Shawn W. Walker€¦ · Shawn W. Walker LouisianaStateUniversity DepartmentofMathematicsand Center...

Introduction to Shape Optimization

Shawn W. Walker

Louisiana State UniversityDepartment of Mathematics and

Center for Computation and Technology (CCT)

IMA Workshop, June 6-10, 2016

Frontiers in PDE-constrained Optimization

Introduction Differential Geometry Surface Derivatives Shape Perturbation Examples Gradient Flows Drag Minimization Conclusion

Table of Contents

1 Introduction

2 Differential Geometry

3 Surface Derivatives

4 Shape Perturbation

5 Examples

6 Gradient Flows

7 Drag Minimization

8 Conclusion

Introduction to Shape Optimization S. W. Walker


Examples of Shape Optimization

Optimal shape of structures (G.Allaire, et al).

Inverse problems (shape detection).

Image processing.

Flow control.

Minimum drag bodies.

X0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Y

0

0.2

0.4

0.6

0.8

1Streamlines



Mathematical Statement

Admissible Set:

Let ΩALL be a “hold-all” domain.

Optimization variable is Ω or Γ.

Denote the admissible set by U .U should have a compactness

property.

Cost Functional:

Let J : U → R be a shape

functional.

Ex: J (Ω) =∫

Ω f(x,Ω) dx.

Ex: J (Γ) =∫

Γg(x,Γ) dS(x).

Ω

ΩALL

Γ

ν



Mathematical Statement

Admissible Set:

Let ΩALL be a “hold-all” domain.

Optimization variable is Ω or Γ.

Denote the admissible set by U .U should have a compactness

property.

Cost Functional:

Let J : U → R be a shape

functional.

Ex: J (Ω) =∫

Ω f(x,Ω) dx.

Ex: J (Γ) =∫

Γg(x,Γ) dS(x).

Ω

ΩALL

Γ

ν

Optimization Problem:

Ω∗ = argminΩ∈U

J (Ω), Γ∗ = argminΓ∈U

J (Γ).



Shape Optimization Example: Drag Minimization

Define Ω = ΩALL \ ΩB.

u = 0

u = ex

u = ex

ν

ν

Ω

ΩB

ΓOΓBΓI

ΓW

ΓW

Navier-Stokes Equations:

(u · ∇)u−∇ · σ = 0, in Ω,

∇ · u = 0, in Ω,

u = 0, on ΓB,

u = ex, on ΓI ,

u = ex, on ΓW ,

σν = 0, on ΓO.





u = 0

u = ex

u = ex

ν

ν

Ω

ΩB

ΓOΓBΓI

ΓW

ΓW


(u · ∇)u−∇ · σ = 0, in Ω,

∇ · u = 0, in Ω,

u = 0, on ΓB,

u = ex, on ΓI ,

u = ex, on ΓW ,

σν = 0, on ΓO.

Cost functional: J (Ω) = −ex ·∫

ΓBσ(u, p)ν (drag force).

Newtonian fluid: σ(u, p) := −pI + 2ReD(u), D(u) := ∇u+(∇u)†

2 .

Reynolds number: Re = ρU0Lµ .



Shape Sensitivity: Simple Example

ν

Ω

∂ΩR

Let Ω be a disk of radius R.

The boundary is ∂Ω with outer normal ν.

Let f = f(x, y) be a smooth function definedeverywhere.

Suppose f also depends on Ω: f = f(x, y; Ω).

Consider the cost functional:

J (Ω) =

∫

Ω

f(x, y; Ω) dx dy




ν

Ω

∂ΩR

Let Ω be a disk of radius R.

The boundary is ∂Ω with outer normal ν.

Let f = f(x, y) be a smooth function definedeverywhere.

Suppose f also depends on Ω: f = f(x, y; Ω).

Consider the cost functional:

J (Ω) =

∫

Ω

f(x, y; Ω) dx dy

What is the sensitivity of J with respect to changing R?

Polar coordinates: J =∫ 2π

0

∫ R

0f(r, θ;R) r dr dθ.

Differentiate:

d

dRJ =

∫ 2π

0

(

d

dR

∫ R

0

f(r, θ;R) r dr

)

dθ

=

∫ 2π

0

∫ R

0

f ′(r, θ;R) r dr dθ +

∫ 2π

0

f(R, θ;R)Rdθ.




V = ν

Ω

∂ΩR

Change back to Cartesian coordinates:

d

dRJ =

∫

Ω

f ′(x, y; Ω) dx dy

+

∫

∂Ω

f(x, y; Ω) dS(x, y),

where f ′ is the derivative with respect to R.




V = ν

Ω

∂ΩR


d

dRJ =

∫

Ω


+

∫

∂Ω



Indeed, f ′ is actually the derivative with respect to deforming Ω!

Here, Ω is deformed by the “flow field” V = ν.




V = ν

Ω

∂ΩR


d

dRJ =

∫

Ω


+

∫

∂Ω



Indeed, f ′ is actually the derivative with respect to deforming Ω!

Here, Ω is deformed by the “flow field” V = ν.

