Optimization of composite structures

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Optimization of composite structures

A topology and shape sensitivity analysis

Gabriel Delgado Keeffe

under the supervision of Gr´egoire Allaire

Ecole Polytechnique Airbus Group Innovations

11th June 2014

1

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Main topics

1) Composite optimal design via a level-set method

(a) Multi-layered composites (b) Multi-phase materials

Material 3 Material 1 Material 2

F

2) Anisotropic topological derivative

C

Ba Ω

C^? a

DJ = lim

a→0

J(a)− J(0) h(a)

Topology optimization. Inverse problems.

2

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Main topics

1) Composite optimal design via a level-set method (a) Multi-layered composites

(b) Multi-phase materials

F

C

Ba Ω

C^? a

DJ = lim

a→0

J(a)− J(0) h(a)

2

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Main topics

F

C

Ba Ω

C^? a

DJ = lim

a→0

J(a)− J(0) h(a)

2

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Main topics

F

C

Ba Ω

C^? a

DJ = lim

a→0

J(a)− J(0) h(a)

2

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Main topics

F

C

Ba Ω

C^? a

DJ = lim

a→0

J(a)− J(0) h(a) Topology optimization.

Inverse problems. 2

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Outline

1 Overview on topology optimization & the level-set method

2 Optimal design of composite materials

3 Topological derivative

4 Conclusions and perspectives

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Overview on topology optimization & the level-set method

Outline

4

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Topology optimization

A topology optimization problem can be cast as follows:

Find the stiffest structure ω ⊂ Ω belonging to an admissible set Uad, which solves the problem

ω⊂Ω,ω∈U

min

^ad

J(ω, u(ω)),

where u(ω) is the displacement field arising in ω due to prescribed excita- tions (f, g).

ω Ω

g g

f u = 0

ω Ω

g g

f u = 0

5

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Topology optimization methods

Density based Boundary variations Homogenization [Murat and Tartar,

1985], [Bendsøe and Kikuchi, 1988]

Level set+Shape sensitivity analysis [Allaire et al., 2002], [Wang et al., 2003]

SIMP[Bendsøe, 1995] Phase field[Chambolle and Bourdin, 2000]

Characteristics

No need of remeshing. Allows naturally topology changes.

Comparative advantages Clear contour of the shape. Easy definition of geometrical parameters.

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Topology optimization methods

Characteristics

No need of remeshing.

Allows naturally topology changes.

Comparative advantages Clear contour of the shape. Easy definition of geometrical parameters.

6

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Topology optimization methods

Characteristics

No need of remeshing.

Allows naturally topology changes.

Comparative advantages Clear contour of the shape.

Easy definition of geometrical parameters.

6

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The Level-set method

Introduced by [Osher and Sethian, 1988].

Definition of the level set function ψ: Ω→ R, with Ω ⊂ R^d(d= 2, 3)







ψ(x) = 0⇔ x ∈ ∂ω, ψ(x) < 0⇔ x ∈ ω, ψ(x) > 0⇔ x ∈ Ω\ω.

Ω ψ < 0 ψ > 0

ψ = 0 Ω\ω

ω

Evolution of the shape ω(t), t∈ R ∂ω Level-set function ψ(x, t)

t₀ t₁ t₂

Shape ω(t)

t₀ t₁ t₂

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The Level-set method







ψ(x) = 0⇔ x ∈ ∂ω, ψ(x) < 0⇔ x ∈ ω, ψ(x) > 0⇔ x ∈ Ω\ω.

Ω ψ < 0 ψ > 0

ψ = 0 Ω\ω

ω

∂ω

Evolution of the shape ω(t), t∈ R Level-set function ψ(x, t)

t₀ t₁ t₂

Shape ω(t)

t₀ t₁ t₂

7

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The Level-set method







ψ(x) = 0⇔ x ∈ ∂ω, ψ(x) < 0⇔ x ∈ ω, ψ(x) > 0⇔ x ∈ Ω\ω.

