Discrete Exterior Calculus and its Connection to FEEC Anil N. - - PowerPoint PPT Presentation

Discrete Exterior Calculus and its Connection to FEEC

Anil N. Hirani University of Illinois at Urbana-Champaign

NSF/CBMS Conference: Finite Element Exterior Calculus, June 13, 2012 Institute for Computational and Experimental Research in Mathematics, Providence, RI

• Computer graphics motivation
• Combinatorial and Geometric / Analytical
• Primal and dual meshes (Delaunay OK)
• Wedge product and A∞ algebras
• Contraction and Lie derivative
• Vector fields via diffeomorphism group
• PyDEC: software and algorithms

• 1. Run-on-one-mesh /

Looks-good philosophy

[Grinspun, H, Desbrun, Schröder 2003]

Who cares ...

• 2. Structural philospophy
• Similar to variational integrators philosophy
• Does it mimic the mathematical structure ?
• Does it lead to :
• New mathematical structures ?
• New algorithms ?

Primal-Dual Meshes and Discrete Exterior Calculus

Duality via subdivision

[Munkres 1984]

Circumcentric duals

σ0, 0-simplex σ1, 1-simplex σ2, 2-simplex σ3, 3-simplex D(σ0) 3-cell D(σ1), 2-cell D(σ2), 1-cell D(σ3), 0-cell

78 9 6 32 98 66 95 51 1 10 33 8 34 3 21 57 11 5 83 45 3 45 −34

d

*

e T

1 Area(T)hα, Ti = 1 Length(e)h⇤α, ei 1 |σp|hα, σpi = 1 | ? σp|h⇤α, ?σpi .

Primal-Dual Complexes

C 0

d0

− − − − → C 1

d1

− − − − → C 2

d2

− − − − → C 3 ? ? y∗0 ? ? y∗1 ? ? y∗2 ? ? y∗3 D3

dT

← − − − − D2

dT

1

← − − − − D1

dT

2

← − − − − D0 C 0

d0

− − − − → C 1

d1

− − − − → C 2 ? ? y∗0 ? ? y∗1 ? ? y∗2 D2

dT

← − − − − D1

dT

1

← − − − − D0

• Either use codifferential (inverse star easy)
• Place variables in primal or dual complex
• Or use FEEC weak form with DEC star

〈σh,τ〉−〈dτ,uh〉 = 0 〈dσh,v〉+〈duh,dv〉+〈v,ph〉 = 〈f ,v〉 〈uh,q〉 = 0

∑ ∗ −dT ∗ ∗d dT ∗d ∏∑σh uh ∏ = ∑ ∗(fh − ph) ∏

QT ∗uh = 0

FEEC DEC

FEEC DEC

FEEC DEC

FEEC DEC

Triangulations

• Well-centered
• Pairwise Delaunay + boundary restriction

2-Well-Centered Tetrahedron (Max θ ≈ 87.6°)

34562 triangles

Theorem

Let L be a triangulation of S2 with m vertices. If L has a vertex v1 of degree d(v1) ≥ m − 3, then L is not Lk u for any interior vertex u in a 2-well-centered tetrahedral mesh in R3.

Corollary

There are at least 9 edges incident to each interior vertex of a 2-well-centered tetrahedral mesh in R3.

Typical results

30 40 50 60 70 80 90

35.89 84.65

10 20 30 40 50 60 70 80 90

18.35 80.82

0.2 0.4 0.6 0.8 1

0.0384 0.9322

Dihedral Angles Face Angles h/R Values

First known dihedral acute triangulation on a cube (has 1370 tetrahedra)

Dihedral acute

Another one

• Kopczynski, Pak, Przytycki, 2010, 2012
• 2715 tetrahedra
• Uses the boundary of 4D 600-cell
• No acute triangulation of n-cube, n >= 4
• No acute triangulations of n-space, n >= 5

Alternatives to WCT

• Producing WCT in 3 dimensions is hard
• WCT is sufficient, but is it necessary ?
• Recent: Pairwise Delaunay+ is enough

Sign rule

L R cτ cρ cλ τ

circumcenter order

Theorem: For a pairwise Delaunay mesh of dimension 2 or 3, duals of internal simplices have positive volume. One sidedness condition for boundary simplices.

