Anthony Gruber anthony.gruber@ttu · 2020-02-03 · Computing stationary solutions to p-Willmore ow...

Computing stationary solutions to p-Willmore flow

Anthony Gruber

[email protected]

Texas Tech University—Costa Rica

April 21, 2020

Anthony Gruber (Texas Tech University) Computational p-Willmore flow April 21, 2020 1 / 41

[email protected]

Source

Source material

A. Gruber, E. Aulisa, “Computational p-Willmore Flow withConformal Penalty” (to appear in ACM Transactions on Graphics).


Outline

1 Introduction

2 The continuous setting

3 The discrete setting

4 Implementation & results


The p-Willmore energy

Let u : M → R3 be a smooth immersion of the closed surface M. Then,the p-Willmore energy is defined as

Wp(u) =

∫M|H|p dµg , p ≥ 1

where H = (1/2)(κ1 + κ2) is the mean curvature of the immersed surfaceu(M).

We will say that the immersion u : M → R3 is p-Willmore provided it isstationary (to first order) with respect to Wp. Alternatively, M is ap-Willmore surface.

Question: How does the p-Willmore functional relate to its namesake?Moreover, how do we visualize p-Willmore surfaces for theoretical andcomputational study?


Review: the Willmore energy

The Willmore energy W2 is the simplest nontrivial model for bendingenergy proposed by Sophie Germain in 1821, involving a symmetric andquadratic combination of κ1 and κ2.

Moreover, the Willmore energy is invariant under conformaltransformations of R3 (Mobius transformations) and punishes deviationfrom roundness. When M is closed,

1

4

∫M

(κ1 − κ2)2 dµg =

∫M

(H2 − K ) dµg =W(u)− 2πχ(M),

by the Gauss-Bonnet theorem.

These tori have identicalWillmore energy!


Invariances of p-Willmore energy?

Note that the conformal invariance of the 2-Willmore energy does notextend to other values of p.

In fact, the p-Willmore energy is generally not even scale-invariant, since adilation u 7→ (1/t)u induces the change (t > 0)∫

MHp dµg 7→

∫M

(tH)p1

t2dµg =

∫Mtp−2Hp dµg

and so the p-Willmore energy depends on the scale factor t when p 6= 2.

This has significant consequences on immersed p-Willmore surfaces forp 6= 2, and is also seen in the gradient flow.

Particularly, an immersed surface can decrease its p-Willmore energy byshrinking uniformly when p = 0 and growing otherwise.


Model problem

We consider the problem of minimizing the p-Willmore energy subject tophysical constraints on surface area and enclosed volume.

More precisely, this means computing an immersion u : M → R3 satisfying

minu

(Wp(u) + λV(u) + νA(u)) ,

s.t.

V0 =

∫M〈u,N〉 dµg := V(u),

A0 =

∫Mdµg := A(u),

where N : M → S2 is the outward-directed unit normal field.

A suitable weak formulation of this will allow for implementation usingpiecewise-linear finite elements.


Why constrain p-Willmore?

Aside from the fact that many useful physical models have minimizerswhich are naturally constrained in these ways, there is the following result.

Theorem: G., Toda, Tran [1]

When p > 2, any p-Willmore surface M ⊂ R3 satisfying H = 0 on ∂M isminimal.

More precisely, let p > 2 and u : M → R3 be a p-Willmore immersion ofthe surface M with boundary ∂M. If H = 0 on ∂M, then H ≡ 0everywhere on M.

Since there are no closed minimal surfaces in R3, this means there are noclosed unconstrained p-Willmore minima when p > 2.

We need additional constraints in order to have (the possibility of)stationary solutions.


Outline

1 Introduction





The p-Willmore equation

It can be shown that any p-Willmore surface satisfies the equation

0 = −p

2∆g

(H|H|p−2

)− pH|H|p−2

(2H2 − K

)+ 2H|H|p.

However, this equation is 4th-order in the immersion u, and so not suitablefor finite-element modeling.

Instead, we must compute a weak expression which:

Is at most first-order in the immersion u.

Does not require a preferential frame in which to calculate derivatives(no moving frame).

Considers general variations ϕ : M → R3, which may have tangentialas well as normal components.

Avoids geometric terms that are not easily discretized, such as K and∇gN.