Ω ψ < 0 ψ > 0

ψ = 0 Ω\ω

ω

t₀ t₁ t₂

Shape ω(t)

t₀ t₁ t₂

7

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The Level-set method







ψ(x) = 0⇔ x ∈ ∂ω, ψ(x) < 0⇔ x ∈ ω, ψ(x) > 0⇔ x ∈ Ω\ω.

Ω ψ < 0 ψ > 0

ψ = 0 Ω\ω

ω

t₀ t₁ t₂

Shape ω(t)

t₀ t₁ t₂

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The Level-set method

If the shape evolves according to a velocity field θ(x, t) defined on Ω with normal componentV(x, t)

(Id + θ)ω x

θ(x, t)

ω Ω

Figure: Deformation of ω Figure: Zoom around x

One can show that the evolution equation satisfied by ψ reads







∂ψ

∂t +V|∇ψ| = 0 x ∈ Ω, t > 0, ψ= ψ0 x∈ Ω, t = 0,

∂ψ

∂n = 0 x∈ ∂Ω, t > 0.

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The Level-set method

If the shape evolves according to a velocity field θ(x, t) defined on Ω with normal componentV(x, t)

(Id + θ)ω x

θ(x, t)

ω Ω

Figure: Deformation of ω Figure: Zoom around x

One can show that the evolution equation satisfied by ψ reads







∂ψ

∂t +V|∇ψ| = 0 x ∈ Ω, t > 0, ψ= ψ0 x∈ Ω, t = 0,

∂ψ

∂n = 0 x∈ ∂Ω, t > 0.

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Shape sensitivity analysis

Definition [Hadamard, 1908], [Simon and Murat, 1976]

Consider the following type of variations of ω⊂ Ω

Id+ θ(ω) := {x + θ(x) for x ∈ ω} , θ ∈ W^1,∞(Ω;R^d) The shape derivative of a function J(ω) is defined as the Fr´echet

derivative in W^1,∞(Ω;R^d) at 0 of the application θ→ J Id + θω, i.e.

J Id+ θω = J(ω) +J⁰(ω)(θ)+ o(θ).

Proposition

Suppose ω and J regular, then there exists a continuous linear form l such that

J⁰(ω)(θ) = l((θ· n)|∂ω)

= Z

∂ω

(θ· n)jds

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Shape sensitivity analysis

J Id+ θω = J(ω) +J⁰(ω)(θ)+ o(θ).

Proposition

J⁰(ω)(θ) = l((θ· n)|∂ω)

= Z

∂ω

(θ· n)jds

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Shape sensitivity analysis

J Id+ θω = J(ω) +J⁰(ω)(θ)+ o(θ).

Proposition

J⁰(ω)(θ) = l((θ· n)|∂ω) = Z

∂ω

(θ· n)jds

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Coupling the level set method and shape derivative

The objective is to find a normal descent directionV^∗ such that

ψ(t₀,·) ≡ ω

ψ(t₀+ ∆t,·) ≡ ω^∆t ⇒ J(ω^∆t)− J(ω) < 0

∂ψ

∂t +V^∗|∇ψ| = 0; t ∈ [t⁰, t0+ ∆t], x∈ Ω

This direction can be deduced from the solution of the variational problem

hV^∗, viH¹(Ω) =−l(v), ∀v ∈ H¹(Ω),

l(v) = J⁰(ω)(vn), Indeed, for∆t small enough

J(ω^∆t)− J(ω) = ∆t× J⁰(ω)(θ^∗) + o(∆t), θ^∗ =V^∗n

= ∆t× l(V^∗) + o(∆t)

= −∆t × kV^∗k²_H¹_(Ω)+ o(∆t) < 0

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Coupling the level set method and shape derivative

ψ(t₀,·) ≡ ω

ψ(t₀+ ∆t,·) ≡ ω^∆t ⇒ J(ω^∆t)− J(ω) < 0

∂ψ

∂t +V^∗|∇ψ| = 0; t ∈ [t⁰, t0+ ∆t], x∈ Ω

hV^∗, viH¹(Ω) =−l(v), ∀v ∈ H¹(Ω),

J(ω^∆t)− J(ω) = ∆t× J⁰(ω)(θ^∗) + o(∆t), θ^∗ =V^∗n

= ∆t× l(V^∗) + o(∆t)

