Evolutionary -Convergence for a Delamination Model Thomas Frenzel, - - PowerPoint PPT Presentation

General Framework & Theory Delamination Model

Evolutionary Γ-Convergence for a Delamination Model

Thomas Frenzel, Alexander Mielke

• Sept. 01, 2016

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

Γ-Convergence

0 = DEε(uε)

?

− → 0 = DE0(u0)

Definition

Let X be a Banach space, Eε : X → (−∞, ∞], we say Eε Γ-converges (strongly or weakly) to E0 and write Eε

Γ

− → E0 if ∀ uε → u0 : lim inf Eε(uε) ≥ E0(u0) ∃ ˆ uε → u0 : lim Eε(ˆ uε) = E0(u0)

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

lower semi continuity

u

− →

u u

− →

u u

− →

u

Γ-limits are lower semi continuous.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

choice of topology

The choice of topology may affect the Γ-limit.

Example

Let a : [0, 1] → R with 0 < a ≤ a ≤ a and denote its 1-periodic extension by aper. Then we define Eε(u) = 1 aper x ε

• u(x)2 dx.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

choice of topology

With aarith = 1

0 a(x) dx and aharm =

1

• a(x)

−1 dx −1 we have the strong and the weak Γ-limit Es(u) = 1 aarithu2 dx and Ew(u) = 1 aharmu2 dx Microstructure is encoded in recovery sequence. For the weak topology we can choose ˆ uε(x) = aper(x/ε)−1aharmu(x).

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

choice of topology

With aarith = 1

0 a(x) dx and aharm =

1

• a(x)

−1 dx −1 we have the strong and the weak Γ-limit Es(u) = 1 aarithu2 dx and Ew(u) = 1 aharmu2 dx Microstructure is encoded in recovery sequence. For the weak topology we can choose ˆ uε(x) = aper(x/ε)−1aharmu(x).

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ-Conv.

convergence of minimizers

Theorem

Let uε be a minimizer of Eε. Then uε → u0 where u0 is a minimizer of E0. In other words, Γ-convergence implies the convergence of solutions to the corresponding Euler-Lagrange-Equations, i.e., 0 = DEε(uε) − → 0 = DE0(u0) The sequence of solutions is a recovery sequence.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ-Conv.

Gradient Systems & Evolutionary Γ-Convergence

microscopic system ε ց 0 macroscopic system ˙ uε = Aε(t, uε) ˙ u = A0(t, u) initial state u0

ε upscaling

− → u0 time evolution ↓ ↓ time t > 0 uε(t) = Sε(t, u0

ε) upscaling

− → u(t) = S0(t, u0)

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ-Conv.

Gradient Systems & Evolutionary Γ-Convergence

A gradient system (X, E, R) induces an evolution equation via D ˙

uR(u, ˙

u) = −DuE(t, u) ⇐ ⇒ ˙ u = DξR∗ u, −DuE(t, u)

• Assume that E satisfies the chain rule, then

E

• T, u(T)
• + J (u, ˙

u) ≤ E

• 0, u(0)
• +

T ∂tE

• t, u(t)
• dt

with J (u, ˙ u) := T R

• ˙

u(t)

• + R∗

− DE(t, u(t))

• dt.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ-Conv.

Gradient Systems & Evolutionary Γ-Convergence

A gradient system (X, E, R) induces an evolution equation via D ˙

uR(u, ˙

u) = −DuE(t, u) ⇐ ⇒ ˙ u = DξR∗ u, −DuE(t, u)

• Assume that E satisfies the chain rule, then

E

• T, u(T)
• + J (u, ˙

u) ≤ E

• 0, u(0)
• +

T ∂tE

• t, u(t)
• dt

with J (u, ˙ u) := T R

• ˙

u(t)

• + R∗

− DE(t, u(t))

• dt.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ-Conv.

Gradient Systems & Evolutionary Γ-Convergence

A gradient system (X, E, R) induces an evolution equation via D ˙

uR(u, ˙

u) = −DuE(t, u) ⇐ ⇒ ˙ u = DξR∗ u, −DuE(t, u)

• Assume that E satisfies the chain rule, then

E

• T, u(T)
• + J (u, ˙

u) = E

• 0, u(0)
• +

T ∂tE

• t, u(t)
• dt

with J (u, ˙ u) := T R

• ˙

u(t)

• + R∗

− DE(t, u(t))

• dt.

