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Efficient and good Delaunay meshes from random Efficient and good Delaunay meshes from points random points M. S. Ebeida et a.l Intro MPS M. S. Ebeida 1 , S. A. Mitchell 1 , Andrew A. Davidson 2 , MPS Anjul Patney 2 , Patrick Knupp 1 and

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Efficient and good Delaunay meshes from random points

  • M. S. Ebeida

et a.l Intro MPS MPS CDT CVM Future Work

Efficient and good Delaunay meshes from random points

  • M. S. Ebeida1, S. A. Mitchell1, Andrew A. Davidson2,

Anjul Patney2, Patrick Knupp1 and John D. Owens2

1Computing Research, Sandia National Laboratories 2Electrical and Computer Engineering, UCDavis

10/24/2011

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Efficient and good Delaunay meshes from random points

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Efficient and good Delaunay meshes from random points

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et a.l Intro MPS MPS CDT CVM Future Work

Collaborators

SNL

  • V. J. Leung.
  • J. E. Bishop
  • M. J. Martinez

UCDavis:

  • A. Patney
  • A. Davidson
  • J. D. Owens
  • M. S. Ebeida et a.l

Efficient and good Delaunay meshes from random points

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Efficient and good Delaunay meshes from random points

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et a.l Intro MPS MPS CDT CVM Future Work

Overview

1 Introduction 2 What is Maximal Poisson-disk sampling? 3 Our solutions for the maximal poisson-disk sampling problem 4 A Conforming Delaunay triangulation method 5 A Conforming Voronoi Meshing method 6 Future Work

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Efficient and good Delaunay meshes from random points

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General Principles in MeshingGenie

... part of Trilinos open source library (http://trilinos.sandia.gov/)

Quality

  • Point Clouds should adequately

represent the associated geometry and physics.

  • Extra points needed to improve

quality / maintain certain connectivity should be kept at a minimum level.

  • Moving points to

non-deterministic locations should also be avoided for generating a provably good mesh.

NACA0012 Image courtesy of http://www.cfd-online.com

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Efficient and good Delaunay meshes from random points

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General Principles in MeshingGenie

... part of Trilinos open source library (http://trilinos.sandia.gov/)

Efficiency, robustness and ... Multi-Grid!

  • Local meshing operations is easier for parallel applications
  • Provable robustness (meshing failure is NOT

ACCEPTABLE in dynamic simulations).

  • Meshing Operations should have minimum storage

requirement.

  • Geometric MG need to be considered.
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et a.l Intro MPS MPS CDT CVM Future Work

What is Maximal Poisson-disk sampling?

Creating a point cloud in given domain satisfying three conditions:

  • Each point is a centre of a disk, with radius r, that

contains no other points: ∀xi ∈ xj ∈ X, xi = xj : ||xi − xj|| ≥ r

  • The point distribution should be bias-free:

∀xi ∈ X, ∀Ω ⊂ D : P (xi ∈ Ω) =

  • Ω

dω

  • Termination is achieved when the domain is completely

saturated: ∀x ∈ D, ∃xi ∈ X : ||x − xi|| < r

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How does Maximal Poisson-disk sampling affect meshing algorithms?

Delaunay Edge length

  • bounded between r

and 2r

  • Connectivity can be

retrieved locally

  • Linear time

complexity

  • Easier parallel

implementation

  • Nice distribution

almost independent

  • f the domain / no.
  • f points
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How does Maximal Poisson-disk sampling affect meshing algorithms?

Moreover

  • Angles between 30◦ and

120◦

  • Nice distribution almost

independent of the domain / no. of points

  • Easier handling of

constrained input.

  • Communication is only

required in case of non-unique solutions.

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Improving the quality via deterministic point insertion:

Chew’s method for example

  • Ignores the information

associated with the input point cloud.

  • Physics is initially ignored

as well.

  • Generates an initial mesh

with low quality.

  • Improves the quality by

inserting points at the centers of large Delaunay circumcircles

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Relationship between the two approaches

Our Approach

  • Points are generated to satisfy a given function F.
  • F is chosen based on (geometry and physics).
  • The extra information enables “nicer”implementation
  • Once the point cloud is generated no more point insertion
  • r movement (termination guarantee).
  • The quality measures would be decided according to F.
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Relationship between the two approaches

Point insertion methods

  • The information associated with the point cloud is

missing/discarded.

