Scientific Computing I Michael Bader Grid Generation and Refinement Basic Types of Scientific Computing I Grids Strcutured Grids Unstrcutured Grids Module 7: Grid Generation Grid Operations Structured Grids Unstructured Michael Bader Grids Adaptive Grids Lehrstuhl Informatik V Block Adaptive Grids Recursively Structured Adaptive Grids Winter 2005/2006
Scientific Grid Generation and Refinement Computing I Michael Bader Numerical treatment of PDE Grid Generation requires approximate description of the and Refinement Basic Types of computational domain Grids Strcutured Grids discretization of the domain: Unstrcutured Grids point discretization (finite differences) Grid Operations cell discretization (finite elements/volumes) Structured Grids Unstructured main tasks: Grids generating and refining grids or meshes; Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids x i,j+1 x i−1,j x i,j x i+1,j x i,j−1 h y h y h x h x
Scientific Grid Generation – Objectives Computing I Michael Bader Objectives: Grid Generation and Refinement accuracy : accurate (and dense) enough to Basic Types of Grids catch the essential physical phenomena Strcutured Grids Unstrcutured Grids boundary approximation : sufficiently Grid Operations detailed to represent boundaries and boundary Structured Grids conditions Unstructured Grids computational efficiency : small overhead for Adaptive Grids Block Adaptive Grids handling of data structures, no loss of Recursively Structured Adaptive Grids performance on supercomputers numerical adequacy : features with a negative impact on numerical efficiency should be avoided (angles, distortions)
Scientific Structured Grids Computing I Michael Bader Grid Generation and Refinement construction of points or elements follows Basic Types of regular process Grids Strcutured Grids geometric (coordinates) and topological Unstrcutured Grids Grid Operations information (neighbour relations) can be Structured Grids derived Unstructured Grids Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids
Scientific Unstructured Grids Computing I Michael Bader Grid Generation and Refinement (almost) no restrictions, maximum flexibilty Basic Types of completely irregular generation, even random Grids Strcutured Grids choice is possible Unstrcutured Grids Grid Operations explicit storage of basic geometric and Structured Grids topological information Unstructured Grids Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids
Scientific Grid Manipulations Computing I Michael Bader Grid Generation grid generation : and Refinement Basic Types of initial placement of grid points or elements Grids Strcutured Grids grid adaption : Unstrcutured Grids Grid Operations need for (additional) grid points often Structured Grids becomes clear only during the computations Unstructured Grids requires possibilities of both refinement and Adaptive Grids coarsening Block Adaptive Grids Recursively Structured grid partitioning : Adaptive Grids standard parallelization techniques are based on some decomposition of the underlying domain
Scientific Grid Operations Computing I Michael Bader Grid Generation and Refinement Basic Types of Typical Operations on the Grid: Grids Strcutured Grids numbering of the nodes/cells Unstrcutured Grids Grid Operations for traversal/processing of the grid Structured Grids for storing the grid Unstructured for partitioning the grid Grids Adaptive Grids identify the neighbours of a node/cell Block Adaptive Grids Recursively Structured neighbouring/adjacent nodes/cells/edges Adaptive Grids due to typical discretization techniques
Scientific Regular Structured Grids Computing I Michael Bader Grid Generation rectangular/ cartesian grids: and Refinement Basic Types of rectangles (2D) or cuboids (3D) Grids Strcutured Grids triangular meshes: Unstrcutured Grids triangles (2D) or tetrahedra (3D) Grid Operations Structured Grids row-major or column-major traversal and Unstructured storage Grids Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids h y h x
Scientific Transformed Structured Grids Computing I Michael Bader transformation of the unit square to the Grid Generation computational domain and Refinement Basic Types of regular grid is transformed likewise Grids Strcutured Grids Unstrcutured Grids (ξ(0),η(1)) (0,1) (1,1) Grid Operations (ξ(1),η(1)) Structured Grids Unstructured Grids (0,0) (1,0) (ξ(1),η(0)) Adaptive Grids (ξ(0),η(0)) Block Adaptive Grids Recursively Structured Adaptive Grids Variants: algebraic : interpolation-based example: Coons patch PDE-based : solve system of PDEs to obtain ( ξ ( x , y ) and η ( x , y )
