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Fast Adaptive Hybrid Mesh Generation Based on Quad-tree Decomposition Mohamed Ebeida Mechanical and Aeronautical Eng. Dept –UCDavis Bay Area Scientific Computing Day 2008 March 29, 2008

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Motivation Structured 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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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 = 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 pts elem. Quad dom % 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!

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