enhances vortex shedding and vortex pairing mechanism of a shear layer. This enhanced mixing in the non-linear region shows a correlation with the growth of perturbation in the linear region
A comparison of interpolation techniques for non-conformal high-order discontinuous Galerkin methods
The capability to incorporate moving geometric features within models for
complex simulations is a common requirement in many fields. Fluid mechanics
within aeronautical applications, for example, routinely feature rotating (e.g.
turbines, wheels and fan blades) or sliding components (e.g. in compressor or
turbine cascade simulations). With an increasing trend towards the
high-fidelity modelling of these cases, in particular combined with the use of
high-order discontinuous Galerkin methods, there is therefore a requirement to
understand how different numerical treatments of the interfaces between the
static mesh and the sliding/rotating part impact on overall solution quality.
In this article, we compare two different approaches to handle this
non-conformal interface. The first is the so-called mortar approach, where flux
integrals along edges are split according to the positioning of the
non-conformal grid. The second is a less-documented point-to-point
interpolation method, where the interior and exterior quantities for flux
evaluations are interpolated from elements lying on the opposing side of the
interface. Although the mortar approach has significant advantages in terms of
its numerical properties, in that it preserves the local conservation
properties of DG methods, in the context of complex 3D meshes it poses notable
implementation difficulties which the point-to-point method handles more
readily. In this paper we examine the numerical properties of each method,
focusing not only on observing convergence orders for smooth solutions, but
also how each method performs in under-resolved simulations of linear and
nonlinear hyperbolic problems, to inform the use of these methods in implicit
large-eddy simulations.Comment: 37 pages, 15 figures, 5 tables, submitted to Computer Methods in
Applied Mechanics and Engineering, revision
Domain decomposition preconditioners for higher-order discontinuous Galerkin discretizations
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, February 2012."September 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 147-155).Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an efficient high-fidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higher-order hybridizable discontinuous Galerkin (HDG) discretization combined with an implicit solution method is proposed as a means to attain engineering accuracy at lower computational cost. Domain decomposition methods are studied for the parallel solution of the linear system arising at each iteration of the implicit scheme. A minimum overlapping additive Schwarz (ASM) preconditioner and a Balancing Domain Decomposition by Constraints (BDDC) preconditioner are developed for the HDG discretization. An algebraic coarse space for the ASM preconditioner is developed based on the solution of local harmonic problems. The BDDC preconditioner is proven to converge at a rate independent of the number of subdomains and only weakly dependent on the solution order or the number of elements per subdomain for a second-order elliptic problem. The BDDC preconditioner is extended to the solution of convection-dominated problems using a Robin-Robin interface condition. An inexact BDDC preconditioner is developed based on incomplete factorizations and a p-multigrid type coarse grid correction. It is shown that the incomplete factorization of the singular linear systems corresponding to local Neumann problems results in a nonsingular preconditioner. The inexact BDDC preconditioner converges in a similar number of iterations as the exact BDDC method, with significantly reduced CPU time. The domain decomposition preconditioners are extended to solve the Euler and Navier- Stokes systems of equations. An analysis is performed to determine the effect of boundary conditions on the convergence of domain decomposition methods. Optimized Robin-Robin interface conditions are derived for the BDDC preconditioner which significantly improve the performance relative to the standard Robin-Robin interface conditions. Both ASM and BDDC preconditioners are applied to solve several fundamental aerodynamic flows. Numerical results demonstrate that for high-Reynolds number flows, solved on anisotropic meshes, a coarse space is necessary in order to obtain good performance on more than 100 processors.by Laslo Tibor Diosady.Ph.D
HOLA: a High-Order Lie Advection of Discrete Differential Forms With Applications in Fluid Dynamics
The Lie derivative, and Exterior Calculus in general, is ubiquitous in the elegant geometric interpretation of many dynamical systems. We extend recent trends towards a Discrete Exterior Calculus by introducing a discrete framework for the Lie derivative defined on differential forms, including a WENO based numerical scheme for its implementation. The usefulness of this operator is demonstrated through the advection of scalar and vector valued fields (arbitrary discrete k-forms) in a desirable intrinsic and metric independent fashion. Examples include Lie advection of fluid flow vorticity, and we conclude with a significant discussion on the conservative Lie advection of fluid mass density for robust free surface flows in computer graphics
Space-time goal-oriented error control and adaptivity for discretizations and reduced order modeling of multiphysics problems
In this thesis, we investigate the use of adaptive methods for the efficient solution of linear multiphysics problems and nonlinear coupled problems. The main ingredients are a posteriori error estimates based on the dual-weighted residual method. By solving an auxiliary adjoint problem, these error estimates can be used to compute local error indicators for spatial and temporal refinements, which can be used for adaptive spatial and temporal meshes for e.g. the Navier-Stokes equations. For interface- and volume-coupled problems, we present a further extension of temporal adaptivity by using different temporal meshes for each subproblem while still being able to assemble the linear system in a monolithic fashion. Since multiphysics problems, like poroelasticity, are expensive to solve for fine discretizations with millions of degrees of freedom, we present a novel online-adaptive model order reduction method called MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates), which merges classical proper orthogonal decomposition based model order reduction with a posteriori error estimates. Thus, we can avoid the costly offline phase of classical model order reduction methods and still achieve high accuracy by enriching the reduced basis on-the-fly in the online phase when the error estimators exceed a given tolerance.German Research Foundation (DFG)/International Research Training Group 2657/Grant Number 43308229/E
The development of a predictive procedure for localised three dimensional river flows
This thesis contains the formulation, development and initial tests of a computer model for the prediction of fully three dimensional turbulent free surface flows typically found at localised areas of river systems. It is the intention that the model will be used to predict flow situations which are fully three dimensional. The model is, therefore, tested against a fully three dimensional test case of flow in a two-stage meandering channel. However, the model is not intended simply to be for computing flows in meandering river channels. Rather the model is intended to be used in a variety of problems which are outlined in the thesis.
