What Makes for a Good Mesh? CS101 Meshing Winter 2007 1 Mesh - - PDF document

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Feb 16, 2023 196 likes •310 views

What Makes for a Good Mesh? CS101 Meshing Winter 2007 1 Mesh Quality What makes a mesh good? application dependent some general principles PL approximation (triangles/tets) faithful to geometry values gradients

SLIDE 1

CS101 – Meshing Winter 2007

What Makes for a Good Mesh?

CS101 – Meshing Winter 2007

Mesh Quality

What makes a mesh good?

application dependent some general principles

PL approximation (triangles/tets) faithful to geometry

values gradients

simulation: numerical error

SLIDE 2

CS101 – Meshing Winter 2007

Judging Quality

Approximation

given: some smooth function

best PL approximation distance: Hausdorff

Leif Kobbelt

CS101 – Meshing Winter 2007

Simplified Setting

Approximating a function

ver a simplicial complex

values at vertices how close is linear interpolant to

underlying function?

element shape and size

SLIDE 3

CS101 – Meshing Winter 2007

Criteria

Setup

f given; g is PL interp. interpolation error gradient interpolation error stiffness matrix

Jonathan Shewchuk

Size very important; shape not so much Size important; angles very important

bad

CS101 – Meshing Winter 2007

Assumptions

Curvature (directional)

assumed to be bounded then over element t

weaker but simpler

SLIDE 4

CS101 – Meshing Winter 2007

Minimum Containment

How to find circle/sphere?

if circumcenter interior to triangle

circumradius

else: half the edge length of edge

with circumcenter on “wrong” side

tet: if circumcenter inside: rcirc else: rmc of triangles (1 or 2) with

circumcenter on wrong side

CS101 – Meshing Winter 2007

Gradients

Values not enough

SLIDE 5

CS101 – Meshing Winter 2007

Gradients

Accuracy important as well

discretization error in FEM mechanics: ∇f represents strains

Jonathan Shewchuk

edge lengths

in circle area circumcircle CS101 – Meshing Winter 2007

Tetrahedra

Error bounds

approximation gradient

Good Bad

Jonathan Shewchuk

SLIDE 6

CS101 – Meshing Winter 2007

Delaunay Optimality

For a given set of vertices

many triangulations for any dimension

Delaunay min. largest rmc

in 2D Delaunay min. largest rcirc

CS101 – Meshing Winter 2007

Stiffness Matrix

Condition number

accuracy of linear algebra

iterative solvers are slower all solvers less accurate

ratio of largest to smallest eigen

value

model problem:

SLIDE 7

CS101 – Meshing Winter 2007

Conditioning

Global stiffness matrix

max eigen value

dominated by single worst element depends on shape

2D indep. of size; 3D largest ele. domin.

min eigen value

relatively independent of shape proportional to areas/volumes of ele.

CS101 – Meshing Winter 2007

Poisson Equation

Specific for each case

max eigen value matters most prefers equilateral triangles small angles bad simpler:

SLIDE 8

CS101 – Meshing Winter 2007

3D Case

Still Poisson equation

need to solve cubic equation… smallest for equilateral tets dihedral angles relevant

small ones ok if opposing edge long

Good Bad

Jonathan Shewchuk

CS101 – Meshing Winter 2007

So Far: Isotropic

Adapt to function

when function is known

r anisotropy in PDE

curvature variation

SLIDE 9

CS101 – Meshing Winter 2007

Anisotropic Meshes

Deform to isotropic space

judge Et careful with gradients

big nasty expressions good element if Et has no large angles

Actually: they are

k, but have to be

carefully aligned CS101 – Meshing Winter 2007

Anisotropic Meshes

Direction matters now

not-/aligned with principal

curvature directions

“equilateral” now with simplices may not be the best

primitives anymore…

SLIDE 10

CS101 – Meshing Winter 2007

Quality Measures

How to optimize a mesh?

use reciprocal differentiability matters

don’t want gradient to vanish for the

near bad elements…

types of quality measures

long catalogues…

CS101 – Meshing Winter 2007

Types of Quality Ms.

Scale invariant

separate size and shape can be misleading…

error depends on size too smaller may be worse shape & OK

Size and shape

all effects in one number more specific to application