Unsigned Bad Boundary Not Delaunay

See also

• H, Nakshatrala, Chaudhry 2008 (Darcy flow)
• Gillette, Bajaj 2010, 2011 (Dual formulations)

Wedge Product

Definition

(α∧β)(e1 ,...,ek+l)

α ∈ §k β ∈ §l α∧β ∈ §k+l

1 k!l! X

σ∈Sk+l

(sgnσ)α ° eσ(1) ,...,eσ(k) ¢ β ° eσ(k+1) ,...,eσ(k+l) ¢

Example

123 132 213 231 312 321 + − − − + +

α ∈ §2 β ∈ §1

(α∧β)(e1 ,e2 ,e3) = 1 2!1! £ α(e1,e2)β(e3)−··· §

Some properties

⇤ α⇤β = (⇧1)klβ⇤α (α⇤β)⇤γ = α⇤(β⇤γ) d(α^β) = (dα)^β+(°1)kα^(dβ)

DEC Wedge (1)

≠ α∧β,σk+lÆ

≠ Æ 1 (k +l)! X

τ∈Sk+l+1

sgn(τ)|σk+l ∩?vτ(k)| |σk+l| (α ^ β)(τ(σk+l))

DEC Wedge (1)

≠ α∧β,σk+lÆ

≠ Æ 1 (k +l)! X

τ∈Sk+l+1

sgn(τ)|σk+l ∩?vτ(k)| |σk+l| (α ^ β)(τ(σk+l))

X (α ^ β)(τ(σk+l))

〈α, £ vτ(0),...,vτ(k) § 〉〈β, £ vτ(k),...,vτ(k+l) § 〉

Example

£ § £ ≠ α∧β,[0,1,2] Æ

1 2 £ °C0 ≠ α,[1,0] Æ≠ β,[0,2] Æ +C0 ≠ α,[2,0] Æ≠ β,[0,1] Æ +C1 ≠ α,[0,1] Æ≠ β,[1,2] Æ °C1 ≠ α,[2,1] Æ≠ β,[1,0] Æ °C2 ≠ α,[0,2] Æ≠ β,[2,1] Æ +C2 ≠ α,[1,2] Æ≠ β,[2,0] Ƨ

α β β α

• Anti-commutative
• Leibniz rule satisfied
• Not natural under pullback
• Not associative in general

Properties

DEC wedge (2)

• |σk+l ∩vτ(k)|

|σk+l|

1 k +l +1

[Castrillon Lopez 2003]

1 (k +l +1)! X

τ2Sk+l+1

sgn(τ)(α ^ β)(τ(σk+l))

instead of

See also

• Scott Wilson 2008
• Uses Whitney forms
• Also lacks associativity

Remark 7.1.4. Lack of associativity: According to Givental [2003] this lack of associativity in general, and a special status for closed forms, is not an accident. Putting the “democratic weighting” aside, the wedge definition works for any simplicial complex (such as singular cochains, for instance). It is known that it is in principal impossible to make a universal definition anti-commutative and associative. This phenomenon has been studied a lot in algebraic topology or homological algebra and gives rise to the concepts of Massey products and homotopy-associative algebras.

