Mesh Generation for High Performance Computing: Current Trends and Applications

Structured Grid Generation

Audience: Contains mostly beginning/intermediate material.

1. Mappings and Invertibility.

The idea of a mapping from a logical to physical domain will be discussed. Discretization of the mapping produces a structured grid. Maps should be invertible to preserve PDE type.

2. Transfinite Interpolation.

Mappings can be constructed by an algebraic grid generation technique known as transfinite interpolation. Lagrange and Hermite interpolation techniques will be described. Advantages and disadvantages of this technique will be given.

3. Grid Quality

The idea of grid quality will be introduced. The importance of properties such as smoothness, orthogonality, skewness, and volume will be presented.

4. Application to Numerical PDE's

Finite volume and finite difference methods use structured grids to numerically solve partial differential equations on complex geometries. To do so requires transformation of the PDE to curvilinear coordinates. An overview of the transformation procedure is given.

5. One-dimensional grid generation

One-dimensional grid generation will be used as an example to introduce basic ideas in grid generation such as grid generation PDE's, optimization, and variational techniques.

6. Connections to Differential Geometry

To understand grid generation techniques in higher dimensions requires knowledge of basic differential geometry. Tangents, normals, and curvature will be discussed, along with the Jacobian matrix and Metric Tensor.

7. Planar Grid Generation

Basic approaches to two-dimensional grid generation, such as algebraic, conformal mapping, elliptic, and hyperbolic, will be presented.

8. Variational Grid Generation

Powerful grid smoothing and adaptive methods can be derived by posing variational principles involving grid quality measures. The mathematical machinery used in this approach will be described.

9. Curve and Surface Grid Generation

Curves and surfaces introduce an additional level of difficulty into the techiniques of structured grid generation. Basic differential relationships on curves and surfaces will be introduced to show how simple curve and surface generators can be derived.

10. Adaptive Grid Generation

Adaptive grid generation goals and procedures are described in terms of the three basic types of adaptive grid generation. The node movement strategy of Anderson will be used as an illustration.

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