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Fast Adaptive Hybrid Mesh Generation Based on Quad-tree Decomposition
Mohamed Ebeida ([email protected])Mechanical and Aeronautical Eng. Dept –UCDavis
Bay Area Scientific Computing Day 2008March 29, 2008
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MotivationStructured Grids
• Relatively simple geometries• Algebraic – Elliptic –
Hyperbolic methods• Line relaxation
solvers • Structured Multigrid
solvers • Adaptation using quad-tree or
oct-tree decomp (FEM)• Grid quality
Unstructured Grids
• Complex geometries • Delaunay point insertion
algorithms / advancing front• re-triangulation mesh points
can move • Agglomeration Multigrid
solvers• Adaptation using quad-tree or
oct-tree (FEM)• Grid quality
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Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)
• Sophisticated Multiblock and Overlapping Structured Grid Techniques are required for Complex Geometries
Motivation
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Motivation• Multigrid solvers
– Multigrid techniques enable optimal O(N) solution complexity
– Based on sequence of coarse and fine meshes– Originally developed for structured grids
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• Agglomeration Multigrid solvers for unstructured meshes
Motivation
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Quad-tree decomposition• Fast• Adaptive• Grid Quality• Line solvers• Hanging
nodes• Multigrid• Complex
geometries
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Our Goals
• A fast technique • Quality• Complex geometries• Adaptive (geometries – solution variables)• Multigrid• Line relaxation solvers• No hanging nodes• Simple optimization steps (3D)• Parallelizable
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Spatial Decomposition
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Strategy
Algorithm
Algorithm 1
Adaptive grid based on the geometries
Algorithm 2
Adaptive grid based on the Simulation
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Algorithm 1 - Geometries
• Start with a coarse Cartesian grid with aspect ratio = 1.0
• Dim: 30x30
Sp = 2.0
256 points
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Algorithm 1 - Geometries
• Perform successive refinements till you reach a level that resolves the curvature of the geometries of the domain
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Algorithm 1 - Geometries
• Level of refinements depend on the curvature of each shape
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Algorithm 1 - Geometries
• Define a buffer zone and delete any element with a node in that zone
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Algorithm 1 - Geometries
• Project nodes on the edge of the buffer zone orthogonally to the geometry
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Algorithm 1 - Geometries
• Move nodes on the edge of the buffer zone orthogonally to the geometry to adjust B.L. elements
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Algorithm 1 - Geometries
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Another way !• Increase the width of the buffer zone and create
boundary elements explicitly better bounds!
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Algorithm 1 - Geometries
• Final mesh
22416 pts
22064 elem.
Quad dom. 94.86%
Min edge length
7.6 x 10
Max A.R. = 64
-6
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Complex geometries
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Testing Algorithm 1 output
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Algorithm 2 – Simulation based
• Use the output of Algorithm 1 as a base mesh for the spatial decomposition
• Run the simulation for n time steps (unsteady) or n iterations (steady)
• Perform Spatial decomposition on the base mesh based on a level set function.
• Map the variables from the grid used in the last simulation
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How about transition elements?• In order to ensure quality, transition
element has to advance one step per spatial decomposition level
x
x
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Results
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Multigrid Levels
• Spatial decomposition allows us to generate prolongation and restriction operators easily
• How about the elements of each grid level?
We already have them
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Multigrid Levels
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Multigrid Results
• For elliptic equations, the application of Multigrid is straight forward once we have the grid levels.
• For convection diffusion equations, line solvers are crucial for good results
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Checking our Goals
• A fast technique • Quality • Complex geometries• Adaptive with a starting coarse grid• Multigrid• Line relaxation solvers• No hanging nodes• Simple optimization steps (3D)• Parallelizable
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Thank you!