We have shown a specific version of a more general formula:

δJ (Ω;V) =

∫

Ω

f ′(x; Ω) dx+

∫

∂Ω

f(x; Ω)V(x) · ν(x) dS(x),

where V is the instantaneous velocity deformation of Ω.



Notation!

Vectors: e.g. x = (x, y, z)†, a = (a1, a2, a3)†, etc., are column

vectors.

Gradients are row vectors:

∇x =

(∂

∂x,∂

∂y,∂

∂z

)

,

∇x =

(∂

∂x1,

∂

∂x2,

∂

∂x3

)

,

where x = (x, y, z)† or x = (x1, x2, x3)†.



Notation!

Integral notation:

∫

Ω

f(x) dx =

∫∫∫

Ω

f(x1, x2, x3) dx1 dx2 dx3,

Sometimes we drop the arguments and differentials:

∫

Ω

f(x) dx ≡∫

Ω

f,

∫

Γ

f(x) dS(x) ≡∫

Γ

f,

∫

Σ

f(x) dα(x) ≡∫

Σ

f.

The measure of a set is denoted | · |, i.e.

|Ω| =∫

Ω

1, |Γ| =∫

Γ

1, |Σ| =∫

Σ

1,

i.e. |Ω| is the volume of Ω, |Γ| is the surface area of Γ, and |Σ| isthe arc-length of the curve Σ.



Parametric Curves

Let α : I → R3 parameterize a

curve Σ = α(I).

Assume |α′(t)| 6= 0 for all t ∈ I.

Arc-length:

α(t) =

∫ t

0

|α′(t)| dt.

Derivative with respect toarc-length α:

d

dα=

1

|α′(t)|d

dt.

-10

-0.5-1

0.5

-0.5

X

1

0Z

Y

0

1.5

0.5

2

0.51

11.5

τ

n

Σ

Example: α(t) = (cos(2πt), sin(2πt), 2t)†, for all t ∈ I = [0, 1].



Parametric Curves

Tangent vector:

τ (t) :=α

′(t)

|α′(t)| , or τ =dα

dα.

Curvature vector:

kn := −d2α

dα2= −dτ

dα,

where n is the unit normal vector.

k is the signed curvature.

Formula in terms of t:

-10

-0.5-1

0.5

-0.5

X

1

0Z

Y

0

1.5

0.5

2

0.51

11.5

τ

n

Σ

k(t)n(t) = − 1

|α′(t)|

(α

′(t)

|α′(t)|

)′

= − α′′(t)

α′(t) ·α′(t)−(

1

|α′(t)|

)′

τ (t)



Mappings

−2

0

2

−2

0

2−2

−1

0

1

2

−2

0

2

−2

0

2−2

−1

0

1

2

−2

0

2

−2

0

2−2

−1

0

1

2

(a) Ellipsoidal Set (b) Rotated Set (c) Deformed Set

x1x1x1 x2x2x2

x3

x3

x3

A mapping Φ = (Φ1,Φ2,Φ3)† can be viewed as a deformation.

Ex: (b) rigid motion: Φ(x) = Rx+ b.

Ex: (c) Φ(x) = (x1 − 1.2 + 1.6 cos(x3π/4), x2, x3)†.



Parametric Representation of a Surface

Think of creating a surface by deforming a flat rubber sheet into acurved sheet.

Let U ⊂ R2 be a “flat” domain (reference domain).

Let X : U → R3 be the

parameterization.

(s1, s2)† in U are the parameters.

Every point x = (x1, x2, x3)† in R

3

is given by

x = X(s1, s2).

Γ = X(U) denotes the surfaceobtained from “deforming” U .

−1

0

1

2 −1

0

1

2

3

0

0.5

1

1.5

2

2.5

3

x1

x2

x3

X

U

Γ



Basic Assumptions

0

0.5

1

1.5

2

2.5

−4

−3

−2

−1

0

1

2

3

4

(a) Non-smooth Surface (b) Degenerate Surface

x1x1x2x2

x3

x3



Basic Assumptions

0

0.5

1

1.5

2

2.5

−4

−3

−2

−1

0

1

2

3

4

(a) Non-smooth Surface (b) Degenerate Surface

x1x1x2x2

x3

x3

Assume that X is smooth andinjective.

Example where X is notsmooth:

X(s1, s2) =

(

s1, s2,

√

s21 + s22

)†

Define the 3× 2 Jacobian matrix

J = [∂s1X, ∂s2X] = ∇sX

Assume J : U → R3×2 has rank 2.

Example where this is not satisfied:

X(s1, s2) = (s1 + s2, (s1 + s2)2, (s1 + s2)

3)†



Parameterization of a Plane

−2−1

01

2

−2

0

2

−2

−1

0

1

2

−2−1

01

2

−2

0

2

−2

−1

0

1

2

x1x1 x2x2

x3

x3

(a) Cartesian parameterization (b) Polar coordinates

Plane: x ∈ R3 : x ·N = 0, N = (N1, N2, N3)

†.Solve for x3: x3 = −1

N3(x1N1 + x2N2).

(a) X(s1, s2) =(

s1, s2,−s1N1+s2N2

N3

)†

, for all (s1, s2)† in U , where

U = R2.

