Talk 2.08 Grid generation and adaptive refinementWednesday, 09/03/2008Summer Academy 2008Numerical Methods in Engineering Goran Rakić, studentHerceg Novi, Montenegro Faculty of Mathematics, Belgrade
● 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
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
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
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 methods defining 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
Thompsons 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
Thompsons Elliptic PDE grid● The system is solved on uniform grid in computational domain which gives coordinate lines in physical domain
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:
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