1. Σ YSTEMS
Introduction to Grid Generation
Delta Pi Systems
Thessaloniki, Greece, 12 December 2011
2. Steps in Grid Generation
1. Define an array of grid points for the domain.
2. Label the grid points and interconnect them in some specified
way to define a discretization of the domain as a union of
mesh cells or elements.
3. Solve the mathematical problem on the discretized domain.
4. Refine or redistribute the grid and return to step 3 to improve
the accuracy of the approximate solution.
5. Display the grid and solution.
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4. A Grid
η y
ξ x
logical space physical space
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5. Logical Space
n Logical space Boundary ∂Ukn
1 U1 = {ξ ∈ E 1 ; 0 ≤ ξ ≤ 1} 2 points
2 U2 = {(ξ, η) ∈ E 2 ; 0 ≤ ξ, η ≤ 1} 4 segments
4 points
3 U3 = {(ξ, η, ζ) ∈ E 3 ; 0 ≤ ξ, η, ζ ≤ 1} 6 faces
12 segments
8 points
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6. Coordinate Maps
Map Coordinates From To
1
X1 x = x(ξ) interval interval
2
X1 x = x(ξ), y = y (ξ) interval curve
3
X1 x = x(ξ), y = y (ξ), z = z(ξ) interval curve
2
X2 x = x(ξ, η), y = y (ξ, η) square region
3
X2 x = x(ξ, η), y = y (ξ, η), z = z(ξ, η) square surface
3
X3 x = x(ξ, η, ζ), y = y (ξ, η, ζ), z = z(ξ, η, ζ) cube volume
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7. Boundary Topology
η
3
4 2
1 ξ
logical space
y permissable y non-permissable
1 4 1 3
2 3 4 2
x x
physical space physical space
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8. Jacobian Matrix
∂xi
Jij = ∂ξj , i = 1, . . . , n, j = 1, . . . , k
Theorem (Inverse Mapping Theorem)
Assume Xk ∈ C1 . Then Xk is locally one-to-one at ξ in the interior
n n
of Uk , if and only if the rank of J is maximal (equals k) at ξ.
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10. Division to subdomains for block grid generation
1. The domain is subdivided into several simple subregions.
2. A mesh for each subregion is generated independently by
means of a map from the uniform grid on the associated
reference domain.
3. Tables in the initial data set define the interconnection of the
subregion meshes.
4. The mesh is smoothed locally.
5. The node points are renumbered to optimize the nodal
adjacency or sparsity pattern.
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