Grid generation and adaptive refinement
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This document discusses various methods for generating computational grids for solving partial differential equations (PDEs). It describes algebraic methods like transfinite interpolation that use known functions to transform domains, as well as PDE-based methods like Thompson's elliptic PDE grid that define grid requirements mathematically. It also covers unstructured grids, adaptive refinement, and moving grids that adjust grid resolution based on solution features.
Grid generation and adaptive refinement
- 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
- 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!
- 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:
- 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