= −∆t × kV^∗k²_H¹_(Ω)+ o(∆t) < 0

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Coupling the level set method and shape derivative

ψ(t₀,·) ≡ ω

ψ(t₀+ ∆t,·) ≡ ω^∆t ⇒ J(ω^∆t)− J(ω) < 0

∂ψ

∂t +V^∗|∇ψ| = 0; t ∈ [t⁰, t0+ ∆t], x∈ Ω

hV^∗, viH¹(Ω) =−l(v), ∀v ∈H¹(Ω),

J(ω^∆t)− J(ω) = ∆t× J⁰(ω)(θ^∗) + o(∆t), θ^∗ =V^∗n

= ∆t× l(V^∗) + o(∆t)

= −∆t × kV^∗k²_H¹_(Ω)+ o(∆t) < 0

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Coupling the level set method and shape derivative

ψ(t₀,·) ≡ ω

ψ(t₀+ ∆t,·) ≡ ω^∆t ⇒ J(ω^∆t)− J(ω) < 0

∂ψ

∂t +V^∗|∇ψ| = 0; t ∈ [t⁰, t0+ ∆t], x∈ Ω

hV^∗, viH¹(Ω) =−l(v), ∀v ∈H¹(Ω), l(v) = J⁰(ω)(vn),

Indeed, for∆t small enough

J(ω^∆t)− J(ω) = ∆t× J⁰(ω)(θ^∗) + o(∆t), θ^∗ =V^∗n

= ∆t× l(V^∗) + o(∆t)

= −∆t × kV^∗k²_H¹_(Ω)+ o(∆t) < 0

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Coupling the level set method and shape derivative

ψ(t₀,·) ≡ ω

ψ(t₀+ ∆t,·) ≡ ω^∆t ⇒ J(ω^∆t)− J(ω) < 0

∂ψ

∂t +V^∗|∇ψ| = 0; t ∈ [t⁰, t0+ ∆t], x∈ Ω

hV^∗, viH¹(Ω) =−l(v), ∀v ∈H¹(Ω), l(v) = J⁰(ω)(vn), Indeed, for∆t small enough

J(ω^∆t)− J(ω) = ∆t× J⁰(ω)(θ^∗) + o(∆t), θ^∗ =V^∗n

= ∆t× l(V^∗) + o(∆t)

= −∆t × kV^∗k²H¹(Ω)+ o(∆t) < 0

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The level set method for topology optimization

Optimization of a triangular car suspension (courtesy of G. Allaire).

Ω

ω

Figure: Iteration 1

Ω

ω

Figure: Iteration 26

Ω

ω

Figure: Iteration 100

Linear elasticity:







−div(Ce(u)) = 0 in Ω

u= 0 onΓD

Ce(u)· n = g onΓ_N

min

ω⊂Ω,|ω|=cJ(ω), J(ω) = Z

ΓN

g·uds

J⁰(ω)(θ) =− Z

∂ω

θ·n(Ce(u) : e(u))dx

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Optimal design of composite materials

Outline

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Multi-phase optimal design

Design variable: shape of each phase (n level-sets→ 2ⁿ materials)

Ψ2 Ψ1

Figure: Level-set functionsΨ1,Ψ2

Ψ1< 0 Ψ₂> 0 Ψ1> 0

Ψ2< 0

Ψ₁> 0 Ψ₂> 0 Ψ₁< 0 Ψ2< 0

Figure: Multi-phase material via superposition ofΨ1,Ψ2

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Multi-layered optimal design

Design variables: shape of each plyΨi (continuous) & stacking sequence ξ (discrete)

Stack center Ψ₁ Ψ₂ Ψ₃ Ψ₄ Ψ₅

ξ

Figure: Multi-layered configuration

Figure: Orthotropic plies. Reinforced fibers in orientations

−45ô,90ô,0ô,45ô

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Multi-layered optimal design

Design variables: shape of each plyΨi (continuous) & stacking sequence ξ (discrete)

Stack center Ψ₁ Ψ₂ Ψ₃ Ψ₄ Ψ₅

ξ

Figure: Multi-layered configuration

Figure: Orthotropic plies.

