Efficient and good Delaunay meshes from points random points M. S. - PowerPoint PPT Presentation

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 John D. Owens 2 CDT CVM 1 Computing Research, Sandia National Laboratories Future Work 2 Electrical and Computer Engineering, UCDavis 10/24/2011 M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Collaborators Efficient and good Delaunay meshes from random points SNL M. S. Ebeida • V. J. Leung. et a.l • J. E. Bishop Intro • M. J. Martinez MPS MPS CDT UCDavis: CVM • A. Patney Future Work • A. Davidson • J. D. Owens M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Overview Efficient and good Delaunay meshes from 1 Introduction random points M. S. Ebeida 2 What is Maximal Poisson-disk sampling? et a.l Intro 3 Our solutions for the maximal poisson-disk sampling problem MPS MPS 4 A Conforming Delaunay triangulation method CDT CVM 5 A Conforming Voronoi Meshing method Future Work 6 Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

General Principles in MeshingGenie ... part of Trilinos open source library (http://trilinos.sandia.gov/) Efficient and good Quality Delaunay meshes from random • Point Clouds should adequately points represent the associated M. S. Ebeida et a.l geometry and physics. • Extra points needed to improve Intro quality / maintain certain MPS MPS connectivity should be kept at a CDT minimum level. CVM • Moving points to Future Work non-deterministic locations should also be avoided for generating a provably good NACA0012 Image courtesy of mesh. http://www.cfd-online.com M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

General Principles in MeshingGenie ... part of Trilinos open source library (http://trilinos.sandia.gov/) Efficient and Efficiency, robustness and ... Multi-Grid! good Delaunay meshes from • Local meshing operations is easier for parallel applications random points • Provable robustness (meshing failure is NOT M. S. Ebeida ACCEPTABLE in dynamic simulations). et a.l • Meshing Operations should have minimum storage Intro requirement. MPS • Geometric MG need to be considered. MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

What is Maximal Poisson-disk sampling? Efficient and Creating a point cloud in given domain satisfying three good Delaunay conditions: meshes from random • Each point is a centre of a disk, with radius r , that points contains no other points: M. S. Ebeida et a.l ∀ x i ∈ x j ∈ X, x i � = x j : || x i − x j || ≥ r Intro MPS • The point distribution should be bias-free: MPS CDT � ∀ x i ∈ X, ∀ Ω ⊂ D : P ( x i ∈ Ω) = d ω CVM Ω Future Work • Termination is achieved when the domain is completely saturated: ∀ x ∈ D , ∃ x i ∈ X : || x − x i || < r M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

How does Maximal Poisson-disk sampling affect meshing algorithms? Efficient and good Delaunay Edge length Delaunay meshes from • bounded between r random points and 2 r M. S. Ebeida et a.l • Connectivity can be retrieved locally Intro MPS • Linear time MPS complexity CDT • Easier parallel CVM implementation Future Work • Nice distribution almost independent of the domain / no. of points M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

How does Maximal Poisson-disk sampling affect meshing algorithms? Efficient and good Delaunay meshes from Moreover random • Angles between 30 ◦ and points M. S. Ebeida 120 ◦ et a.l • Nice distribution almost Intro independent of the domain MPS / no. of points MPS CDT • Easier handling of CVM constrained input. Future Work • Communication is only required in case of non-unique solutions. M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Improving the quality via deterministic point insertion: Efficient and good Delaunay Chew’s method for example meshes from random • Ignores the information points associated with the input M. S. Ebeida et a.l point cloud. Intro • Physics is initially ignored MPS as well. MPS • Generates an initial mesh CDT with low quality. CVM Future Work • Improves the quality by inserting points at the centers of large Delaunay circumcircles M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Relationship between the two approaches Efficient and good Our Approach Delaunay meshes from random • Points are generated to satisfy a given function F . points • F is chosen based on (geometry and physics). M. S. Ebeida et a.l • The extra information enables “nicer”implementation Intro • Once the point cloud is generated no more point insertion MPS or movement (termination guarantee). MPS • The quality measures would be decided according to F . CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Relationship between the two approaches Efficient and good Point insertion methods Delaunay meshes from • The information associated with the point cloud is random points missing/discarded. M. S. Ebeida • A sequence of Delaunay meshes is generated trying to find et a.l a maximal distribution for an unknown implicit function, Intro F associated with an input desired quality measure. MPS • Solution may not exist, and if it exists, it may not be MPS desirable! CDT CVM • Low quality meshes can cause robustness issues esp. in 3D. Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

A simple demonstration using an extremely nice point cloud Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

A simple demonstration using an extremely nice point cloud Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

A simple demonstration using an extremely nice point cloud Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

A simple demonstration using an extremely nice point cloud Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Convinced? If so, let’s see how we solved Intro the uniform case :) MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Classical dart throwing Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

The maximal Poisson-disk sampling Efficient and Challenges of classical dart throwing: good Delaunay meshes from • An efficient method to retrieve conflicts. random points • Filling the small gaps between the disks. M. S. Ebeida et a.l • Detection of the termination condition. Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

The maximal Poisson-disk sampling: Existing solutions Efficient and good Approaches Delaunay meshes from • Quadtrees methods. random points • Advancing front methods. M. S. Ebeida et a.l • Tiles to improve Intro parallelism. MPS MPS Common issues: CDT • Not strictly “unbiased”. CVM • Not maximal: dependent Future Work on finite precision. • Memory or run-time complexity. M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Our first solution: SIGGRAPH2011 Efficient and • The first provably bias-free, maximal, E ( n log n ) time good Delaunay O ( n ) space. meshes from random points • Utilizes a Cartesian background grid M. S. Ebeida • Dynamic linear representation of the voids in the domain. et a.l Intro MPS MPS CDT CVM Future Work M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Our first MPS solution: Results Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work The generated point cloud have a desired blue noise property. M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Our first MPS solution: Results Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work 1M 2D points in less than 10 seconds using a modern laptop. 8 M 2D points using 2GB of memory M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points

Our second MPS solution (Eurographics2012 - in preparation) Efficient and good Delaunay meshes from random points M. S. Ebeida et a.l Intro MPS MPS CDT CVM Future Work 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!). M. S. Ebeida et a.l Efficient and good Delaunay meshes from random points