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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 Unstructured 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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Motivation Sophisticated Multiblock and Overlapping Structured Grid Techniques are required for Complex Geometries Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)

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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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Motivation Agglomeration Multigrid solvers for unstructured meshes

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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**

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 % 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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