Mesh Generation

Mesh Generation

Pierre Alliez

INRIA Sophia Antipolis - Mediterranee

Definitions

 Mesh: Cellular complex partitioning an input domain into elementary cells.

 Elementary cell: Admits a bounded description.

 Cellular complex: Two cells are disjoint or share a lower dimensional face.

Variety

 Domains: 2D, 3D, nD, surfaces, manifolds, periodic vs non-periodic, etc.

Variety

 Elements: simplices (triangles, tetrahedra), quadrangles, polygons, hexahedra, arbitrary cells.

…

Variety

 Structured mesh: All interior nodes have an equal number of adjacent elements.

 Unstructured mesh: Any number of elements can meet at a single vertex.

Variety

 Anisotropic vs Isotropic

Variety

 Element distribution: Uniform vs adapted

Variety

 Grading: Smooth vs fast

Required Properties

 control/optimization over:

 #elements

 shape

 orientation

 size

 grading

 … (application dependent)

Mesh Generation

 Input:

 Domain boundary + internal constraints

 Constraints

 Sizing

 Grading

 Shape

 Topology

 …

2D Delaunay Refinement

2D Triangle Mesh Generation

Input:

 PSLG C (planar straight line graph)

 Domain  bounded by edges of C

Output:

 triangle mesh T of  such that

 vertices of C are vertices of T

 edges of C are union of edges in T

 triangles of T inside  have controlled size and quality

Key Idea

 Break bad elements by inserting circumcenters (Voronoi vertices) [Chew, Ruppert,

Shewchuk,...]

“bad” in terms of size or shape

Basic Notions

C: PSLG describing the constraints T: Triangulation to be refined

Respect of the PSLG

 Edges a C are split until constrained subedges are edges of T

 Constrained subedges are required to be Gabriel edges

 An edge of a triangulation is a Gabriel edge if its smallest circumcirle encloses no vertex of T

 An edge e is encroached by point p if the smallest circumcirle of e encloses p.

Refinement Algorithm

C: PSLG bounding the domain to be meshed.

T: Delaunay triangulation of the current set of vertices T_|: T  

Constrained subedges: subedges of edges of C

Initialise with T = Delaunay triangulation of vertices of C Refine until no rule apply

 Rule 1

if there is an encroached constrained subedge e insert c = midpoint(e) in T (refine-edge)

 Rule 2

if there is a bad facet f in T_|

c = circumcenter(f)

if c encroaches a constrained subedge e refine-edge(e).

else

insert(c) in T

2D Delaunay Refinement

PSLG

Background

Constrained Delaunay Triangulation

Pseudo-dual: Bounded Voronoi Diagram

constrained Bounded Voronoi diagram

“blind” triangles

Delaunay Edge

An edge is said to be a Delaunay edge, if it is inscribed in an empty circle

Gabriel Edge

An edge is said to be a Gabriel edge, if its diametral circle is empty

Conforming Delaunay Triangulation

A constrained Delaunay triangulation is a conforming Delaunay triangulation, if every constrained edge is a Delaunay edge

non conforming conforming

Conforming Gabriel Triangulation

A constrained Delaunay triangulation is a conforming Gabriel triangulation, if every constrained edge is a Gabriel edge

non conforming conforming Gabriel

Steiner Vertices

Any constrained Delaunay triangulation can be refined into a conforming Delaunay or Gabriel triangulation by adding Steiner vertices.

non conforming conforming Gabriel

Delaunay Refinement

Rule #1: break bad elements by inserting circumcenters (Voronoi vertices)‏

 “bad” in terms of size or shape (too big or skinny)‏

Picture taken from [Shewchuk]

Delaunay Refinement

Rule #2: Midpoint vertex insertion

A constrained segment is said to be encroached, if there is a vertex inside its diametral circle

Picture taken from [Shewchuk]

Delaunay Refinement

Encroached subsegments have priority over skinny triangles

Picture taken from [Shewchuk]

Surface Mesh Generation

Mesh Generation

Key concepts:

 Voronoi/Delaunay filtering

 Delaunay refinement

Voronoi Filtering

 The Voronoi diagram restricted to a curve S, Vor_|S(E), is the set of edges of Vor(E) that

intersect S.

S

Delaunay Filtering

 The restricted Delaunay triangulation restricted to a curve S is the set of edges of the Delaunay triangulation whose dual edges intersect S.

