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 2008March 29, 2008

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

Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)

• Sophisticated Multiblock and Overlapping Structured Grid Techniques are required for Complex Geometries

Motivation

Motivation• Multigrid solvers

– Multigrid techniques enable optimal O(N) solution complexity

– Based on sequence of coarse and fine meshes– Originally developed for structured grids

• Agglomeration Multigrid solvers for unstructured meshes

Motivation

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

Algorithm 2

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!