(b) X(r, θ) =(

r cos θ, r sin θ,−r cos θN1+sin θN2

N3

)†

, where

U = (0,∞)× (0, 2π).



Regular Surface; Local Charts

A regular surface is built from many maps and reference domains(U,X).

We call (U,X) a local chart.

For a general surface, one needs an atlas of local charts: (Ui,Xi).

x1

x2

x3

s1

s2

U1U2

U3

Γ

X

X

X



Basic Properties

The local charts (U,X) that make up a regular surface must besufficiently smooth.

The surface must have a well-defined tangent plane.

Tangent plane is spanned by ∂s1X, ∂s2X.

−1−0.5

00.5

1

−1−0.5

00.5

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1x2

x3 Notation:

Let Tx(Γ) denote the tangent planeof the surface Γ at the point x.



First Fundamental Form

Deforming the reference domain distorts it.

Need a characterization of the distortion.

Define the coefficients of the First Fundamental Form:

gij = ∂siX · ∂sjX, for 1 ≤ i, j ≤ 2.

10.5

0-0.5

-1-1

-0.5

0

0.5

0.5

0

-0.5

-1

1

1

0.50

-0.5-1-1

-0.5

0

0.5

-1

-0.5

0

0.5

1

1

x1x1 x2x2

x3

x3

(a) Top-Half Of Unit Sphere (b) Side-Half Of Unit Sphere



Example: Plane

Recall the cartesian parameterization of a plane:

X(s1, s2) =(

s1, s2,− s1N1+s2N2

N3

)†

.

Compute: ∂s1X =(

1, 0,−N1

N3

)†

, ∂s2X =(

0, 1,−N2

N3

)†

.

First fundamental form coefficients are:

g11 = ∂s1X · ∂s1X = 1 + (N1/N3)2,

g12 = ∂s1X · ∂s2X = g21 = (N1/N3)(N2/N3),

g22 = ∂s2X · ∂s2X = 1 + (N2/N3)2,

which are all constant.



Example: Sphere

Consider the parameterization of part of a sphere:

(x1, x2, x3)† ∈ R

3 : x21 + x2

2 + x23 = 1.

Top-half: X1(s1, s2) =(

s1, s2,+√

1− (s21 + s22))†

,

for all (s1, s2)† in U = (s1, s2)† ∈ R

2 : s21 + s22 < 1.

Compute: ∂s1X(s1, s2) =(

1, 0,−s1(1− (s21 + s22))−1/2

)†

,

∂s2X(s1, s2) =(

0, 1,−s2(1− (s21 + s22))−1/2

)†

.

First fundamental form coefficients are:

g11 = ∂s1X · ∂s1X =1− s22

1− (s21 + s22),

g12 = ∂s1X · ∂s2X = g21 =s1s2

1− (s21 + s22),

g22 = ∂s2X · ∂s2X =1− s21

1− (s21 + s22),

which blow-up as (s21 + s22) → 1.



Metric Tensor

The coefficients gij can be grouped into a 2× 2 matrix denoted g:

g =

[g11 g12g21 g22

]

= (∇sX)†(∇sX),

which is a symmetric matrix because g12 = g21.g is positive definite for a regular surface.The inverse matrix:

g−1 =

[g11 g12

g21 g22

]

=1

det(g)

[g22 −g12−g21 g11

]

;

note the superscript indices.Of course, we have the following property because g g−1 = I:

δij =

2∑

k=1

gikgkj =

2∑

k=1

gikgjk,

where δij is the “Kronecker delta”:

δij =

0, if i 6= j,1, if i = j.



Normal Vector

If X parameterizes a surface, then the normal vector ν is given by

ν(s) = ν(s1, s2) =∂s1X(s)× ∂s2X(s)

|∂s1X(s)× ∂s2X(s)| =∂s1X(s)× ∂s2X(s)√

det g

Choice of parameterization induces an orientation of the surface.

−2

−1

0

1

2 −2−1

01

2

−2

−1

0

1

2

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

x1

x1 x2x2

x3

x3

(a) Orientable (b) Non-orientable



Surface Area

Let (U,X) be a local chart of Γ, with R = X(U) ⊂ Γ.

The area of R is given by

|R| :=∫

R

1 =

∫∫

U

|∂s1X×∂s2X| ds1 ds2 =

∫∫

U

√

det g(s1, s2) ds1 ds2,

where g = g(s1, s2) is the metric tensor.

Standard change of variable formula for surface integrals.



Deviating From The Tangent Plane

Curved surface Γ.

Tangent plane TP (Γ) in green.

Let P = X(s1, s2) andQ = X(s1 + ℓ1, s2 + ℓ2).

distance: dist(Q, TP (Γ)) =

1

2

2∑

i,j=1

ℓiℓj ∂si∂sjX · ν︸︷︷︸

=hij

+ H.O.T.

where ν is the normal vector of Γat P .

−1−0.5

00.5

1−1

−0.5

0

0.5

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

x3

P

Q

hij are the coefficients of the second fundamental form.



Second Fundamental Form

Define the coefficients of the second fundamental form:

hij = ν · ∂si∂sjX, for 1 ≤ i, j ≤ 2,

Since ν · ∂sjX = 0 for j = 1, 2, we can differentiate to see that

∂siν · ∂sjX+ ν · ∂si∂sjX = 0, for 1 ≤ i, j ≤ 2.

