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

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

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

Motivation Multigrid solvers Multigrid techniques enable optimal O(N) solution complexity Based on sequence of coarse and fine meshes Originally developed for structured grids

Motivation Agglomeration Multigrid solvers for unstructured meshes

Quad-tree decomposition Fast Adaptive Grid Quality Line solvers Hanging nodes Multigrid Complex geometries

Our Goals A fast technique Quality Complex geometries Adaptive (geometries – solution variables) Multigrid Line relaxation solvers No hanging nodes Simple optimization steps (3D) Parallelizable

Spatial Decomposition

Strategy Algorithm Algorithm 1 Adaptive grid based on the geometries Adaptive grid based on the Simulation

Algorithm 1 - Geometries Start with a coarse Cartesian grid with aspect ratio = 1.0 Dim: 30x30 Sp = 2.0 256 points

Algorithm 1 - Geometries Perform successive refinements till you reach a level that resolves the curvature of the geometries of the domain

Algorithm 1 - Geometries Level of refinements depend on the curvature of each shape

Algorithm 1 - Geometries Define a buffer zone and delete any element with a node in that zone

Algorithm 1 - Geometries Project nodes on the edge of the buffer zone orthogonally to the geometry

Algorithm 1 - Geometries Move nodes on the edge of the buffer zone orthogonally to the geometry to adjust B.L. elements

Algorithm 1 - Geometries

Another way ! Increase the width of the buffer zone and create boundary elements explicitly  better bounds!

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

Complex geometries

Testing Algorithm 1 output

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

How about transition elements? In order to ensure quality, transition element has to advance one step per spatial decomposition level x x

Results

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

Multigrid Levels

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

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

Thank you!