Meshing For surface representation Meshing complex geometry with - - PDF document

CS 101

Meshing

CS 101

Meshing

Peter Schröder Mathieu Desbrun

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Meshing

An essential preprocessing step

For surface representation

complex geometry with a few basic geometric primitives

For simulations

realistic/accurate animation (PDE) domain discretization (space, time)

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The World of Meshing

Different crowds involved

Theory: Computational Geometry

quality and complexity uber alles

Practice: Engineers

efficiency and robustness

Both are interesting… This class will cover both aspects

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Today’s Show

What’s a Mesh?

Formal, basic definitions

vertex, edge, simplex, manifold, etc…

How should a Mesh look like?

Not every mesh is equal…

basic requirements often required

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What’s a Mesh?

Formally

abstract simplicial complex K

singletons, pairs, triples,… of integers

containment property partial order, face, coface…

abstract simplices

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Simplicial Complex

Topological realization

identify V with unit vectors in RN subset topology of ambient space closure, star, and link

convex hull of vertex images

make subcomplex incidence

“1-ring”

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Topological Structure

2-manifold (with boundary)

every point has an open, (half-)

disklike subset surrounding it

|K| 2-manifold iff |St v| ~ R2

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Topological Invariants

Euler characteristic

for surfaces: F-E+V=χ=2(1-g)

not required to be simplicial

more generally for simplicial

complexes

proof by

induction (shelling)

relation to Hopf index theorem

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Simplicial Complex

Geometric realization

the concrete embedding πv(|K|)

piecewise linear vertex images specify everything

we may allow self intersections

derivative map injective (typically)

immersion CS101 - Meshing

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Other Views

Topological realization

mesh in space given by PL basis

functions with vertices as coefficients

Not mentioned

cell complexes—but rather obvious quad meshes quite useful

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What is Meshing?

General Idea:

breaking up a physical domain

2d domain in 2d, or in 3d, or 3d domain

into simpler subdomains—elements

simplices or not (triangle, quads, …) “flat” 2D “flat” 3D

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Brief Glossary

Node, element: Structured/Unstructured Mesh:

regular valence and degree

Isotropic/Anisotropic Mesh:

without/with stretched elements

Graded Mesh:

w/ elements varying in size

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Brief Glossary

Node, element: Structured/Unstructured Mesh:

regular valence and degree

Isotropic/Anisotropic Mesh:

without/with stretched elements

Graded Mesh:

w/ elements varying in size

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Why (Un)Structured?

Structure brings:

simpler data

structure

better

compression

reuse of DSP algos

smoothing wavelets

Unstructured Unstructured meshing meshing Structured Structured meshing meshing

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Brief Glossary

Node, element: Structured/Unstructured Mesh:

regular valence and degree

Isotropic/Anisotropic Mesh:

without/with stretched elements

Graded Mesh:

w/ elements varying in size

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Why Anisotropy?

# V # V # F # F

Isotropic Isotropic meshing meshing Anisotropic Anisotropic meshing meshing

# V # V # F # F

Anisotropic Anisotropic quad quad meshing meshing

# V # V # F # F CS101 - Meshing

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Brief Glossary

Node, element: Structured/Unstructured Mesh:

regular valence and degree

Isotropic/Anisotropic Mesh:

without/with stretched elements

Graded Mesh:

w/ elements varying in size

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Why Gradation?

Allows for better capture

• f details

for same vertex budget

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Future Lectures

Some themes will review:

Delaunay/Voronoi

inside out! Very cool theory, very

useful in practice too.

Resampling

simplification, mesh improvement, …

Reconstruction

where do meshes come from??