Numerical Grid Generation, Foundations and Appliations
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TOC
# Introduction
# Boundary-Conforming Coordinate Systems
1. Basic Concepts
2. Generalization
1. Boundary-value Problem — Physical Region
2. Boundary-value Problem — Transformed Region
3. Transformed Region Configurations
1. Simply-connected Regions
2. Multiply-connected Regions
3. Embedded Regions
4. Other Configurations
5. Three-dimensional Regions
4. Composite Grids
1. Simply-connected Regions
2. Multiply-connected Regions
3. Embedded Regions
4. Three-dimensional Regions
5. Overlaid Grids
5. Branch Cuts
1. Point Correspondence
2. Derivative Correspondence
6. Implementation
# Transformation Relations
1. Base Vectors
1. Covariant
2. Contravariant
3. Orthogonality
2. Differential Elements
1. Covariant metric tensor
2. Arc length element
3. Surface area element
4. Volume element
3. Derivative Operators
1. Divergence
2. Curl
3. Gradient
4. Laplacian
4. Relations Between Covariant and Contravariant Metrics
1. Base vectors
2. Metric Tensors
5. Restatement of Derivative Operators
1. Conservative
2. Non-conservative
6. Normal and Tangential Derivatives
1. Tangent to coordinate lines
2. Normal to coordinate surfaces
3. Normal to coordinate lines and tangent to coordinate surfaces
7. Integrals
1. Surface integral
2. Volume integral
3. Line integrals
8. Two-Dimensional Forms
1. Metric elements
2. Transformation relations
9. Time derivatives
1. First Derivative
2. Convective terms
# Numerical Implementation
1. Transformed Eqations
2. Discrete Representation of Derivatives
3. Special Points
4. Metric Identities
5. Implementation Procedure
# Truncation Error
1. Order On Nonuniform Spacing
1. Order with fixed distribution function
2. Order with fixed number of points
2. Effect of Numerical Metric Coefficients
3. Evaluation of Distribution Functions
4. Two-Dimensions Forms
# Elliptic Generation Systems
1. Generation Equations
1. Laplace system
2. Poisson system
3. Effect of boundary point distribution
4. General Poisson-type systems
5. Other systems
2.
1. Attraction to coordinate lines/points
2. Attraction to lines/points in space
3. Evaluation along a coordinate line
4. Evaluation on a coordinate surface
5. Evaluation from boundary point distribution
6. Iterative determination
3. Surface Grid Generation Systems
1. Surface grid generation
2. Three-dimensional grids
4. Implementation
1. Difference equations
2. Control functions
3. Surface generation systems
# Parabolic and Hyperbolic Generation Systems
1. Hyperbolic Grid Generation
2. Parabolic Grid Generation
# Algebraic Generation Systems
1. Unidirectional Interpolation
1. Lagrange interpolation
2. Hermite interpolation
3. Other forms of polynomial interpolation
4. Splines
5. Tension splines
6. B-Splines
7. Multi-surface interpolation
8. Uniformity
9. Functions other than polynomials
2. Multi-Directional Interpolation
1. Transfinite interpolation
2. Projectors
# Orthogonal Systems
1. General Formulation
2. Two-Dimensional Orthogonal Coordinates
1. Conformal systems
2. Other systems
3. Systems based on first-order equations
3. Three-Dimensional Orthogonal Coordinates
4. Nearly-Orthogonal Systems
# Conformal Mapping
1. Construction by Finite-Differences
2. Schwarz-Christoffel Transformation
3. Construction from Integral Equations
4. Elementary Complex Transformations
# Adaptive Grids
1. One-Dimensional Adaption
1. Equidistribution
2. Equidistribution by transformation
3. Weight functions
2. Multiple-Dimensional Adaption
1. Adaption along fixed lines
2. Uncoupled adaption
3. Coupled adaption
4. Weight functions
3. Variational Approach
1. Variational formulation
2. Euler equations
3. Brackbill-Saltzman construction
4. Applications
5. Extensions
4. Other Approaches
1. Attraction-Repulsion
2. Reaction analogy
3. Moving finite elements
5. Correlations
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