Reinforced fibers in orientations

−45ô,90ô,0ô,45ô

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The buckling problem

Figure: Buckling under compression

Figure: Buckled wing

Ω ε

Γ_N g

Γ_D

0

−h +h











Ω : Hold-all domain g: in-plane loads ΓN : Neumann BC Γ_D : Dirichlet BC ε: Thickness of each ply 2h : Total thickness laminate A : extensional stiffness D : bending stiffness A = 2ε

N

X

i=1

χⁱAⁱ+ (1− χⁱ)A⁰

, χⁱ: characteristic function

D = 2ε³ 3

N

X

i=1

i³− (i − 1)³

The multi-layered design problem becomes a multi-phase design problem.

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The buckling problem

Ω ε

Γ_N g

Γ_D

0

−h +h











Ω : Hold-all domain g: in-plane loads ΓN : Neumann BC Γ_D : Dirichlet BC ε: Thickness of each ply 2h : Total thickness laminate A : extensional stiffness D : bending stiffness

A = 2ε

N

X

i=1

, χⁱ: characteristic function

D = 2ε³ 3

N

X

i=1

i³− (i − 1)³

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The buckling problem

Ω ε

Γ_N g

Γ_D

0

−h +h











N

X

i=1

, χⁱ : characteristic function

D = 2ε³ 3

N

X

i=1

i³− (i − 1)³

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The buckling problem

Ω ε

Γ_N g

Γ_D

0

−h +h











N

X

i=1

, χⁱ : characteristic function

D = 2ε³ 3

N

X

i=1

i³− (i − 1)³

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The linearized buckling problem

Definition (von K´arm´an plate model)

Let w ∈ H²(Ω) be the vertical displacement. Consider the eigenvalue problem











∇² : (D∇²w) = λAe(u) : ∇²w in Ω,

w= 0,∇w · n = 0 onΓD,

(D∇²w)_nn= 0 onΓ_N,

∇ · (D∇²w)· n + _∂τ^∂ (D∇²w)nτ =λ2hg· ∇w onΓ_N,

Let u∈ H¹(Ω;R²) be the in-plane displacementand solves







−div(Ae(u)) = 0 in Ω, u= 0 onΓ_D, Ae(u) · n = 2hg, onΓN.

We chose λ= λ1, the smallest positive eigenvalue, as the critical buckling load.

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The linearized buckling problem











∇² : (D∇²w) =λAe(u) : ∇²w in Ω,

w= 0,∇w · n = 0 onΓD,







We chose λ= λ1, the smallest positive eigenvalue, as the critical buckling load.

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The linearized buckling problem











∇² : (D∇²w) =λAe(u) : ∇²w in Ω,

w= 0,∇w · n = 0 onΓD,







We chose λ= λ1, the smallest positive eigenvalue, as the critical

buckling load. ¹⁶

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The linearized buckling problem











∇² : (D∇²w) =λAe(u) : ∇²w in Ω,

w= 0,∇w · n = 0 onΓD,







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The linearized buckling problem











∇² : (D∇²w) =λAe(u) : ∇²w in Ω,

w= 0,∇w · n = 0 onΓD,

∇ · (D∇²w)· n +_∂τ^∂ (D∇²w)nτ =λ2hg· ∇w onΓ_N, Let u∈ H¹(Ω;R²) be the in-plane displacementand solves







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Elastic transmission conditions (ETC)

Sharp interface formulation

A = A¹+ H(ψ)(A⁰− A¹)

A¹,A⁰: Elastic phases H : Heaviside function.

Smooth interface formulation

A = A¹+ H(dω)(A⁰− A¹)

H: Regular approximation H

: Interpolation width.