(2D)

Delaunay Filtering

Dual Voronoi edge

Delaunay triangulation

restricted to surface S

facet

Voronoi edge  surface S

Delaunay Refinement

Steiner point

Bad facet = big or

badly shaped or

large approximation error

Surface Mesh Generation Algorithm

repeat {

pick bad facet f

insert furthest (dual(f)  S) in Delaunay triangulation update Delaunay triangulation restricted to S

}

until all facets are good

Surface Meshing at Work

Isosurface from 3D Grey Level Image

input

Output Mesh Properties

Termination Parsimony

Output mesh properties:

 Well shaped triangles

 Lower bound on triangle angles

 Homeomorphic to input surface

 Manifold

 not only combinatorially, i.e., no self-intersection

 Faithful Approximation of input surface

 Hausdorff distance

 Normals

Delaunay Refinement vs Marching Cubes

Delaunay refinement Marching cubes in octree

Guarantees

Volume Mesh Generation

Volume Meshing

 Couple the latter algorithm with 3D Delaunay refinement

(insert circumcenters of “bad” tetrahedra)

 Remove slivers at post-processing with sliver exudation

More Delaunay Filtering

Delaunay triangulation

restricted to domain

Dual Voronoi vertex inside domain

(“oracle”)‏

Delaunay Filtering

domain boundary

restricted Delaunay triangulation

3D Restricted Delaunay Triangulation

Delaunay Refinement

Steiner point

Volume Mesh Generation Algorithm

repeat {

pick bad simplex

if(Steiner point encroaches a facet)‏

refine facet else

refine simplex

update Delaunay triangulation restricted to domain

}

until all simplices are good Exude slivers

Delaunay Refinement

Example

Example

Multi-Domain Volume Mesh

Tetrahedron Zoo

Sliver

4 well-spaced vertices near the equator of their circumsphere

Slivers

Sliver Exudation

 Delaunay triangulation turned into a regular triangulation with null weights.

 Small increase of weights triggers edge-facets flips to remove slivers.

Sliver Exudation Process

 Try improving all tetrahedra with an aspect ratio lower than a given bound

 Never flips a boundary facet

distribution of aspect ratios

Example Sliver Exudation

Piecewise Smooth Surfaces

Input: Piecewise smooth complex

More Delaunay Filtering

primitive dual of test against

Voronoi vertex tetrahedron inside domain

Voronoi edge facet intersect domain boundary Voronoi face edge intersect crease

Delaunay Refinement

 Steiner points

Enrich Set of Rules…

Example

init 5 20

50 end slivers

Summary

 Meshes

 Definition, variety

 Background

 Voronoi

 Delaunay

 constrained Delaunay

 restricted Delaunay

 Generation

 2D, 3D, Delaunay refinement

Mesh Optimization

Pierre Alliez

INRIA Sophia Antipolis - Mediterranee

Goals

 3D simplicial mesh generation

 optimize shape of elements

 for matrix conditioning

 isotropic

 control over sizing

 dictated by simulation

 constrained by boundary

 low number of elements desired

 more elements = slower solution time

Popular Meshing Approaches

 advancing front

 specific subdivision

 octree

 lattice (e.g. body centered cubic)

 Delaunay

 refinement

 sphere packing

 spring energy

 Laplacian

 non-zero rest length

 aspect / radius ratios

 dihedral / solid angles

 max-min/min-max

 volumes

 edge lengths

 containing sphere radii [Freitag Amenta Bern Eppstein]

 sliver exudation

[Edelsbrunner Goy]

combined with local optimizations

Variational?

 Design one energy function such that good

solutions correspond to low energy ones (global minimum in general a mirage).

 Solutions found by optimization techniques.

Example Energy in 2D

 





k

j x R

j

j

dx x

x E

..

1

||

||

Lloyd Iteration

demo

2D Optimized Triangle Meshing

2D Optimized Triangle Meshing

Delaunay refinement

Termination

 shape criterion: radius-edge ratio

 in 2D: max 2 (implies min 20.7º)

 in 3D: max 2 (nothing similar on dihedral angles)

[Chew, Ruppert, Shewchuk, ...]

Delaunay refinement

+ greedy (fast)

+ easy incorporation of sizing field + allows boundary conforming

 possibly with Steiner points

 even for sharp angles on boundary [Teng]

+ guaranteed bounds on radius-edge ratio - blind to slivers

- and experimentally...produces slivers

Background

Delaunay Triangulation

 Duality on the paraboloid: Delaunay

triangulation obtained by projecting the lower part of the convex hull.