[H. 2003]

A-infinity algebras

• Introduced by Stasheff 1963 in topology
• Kontsevich 1994 (ICM) used in physics
• Dolotin, Morozov, Shakirov 2008
• Keller’s survey articles

d2 = 0 d∧+∧d = 0 ∧2 = 0

= ∧ + ∧ + − ∧2(α,β,γ) = ∧(∧(α,β),γ)−∧(α,∧(β,γ)) = (α∧β)∧γ−α∧(β∧γ)

(d+∧)2 = 0

(d+∧+m(3) +m(4) +...)2 = 0

d2 = 0 d∧+∧d = 0 dm(3) +m(3) d+∧2 = 0 dm(4) +m(4) d+∧m(3) +m(3)∧ = 0 dm(5) +m(5) d+∧m(4) +m(4) ∧+ ° m(3)¢2 = 0

Contraction and Lie derivative

Definitions and CMF

iX α := α(X,...) L X α := d dt ⇧ ⇧ ⇧

t=0ϕ⌥ t α

L X α = iX dα+diX α

Cartan Magic Formula

Proxies versus forms

α a 1-form in R3 (L X α)⇥ = L X α⇥

β a 2-form in R3 ⇤ ⌥

• L X β

⇥⌅⇥ = L X

• ⌥β

⇥⇥

α = xdx, X = x ∂ ∂x L X α = dix ∂

∂x xdx = dx2 = 2xdx

L X α] = ∑ x ∂ ∂x ,x ∂ ∂x ∏ = 0

Euler equation

ut +u ·⌅u = ⇧⌅p divu = 0 u ·n = 0

• n ∂⇤,

u(x,0) = u0(x)

u

t +L u u ⇧ 1

2 d⇤u⇤2 = ⇧dp δu = 0 iu µ = 0

• n ∂⇤,

u(x,0) = u0(x)

L u u ⇧ 1 2 d⇤u⇤2 = diu u +iu du ⇧ 1 2 diu u = 1 2 diu u +iu du = 1 2 diu u +⌥ ⌃ ⌥du ⇥u⌥

curlu ⌥u

DEC contraction / Lie (1)

iX α = (⌃1)k(n⌃k) ⌃ α⇤X⌥

Then define Lie derivative via Cartan formula [H. 2003]

Extrusion and related

• Bossavit 2003
• H. 2003
• Jinchao Xu
• Heumann and Hiptmair 2008, 2011
• Mullen et al. 2011

⇥

c

iX α = d dt ⇧ ⇧ ⇧

t=0

⇥

Htc

α ⇧ ⇥

c

L X β = d dt ⇧ ⇧ ⇧

t=0

⇥

Stc

β

DEC contraction / Lie (2)

Fluids and diffeomorphisms

• Euler equation as evolution on group of

volume preserving diffeomorphisms

• Lie algebra is div-free vector fields
• V. Arnold 1966, Ebin and Marsden 1970
• Now important in brain image matching

Structure preservation

f ,h ⇥0, ϕ Diffvol(⇤) f ϕ = f

if f is constant

⌅f ϕ,h ϕ⇧ = ⌅f ,h⇧

Discrete

D(⇤) = {q GL(N)+⇧ ⇧ ⇧

• j

qi j = 1 ⇥i, qTV q = V }

d(⇤) = {A gl(N)

⇧ ⇧ ⇧

• j

Ai j = 0 ⇥i, ATV +V A = 0}

[Gawlik et al.]

Discrete

D(⇤) = {q GL(N)+⇧ ⇧ ⇧

• j

qi j = 1 ⇥i, qTV q = V }

d(⇤) = {A gl(N)

⇧ ⇧ ⇧

• j

Ai j = 0 ⇥i, ATV +V A = 0} q(t)TV q(t) = V d dt ⇧ ⇧ ⇧

t=0q(t)TV q(t) = 0

˙ q(0)TV +V ˙ q(0) = 0 [Gawlik et al.]

Software

PyDEC

Features

• Complexes : simplicial, cubical, Rips, abstract
• DEC and lowest order FEEC

Examples

1.000 3.000 3.931 4.759 5.345 0.000 4.311 5.621

• 0.667

0.000 0.667

2 1 1 2 1 1 1 2 1 1 1 1 1

Outlook

• Vector fields, Lie derivative, contraction
• A-infinty and wedge products
• Higher order DEC
• Analysis
• Pairwise Delauany meshing for surfaces