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Introduction to Grid Generation

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Introduction to Grid Generation

  1. 1. Σ YSTEMSIntroduction to Grid Generation Delta Pi SystemsThessaloniki, Greece, 12 December 2011
  2. 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. Σ YSTEMS
  3. 3. Transformation, map, or coordinate system η y ξ x logical space physical space Σ YSTEMS
  4. 4. A Grid η y ξ x logical space physical space Σ YSTEMS
  5. 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 Σ YSTEMS
  6. 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 Σ YSTEMS
  7. 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 Σ YSTEMS
  8. 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. Σ YSTEMS
  9. 9. Block-Structured Grids 5 4 6 1 2 3 7 Σ YSTEMS
  10. 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. Σ YSTEMS
  11. 11. Schematic of different components Geometric Grid Generation/ Analysis Modeling Computation Graphics Refinement 1. Primitive lines and circles 2. Parametric curves (e.g., Bezier curves) and interpolated curves 3. Piecewise-composite curves Σ YSTEMS
  12. 12. Open source tools for grid generation and visualization: ◮ Gmsh ◮ Netgen ◮ ParaView Σ YSTEMS
  13. 13. Contact us Delta Pi Systems Thessaloniki, Greece http://www.delta-pi-systems.eu Σ YSTEMS

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