Delaunay Triangulation

Project the 2D point set

‏

onto the 3D paraboloid

‏

z=x²+y²

‏

Compute the 3D

‏

lower convex hull

‏

z=x²+y²

‏

Project the 3D facets

‏

back to the plane.

‏

z=x²+y²

‏

Proof

 The intersection of a plane with the paraboloid is an ellipse

whose projection to the plane is a circle.

 s lies within the circumcircle of p, q, r iff s’ lies on the lower side of the plane passing through p’, q’, r’.

 p, q, r  S form a Delaunay

triangle iff p’, q’, r’ form a face of the convex hull of S’.

q p

r r’

q’ p’

s s’

Voronoi Diagram

 Given a set S of points in the plane, associate with each point p=(a,b)S the plane

tangent to the paraboloid at p:

z = 2ax+2by-(a2+b2).

 VD(S) is the projection to the (x,y) plane of the 1-skeleton of the convex polyhedron formed from the intersection of the halfspaces above these planes.

q p q’ p’

First Idea: Lloyd Algorithm

(after Lloyd relaxation)

...back to primal ?

Centroidal Voronoi Tessellation

distribution of radius ratios

1 0

Occurrences

220 “slivers”‏(tets‏with‏radius‏ratio‏<‏0.2)

5K sites 30K tets

Tetrahedra Zoo

well-spaced points generate only round or sliver tetrahedra

Key Idea

 adopt the "function approximation" point of view [Chen 04] Optimal Delaunay Triangulation

 1D: f(x)=x² centered at any vertex

 minimize the L¹ norm between f and PWL interpolation

Key Idea

 3D: x² (graph in IR⁴)

 approximation theory:

 linear interpolation: optimal shape of the element related to the Hessian of f [Shewchuk]

 Hessian(x²) = Id

 regular tetrahedron best

 note: FE ~ mesh that best interpolates a function + matrix conditioning

Key Question

 which mesh best approximates the paraboloid?

 (PWL interpolates)

 Answers:

 for fixed point locations

 Delaunay (lifts to lower facets of convex hull)

 for fixed connectivity

 quadratic energy

 closed form for local optimum

Function Approximation

 Given:

 triangulation 

 bounded domain  in IRⁿ

 Consider function approximation error:



 ||

||

) ,

,

(

T L

f

f p

f T Q

linear interpolation

Function Approximation

Theorem [Chen 04]:

[d’Azevedo-Simpson 89] in IR², p =  [Rippa 92] in IR², 1  p  

[Melissaratos 93] in IR^D, 1  p  









Q T x p p p

x DT

Q

PV

T

( , || || , ), 1 min

) ,

||

||

,

(

set of all triangulations with a given set V = convex hull of V

Isotropic function

Function Approximation

Let us V vary Problem:

 find triangulation T* such that:

Proof:

 existence

 necessary condition for p=1









Q T f p p p

f T

Q

PN

T

inf ( , , ), 1 )

,

*, (

set of all triangulations with a at most N vertices

Function Approximation









Q T x p p p

x DT

Q

PV

T

( , || || , ), 1 min

) ,

||

||

,

(

set of all triangulations with a given set V = convex hull of V

Isotropic function

(2D)

Function Approximation

 x_i: vertex





 |A|: Lebesgue measure of set A in IRⁿ

i

x_i

Function Approximation



 ||

||

) ,

,

(

T L

I

f

f p

f T Q

p p

T

I

x f x dx

f p

f T Q

/ 1

,

( ) ( ) |

| )

, ,

( 



 



 

 









 f x f x dx f

T

Q ( , , 1 ) (

( ) ( ))

 

 



 f x dx f x dx f

T

Q ( , , 1 )

( ) ( )

f

f

Function Approximation

 

 



 f x dx f x dx f

T

Q( , ,1) _I,T ( ) ( )

 

 



 f x dx f x dx

T

T

I_, ( ) ( )

t t

  

 





 



 





  f x k f x dx

n _T

n

k

) ( )

, (

| 1 |

1 ¹

t t 1 t

 

 



 

 f x f x dx

n _{xi T} ( ⁱ ) | ⁱ | ( ) 1

1

n+1 overlaps

Function Approximation

 

 





 





i i i

dx x

f x

n f f

Q

x

i i

i ( ) | | ( )

1 ) 1

1 , , (

restrict to patch _i incident to vertex x_i

i x_i

  

