Adaptive Finite Element Solution Algorithm for the Euler Equations

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.