A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented

Development of a finite element method for light activated polymers

Traditional Shape Memory Polymers (SMPs) belong to a class of smart materials which have shown promise for a wide range of applications. They are characterized by their ability to maintain a temporary deformed shape and return to an original parent permanent shape. The first SMPs developed responded to changes in temperature by exploiting the difference in modulus and chain mobility through the glass transition temperature. However, in recent years, new SMPs have been developed that respond to other stimuli besides temperature; these can include electricity, magnetism, changes in chemical concentration, and even light. In this thesis, we consider the photo-mechanical behavior of Light Activated Shape Memory Polymers (LASMPs), focusing on the numerical aspects. The mechanics behind LASMPS is rather abstract and cumbersome, even for simple geometries. In order to move these materials out of the lab and into the more modern engineering design framework of commercial design and engineering software, robust numerical methods must be developed in order to implement sound and accurate simulations. The photo-mechanical theory is summarized and some constitutive laws that govern LASMPS are described. Implementation of the multiphysics governing equations takes the form of a user defined element subroutine within the commercial software package ABAQUS/STANDARD. Simulations are carried out with varied geometries and symmetries, for example plane-strain, axisymmetric, and three-dimensional geometries under complex photo-mechanical loadings

Numerical investigation of separated transonic turbulent flows with a multiple-time-scale turbulence model

A numerical investigation of transonic turbulent flows separated by curvature and shock wave - boundary layer interaction is presented. The free stream Mach numbers considered are 0.4, 0.5, 0.6, 0.7, 0.8, 0.825, 0.85, 0.875, 0.90, and 0.925. In the numerical method, the conservation of mass equation is replaced by a pressure correction equation for compressible flows and thus incremental pressure is solved for instead of density. The turbulence is described by a multiple-time-scale turbulence model supplemented with a near-wall turbulence model. The present numerical results show that there exists a reversed flow region at all free stream Mach numbers considered whereas various k-epsilon turbulence models fail to predict such a reversed flow region at low free stream Mach numbers. The numerical results also show that the size of the reversed flow region grows extensively due to the shock wave - turbulent boundary layer interaction as the free stream Mach number is increased. These numerical results show that the turbulence model can resolve the turbulence field subjected to extra strains caused by the curvature and the shock wave - turbulent boundary layer interaction and that the numerical method yields a significantly accurate solution for the complex compressible turbulent flow

Computational methods for internal flows with emphasis on turbomachinery

Current computational methods for analyzing flows in turbomachinery and other related internal propulsion components are presented. The methods are divided into two classes. The inviscid methods deal specifically with turbomachinery applications. Viscous methods, deal with generalized duct flows as well as flows in turbomachinery passages. Inviscid methods are categorized into the potential, stream function, and Euler aproaches. Viscous methods are treated in terms of parabolic, partially parabolic, and elliptic procedures. Various grids used in association with these procedures are also discussed

Some Observations on the Interaction Between Linear and Nonlinear Stabilization for Continuous Finite Element Methods Applied to Hyperbolic Conservation Laws

We discuss stability and accuracy of stabilized finite element methods for hyperbolic problems. In particular we focus on the interaction between linear and nonlinear stabilizations. First we show that the combination of linear and nonlinear stabilization can be designed to be invariant preserving. Then we show that such a combined method allows for the classical O(hk+12) error estimates for smooth solutions of space semidiscretized formulations of the linear transport equation. Based on these ideas we propose a Runge–Kutta finite element method for the compressible Euler equations using entropy viscosity to ensure stability at shocks and gradient jump penalty to prevent propagation of high frequency content into the smooth part of the solution. In a numerical example we show that the method predicts the shock structure accurately, without high frequency pollution of the smooth parts

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

On the Advective Component of Active Flux Schemes for Nonlinear Hyperbolic Conservation Laws

A new class of numerical methods called Active Flux (AF) is investigated for nonlinear hyperbolic conservation laws. The AF method is designed specifically to address the aspect that most modern compressible flow methods fail to do; the multidimensionality aspect. It addresses the shortcoming by employing a two stage update process. In the first stage, a nonconservative form of the system is introduced to provide the flexibility to pursue distinct numerical approaches for flow processes with differing physics. Because each process is treated separately, the numerical method can be appropriately formed to reflect each type of physics and to provide the maximal stability. The method is completed with the conservation update to produce a third-order accurate scheme. The AF advection scheme is founded on the characteristic tracing method, a semi-Lagrangian method, which has long been used for developing numerical methods for hyperbolic problems. The first known AF method for advection, Scheme V by van Leer, is revisited as a part of the development of the scheme. Details of Scheme V are examined closely, and new improvements are made for the multidimensional nonlinear advection scheme. A detailed study of the nonlinear system of equations is made possible by the pressureless Euler system, which is the advective component of the Euler system. It serves as a stepping stone for the Euler system, and all necessary details of the nonlinear system are explored. Lastly, an extension to the Euler system is presented where a novel nonlinear operator splitting method is introduced to correctly blend the contributions of the nonlinear advection and acoustic processes. The AF method, as a result, produces a maximally stable, third-order accurate method for the multidimensional Euler system. Some guiding principles of limiting are presented. Because two types of flow feature are separately treated, the limiting process must also be kept separate. Advective problems obeying natural bounding principles are treated differently from acoustic problems with no explicit bounding principles. Distinct limiting approaches are explored along with discussions.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138695/1/jmaeng_1.pd

Das unstetige Galerkinverfahren für Strömungen mit freier Oberfläche und im Grundwasserbereich in geophysikalischen Anwendungen

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory. Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing. This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.Für viele geophysikalische Anwendungen spielen Strömungen mit freier Oberfläche und im Grundwasserbereich oder sogar die Kopplung dieser beiden eine zentrale Rolle. Oftmals charakteristisch für diese Anwendungsszenarien sind große Rechengebiete und lange Simulationszeiten. Folglich ist das Berechnen akkurater Lösungen mit beträchtlichem Rechenaufwand verbunden und der Einsatz effizienter Lösungsverfahren sowie von Techniken des Hochleistungsrechnens obligatorisch, um Ergebnisse innerhalb eines annehmbaren Zeitrahmens zu erhalten. Unstetige Galerkinverfahren stellen eine Gruppe numerischer Verfahren zum Lösen von Differentialgleichungen dar, und kombinieren Eigenschaften von Methoden der Finiten Volumen- und Finiten Elementeverfahren. Sie ermöglichen hohe Approximationsordnungen, bieten einen hohen Grad an Flexibilität und sind für paralleles Rechnen gut geeignet. Diese Dissertation besteht aus acht Artikeln und einer erweiterten Zusammenfassung, in diesen die Anwendung unstetiger Galerkinverfahren auf mathematische Modelle inklusive solcher für Strömungen mit freier Oberfläche und im Grundwasserbereich beschrieben wird. Die behandelten Themen umfassen Diskretisierungs- und Implementierungsaspekte, die Parallelisierung der Methode sowie eine diskrete Stabilitätsanalyse des gekoppelten Modells