Hence, we can write

hij = −∂siν · ∂sjX = ν · ∂si∂sjX, for 1 ≤ i, j ≤ 2.



Second Fundamental Form

Define the coefficients of the second fundamental form:

hij = ν · ∂si∂sjX, for 1 ≤ i, j ≤ 2,

Since ν · ∂sjX = 0 for j = 1, 2, we can differentiate to see that

∂siν · ∂sjX+ ν · ∂si∂sjX = 0, for 1 ≤ i, j ≤ 2.

Hence, we can write

hij = −∂siν · ∂sjX = ν · ∂si∂sjX, for 1 ≤ i, j ≤ 2.

The coefficients hij can be grouped into a 2× 2 matrix denoted h:

h =

[h11 h12

h21 h22

]

,

which is a symmetric matrix because h12 = h21.

h is not necessarily positive definite.

Alternative formula:

h = −(∇sν)†(∇sX).



Shape Operator and Curvature

Define the 2× 2 matrix S := −hg−1.

This is the shape operator of the surface.

The eigenvalues of S are the principle curvatures κ1 and κ2.

κ1 and κ2 do not depend on the parameterization.



Shape Operator and Curvature

Define the 2× 2 matrix S := −hg−1.

This is the shape operator of the surface.

The eigenvalues of S are the principle curvatures κ1 and κ2.

κ1 and κ2 do not depend on the parameterization.

Definitions of Curvature:

Summed curvature: κ := κ1 + κ2.

Vector curvature: κ := κν.

Gauss curvature: κG := κ1κ2.

Formulas:

Summed curvature:

κ := trace S = −2∑

i,j=1

gijhij

Gauss curvature:

κG := detS =det(−h)

det g=

deth

det g



Classification of Surfaces

−1

0

1

−10

1

−1

−0.5

0

0.5

1

1.5

2

−1

0

1

−10

1

−1

−0.5

0

0.5

1

1.5

2

−1

0

1

−10

1

−1

−0.5

0

0.5

1

1.5

2

x1x1x1 x2x2x2

x3

x3

x3

(a) Elliptic (κG > 0) (b) Parabolic (κG = 0) (c) Hyperbolic (κG < 0)

The Gauss curvature provides a way to classify (qualitatively) localsurface geometry.



Functions on Surfaces

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.46

x1x1x2x2

x3

x3

(a) f1 (b) f2

Let f : Γ → R be a function defined on Γ.

If (U,X) is a local chart, then define f := f X : U → R.

We say f is smooth i.f.f. f is smooth in the usual sense.

In practice, we first define f : U → R, then form f := f X−1.



Tangent Vector Fields

A tangential vector field v : Γ → R3 is such that v(x) ∈ Tx(Γ) for

all x in Γ.

Using a local chart (U,X), we can find q1, q2 such that

v = q1(s1, s2)∂s1X(s1, s2)+q2(s1, s2)∂s2X(s1, s2), for all (s1, s2)† in U,

where ∂s1X, ∂s2X is a tangent basis.

We say v is smooth, i.f.f. q1, q2 are smooth.



Tangential Directional Derivative

Let ω : Γ → R be a surface function.

Denote the tangential directional derivative of ω at P in Γ, in thedirection v in TP (Γ), by Dvω(P ).

Suppose α(t) is a parameterization of acurve (red) contained in Γ such thatα(0) = P and α

′(0) = v (blue), then

Dvω(P ) :=d

dtω(α(t))

∣∣∣t=0

−1−0.5

00.5

1

−1−0.5

00.5

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1x2

x3



Explicit Calculation

Introduce a local chart: (U,X).

Take α(t) = X s(t), where s : I → U parameterizes a curve in U .

Define ω = ω X : U → R.

Note: ω(α(t)) = ω X s(t) = ω s(t).







Note: ω(α(t)) = ω X s(t) = ω s(t).Therefore, the chain rule gives

d

dtω(α(t)) =

d

dtω(s(t)) = (s′(t) · ∇s)ω(s) = ∇ωs′,

where ∇ω is a 1× 2 row vector and s′ is a 2× 1 column vector.







Note: ω(α(t)) = ω X s(t) = ω s(t).Therefore, the chain rule gives

d

dtω(α(t)) =

d

dtω(s(t)) = (s′(t) · ∇s)ω(s) = ∇ωs′,

where ∇ω is a 1× 2 row vector and s′ is a 2× 1 column vector.

Expanding further with the metric tensor g = (∇sX)†(∇sX), we obtain

d

dtω(α(t)) = ∇ωg−1gs′,

= ∇ωg−1(∇sX)†(∇sX)s′

= ∇ωg−1(∇sX)†α′(t).



Surface Gradient Operator

Hence, we obtain

d

dtω(α(t))

∣∣∣t=0

= ∇ωg−1(∇sX)†α′(t)∣∣∣t=0

⇒ Dvω(P ) = ∇ωg−1(∇sX)†︸︷︷︸

1×3 row vector

· v.




