Summer Academy 2008, Numerical Methods in Engineering, Herceg Novi, Montenegro

1. Talk 2.08
Grid generation
and adaptive refinement
Wednesday, 09/03/2008
Summer Academy 2008
Numerical Methods in Engineering Goran Rakić, student
Herceg Novi, Montenegro Faculty of Mathematics, Belgrade

2. ● The solution of PDE
can be simplified by a
well-constructed grid.
● Grid which is not well
suited to the problem
can lead to instability
or lack of convergence

4. Logical and physical domain

5. Requirements for transformation
● Jacobian of the transformation should be
non-zero to preserve properties of hosted equations
(one to one mapping) where Jacobian matrix is:
● Smooth, orthogonal grids (or grids without small
angles) usually result in the smallest error.

6. Additional requirements
● Grid spacing in physical domain should
correlate with expected numerical error

7. Continuum and discrete grids
● Evaluating continumm boundary conforming
transformation in discrete points of logical
space gives discrete grid in physical space

8. Quick overview
● Structured grids
● Unstructured grids
● Special grids (multiblock, adaptive,...)

9. Algebraic methods
● Known functions are used in one, two, or three
dimensions for transformation
● Interpolation between pair of boundaries
● If boundaries are given as data points,
approximation must be used to fit function to
data points first.

10. Bilinear maps
● Combining normalization and translation for
transforming any quadralateral physical domain
to rectangle to create bilinear maps
● One dimension:

11. Bilinear maps in two dimensions
● Two dimensions (vector form):

12. Special coordinate systems
● Polar, Spherical and Cylindrical
● Parabolic Cylinder coordinates
● Elliptic Cylinder coordinates
● ...
● And not to forgot, Cartesian grids
...where we all start from

13. Transfinite interpolation (TFI)
● Rapid computation (compare to PDE methods)
● Easy to control point locations
● Using Lagrange polynomials for blending:
ξ, ξ-1, η, η-1

15. Boundary parametrization... done

16. Let's fix ξ and let η go from 0..1:
Now add ξ direction:
Left boundary
Hmm, something is wrong when moving both ξ and η:
ξ = 1, right boundary

17. Ta da!

18. TFI examples (1/2)

19. TFI examples (2/2)
1
0 1

20. Topology of a hole
● Transformation preserves holes
● But with little magic...

22. PDE methods for grid generation
● Algebraic methods (affine trans., bilinear, TFI)
defining a grid geometrically
● PDE methods
defining requirements for grid mathematically

23. PDE methods for grid generation
● We have to construct system of PDEs whose
solutions are boundary conforming grid
coordinate lines with specified line spacing
● Solving the system gives grid
● For large grids the computing time is
considerable

24. Thompson's Elliptic PDE grid
● ξ = F(x,y) and η = G(x,y) are unknowns in
Poisson eq with condition so x,y boundaries are
mapped to boundaries of computational domain
where P and Q defines grid point spacing
● Then instead solving ξ and η we change
independent and dependent variables

25. Thompson's Elliptic PDE grid
● The system is solved on uniform grid in
computational domain which gives coordinate
lines in physical domain

26. Example copied from the book

27. Example copied from the book
Boundary:

28. PDE methods for grid generation
● Hyperbolic – when wall boundaries are well
defined, but far field boundary is left
● Can be used to smooth out metric
discontinuities in the TFI

29. This slide is intentionally left blank.

30. Unstructured grids
● Field is in rapid expansion
● Faster to generate on complex domains
● Easy local refinement
● Complex data structure (link matrix or else)
● Can be generated more automatically even on
complex domains, compared to structured grids

31. Delaunay triangulation
● Simple criteria to connect points to form
conforming, non intersecting unstructured grid

32. Delaunay triangulation algorithm
● Nice incremental algorithm
● Introduce new point, locally break triangulation
and then retriangulate affected part
● Flipping algorithm:

33. Point generation?

34. Advancing front generation
● Construct a grid from boundary informations
● Connect boundary points to create edges
(called “front”)
● Select any edge in front and create its
perpendicular bisector. On a bisector pick a
point at the distance d inside the domain
● In that point, create a circle of radius r, order
any points inside circle by distance from center
and for each create triangles with edge vertices
● Pick up the first triangle that is not intersecting
edges, and update front (connect, remove edges)

36. Overlapping (Chimera-) grids
● Built using partially overlapping blocks
● Boundary conditions are exchanged between
domains using interpolation
● Can combine structured and unstructured
sub-grids

38. Adaptive grid refinement
● We want to reduce error without unnecessary
computational costs
● Regions of rapid variations of solution needs
better resolution
● Using AGR we can discretize huge domains
(astrophysics) and/or domains with non-uniform
variations across regions of interest
● Save both memory and CPU time
● Trivial to implement for unstructured grids

39. Moving grids
● Solution adaptive methods for time-depended
PDEs where regions of “rapid variations” moves
in time (like Burgers' flow equation)
● Let grid points move with “whatever fronts are
present” keeping number of grid points constant

40. Moving grids math
● Transform PDEs to include time changing grid
transformation
● When discretized, time depending grid points
are also unknowns so one has to find both
so more equations must be added.

41. Moving grids math (cont.)
● New equations should connect grid points
changing position with equidistribution principle
of error in computed PDE solution
● Having an error-monitor function we want it
to be equal over average on all grid sections
● They also must prevent rapid grid movement

42. Moving grid example without any
real number-crunching shown

43. Cheating the “Summary” question
● No method that fits all
● In structured domains, algebraic methods are
preferred for speed and simplicity
● Usually implemented in multi disciplinary
software packages that goes with CAD
interface, surface editing and visualization tools
● Multi-block