The Reynolds Averaged Navier-Stokes equations form the basis of the physical system. The Reynolds stresses are represented by two different stress-strain relationships: (1) a linear relationship and (2) a non-linear relationship. These relationships rely on an eddy viscosity and a turbulence time-scale which are calculated from two characterising turbulence quantities, a velocity squared scale, k, and an inverse length scale, . These quantities are computed from differential transport equations. Non-linear stress-strain relationships are relatively new and, it has been argued by their originators, require application to several different problems to fully ascertain their potential for future use. The author addresses this demand by applying them to two new problems. These are flow in a plenum chamber and open channel flow over a backward facing step.
The equations are solved by an operator splitting method which, it is argued, allows for an accurate and realistic treatment of the troublesome advection terms at low spatial resolutions. This is thought to be essential since for three dimensional problems owing to computer time limitations achieving grid independent solutions with low order schemes is at present very difficult. The advantage of the present approach is demonstrated with reference to a simple one dimensional analogue
Adaptive high-resolution finite element schemes
The numerical treatment of flow problems by the finite element method
is addressed. An algebraic approach to constructing high-resolution
schemes for scalar conservation laws as well as for the compressible
Euler equations is pursued. Starting from the standard Galerkin
approximation, a diffusive low-order discretization is constructed by
performing conservative matrix manipulations. Flux limiting is
employed to compute the admissible amount of compensating
antidiffusion which is applied in regions, where the solution is
sufficiently smooth, to recover the accuracy of the Galerkin finite
element scheme to the largest extent without generating non-physical
oscillations in the vicinity of steep gradients. A discrete Newton
algorithm is proposed for the solution of nonlinear systems of
equations and it is compared to the standard fixed-point defect
correction approach. The Jacobian operator is approximated by divided
differences and an edge-based procedure for matrix assembly is devised
exploiting the special structure of the underlying algebraic flux
correction (AFC) scheme. Furthermore, a hierarchical mesh adaptation
algorithm is designed for the simulation of steady-state and transient
flow problems alike. Recovery-based error indicators are used to
control local mesh refinement based on the red-green strategy for
element subdivision. A vertex locking algorithm is developed which
leads to an economical re-coarsening of patches of subdivided
cells. Efficient data structures and implementation details are
discussed. Numerical examples for scalar conservation laws and the
compressible Euler equations in two dimensions are presented to assess
the performance of the solution procedure.In dieser Arbeit wird die numerische Simulation von skalaren
Erhaltungsgleichungen sowie von kompressiblen Eulergleichungen mit
Hilfe der Finite-Elemente Methode behandelt. Dazu werden
hochauflösende Diskretisierungsverfahren eingesetzt, welche auf
algebraischen Konstruktionsprinzipien basieren. Ausgehend von der
Galerkin-Approximation wird eine Methode niedriger Ordnung
konstruiert, indem konservative Matrixmodifikationen durchgeführt
werden. Anschließend kommt ein sog. Flux-Limiter zum Einsatz, der in
Abhängigkeit von der lokalen Glattheit der Lösung den zulässigen
Anteil an Antidiffusion bestimmt, die zur Lösung der Methode niedriger
Ordnung hinzuaddiert werden kann, ohne dass unphysikalische
Oszillationen in der Nähe von steilen Gradienten entstehen. Die
resultierenden nichtlinearen Gleichungssysteme können entweder mit
Hilfe von Fixpunkt-Defektkorrektur-Techniken oder mittels diskreter
Newton-Verfahren gelöst werden. Für letztere wird die Jacobi-Matrix
mit dividierten Differenzen approximiert, wobei ein effizienter,
kantenbasierter Matrixaufbau aufgrund der speziellen Struktur der
zugrunde liegenden Diskretisierung möglich ist. Ferner wird ein
hierarchischer Gitteradaptionsalgorithmus vorgestellt, welcher sowohl
für die Simulation von stationären als auch zeitabhängigen Strömungen
geeignet ist. Die lokale Gitterverfeinerung folgt dem bekannten
Rot-Grün Prinzip, wobei rekonstruktionsbasierte Fehlerindikatoren zur
Markierung von Elementen zum Einsatz kommen. Ferner erlaubt das
sukzessive Sperren von Knoten, die nicht gelöscht werden können, eine
kostengünstige Rückvergröberung von zuvor unterteilten Elementen. In
der Arbeit wird auf verschiedene Aspekte der Implementierung sowie auf
die Wahl von effizienten Datenstrukturen zur Verwaltung der
Gitterinformationen eingegangen. Der Nutzen der vorgestellten
Simulationswerkzeuge wird anhand von zweidimensionalen
Beispielrechnungen für skalare Erhaltungsgleichungen sowie für die
kompressiblen Eulergleichungen analysiert
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