Hence, we obtain

d

dtω(α(t))

∣∣∣t=0

= ∇ωg−1(∇sX)†α′(t)∣∣∣t=0

⇒ Dvω(P ) = ∇ωg−1(∇sX)†︸︷︷︸

1×3 row vector

· v.

In general,

Dvω X = ∇ωg−1(∇sX)† · v, for arbitrary v,

evaluated at s with P = X(s).




Hence, we obtain

d

dtω(α(t))

∣∣∣t=0

= ∇ωg−1(∇sX)†α′(t)∣∣∣t=0

⇒ Dvω(P ) = ∇ωg−1(∇sX)†︸︷︷︸

1×3 row vector

· v.

In general,

Dvω X = ∇ωg−1(∇sX)† · v, for arbitrary v,

evaluated at s with P = X(s).

Denote the surface gradient of ω at P in Γ by ∇Γω(P ) in TP (Γ).

Define it by ∇Γω(P ) · v = Dvω(P ), for all v ∈ TP (Γ).

Thus,

(∇ΓωX)·v = DvωX = ∇ωg−1(∇sX)†·v, for all v ∈ TX(s)(Γ).

Therefore, ∇Γω X = ∇sωg−1(∇sX)†.



Surface Gradient Example

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

x1x1x2x2

x3

x3

(a) ∇Γf1 (b) ∇Γf2

If ϕ = (ϕ1, ϕ2, ϕ3), then

(∇Γϕ) X =

(∇Γϕ1) X(∇Γϕ2) X(∇Γϕ3) X

, a 3× 3 matrix.



Other Surface Operators; Curvatures

Surface Operators:

Identity map: idΓ(x) = x for all x in Γ.

Tangent space projection: ∇ΓidΓ = I− ν ⊗ ν.

Surface divergence: ∇Γ ·ϕ = trace(∇Γϕ).

Surface Laplacian (Laplace-Beltrami): ∆Γω := ∇Γ · ∇Γω.

Alternate curvature formulas:

Curvature vector: −∆ΓidΓ = κν.

Gauss curvature:

κG = ν · ∂s1ν × ∂s2ν√

det(g), κGν =

∂s1ν × ∂s2ν√

det(g).

Surface divergence of the normal vector: ∇Γ · ν = κ.



Integration by Parts on Surfaces

Let Γ be a surface with boundary ∂Γ.

ν is the oriented normal vector of Γ.

τ is the positively oriented tangent vector of ∂Γ.

Let ω : Γ → R be differentiable.

Integration by parts:

∫

Γ

∇Γω =

∫

Γ

ωκν +

∫

∂Γ

ω(τ × ν)

(τ × ν) is the “outer normal” of the boundary ∂Γ.

Another integration by parts formula:

∫

Γ

∇Γϕ · ∇Γη = −∫

Γ

ϕ∆Γη +

∫

∂Γ

ϕ(τ × ν) · ∇Γη,

where ϕ, η : Γ → R are differentiable.



Shape Functionals

Recall shape functionals:

E(Ω) =∫

Ω

f(x,Ω), J (Γ) =

∫

Γ

f(x,Ω), B(Γ) =∫

Γ

g(x,Γ).

Note: Γ ⊂ Ω.

Application: optimization.

Ω∗ = argminΩ

E(Ω), Γ∗ = argminΓ

J (Γ), Γ∗ = argminΓ

B(Γ).



Perturbing the Domain

x1

x2 x3

x4x′1

x′2

x′3

x′4

ǫV ǫV

Ω Ωǫ

Γ

Γǫ

Perturbation of the identity:

Mapping the bulk domain:

Φǫ(x) := x+ ǫV(x), for all x in ΩALL,

where Ωǫ = Φǫ(Ω).

Mapping an embedded surface:

Xǫ X−1(x) := x+ ǫV(x), for all x in Γ ⊂ ΩALL,

where Γǫ = Xǫ X−1(Γ).



Perturbing Functions

Material derivative:

f(Ω;V)(x) ≡ f(x) := limǫ→0

fǫ(Φǫ(x)) − f(x)

ǫ, for all x in Ω,

Extend f to all of ΩALL such that

fǫ(Φǫ(x)) = f(ǫ,Φǫ(x)), for all x ∈ Ω, ǫ ∈ [0, ǫmax].

Differentiate:

f(x) =d

dǫf(ǫ,Φǫ(x))

∣∣∣ǫ=0

=∂

∂ǫf(0,x) +∇f(0,x) · d

dǫΦǫ(x)

∣∣∣ǫ=0

=∂

∂ǫf(0,x)

︸︷︷︸

=:f ′(Ω;V)(x)

+(V(x) · ∇) f(0,x)︸︷︷︸

=f(x)

.

Shape derivative: f ′(x) ≡ f ′(Ω;V)(x).