The variational framework

To compute a usable expression for δWp, we adopt the framework ofDziuk and Elliott [2].

Consider a parametrization X0 : V ⊂ R2 → R3 of (a portion of) thesurface M, and let u0 : M → R3 be identity on M, so u ◦ X = X .

A variation of the immersion u is a smooth function ϕ : M → R3 and a1-parameter family u(x , t) : M × (−ε, ε)→ R3 such that u(x , 0) = u0 and

u(x , t) = u0(x) + tϕ(x).

Note that this pulls back to a variation X : V × (−ε, ε)→ R3,

X (v , t) = X0(v) + tΦ(v),

where Φ = ϕ ◦ X . Note further that (since u is identity on X (t)) the timederivatives are related by

u =d

dtu(X , t) = ∇R3u · X + ut = X .


Calculating the first variation

To study p-Willmore flow with area and volume constraints, our goal is todevelop a suitable weak-form expression of the equation

u = −δWp(u)− λδV(u)− νδA(u).

First, note that the components of the induced metric on M are

gij =⟨∂xiX , ∂xjX

⟩= 〈Xi ,Xj〉

so that the surface gradient of a function f defined on M can be expressedon V as (Einstein summation assumed)

(∇g f ) ◦ X = g ijFiXj ,

where F = f ◦ X and g ikgkj = δij . It follows that for f , ϕ : M → R3,

〈df , dϕ〉g ◦ X = g ijFiΦj ,

and the Laplace-Beltrami operator on M is

(∆g f ) ◦ X = (divg∇g f ) ◦ X =1√

det g∂j(√

det gg ijFi).


Calculating the first variation (2)

The trick to finding a good expression for the variation of Wp is to exploitthe geometry of the problem.

In particular, there is the identity ∆gu = 2HN for (twice) the meancurvature vector, so Dziuk noticed in [3] that the 4th-order Willmoreequation can be split by defining Y = ∆gu.

Weakly, we have

0 =

∫M〈Y , ψ〉 dµg +

∫M〈du, dψ〉g dµg ,

for all ψ ∈ H1(M;R3).

With this idea, the (2p-scaled) p-Willmore functional becomes

Wp(M) =

∫M|Y |p dµg ,

which involves no explicit derivatives of u.Anthony Gruber (Texas Tech University) Computational p-Willmore flow April 21, 2020 13 / 41

Calculating the first variation (3)

Problem (Closed surface p-Willmore flow with constraint)

Let p ≥ 1, D(ϕ) = ∇gϕ+ (∇gϕ)T , and W := |Y |p−2Y . Determine a family u : M × (0,T ]→ R3 ofsurface immersions with M(t) = u(M, t) such that M(0) has initial volume V0, initial surface area A0,and for all t ∈ (0,T ] the equations

0 =

∫M〈u, ϕ〉 dµg +

∫Mν 〈du, dϕ〉g dµg +

∫Mλ 〈ϕ,N〉 dµg

+

∫M

((1− p)|Y |p − p divgW ) divgϕ dµg

+

∫Mp(〈D(ϕ)du, dW 〉g − 〈dϕ, dW 〉g

)dµg , (1)

0 =

∫M〈Y , ψ〉 dµg +

∫M〈du, dψ〉g dµg ,

0 =

∫M

⟨W − |Y |p−2Y , ξ

⟩dµg ,

A0 =

∫M

1 dµg , (2)

3V0 =

∫M〈u,N〉 dµg , (3)

are satisfied for some piecewise-constant λ, ν and all ϕ,ψ, ξ ∈ H1(M(t);R3

).


Continuous stability

This weak formulation can be shown to monotonically decrease thep-Willmore energy.

Theorem: Aulisa, G.

The closed surface p-Willmore flow is energy decreasing for p ≥ 1, i.e.

0 =

∫M(t)|u|2 dµg +

d

dt

∫M(t)

(|Y |p + λ 〈u,N〉+ ν) dµg

for all t ∈ (0,T ] and any piecewise-constant λ, ν.

This very nice property is what allows us to compute p-Willmoresurfaces using the gradient flow.

It is important to make sure that this stability property is preserved byour chosen discretization.

We now have a beautiful way to compute any p-Willmore surface that isstationary under the constrained p-Willmore flow, right? ......


Mesh degeneration

Unfortunately, discrete computational flows typically involve some degreeof mesh sliding, which can artificially break the simulation.

