Adaptive Finite Element Solution Algorithm for the Euler Equations

Softcover Nachdruck of the original 1. Auflage 1991
ISBN: 978-3-528-07632-0
Verlag: Vieweg+Teubner Verlag

Buch, Englisch, Band 32, 166 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 289 g

Reihe: Notes on Numerical Fluid Mechanics and Multidisciplinary Design

This monograph is the result of my PhD thesis work in Computational Fluid Dynamics at the Massachusettes Institute of Technology under the supervision of Professor Earll Murman. A new finite element al­ gorithm is presented for solving the steady Euler equations describing the flow of an inviscid, compressible, ideal gas. This algorithm uses a finite element spatial discretization coupled with a Runge-Kutta time integration to relax to steady state. It is shown that other algorithms, such as finite difference and finite volume methods, can be derived using finite element principles. A higher-order biquadratic approximation is introduced. Several test problems are computed to verify the algorithms. Adaptive gridding in two and three dimensions using quadrilateral and hexahedral elements is developed and verified. Adaptation is shown to provide CPU savings of a factor of 2 to 16, and biquadratic elements are shown to provide potential savings of a factor of 2 to 6. An analysis of the dispersive properties of several discretization methods for the Euler equations is presented, and results allowing the prediction of dispersive errors are obtained. The adaptive algorithm is applied to the solution of several flows in scramjet inlets in two and three dimensions, demonstrat­ ing some of the varied physics associated with these flows. Some issues in the design and implementation of adaptive finite element algorithms on vector and parallel computers are discussed.

Zielgruppe

Research

Autoren/Hrsg.

Shapiro, Richard A.

Inhaltsverzeichnis

1 Introduction.- 1.1 Research Goals.- 1.2 Overview of Thesis.- 1.3 Survey of Finite Element Methods for the Euler Equations.- 2 Governing Equations.- 2.1 Euler Equations.- 2.2 Non-Dimensionalization of the Equations.- 2.3 Auxiliary Quantities.- 2.4 Boundary Conditions.- 3 Finite Element Fundamentals.- 3.1 Basic Definitions.- 3.2 Finite Elements and Natural Coordinates.- 3.3 Typical Elements.- 4 Solution Algorithm.- 4.1 Overview of Algorithm.- 4.2 Spatial Discretization.- 4.3 Choice of Test Functions.- 4.4 Boundary Conditions.- 4.5 Smoothing.- 4.6 Time Integration.- 4.7 Consistency and Conservation.- 5 Algorithm Verification and Comparisons.- 5.1 Introduction.- 5.2 Verification and Comparison of Methods.- 5.3 Effects of Added Dissipation.- 5.4 Biquadratic vs. Bilinear.- 5.5 Three Dimensional Verification.- 5.6 Summary.- 6 Adaptation.- 6.1 Introduction.- 6.2 Adaptation Procedure.- 6.3 Adaptation Criteria.- 6.4 Embedded Interface Treatment.- 6.5 Examples of Adaptation.- 6.6 CPU Time Comparisons.- 7 Dispersion Phenomena and the Euler Equations.- 7.1 Introduction.- 7.2 Difference Stencils.- 7.3 Linearization of the Equations.- 7.4 Fourier Analysis of the Linearized Equations.- 7.5 Numerical Verification.- 7.6 Conclusions.- 8 Scramjet Inlets.- 8.1 Introduction.- 8.2 Two-Dimensional Test Cases.- 8.3 Three-Dimensional Results.- 9 Summary and Conclusions.- 9.1 Summary.- 9.2 Contributions of the Thesis.- 9.3 Conclusions.- 9.4 Areas for Further Exploration.- A Computational Issues.- A.1 Introduction.- A.2 Vectorization and Parallelization Issues.- A.3 Computer Memory Requirements.- A.4 Data Structures for Adaptation.- B Scramjet Geometry Definition.- References.- List of Symbols.