 

  

 

 













 

i i

j k j k i

dx x

f x

n f x

f

n x ⁱ

i x

x x

k

j ( ) ( )

1

| ) |

(

| ) ( 1 |

1

t t ,

t

constant

 



  















 

i

j k j k i

i i

x x x

k j

ODT x f x f x

E

t t

t ( ) | ( ) | | ( )

|

,

minimize

x_k

Function Approximation

 



  







i

j k j k i

i i

x x x

k j

ODT x f x f x

E

t t

t ( )| ( )) | | ( ) (|

,

 



   











 





i

j xk j xk xi

k i

j i

i x f x

x f

t t

t

,

* | | ( ) ( )

|

| ) 1

(

||2

||

)

(x x

f 

if

minimum if E_ODT  0

 



   











 



i

j xk j xk xi

k i

j i

i x x

x

t t

t

,

* 2

||

||

) (

|

| |

| 2

1

Geometric Interpretation



 

i j

j j

i

i

c

x

t

t |

| |

|

*

1

circumcenter

 Note: optimal location depends only on the 1- ring neighbors, not on the current location. If all incident vertices lie on a common sphere, optimal location is at sphere center.

demo

Optimization

 alternate updates of

 connectivity

 vertex location

 both steps minimize the same energy

 as for Lloyd iteration

 for convex fixed boundary

 energy monotonically decreases

 convergence to a (local) minimum

Underlaid vs Overlaid Approximant

 CVT

 partition

 approximant

 compact Voronoi cells

 isotropic sampling

 ODT

 overlapping decomposition

 PWL interpolant

 compact simplices

 isotropic meshing

Optimization

Alternate updates of

 connectivity (Delaunay triangulation)

 vertex locations ^demo

Optimal Delaunay Triangulation

distribution of radius ratios 3 “slivers”,‏each‏with‏two‏vertices‏on‏boundary

Sizing Field

Goal reminder:

 shape of elements

 boundary approximation

 minimize #elements

 note: not independent (well-shape elements force K- Lipschitz sizing field [Ruppert, Miller et al.])

Proposal:

 size <= lfs (local feature size) on boundary

 sizing field = maximal K-Lipschitz

 ^{|| || ( )} 

inf )

( x K x y lfs y

y

 





Sizing Field

 ^{|| || ( )} 

inf )

( x K x y lfs y

y

 





K=1

Sizing Field: Examples

1 0.5

K=0.1

3 10 100

Algorithm

 read input boundary 

 setup data structure & preprocessing

 compute sizing field

 generate initial sites inside 

 do

- Delaunay triangulation of {x_i}

- move sites to optimal locations {x_i^*}

until convergence or stopping criterion

 extract interior mesh

Input Boundary 

 surface triangle mesh

 Requirements:

 intersection free

 closed

 restricted Delaunay triangulation of the input

vertices [Oudot-Boissonnat, Cohen-Steiner et al.]

Input Boundary 

[Oudot-Boissonnat]

Input Boundary 

Setup & Preprocessing

 Insertion of input mesh vertices to a 3D Delaunay triangulation

 control mesh

 used to answer inside/outside queries

?

?

Sizing Field: Poles

[Amenta-Bern 98]

control mesh

poles

Sizing Field: lfs

Approximation: distance to set of poles

CGAL orthogonal search in a kD-tree [Tangelder-Fabri]

Sizing Field: Examples

Approximation: fast marching from boundary Representation: regular grid or balanced octree

Optimization

 classification boundary/interior vertices by localization of boundary in Voronoi cells

 CVT on boundary

 ODT inside

Optimization: init

0

Distribution of radius ratios

Optimization: step 1

1

Optimization: step 2

2

Optimization: step 50

50

Optimization: step 50

Optimization: step 50

Interior Mesh Extraction

 Delaunay triangulation tessellates the convex hull

Hand: lfs

Hand: Sizing

Hand: Sizing

Hand: Sizing

Hand: Radius Ratios

Stanford Bunny

Stanford Bunny

Stanford Bunny

Stanford Bunny (uniform)

Torso

Torso

Torso

Torso

Torso

Joint

Conclusion

 Meshes

 Definition, variety

 Background

 Voronoi

 Delaunay

 constrained Delaunay

 restricted Delaunay

 Generation

 2D, 3D, Delaunay refinement

 Optimization

 2D, 3D

 Lloyd iteration, function approximation approach