In order to prevent this, we now formulate a secondary system which willregularize the evolving surface at each time step.

This will help prevent failure which is not reflected by the continuous flow.


Conformal correction (1)

To correct mesh sliding at each time step, the goal is to enforce“Cauchy-Riemann equations” on the tangent bundle TM.

Let u : M → ImH ∼= R3 be an oriented immersion of M, and J be acomplex structure (rotation operator J2 = −IdTM) on TM. Then, if∗α = α ◦ J is the usual Hodge star on forms,

Thm: Kamberov, Pedit, Pinkall [4]

The immersion u is conformal if and only if there is a Gauss mapN : M → ImH such that ∗du = N du.

Note that,

N ⊥ du(v) for all tangent vectors v ∈ TM.

v ,w ∈ ImH −→ vw = −v · w + v × w .

Conclusion: conformality may be enforced by requiring∗du(v) = N × du(v) on a basis for TM!



How is this accomplished on the parametrization domain V ? Choosex1, x2 as coordinates on V , then:

∂1 := ∂x1 and ∂2 := ∂x2 are an ON basis for TV .

dX (∂1) := X1 and dX (∂2) := X2 are a (not usually ON) basis for TM.

J ∂1 = ∂2, J ∂2 = −∂1 on TV .

du(Xi ) = dX (∂i ) since u is identity on M.

We proceed through minimization. Define the conformal distortionfunctional,

CD(u) =1

2

∫M|du J − N × du|2 dµg ,

and notice that

|du J − N × du|2 ◦ X= g ij 〈(dX J(∂i )− N × dX (∂i )) , (dX J(∂j)− N × dX (∂j))〉 .



Careful rearrangement of the variation of this expression (assuming no

change in the metric!!) yields the dyadic product⟨Q, Jacϕ

⟩, where

(indices 1 ≤ i ≤ 3, mod 3),

Q i1 = g22 Wi + g11 (ni+1 Vi+2 − ni+2 Vi+1)

+ g12 (ni+2 Wi+1 − ni+1 Wi+2 − Vi ) ,

Q i2 = g11 Vi + g22 (ni+2 Wi+1 − ni+1 Wi+2)

+ g12 (ni+1 Vi+2 − ni+2 Vi+1 −Wi ) .

Here, the ni represent the components of N on (TV )⊥ and V ,W arevectors involving quadratic combinations of the ni and X i

j = 〈dX (∂j), ei 〉.

Since (at least formally) there is a tensor Q ∈ T ∗V ⊗ TM such thatQ Ij = gkjQ I

k , we conclude

δCD(u)ϕ =

∫M〈Q, dϕ〉g dµg .



Minimizing CD(u) directly will not respect the current surface geometry;we also need a constraint.

Idea: If u(x) (old) and u(x) (new) are close, then (u − u)(x) should beorthogonal to N(x) (to first order).

More formally, given ε > 0 and an immersion u with outer normal field N,the mesh regularization problem is to find a function v : M → R3 and aLagrange multiplier ρ : M → R, so that the new immersion u = u + v isthe solution to

minv

(CD(u + v) + ε|ρ|2A(u) + ρ

∫M〈v ,N〉 dµg

).

This provides something close to a tangential reparametrization of M.

Note the penalty term in the minimization, which is beneficial for saddlepoint problems involving a mixture of linear and constant finite-elements.



Formulated weakly, the goal of this procedure is find a new immersionu = u + v and a multiplier ρ which satisfy the system

0 =

∫Mρ 〈ϕ,N〉 dµg +

∫M〈Q, dϕ〉g dµg ,

0 =

∫Mψ 〈v ,N〉 dµg + ε

∫Mψρ dµg ,

for all ϕ,ψ ∈ H1(M;R3).

As can be seen, this works quite well in keeping meshes well-behaved asthey evolve.


Conformal to what?

To make full use of the conformal penalty regularization procedure, it isimportant to specify a reference triangulation.

In practice this is done by letting the largest angle of each element adjustthe others.

