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

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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 

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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... 

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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 

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 

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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) 

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

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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