A⁰ A¹

H

Transmission conditions on ψ= 0 ([[·]] denotes the jump through ∂ω):







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[−λ1(Ae(u) : ∇w) · n + div(D∇²w)· n + _∂τ^∂ (D∇²w)nτ]] = 0

[[u]] = 0, [[Ae(u) · n]] = 0

Signed distance function:

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω,

ψ = 0 Ω

ω

A⁰ A¹

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Elastic transmission conditions (ETC)

A = A¹+ H(ψ)(A⁰− A¹)

A = A¹+ H(dω)(A⁰− A¹)

A⁰ A¹

H







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[u]] = 0, [[Ae(u) · n]] = 0 Signed distance function:

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω, ψ = 0

Ω

ω

A⁰ A¹

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Elastic transmission conditions (ETC)

A = A¹+ H(ψ)(A⁰− A¹)

A = A¹+ H(dω)(A⁰− A¹)

A⁰ A¹

H







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[u]] = 0, [[Ae(u) · n]] = 0

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω, ψ = 0

Ω

ω

A⁰ A¹

17

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Elastic transmission conditions (ETC)

A = A¹+ H(ψ)(A⁰− A¹)

A = A¹+ H(dω)(A⁰− A¹)

A⁰ A¹

H







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[u]] = 0, [[Ae(u) · n]] = 0

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω, ψ = 0

Ω

ω

A⁰ A¹

17

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Elastic transmission conditions (ETC)

A = A¹+ H(ψ)(A⁰− A¹)

A = A¹+ H(d_ω)(A⁰− A¹)

A⁰ A¹

H







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[u]] = 0, [[Ae(u) · n]] = 0

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω, ψ = 0

Ω

ω

A⁰ A¹

17

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Elastic transmission conditions (ETC)

A = A¹+ H(ψ)(A⁰− A¹)

A = A¹+ H(d_ω)(A⁰− A¹)

A⁰ A¹

H







[[w]] = 0, [[∇w · n]] = 0, [[(D∇²w)nn]] = 0,

[[u]] = 0, [[Ae(u) · n]] = 0

dω(x) =







−d(x, ∂ω) if x ∈ ω,

0 if x∈ ∂ω,

d(x, ∂ω) if x∈ Ω\¯ω,

ψ = 0 Ω

ω

A⁰ A¹

17

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Composite optimization problem

Definition (Lightweight design problem) Find the best composite structure(O, ξ), with:

Collection of ply-shapes: O = {ωⁱ}i=1...N Stacking sequence: ξ

O∈ Uminad,ξ∈Y {V (O)|G(O, ξ) ≤ 0} , where:

Total weight: V(O) Failure constraint: G(O, ξ) Admissible shapes: Uad Admissible stacking: Y Related works:

Stacking sequence optimization: G¨urdal, Haftka& co-workers (90’s-) Fiber orientation tailoring: Lund& co-workers (2005-).

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

Figure: Each ply has its own shape ω_i

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

Figure: Transversal cut. Multi-phase structure.

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Composite optimization problem

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Composite optimization problem

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Composite optimization problem

O∈ Uminad,ξ∈Y {V(O)|G(O, ξ) ≤ 0} , where:

Total weight: V(O) Failure constraint: G(O, ξ)

Admissible shapes: Uad Admissible stacking: Y Related works:

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Composite optimization problem

Total weight: V(O) Failure constraint: G(O, ξ) Admissible shapes: Uad Admissible stacking: Y

Related works:

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Composite optimization problem

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Composite optimization problem

Definition (Lightweight design problem)

Find the best composite structure(O, ξ), with:

O∈ Umin_ad,ξ∈Y {V (O)|G(O, ξ) ≤ 0} , where:

Total weight: V(O) Failure constraint: G(O, ξ) Admissible shapes: Uad Admissible stacking: Y

Related works:

ω2

ω₃ ω4 ω5

ε

ω1 Π = 0

A¹ A² A³ A⁵ A⁰

A² A5

A4

A⁰

18

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Bi-level formulation

Proposition

The mixed lightweight design composite problem can be equivalently written as

O∈Umin_ad{V (O)|M(O) ≤ 0} , where M constraint margin function M(O) := min

ξ∈Y G(O, ξ).