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ-Conv.

Well-prepared evolutionary Γ-convergence

Eε

• T, uε(T)
• + Jε(uε, ˙

uε) ≤ Eε

• 0, uε(0)
• +

T

0 ∂tEε

• t, uε(t)
• dt

↓ Γ- lim inf ↓ Γ- lim inf ↓ assptn. ↓ assptn. E0

• T, u0(T)
• + J0(u0, ˙

u0) ≤ E0

• 0, u0(0)
• +

T

0 ∂tE0

• t, u0(t)
• dt

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Delamination Model

f (t)

Φ

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Delamination Model

f (t)

Φ

Eε(t, u) = 1 1 2u′ 2 + εαΦ u ε

• dx − f (t)u(0),

R( ˙ u) = 1 1 2 ˙ u′2 dx,

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Delamination Model

f (t)

Φ

˙ u(t, x) = −u(t, x) + f (t)(1 − x) − εα−1

1

• w(x, y)Φ′

u(t, y) ε

• dy

u(t, 1) = 0 u(0, x) = u0(x)

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Apriori Estimates

The EDB formulation Eε

• T, uε(T)
• + Jε(uε, ˙

uε) ≤ Eε

• 0, uε(0)
• +

T ∂tEε

• t, uε(t)
• dt

leads to bounds in L∞ 0, T; H1(0, 1)

• and

H1 0, T; H1(0, 1)

• resulting in convergence of uε → u (up to a subsequence) in

strongly in C

• 0, T; Cβ(0, 1)
• and

weakly in H1 0, T; H1(0, 1)

• and uε(t) ⇀ u(t) a.e. in H1(0, 1)

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Γ-Convergence

It is easy to verify that with respect to the weak H1(0, 1) topologie we have Eε(t, u) = 1 1 2u′ 2

ε

+ εαΦ uε ε

• dx − f (t)uε(0)

Γ

− → E0(t, u) = 1 1 2u′ 2 dx − f (t)u(0).

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Γ-Convergence

The difficult part is the Γ-convergence (w.r.t. weak H1 0, T; H1(0, 1)

• topology) of

Jε(uε, ˙ uε) = T R

• ˙

uε(t)

• + R∗

− DEε(t, uε(t))

• dt

= T 1 1 2 ˙ u′

ε 2 + 1

2(u′

ε + f (t) − µε(uε))2 dx dt

J0(u, ˙ u) = T 1 1 2 ˙ u′

2 + 1

2

• − ˆ

µ(u0) 2 dx dt

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Γ-Convergence

The difficult part is the Γ-convergence (w.r.t. weak H1 0, T; H1(0, 1)

• topology) of

Jε(uε, ˙ uε) = T R

• ˙

uε(t)

• + R∗

− DEε(t, uε(t))

• dt

= T 1 1 2 ˙ u′

ε 2 + 1

2(u′

ε + f (t) − µε(uε))2 dx dt

J0(u, ˙ u) = T 1 1 2 ˙ u′

2 + 1

2

• u′

0 + f (t) − ˆ

µ(u0) 2 dx dt

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Γ-Convergence

The difficult part is the Γ-convergence (w.r.t. weak H1 0, T; H1(0, 1)

• topology) of

Jε(uε, ˙ uε) = T R

• ˙

uε(t)

• + R∗

− DEε(t, uε(t))

• dt

= T 1 1 2 ˙ u′

ε 2 + 1

2(u′

ε + f (t) − µε(uε))2 dx dt

J0(u, ˙ u) = T 1 1 2 ˙ u′

2 + 1

2

• DE0(u0) − ˆ

µ(u0) 2 dx dt

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Investigation of Limit System

We arrived at DR( ˙ u) = −DE0(t, u) + h(t, u) But we also have DR( ˙ u) = −DE0(t, u) ⇔ DR( ˙ u) = −DE0(t, u) + h(t, u) ˙ u(t, x) = −u(t, x) + f (t)(1 − x)

Evolutionary Γ-Convergence for a Delamination Model

General Framework & Theory Delamination Model

Outlook

x0(t1) x0(t2) 1 2 3

x u(t1),u(t2) Fracture law ˙ x0(t) = const(Φ) f (t)3

Evolutionary Γ-Convergence for a Delamination Model