  • A sequence of Delaunay meshes is generated trying to find

a maximal distribution for an unknown implicit function, F associated with an input desired quality measure.

  • Solution may not exist, and if it exists, it may not be

desirable!

  • Low quality meshes can cause robustness issues esp. in 3D.
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A simple demonstration using an extremely nice point cloud

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et a.l Intro MPS MPS CDT CVM Future Work

A simple demonstration using an extremely nice point cloud

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Efficient and good Delaunay meshes from random points

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et a.l Intro MPS MPS CDT CVM Future Work

A simple demonstration using an extremely nice point cloud

  • M. S. Ebeida et a.l

Efficient and good Delaunay meshes from random points

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A simple demonstration using an extremely nice point cloud

Convinced? If so, let’s see how we solved the uniform case :)

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Classical dart throwing

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The maximal Poisson-disk sampling

Challenges of classical dart throwing:

  • An efficient method to retrieve conflicts.
  • Filling the small gaps between the disks.
  • Detection of the termination condition.
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The maximal Poisson-disk sampling: Existing solutions

Approaches

  • Quadtrees methods.
  • Advancing front methods.
  • Tiles to improve

parallelism. Common issues:

  • Not strictly “unbiased”.
  • Not maximal: dependent
  • n finite precision.
  • Memory or run-time

complexity.

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Our first solution: SIGGRAPH2011

  • The first provably bias-free, maximal, E(n log n) time

O(n) space.

  • Utilizes a Cartesian background grid
  • Dynamic linear representation of the voids in the domain.
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Our first MPS solution: Results

The generated point cloud have a desired blue noise property.

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Our first MPS solution: Results

1M 2D points in less than 10 seconds using a modern laptop. 8 M 2D points using 2GB of memory

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Our second MPS solution (Eurographics2012 - in preparation)

Almost same speed as Solution I (at least in 2D). 24/8 M 2D/3D points using 2GB of memory Generic code in any d-dimensional space (if you can afford it!).

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Our second MPS solution: Results

Our 3D Our 2D Gamito 3D Gamito 2D Ebeida 2D Our 4D Gamito 4D 4 8 16 32 64 128 256 512 1024 2048 0.01 0.02 0.04 0.08 0.16 0.32 0.64 1.28 2.56 5.12 10.24 20.48 40.96 Time (seconds) Number of output points (millions)

Run‐Time by Algorithm and Dimension

Our 2D Our 3D Our 4D Gamito 2D Gamito 3D Gamito 4D Ebeida 2D

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Our second MPS solution: Results

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Efficient and good Delaunay meshes from random points

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Question? ... So we now have a random point cloud Can we use the properties associated with its distance function to improve the Delaunay/Voronoi meshing algorithms?

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  • 1. An Indirect method using a novel CDT

algorithm (SIAM-GD 2011)

We were able to process 1 Million points in 2.7 seconds using a modern laptop.

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A linear CDT algorithm: Protecting the domain boundaries

Two Approaches

  • Protecting with close

disks: Sample the boundary edges using a smaller distribution radius.

  • Protecting with interior

disks: Sample the boundary edges with the same distribution radius and then introduce some interior points to avoid large boundary angles

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A linear CDT algorithm: Examples

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A linear CDT algorithm: Quality

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A linear CDT algorithm: Quality

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A linear CDT algorithm: Quality

0.1 0.2 0.3 0.4

Fraction

Internal Triangle Angles

1E6 points 1E5 points 1E4 points 1E3 points 1E2 points

0.1 0.2 0.3 0.4

Fraction

Internal Edges

1E6 points 1E5 points 1E4 points 1E3 points 1E2 points

0.3 0.4

Boundary Triangle Angles

1E6 points i

0.1

Angle Range (degrees) 1E3 points 1E2 points

0.3 0.4

Boundary Edges

1E6 points 1E5 points

0.1

Length Ratio to r

0.1 0.2 0.3

Fraction

Boundary Triangle Angles

1E6 points 1E5 points 1E4 points 1E3 points 1E2 points

0.1 0.2 0.3

Fraction

y g

1E5 points 1E4 points 1E3 points 1E2 points Angle Range (degrees) Length Ratio to √3r/2

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A linear CDT algorithm: Memory Consumption

