Adaptive Finite Element Solution Algorithm for the Euler Equations (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.

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.- 2.4.1 Solid Surface Boundary Conditions.- 2.4.2 Open Boundary Conditions.- 3 Finite Element Fundamentals.- 3.1 Basic Definitions.- 3.2 Finite Elements and Natural Coordinates.- 3.2.1 Properties of Interpolation Functions.- 3.2.2 Natural Coordinates and Derivative Calculation.- 3.3 Typical Elements.- 3.3.1 Bilinear Element.- 3.3.2 Biquadratic Element.- 3.3.3 Trilinear Element.- 4 Solution Algorithm.- 4.1 Overview of Algorithm.- 4.2 Spatial Discretization.- 4.3 Choice of Test Functions.- 4.3.1 Test Functions for Galerkin Method.- 4.3.2 Test Functions for Cell-Vertex Method.- 4.3.3 Test Functions for Central Difference Method.- 4.4 Boundary Conditions.- 4.4.1 Solid Surface Boundary Condition.- 4.4.2 Open Boundary Condition.- 4.5 Smoothing.- 4.5.1 Conservative, Low-Accuracy Second Difference.- 4.5.2 Non-Conservative, High-Accuracy Second Difference.- 4.5.3 Combined Smoothing.- 4.5.4 Smoothing on Biquadratic Elements.- 4.6 Time Integration.- 4.7 Consistency and Conservation.- 4.7.1 Making Artificial Viscosity Conservative.- 5 Algorithm Verification and Comparisons.- 5.1 Introduction.- 5.2 Verification and Comparison of Methods.- 5.2.1 5° Converging Channel.- 5.2.2 15° Converging Channel.- 5.2.3 4% Circular Arc Bump.- 5.2.4 10% Circular Arc Bump.- 5.2.5 10% Cosine Bump.- 5.2.6 CPU Comparison and Recommendations.- 5.2.7 Verification of Conservation.- 5.3 Effects of Added Dissipation.- 5.4 Biquadratic vs. Bilinear.- 5.4.1 5° Channel Flow.- 5.4.2 4% Circular Arc Bump.- 5.4.3 10% Cosine Bump.- 5.5 Three Dimensional Verification.- 5.6 Summary.- 6 Adaptation.- 6.1 Introduction.- 6.2 Adaptation Procedure.- 6.2.1 Placement of Boundary Nodes.- 6.2.2 How Much Adaptation?.- 6.3 Adaptation Criteria.- 6.3.1 First-Difference Indicator.- 6.3.2 Second-Difference Indicator.- 6.3.3 Two-Dimensional Directional Adaptation.- 6.4 Embedded Interface Treatment.- 6.4.1 Two-Dimensional Interface.- 6.4.2 Three-Dimensional Interface.- 6.5 Examples of Adaptation.- 6.5.1 Multiple Shock Reflections.- 6.5.2 4% Circular Arc Bump.- 6.5.3 10% Circular Arc Bump.- 6.5.4 3-D Channel.- 6.5.5 Distorted Grid.- 6.6 CPU Time Comparisons.- 7 Dispersion Phenomena and the Euler Equations.- 7.1 Introduction.- 7.2 Difference Stencils.- 7.2.1 Some Properties of the Galerkin Stencil.- 7.2.2 Some Properties of the Cell-Vertex Stencil.- 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.- 8.2.1 M? = 5, 0° Yaw.- 8.2.2 M? = 5, 5° Yaw.- 8.2.3 M? = 2, 0° Yaw.- 8.2.4 M? = 3, 0° Yaw.- 8.2.5 M? = 3, 7° Yaw.- 8.2.6 Inlet Performance and Total Pressure Loss.- 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.3.1 Two-Dimensional Memory Requirements.- A.3.2 Three-Dimensional Memory Requirements.- A.4 Data Structures for Adaptation.- A.4.1 Finding the Children of an Element.- A.4.2 Finding The Adjacent Element.- B Scramjet Geometry Definition.- References.- List of Symbols.