Scientific Composite Structured Grids Computing I Michael Bader subdivide (complicated) domain into Grid Generation and Refinement subdomains of simpler form Basic Types of Grids Strcutured Grids Unstrcutured Grids Grid Operations Structured Grids Unstructured and use regular meshs on each subdomain Grids Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids at interfaces: conforming at interface (“glue” required?) overlapping grids ( chimera grids)
Scientific Block Structured Grids Computing I Michael Bader Grid Generation and Refinement Basic Types of Grids Strcutured Grids Unstrcutured Grids Grid Operations Structured Grids subdivision into logically rectangular Unstructured Grids subdomains Adaptive Grids (with logically rectangular local grids) Block Adaptive Grids Recursively Structured Adaptive Grids subdomains fit together in an unstructured way, but continuity is ensured (coinciding grid points) popular in computational fluid dynamics
Scientific Delauney Triangulation Computing I Michael Bader assume: grid points are already obtained – Grid Generation how to define elements? and Refinement Basic Types of based on Voronoi diagrams Grids Strcutured Grids Voronoi region around each given grid point: Unstrcutured Grids V i = { P : � P − P i � < � P − P j � ∀ j � = i } Grid Operations Structured Grids V Unstructured 3 Grids P P 3 2 Adaptive Grids Block Adaptive Grids Recursively Structured V 1 Adaptive Grids P V P 1 2 4 V 4 P P 6 5 V 5 V 6
Scientific Delauney Triangulation (2) Computing I Michael Bader Grid Generation Algorithm and Refinement Voronoi region around each given grid point: Basic Types of 1 Grids V i = { P : � P − P i � < � P − P j � ∀ j � = i } Strcutured Grids Unstrcutured Grids connect points that are located in adjacent 2 Grid Operations Voronoi regions Structured Grids Unstructured leads to set of disjoint triangles (tetrahedra in 3 Grids 3D) Adaptive Grids Block Adaptive Grids Applications: Recursively Structured Adaptive Grids closely related to FEM, typically triangles/tetrahedra very widespread
Scientific Point Generation Computing I Michael Bader Grid Generation and Refinement how do we get the grid points? Basic Types of Grids start with a regular grid and refine/modify Strcutured Grids Unstrcutured Grids boundary-based: Grid Operations Structured Grids start with some boundary point distribution, Unstructured generate Delaunay triangulation, Grids and subdivide (following suitable rules) Adaptive Grids Block Adaptive Grids if helpful, add point or lines sources Recursively Structured Adaptive Grids (singularities, bound. layers) Advancing Front methods
Scientific Advancing Front Methods Computing I Michael Bader Grid Generation and Refinement Basic Types of advance a front step-by-step towards interior Grids Strcutured Grids starting from the boundary ( starting front ) Unstrcutured Grids Grid Operations Structured Grids Unstructured Grids Adaptive Grids Block Adaptive Grids P P P P 1 Recursively Structured 1 P P 2 2 Adaptive Grids
Scientific Advancing Front Methods (2) Computing I Michael Bader Advancing algorithm: Grid Generation and Refinement choose an edge on the current front, say PQ 1 Basic Types of Grids create a new point R at equal distance d from 2 Strcutured Grids Unstrcutured Grids P and Q Grid Operations determine all grid points lying within a circle 3 Structured Grids around R, radius r Unstructured Grids order these points w.r.t. distance from R 4 Adaptive Grids Block Adaptive Grids for all points, form triangles with P and Q; 5 Recursively Structured Adaptive Grids select one of these triangles add triangle to grid (unless: intersections, . . . ) 6 update triangulation and front line: add new 7 cell, update edges
Scientific Adaptive Grids Computing I Michael Bader Characterization of adaptive grids: Grid Generation and Refinement size of grid cells varies considerably Basic Types of to locally improve accuracy Grids Strcutured Grids sometimes requirement from numerics Unstrcutured Grids Grid Operations Structured Grids Unstructured Grids Adaptive Grids Block Adaptive Grids Recursively Structured Adaptive Grids Challenge for structured grids: efficient storage/traversal retrieve structural information (neighbours, etc.)
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