Scientific Computing I Module 7: Grid Generation Michael Bader - - PowerPoint PPT Presentation

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Scientific Computing I

Module 7: Grid Generation

Michael Bader

Winter 2009/2010

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Grid Generation and Refinement

Numerical treatment of PDE

• requires approximate description of the computational domain
• discretization of the domain:
• point discretization (finite differences)
• cell discretization (finite elements/volumes)
• main tasks:

generating and refining grids or meshes;

xi,j xi−1,j xi+1,j xi,j+1 xi,j−1 hx hy hx hy

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Grid Generation – Objectives

Objectives:

• accuracy: accurate (and dense) enough to catch the essential

physical phenomena

• boundary approximation: sufficiently detailed to represent

boundaries and boundary conditions

• computational efficiency: small overhead for handling of data

structures, no loss of performance on supercomputers

• numerical adequacy: features with a negative impact on

numerical efficiency should be avoided (angles, distortions)

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Structured Grids

• construction of points or elements follows regular process
• geometric (coordinates) and topological information (neighbour

relations) can be derived (i.e. are not stored)

• memory addresses can be easily computed

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Unstructured Grids

• (almost) no restrictions, maximum flexibilty
• completely irregular generation, even random choice is possible
• explicit storage of basic geometric and topological information
• usually complicated data structures

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Grid Manipulations

grid generation:

• initial placement of grid points or elements

grid adaption:

• need for (additional) grid points often becomes clear only during

the computations

• requires possibilities of both refinement and coarsening

grid partitioning:

• standard parallelization techniques are based on some

decomposition of the underlying domain

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Grid Operations

Typical Operations on the Grid:

• numbering of the nodes/cells
• for traversal/processing of the grid
• for storing the grid
• for partitioning the grid
• identify the neighbours of a node/cell
• neighbouring/adjacent nodes/cells/edges
• due to typical discretization techniques

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Regular Structured Grids

• rectangular/cartesian grids:

rectangles (2D) or cuboids (3D)

• triangular meshes:

triangles (2D) or tetrahedra (3D)

• row-major or column-major traversal and storage

hx hy

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Transformed Structured Grids

• transformation of the unit square to the computational domain
• regular grid is transformed likewise

(0,0) (1,0) (0,1) (1,1) (ξ(0),η(0)) (ξ(1),η(1)) (ξ(0),η(1)) (ξ(1),η(0))

Variants:

• algebraic: interpolation-based
• PDE-based: solve system of PDEs to obtain ξ(x, y) and η(x, y)

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Composite Structured Grids

• subdivide (complicated) domain into subdomains of simpler form
• and use regular meshs on each subdomain
• at interfaces:
• conforming at interface (“glue” required?)
• overlapping grids (chimera grids)

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Block Structured Grids

• subdivision into logically rectangular subdomains

(with logically rectangular local grids)

• subdomains fit together in an unstructured way, but continuity is

ensured (coinciding grid points)

• popular in computational fluid dynamics

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Delaunay Triangulation

• assume: grid points are already obtained – how to define

elements?

• based on Voronoi diagrams
• Voronoi region around each given grid point:

Vi = {P : P − Pi < P − Pj ∀j = i}

P P P P P P

1 2 3 4 5 6

V V V V V V

1 2 3 4 5 6

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Delaunay Triangulation (2)

Algorithm

• 1. Voronoi region around each given grid point:

Vi = {P : P − Pi < P − Pj ∀j = i}

• 2. connect points that are located in adjacent Voronoi regions
• 3. leads to set of disjoint triangles (tetrahedra in 3D)

Applications:

• closely related to FEM, typically triangles/tetrahedra
• very widespread

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Point Generation

how do we get the grid points?

• start with a regular grid and refine/modify
• boundary-based:
• start with some boundary point distribution,
• generate Delaunay triangulation,
• and subdivide (following suitable rules)
• if helpful, add point or lines sources (singularities, bound. layers)
• Advancing Front methods

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Advancing Front Methods

• advance a front step-by-step towards interior
• starting from the boundary (starting front)

P P

1 2

P P P

1 2

P

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Advancing Front Methods (2)

Advancing algorithm:

• 1. choose an edge on the current front, say PQ
• 2. create a new point R at equal distance d from P and Q
• 3. determine all grid points lying within a circle around R, radius r
• 4. order these points w.r.t. distance from R
• 5. for all points, form triangles with P and Q;

select one of these triangles

• 6. add triangle to grid (unless: intersections, . . . )
• 7. update triangulation and front line: add new cell, update edges

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Adaptive Grids

Characterization of adaptive grids:

• size of grid cells varies considerably
• to locally improve accuracy
• sometimes requirement from numerics

Challenge for structured grids:

• efficient storage/traversal
• retrieve structural information (neighbours, etc.)

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Block Adaptive Grids

• retain regular structure
• refinement of entire blocks
• similar to block structured grids
• efficient storage and processing
• but limited w.r.t. adaptivity

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Recursively Structured Adaptive Grids

• based on recursive subdivision of parent cell(s)
• leads to tree structures
• quadtree/octree or substructuring of triangles:
• efficient storage; flexible adaptivity
• but complicated processing (recursive algorithms)

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