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