MPS ‐ Phase I MPS ‐ Phase II Time Memory Consumption (MB) 1970 660 CDT 1733

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A linear CDT algorithm: Speed Comparison

100k points / second 10 15 20 25 Time (seconds)

Speed Comparison

mps ‐ serial cdt ‐ serial cdt ‐ GPU Triangle 374k points / second 735k points / second 5 0.00E+0 5.00E+5 1.00E+6 1.50E+6 2.00E+6 2.50E+6 Number of points

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An indirect CVM algorithm: Examples

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Constraint Voronoi Meshing: Example II Joseph E. Bishop and Mario J. Martinez

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A direct method using Edge/Face trimming (IMR2011)

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A direct method using Edge/Face trimming: handling internal interfaces

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A direct method using Edge/Face trimming: handling non-convex cells

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et a.l Intro MPS MPS CDT CVM Future Work

A direct method using Edge/Face trimming

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Efficient and good Delaunay meshes from random points

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et a.l Intro MPS MPS CDT CVM Future Work

A direct method using Edge/Face trimming

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Efficient and good Delaunay meshes from random points

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et a.l Intro MPS MPS CDT CVM Future Work

A direct method using Edge/Face trimming

  • M. S. Ebeida et a.l

Efficient and good Delaunay meshes from random points

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A direct method using Edge/Face trimming

4000 8000 12000 16000 20000 24000 44 57 69 82 95 107 120 133 146 158

Number of Cells Dihedral Angle (degrees)

Interior‐Interior Facet Dihedrals

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A direct method using Edge/Face trimming

200 400 600 800 1000 1200 44 54 64 75 85 95 105 116 126 136

Number of Cells Dihedral Angle (degrees)

Border‐Interior Facet Dihedrals

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A direct method using Edge/Face trimming

200 400 600 800 1000 1.40 1.51 1.61 1.72 1.83 1.93 2.04 2.15 2.25 2.36

Number of Cells Aspect Raio Ro/Ri

Interior Cell Aspect Ra7o

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A direct method using Edge/Face trimming

20 40 60 80 1.58 1.70 1.81 1.92 2.03 2.14 2.25 2.36 2.47 2.59

Number of Cells Aspect Raio Ro/Ri

Fringe Cell Aspect Ra8o

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Here is what we can do next:

  • 1. Sampling (Meshing) of curved surfaces:

Existing Approaches

  • Not strictly “unbiased”.
  • Not maximal: dependent
  • n finite precision.
  • Memory or run-time

complexity.

  • Geodesic approximations.
  • The extra step (Delaunay

/ Voronoi meshing) is missing.

Image courtesy of Li-Yi Wei and Rui Wang (SIGGRAPH 2011)

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Here is what we can do next:

  • 2. Adaptive / Anisotrpoic Sampling (Meshing):

Existing Approaches

  • The problem is not well defined.
  • Ambiguous algorithms.
  • Adaptive: r is not constant.
  • Anisotropic: Ellipses are utilized

instead of circles.

  • Missing analysis of the associated

Delaunay/Voronoi Meshes.

  • Parallel implementations using

MPI.

Image courtesy of Li-Yi Wei and Rui Wang (SIGGRAPH 2011)

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Here is what we can do next:

  • 3. Voronoi Meshing with no short edges (A unique

Delaunay Mesh) :

  • Constrained sampling to avoid

more than d+1 points on the same circumcircle

  • Short edges affect the selection
  • f time steps in unsteady

simulations.

  • Edge collapse will result in

non-planar faces

  • Unique Delaunay meshes would

require no communication

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Here is what we can do next:

  • 4. Unstructured Geometric Multigrid based on MPS:

Proposed Algorithm

  • Construct a sequence of meshes

by scaling F.

  • Each grid will have a guaranteed

quality

  • Discertized systems will maintain

sparsity.

  • Retrieve the

prolongation/restriction

  • perators using the Voronoi

relations between successive levels

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Here is what we can do next:

  • 4. Unstructured Geometric Multigrid based on MPS:

Other Approaches

  • Structured MG: Undesired

distribution.

  • Unstructured MG:

Complicated Agglomeration techniques.

  • Algebraic MG: Coarse

levels loose sparsity.

Image courtesy of Dimitri Mavriplis(Unstructured Grid Technology for CFD, 2008. Organized By: PET Program DoD)

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Questions?!

Thank you!

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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