Perturbing Functions

Relationship between material derivative and shape derivatives:

Perturbing functions f defined on Ω:

f(x) = f ′(x) +V(x) · ∇f(x)

Perturbing functions g defined on Γ:

g(x) = g′(x) +V(x) · ∇Γg(x)

Other examples:

Local perturbation of Γ:

˙idΓ = V, id′Γ = (V · ν)ν

Perturb the normal vector:

ν = −(∇ΓV)†ν, ν′ = −∇Γ (V · ν)†

Perturb the summed curvature:

κ = −ν · (∆ΓV)− 2(∇ΓV) : (∇Γν), κ′ = −∆Γ (V · ν)



Perturbing Functionals

Consider the perturbed functionals:

Eǫ =∫

Ωǫ

f(Ωǫ), Jǫ =

∫

Γǫ

f(Ωǫ), Bǫ =

∫

Γǫ

g(Γǫ), for all ǫ ≥ 0,

and define the corresponding shape perturbations:

δE(Ω) ·V ≡ δE(Ω;V) :=d

dǫEǫ∣∣∣ǫ=0+

,

δJ (Γ) ·V ≡ δJ (Γ;V) :=d

dǫJǫ

∣∣∣ǫ=0+

,

δB(Γ) ·V ≡ δB(Γ;V) :=d

dǫBǫ

∣∣∣ǫ=0+

.



Perturbing Functionals

δE(Ω;V) =

∫

Ω

f(Ω;V) +

∫

Ω

f(Ω)(∇ ·V) =

recall intro. example︷︸︸︷∫

Ω

f ′(Ω;V) +

∫

∂Ω

f(Ω)(V · ν),

δJ (Γ;V) =

∫

Γ

f(Ω;V) + f(∇Γ ·V)

=

∫

Γ

f ′(Ω;V) + (V · ∇)f + f(∇Γ ·V)

=

∫

Γ

f ′(Ω;V) +[(ν · ∇)f + fκ

](V · ν) +

∫

∂Γ

f(τ × ν) ·V,

δB(Γ;V) =

∫

Γ

g(Γ;V) + g(∇Γ ·V)

=

∫

Γ

g′(Γ;V) + (V · ∇Γ)g + g(∇Γ ·V)

=

∫

Γ

g′(Γ;V) + gκ(V · ν) +∫

∂Γ

g(τ × ν) ·V.



Structure Theorem

Hadamard-Zolesio structure theorem.

Ω is a “flat” domain.

Let J = J (Ω) be a generic shape functional, whose shapeperturbation exists.

If everything is sufficiently smooth, then one can always write

δJ (Ω;V) =

∫

∂Ω

η(V · ν),

for some function η in L1(∂Ω).

Example:

δE(Ω;V) =

∫

Ω

f ′(Ω;V) +

∫

∂Ω

f(Ω)(V · ν).

Rewriting∫

Ω f ′(Ω;V) as an integral over ∂Ω requires using anadjoint PDE.



Minimal Surfaces

10.5

01

0.5

0

0.5

0

10.5

010.5

0.5

00

10.5010.5

0

0.5

0

(a) u = ga, on ∂Ω (b) u = gb, on ∂Ω (c) u = gc, on ∂Ω

xxx yyy

z

Let Γ be a surface with boundary ∂Γ ≡ Σ.

Perimeter functional: J (Γ) =∫

Γ 1 dS.

What is the first order optimality condition?

Apply shape perturbation formula with f = 1:

δJ (Γ;V) =

∫

Γ

f ′ +[(ν · ∇)f + fκ

](V · ν) +

∫

Σ

f(τ × ν) ·V,

=

∫

Γ

κν ·V = 0,

for all smooth perturbations V that vanish on Σ.

First order condition is: κ = 0.Introduction to Shape Optimization S. W. Walker


Surface Tension

Γs

ΓΓ

b

bbs

bs

νν

Σ τΣ

θclex

ex

ey

ezez

Γs,g

Let γ be a surface tension coefficient.

Consider the surface energy functional: J (Γ) =∫

Γγ dS(x).

Differentiating the energy gives the “force”:

δJ (Γ;V) =

∫

Γ

γκν ·V +

∫

Σ

γb ·V,

where V is a perturbation of the surface Γ. Note: b = τΣ × ν.

δJ (Γ;V) is the relative force exerted by surface tension againstdeforming the surface by V.

Integral over Σ are the contact line forces.



Shape Optimization

Consider a shape functional J = J (Γ) defined over a set ofadmissible shapes U .The optimization problem is:

find Γ∗ ∈ U , such that J (Γ∗) = minΓ∈U

J (Γ).

Can we evolve Γ = Γ(t) with a “velocity field” such that

J (Γ(t2)) < J (Γ(t1)), whenever t1 < t2.

Note: “time” here is a pseudotime that corresponds to our flow“velocity.”



Variational Problem

Define a “velocity” space:

V(Γ) =

V :

∫

Γ

|V|2 < ∞

, with norm ‖V‖V(Γ) :=(∫

Γ

|V|2)1/2

Define a bilinear form b : V(Γ)× V(Γ) → R such thatb(V,Y) =

∫

ΓV ·Y.

Note that ‖V‖V(Γ) =√

b(V,V).

We now define the velocity field as follows: at each time t ≥ 0, findV(t) in V(Γ(t)) that solves the following variational problem:

b(V(t),Y) = −δJ (Γ(t);Y), for all Y ∈ V(Γ(t)).

This ensures we decrease the energy because

δJ (Γ(t);V(t)) = −b(V(t),V(t)) = −‖V(t)‖2V(Γ) < 0.