Optimization algorithm

Let O⁰ ∈ Uad be an initial feasible point. For k ≥ 0, iterate until convergence

(

1) Solve ξ^k= argmin_ξ∈YG(O^k, ξ).

2) Find O^k+1 such that V(O^k+1) < V (O^k) and G(O^k+1, ξ^k)≤ 0.

1) Integer programming solver 2) Level-set method.

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

(

19

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

(

2) Find O^k+1 such that V(O^k+1) < V (O^k) and G(O^k+1, ξ^k)≤ 0. 1) Integer programming solver 2) Level-set method.

19

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

( 1) Solve ξ^k= argmin_ξ∈YG(O^k, ξ).

2) Find O^k+1 such that V(O^k+1) < V (O^k) and G(O^k+1, ξ^k)≤ 0. 1) Integer programming solver 2) Level-set method.

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

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Bi-level formulation

Proposition

ξ∈Y G(O, ξ).

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Multi-phase shape sensitivity analysis

Introduce the general criterion J(ω) =R

Ωj(x, u) dx

Some formulae of J⁰(ω)(θ) in engineering literature are incorrect. In a multi-phase framework, three shape derivative formulae arise: Proposition [Allaire, Dapogny, Delgado & Michailidis (2014)]

I. Continuous sharp interface The shape derivative of J reads

J⁰(ω)(θ) =− Z

∂ω

D(u, p) θ· n ds, with

D(u, p) =−σ(p)nn : [[e(u)_nn]]− 2σ(p)nτ : [[e(u)_nτ]] + [[σ(u)_{τ τ}]] : e(p)_{τ τ}. where σ(v) =A e(v) and p is the adjoint state, solution of

−div (Ae(p)) = −j⁰(x, u) in Ω, Ae(p) · n = 0 on ΓN, p= 0 on ΓD.

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Multi-phase shape sensitivity analysis

Ωj(x, u) dx

Some formulae of J⁰(ω)(θ) in engineering literature are incorrect.

In a multi-phase framework, three shape derivative formulae arise: Proposition [Allaire, Dapogny, Delgado & Michailidis (2014)]

J⁰(ω)(θ) =− Z

∂ω

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Multi-phase shape sensitivity analysis

Ωj(x, u) dx

In a multi-phase framework, three shape derivative formulae arise:

Proposition [Allaire, Dapogny, Delgado & Michailidis (2014)] I. Continuous sharp interface

The shape derivative of J reads J⁰(ω)(θ) =−

Z

∂ω

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Multi-phase shape sensitivity analysis

Ωj(x, u) dx

In a multi-phase framework, three shape derivative formulae arise:

Proposition [Allaire, Dapogny, Delgado & Michailidis (2014)]

J⁰(ω)(θ) =− Z

∂ω

−div (Ae(p)) = −j⁰(x, u) in Ω, Ae(p) · n = 0 on Γ^N, p= 0 on ΓD.

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Multi-phase shape sensitivity analysis

Proposition [Allaire, Dapogny, Delgado & Michailidis (2014)]

II. Discrete sharp interface

Let Ω_h be a conformal simplicial mesh of Ω and u_h, p_h the finite element approximations of u, p, respectively. Suppose that ∂ω is never aligned with the face of an element of Ωh. Then the discrete shape derivative of J_h(ω) =R

Ωhj(x, u_h)dx reads J_h⁰(ω)(θ) =−

Z

∂ω

[[A]]e(uh) : e(p_h) θ· n ds.

∂ω

A^∗= ρA⁰+ (1− ρ)A¹

A¹

A⁰

σ⁰(u⁰_h)· n⁰ 6= σ¹(u¹_h)· n¹ u⁰_h = u¹_h

A^∗ = ρA⁰+ (1− ρ)A¹

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