Algorithm 1 Generation of target angles

Require: Reference triangulation T of the closed surface M.for T ∈ T do

for vertex 1 ≤ i ≤ 3 doCompute mi = # of adjacent elementsαi ← αi/mi

end forDetermine maximum vertex angle αi .if αi > αj for all j 6= i then

for vertices j 6= i do

αj ← αj (π − αi ) /(∑

k 6=i αk

)end for

elsefor vertices 1 ≤ j ≤ 3 do

αj ← αj π /(∑3

k=1 αk

).

end forend if

end for


Outline

1 Introduction





The spatial discretization

We assume the smooth surface M is polygonally approximated bynondegenerate simplices Th, so that

Mh =⋃

Th∈Th

Th.

Denoting the nodes of this triangulation by {aj}Nj=1, the standard nodalbasis {φi} on Mh(t) satisfies φi (aj , t) = δij .

The space of piecewise-linear finite elements on Mh(t) is then

Sh(t) = Span{φi} = {φ ∈ C 0(Mh(t)) : φ|Th∈ P1(Th),Th ∈ Th},

where P1(Th) denotes the space of linear polynomials on Th.

Note that in practice we allow not only triangulations, but alsoquadrangulations of the continuous surface M.


What about the temporal discretization?

A good discretization of the continuous systems discussed should:

Preserve (at least empirically) the energy-decrease of the p-Willmoreflow.

Be robust to noise and other numerical artifacts.

Be relatively fast to implement.

The simplest thing to do is to linearize the problem at each time step,effectively pushing the nonlinearities into the time domain. This is thestandard strategy of Dziuk and Elliott in [2].

Instead, we choose our discretization “as centrally as possible”. Moreprecisely, let τ > 0 be a fixed temporal stepsize, and denote ukh = uh(·, kτ).

Then, Mk+ 1

2h is the image of the immersion u

k+ 12

h = (1/2)(ukh + uk+1

h

),

and for any field quantity F ,

Fk+ 1

2h =

1

2

(F kh + F k+1

h

).


Discrete p-Willmore flow

Problem

Let u,Y ,W , λ, ν be as in Problem 1. Given the discrete data ukh ,Ykh ,W

kh at time t = kτ , the p-Willmore flow problem

is to find functions uk+1h ,Y k+1

h ,W k+1h , λh, νh which satisfy the system of equations

0 =

∫M

k+ 12

h

⟨Y

k+ 12

h , ψh

⟩dµgh +

∫M

k+ 12

h

⟨duk+1

h , dψh

⟩ghdµgh ,

0 =

∫M

k+ 12

h

⟨(W

k+ 12

h −∣∣∣∣Y k+ 1

2h

∣∣∣∣p−2

Yk+ 1

2h

), ξh

⟩dµgh , (4)

0 =

∫M

k+ 12

h

⟨du

k+ 12

h ,(duk+1

h − dukh

)⟩gh

dµgh , (5)

0 =

∫M

k+ 12

h

⟨(uk+1h − ukh

),N

k+ 12

h

⟩dµgh , (6)

0 =

∫M

k+ 12

h

⟨(uk+1h − ukh

), ϕh

⟩τ

dµgh +

∫M

k+ 12

h

λh

⟨ϕh,N

k+ 12

h

⟩dµgh

+

∫M

k+ 12

h

νh

⟨du

k+ 12

h , dϕh

⟩gh

dµgh + (1− p)

∫M

k+ 12

h

∣∣∣∣Y k+ 12

h

∣∣∣∣p ⟨duk+ 12

h , dϕh

⟩gh

dµgh

− p

∫M

k+ 12

h

(divghW

k+ 12

h

)⟨du

k+ 12

h , dϕh

⟩gh

dµgh − p

∫M

k+ 12

h

⟨dW k+1

h , dϕh

⟩ghdµgh

+ p

∫M

k+ 12

h

⟨D(ϕh)dukh , dW

kh

⟩ghdµgh , (7)

for all ϕh, ψh, ξh ∈ Sh.


Discrete conformal penalty regularization

Problem (Discrete conformal penalty regularization)

Let ε > 0 be fixed, let u, u,N, ρ be as before, and let N = (1/2)(N + N

). Given

uk+1h ,Nk+1

h , solving the discrete conformal penalty regularization problem means

finding functions uk+1h , ρh which satisfy the system

0 =

∫Mk+1

h

ρh

⟨ϕh, N

k+1h

⟩dµgh +

∫Mk+1

h

⟨Qk+1

h , dϕh

⟩ghdµgh ,

0 =

∫Mk+1

h

ψh

⟨(uk+1h − uk+1

h

), Nk+1

h

⟩dµgh + ε

∫Mk+1

h

ψh ρh dµgh ,

for all ϕh, ψh ∈ Sh and where⟨Qk+1

h , dϕh

⟩gh

refers to the discretization on the

known surface Mk+1h of the analogous continuous quantity, which involves

components of the known normal Nk+1h and derivatives of the unknown immersion

uk+1h , computed with respect to Mk+1

h .