L2-Gradient Descent

Discretize time:

Start with an initial domain Γ0.

FOR i = 0, 1, 2, ..., find Vi in V(Γi) such that

b(Vi,Y) = −δJ (Γi;Y), for all Y ∈ V(Γi).

Define perturbation of the identity:

X∆t (Xi)−1(x) := idΓi(x) + ∆tVi(x), for all x in Γi.

where Xi is a parameterization of Γi, and ∆t is a “time-step”.

Update the domain: Γi+1 = X∆t (Xi)−1(Γi).

If the time step ∆t is small enough, then the sequence J (Γi)i≥0

is decreasing, i.e.

J (t0) > J (t1) > J (t2) > · · · > J (ti) > J (ti+1) > · · ·

Choosing a different space V(Γ) can improve the convergence rate.



Mean Curvature Flow

Assume Γ is a closed surface.

Consider the perimeter functional: J (Γ) =∫

Γ1 dS(x).

For each t ≥ 0, find V(t) such that

∫

Γ(t)

V(t) ·Y = −δJ (Γ(t);Y) = −∫

Γ(t)

κ(t)ν(t) ·Y,

for all smooth perturbations Y.



Mean Curvature Flow

Assume Γ is a closed surface.

Consider the perimeter functional: J (Γ) =∫

Γ1 dS(x).

For each t ≥ 0, find V(t) such that

∫

Γ(t)

V(t) ·Y = −δJ (Γ(t);Y) = −∫

Γ(t)

κ(t)ν(t) ·Y,

for all smooth perturbations Y.

Continuing, we get∫

Γ(t)

[V(t) + κ(t)ν(t)

]·Y = 0, for all smooth Y.

Therefore, V(t,x) = −κ(t,x)ν(t,x), for all x ∈ Γ(t).

The surface Γ(t) moves with normal velocity:

V · ν = −κ.

Mean curvature flow is the L2-gradient flow of perimeter.





u = 0

u = ex

u = ex

ν

ν

Ω

ΩB

ΓOΓBΓI

ΓW

ΓW


(u · ∇)u−∇ · σ = 0, in Ω,

∇ · u = 0, in Ω,

u = 0, on ΓB,

u = ex, on ΓI ,

u = ex, on ΓW ,

σν = 0, on ΓO.

Cost functional: J (Ω) = −ex ·∫

ΓBσ(u, p)ν (drag force).

Newtonian fluid: σ(u, p) := −pI + 2ReD(u), D(u) := ∇u+(∇u)†

2 .

Reynolds number: Re = ρU0Lµ .



Rewrite Cost Functional

Define ϕ := u− ex : ⇒ ∇ · ϕ = 0, in Ω,

ϕ = −ex, on ΓB,

ϕ = 0, on ΓI ∪ ΓW .

Manipulate the cost:

J (Ω) = −ex ·∫

ΓB

σ(u, p)ν =

∫

ΓB

ϕ · σ(u, p)ν

=

∫

∂Ω

ν · σϕ =

∫

Ω

∇ · (σϕ) =∫

Ω

[∇ · σ] ·ϕ+ σ : ∇ϕ

=

∫

Ω

[∇ · σ] ·ϕ− pI : ∇u+2

ReD(u) : ∇u

=

∫

Ω

[(u · ∇)u] · ϕ+2

Re

∫

Ω

D(u) : D(u)

Perturbing bulk domains is usually easier than surface domains.



Differentiate The Cost

J (Ω) =

∫

Ω

[(u · ∇)u] ·ϕ+2

Re

∫

Ω

D(u) : D(u)

Apply shape perturbation formula:

δJ (Ω;V) =

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′]

·ϕ+

∫

Ω

[(u · ∇)u] ·ϕ′

+

∫

∂Ω

(V · ν)[(u · ∇)u] ·ϕ+

+4

Re

∫

Ω

D(u′) : D(u) +2

Re

∫

∂Ω

(V · ν)|D(u)|2


Differentiate The Cost

J (Ω) =

∫

Ω

[(u · ∇)u] ·ϕ+2

Re

∫

Ω

D(u) : D(u)

Apply shape perturbation formula:

δJ (Ω;V) =

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′]

·ϕ+

∫

Ω

[(u · ∇)u] ·ϕ′

+

∫

∂Ω

(V · ν)[(u · ∇)u] ·ϕ+

+4

Re

∫

Ω

D(u′) : D(u) +2

Re

∫

∂Ω

(V · ν)|D(u)|2

=

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′]

·ϕ+2

Re

∫

Ω

D(u′) : ∇ϕ

+

∫

Ω

[(u · ∇)u] · u′ +2

Re

∫

Ω

D(u) : ∇u′+

+

∫

ΓB

(V · ν)[(u · ∇)u] · ϕ+2

Re

∫

ΓB

(V · ν)|D(u)|2,

where we used the fact that V = 0 on ∂Ω \ ΓB .



