grid generation and adaptive refinement
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Talk 2.08
Grid generationand adaptive refinement
Goran Rakić, studentFaculty of Mathematics, Belgrade
Wednesday, 09/03/2008
Summer Academy 2008Numerical Methods in EngineeringHerceg Novi, Montenegro
● 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
Logical and physical domain
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.
Additional requirements
● Grid spacing in physical domain should correlate with expected numerical error
Continuum and discrete grids
● Evaluating continumm boundary conforming transformation in discrete points of logical space gives discrete grid in physical space
Quick overview
● Structured grids
● Unstructured grids
● Special grids (multiblock, adaptive,...)
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.
Bilinear maps
● Combining normalization and translation for transforming any quadralateral physical domain to rectangle to create bilinear maps
● One dimension:
Bilinear maps in two dimensions
● Two dimensions (vector form):
Special coordinate systems
● Polar, Spherical and Cylindrical● Parabolic Cylinder coordinates● Elliptic Cylinder coordinates● ...● And not to forgot, Cartesian grids
...where we all start from
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
Boundary parametrization... done
Let's fix ξ and let η go from 0..1:
Now add ξ direction:
Hmm, something is wrong when moving both ξ and η:Left boundary
ξ = 1, right boundary
Ta da!
TFI examples (1/2)
TFI examples (2/2)
1
0 1
Topology of a hole
● Transformation preserves holes● But with little magic...
PDE methods for grid generation
● Algebraic methods (affine trans., bilinear, TFI)defining a grid geometrically
● PDE methodsdefining requirements for grid mathematically
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
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
Thompson's Elliptic PDE grid
● The system is solved on uniform grid in computational domain which gives coordinate lines in physical domain
Example copied from the book
Example copied from the book
Boundary:
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
This slide is intentionally left blank.
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
Delaunay triangulation
● Simple criteria to connect points to form conforming, non intersecting unstructured grid
Delaunay triangulation algorithm
● Nice incremental algorithm● Introduce new point, locally break triangulation
and then retriangulate affected part
● Flipping algorithm:
Point generation?
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)
Overlapping (Chimera-) grids
● Built using partially overlapping blocks● Boundary conditions are exchanged between
domains using interpolation● Can combine structured and unstructured
sub-grids
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
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
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.
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
Moving grid example without any real number-crunching shown
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
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