Linear vs. Nonlinear regularization

Since the only nonlinearity in the discrete conformal penalty regularizationcomes from N, it is easy to adapt our method to require only a linear solveby instead using the known N.

Below is a performance comparison on a cow with 34.5k triangles. Originalmesh (left), linear algorithm (middle), nonlinear algorithm (right).


Outline

1 Introduction





Main algorithm

Fist, note the following (linear) systems used to generate the initialcurvature data.

0 =

∫Mk

h

⟨Y kh , ψh

⟩dµgh +

∫Mk

h

⟨dukh , dψh

⟩ghdµgh , (8)

0 =

∫Mk

h

⟨(W k

h −∣∣∣Y k

h

∣∣∣p−2Y kh

), ξh

⟩dµgh . (9)

Algorithm 2 p-Willmore flow with conformal penalty

Require: Closed, oriented surface immersion u0h : M0

h → R3; real numbers ε, τ > 0, integerkmax ≥ 1.while 0 ≤ k ≤ kmax do

Solve (8) for Y kh

Solve (9) for W kh

Solve the p-Willmore flow problem for uk+1h ,Y k+1

h ,W k+1h , λh, µh

Solve the mesh regularization problem for uk+1h , ρh

uk+1h = uk+1

hk = k + 1

end while


Some implementation details

The nonlinear systems in our algorithm are solved using two-step Newtoniteration.

If R(vh) = 0 represents (formally) one such system and v0h is an initial

guess, this involves computing

vih = vi−1h − J −1

(vi−1h

)R(vi−1h

)for i ≥ 1,

where J (vh) is the Jacobian of the residual R in vh, evaluated through

J(vih)

=∂R∂vh

(vih).

The necessary integrals are computed using a 7th-order tensor productquadrature rule, and the Jacobian evaluation is done using the automaticdifferentiation library ADEPT.

This allows for arbitrarily accurate derivative calculations, which improvesstability of the simulations.


Unconstrained flow comparison on a letter C

MCF(0-Willmore)

Willmoreflow(2-Willmore)

4-Willmoreflow


Constrained 2-Willmore flow

Area and volume constrained 2-Willmore flow of a rabbit-dog.

Volume constrained 2-Willmore flow of a genus 4 statue mesh.


Volume-constrained flow comparison on a rabbit-dog

MCF(0-Willmore)

Willmoreflow(2-Willmore)

4-Willmoreflow


Mesh edit of a cartoon armadillo


Area preserving 2-Willmore flow of a cow


Area and volume preserving 2-Willmore flow of a cow


Almost isometric 2-Willmore flow of a torus knot


Almost-isometric 2-Willmore flow of a torus knot (again)


Future work

Challenges:

Find a reasonable way to conformally-correct on the surface itself, notjust its tangent space (higher-order approximation).

Stabilize the p-Willmore flow for higher values of p.

Ideas to investigate:

Build conformality directly into the flow equations.

Compute a time-dependent holomorphic 1-form basis for T ∗M anduse it to conformally parametrize at each step.

Extend these ideas to other curvature flows of interest (Ricci flow,Yamabe flow, etc.)


References

A. Gruber, M. Toda, and H. Tran.On the variation of curvature functionals in a space form withapplication to a generalized willmore energy.Annals of Global Analysis and Geometry, May 2019.

G. Dziuk and C.M. Elliott.Finite element methods for surface pdes.Acta Numerica, 22:289–396, 2013.

G. Dziuk.Computational parametric Willmore flow.Numerische Mathematik, 111(1):55, 2008.

G. Kamberov, F. Pedit, and U. Pinkall.Bonnet pairs and isothermic surfaces.Duke Math. J., 92(3):637–644, 04 1998.


Anthony Gruber anthony.gruber@ttu · 2020-02-03 · Computing stationary solutions to p-Willmore ow...

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