Differentiate The PDE

δJ (ΓB ;V) =

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′]

·ϕ+

∫

Ω

σ(u′, p

′) : ∇ϕ

+

∫

Ω

[(u · ∇)u] · u′ +

∫

Ω

σ(u, p) : ∇u′+

+

∫

ΓB

(V · ν)[(u · ∇)u] · ϕ+2

Re

∫

ΓB

(V · ν)|D(u)|2


Differentiate The PDE

δJ (ΓB ;V) =

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′]

·ϕ+

∫

Ω

σ(u′, p

′) : ∇ϕ

+

∫

Ω

[(u · ∇)u] · u′ +

∫

Ω

σ(u, p) : ∇u′+

+

∫

ΓB

(V · ν)[(u · ∇)u] · ϕ+2

Re

∫

ΓB

(V · ν)|D(u)|2

int. by parts =

∫

Ω

[

(u′ · ∇)u+ (u · ∇)u′ −∇ · σ′]

·ϕ+

∫

∂Ω

σ′ν ·ϕ

+

∫

Ω

[

(u · ∇)u−∇ · σ]

· u′ +

∫

∂Ω

σν · u′+

+

∫

ΓB

(V · ν)[(u · ∇)u] · ϕ+2

Re

∫

ΓB

(V · ν)|D(u)|2

(u · ∇)u−∇ · σ = 0, in Ω,

∇ · u = 0, in Ω,

u = 0, on ΓB ,

u = ex, on ΓI ∪ ΓW ,

σν = 0, on ΓO.

(u′ · ∇)u+ (u · ∇)u′ −∇ · σ′ = 0, in Ω,

∇ · u′ = 0, in Ω,

u′ =??, on ΓB ,

u′ = 0, on ΓI ∪ ΓW ,

σ′ν = 0, on ΓO .



Use The Adjoint Variable

The shape perturbation reduces to

δJ (ΓB ;V) =

∫

ΓB

σ′ν ·ϕ+

∫

ΓB

σν · u′ +2

Re

∫

ΓB

(V · ν)|D(u)|2



Use The Adjoint Variable

The shape perturbation reduces to

δJ (ΓB ;V) =

∫

ΓB

σ′ν ·ϕ+

∫

ΓB

σν · u′ +2

Re

∫

ΓB

(V · ν)|D(u)|2

Next, we need the adjoint PDE, i.e. find (r, π) such that:

−[∇r]†u− (u · ∇)r−∇ · S(r, π) = 0, in Ω,

∇ · r = 0, in Ω,

r = ϕ = −ex, on ΓB,

r = 0, on ΓI ∪ ΓW ,

Sν = 0, on ΓO,

where S(r, π) = −πI+ 2

ReD(r).

Thus, after lots of integration by parts, etc.,

δJ (ΓB ;V) =

∫

ΓB

Sν · u′ +

∫

ΓB

σν · u′ +2

Re

∫

ΓB

(V · ν)|D(u)|2



Simplify

Since u = 0 on ΓB, regardless of the shape of ΓB, we have that

u = 0, on ΓB.

Therefore, u′ = −(V · ∇)u = −[∇u]V, on ΓB.

Hence, the shape perturbation simplifies to

δJ (ΓB;V) = −∫

ΓB

(Sν + σν) · [∇u]V +2

Re

∫

ΓB

(V · ν)|D(u)|2

= −∫

ΓB

(Sν + σν) · [∇u]ν

(V · ν)

+2

Re

∫

ΓB

(V · ν)|D(u)|2

=

∫

ΓB

η(V · ν).

This satisfies the structure theorem.

η is sometimes called the shape gradient.



Shape Gradient Descent

Start with an initial domain ΓiB.

FOR i = 0, 1, 2, ..., do the following.

Solve for (u, p) and (r, π) on Ωi.

Evaluate ηi.

Find Vi in V(ΓiB) such that

b(Vi,Y) = −δJ (ΓiB;Y), for all Y ∈ V(Γi

B).

Update the domain using Vi to obtain Γi+1B (line-search!).

Choice of V(ΓB) can improve the convergence rate, e.g.

V(ΓB) =

V :

∫

ΓB

|V|2 +∫

ΓB

|∇ΓBV|2 < ∞

.

Min Drag Movie



Remarks

Issues we did not talk about:

Existence of a minimizer; mostly the same as usual.

Can restrict the admissible set to give existence.

In general, can be quite difficult. Even the minimal surface problemis not completely understood.

Lower-semicontinuity of the functional.

Optimize-then-discretize Vs. Discretize-then-optimize, e.g. the issueof inconsistent gradients.

Is the solution regular enough?



Book On Shape Differential Calculus

Moving interfaces often involvecurvature.

Can be understood in the contextof shape differentiation.

Shape gradients necessary for shapeoptimization.

SIAM Book Series: Advances in

Design and Control, vol 28, July 2015.

Book is written at theundergraduate level.

Also useful for first year graduatestudents.

First book to make this materialaccessible to a wider audience.



Summary

Provided a “crash course” on differential geometry.

Defined surface derivatives.

Introduced shape differential calculus.

Brief introduction to gradient flows.

Worked an example on drag minimization.

Special thanks to

through the following grants: DMS-1418994, DMS-1555222 (CAREER)


Shawn W. Walker€¦ · Shawn W. Walker LouisianaStateUniversity DepartmentofMathematicsand Center...

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