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A multi-phase transition model for dislocations with interfacial microstructure Simone Cacace Adriana Garroni Mathematics Department G.Castelnuovo University of Rome La Sapienza CANUM 29 May 2008 Simone Cacace - Adriana Garroni A

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A multi-phase transition model for dislocations with interfacial microstructure

Simone Cacace Adriana Garroni

Mathematics Department ”G.Castelnuovo” University of Rome ”La Sapienza”

CANUM 29 May 2008

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Summary

Dislocations in Crystals

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Summary

Dislocations in Crystals The Multi-Phase Model by Koslowski and Ortiz

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Summary

Dislocations in Crystals The Multi-Phase Model by Koslowski and Ortiz Γ-convergence of the Energy Functional

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Summary

Dislocations in Crystals The Multi-Phase Model by Koslowski and Ortiz Γ-convergence of the Energy Functional Numerical Approximation (Finite Elements)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Crystals

Materials with a regular atomic structure (crystal lattice)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Dislocations

One-dimensional Defects of the crystal lattice responsible for plastic properties of the material

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Dislocations

One-dimensional Defects of the crystal lattice responsible for plastic properties of the material When an external force is applied to a crystal, dislocations make easy the relative slip of the atoms

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Dislocations

One-dimensional Defects of the crystal lattice responsible for plastic properties of the material When an external force is applied to a crystal, dislocations make easy the relative slip of the atoms

Simone Cacace 2007 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Crystallographic Slip

The specific geometry of a crystal

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems) Cubic Lattice

  • ✁
✂ ✂ ✄ ✄ ☎ ☎ ✆ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞ ☞ ✌ ✌ ✍ ✎

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 14

Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems) Cubic Lattice

  • ✁
✁ ✂ ✂ ✄ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ✡ ☛ ☛ ☞ ✌ ✍ ✎

b b2

1

S

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 15

Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems) Cubic Lattice

  • ✁
✁ ✂ ✂ ✄ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ✡ ☛ ☛ ☞ ✌ ✍ ✎

b b2

1

S

Slip Field = u1b1 + u2b2 u1, u2 : S → R

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 16

Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems) Cubic Lattice

  • ✁
✁ ✂ ✂ ✄ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ✡ ☛ ☛ ☞ ✌ ✍ ✎

b b2

1

S

Slip Field = u1b1 + u2b2 u1, u2 : S → R In general the slip of the atoms is Not Uniform

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 17

Crystallographic Slip

The specific geometry of a crystal constrains plastic deformations along planes (Slip Planes) identifies the directions along which the slip of the atoms is preferred (Slip Systems) Cubic Lattice

  • ✁
✁ ✂ ✂ ✄ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ✡ ☛ ☛ ☞ ✌ ✍ ✎

b b2

1

S

Slip Field = u1b1 + u2b2 u1, u2 : S → R In general the slip of the atoms is Not Uniform = ⇒ Dislocations

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs) N slip systems active on S (given by the geometry of the crystal lattice)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs) N slip systems active on S (given by the geometry of the crystal lattice) Slip Field =

N

  • i=1

uibi u = (u1, ..., uN) : S → RN

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs) N slip systems active on S (given by the geometry of the crystal lattice) Slip Field =

N

  • i=1

uibi u = (u1, ..., uN) : S → RN Coordinate system:

Q

x x x1

2 3 Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 23

The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs) N slip systems active on S (given by the geometry of the crystal lattice) Slip Field =

N

  • i=1

uibi u = (u1, ..., uN) : S → RN Coordinate system:

Q

x x x1

2 3

Slip plane − → S = {x3 = 0}

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 24

The 2D Multi-Phase Model by Koslowski and Ortiz

We consider an infinite, periodic, elastic crystal with a single slip plane S (on which the plastic distorsion occurs) N slip systems active on S (given by the geometry of the crystal lattice) Slip Field =

N

  • i=1

uibi u = (u1, ..., uN) : S → RN Coordinate system:

Q

x x x1

2 3

Slip plane − → S = {x3 = 0} Periodicity Cell − → Q × R Q = (−1/2, 1/2)2 ⊂ R2

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy of the Crystal

Given a slip field u, the energy of the crystal can be written as the sum of two terms E(u) = E elastic(u) + E core(u)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy of the Crystal

Given a slip field u, the energy of the crystal can be written as the sum of two terms E(u) = E elastic(u) + E core(u) E elastic: the long range elastic energy induced by the slip field

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy of the Crystal

Given a slip field u, the energy of the crystal can be written as the sum of two terms E(u) = E elastic(u) + E core(u) E elastic: the long range elastic energy induced by the slip field E core: the short range interatomic energy which penalizes slips not compatible with the crystal lattice represents the distorsion of the lattice around the dislocations (cores)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Long Range Elastic Energy

E elastic(u) =

  • Q
  • Q

(u(x) − u(y))TJ(x − y)(u(x) − u(y)) dx dy

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Long Range Elastic Energy

E elastic(u) =

  • Q
  • Q

(u(x) − u(y))TJ(x − y)(u(x) − u(y)) dx dy The kernel J depends on the crystal lattice and satisfies the following properties:

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Long Range Elastic Energy

E elastic(u) =

  • Q
  • Q

(u(x) − u(y))TJ(x − y)(u(x) − u(y)) dx dy The kernel J depends on the crystal lattice and satisfies the following properties: J(t) ∈ MN×N

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Long Range Elastic Energy

E elastic(u) =

  • Q
  • Q

(u(x) − u(y))TJ(x − y)(u(x) − u(y)) dx dy The kernel J depends on the crystal lattice and satisfies the following properties: J(t) ∈ MN×N J defines a positive quadratic form which is equivalent to the square

  • f the H

1 2 seminorm:

c1 |t|3 |ξ|2 ≤ ξTJ(t)ξ ≤ c2 |t|3 |ξ|2

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Core Energy of Dislocations

It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u ∈ ZN.

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Core Energy of Dislocations

It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u ∈ ZN. E core(u) = 1 ε

  • Q

dist2(u(x), ZN) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Core Energy of Dislocations

It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u ∈ ZN. E core(u) = 1 ε

  • Q

dist2(u(x), ZN) dx ε ∼ distance between the atoms of the crystal

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Core Energy of Dislocations

It is described by a piecewise quadratic potential which penalizes distorsions of the crystal lattice induced by slip fields u ∈ ZN. E core(u) = 1 ε

  • Q

dist2(u(x), ZN) dx ε ∼ distance between the atoms of the crystal

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy Functional

Eε(u) =

  • Q
  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y)) dx dy+1 ε

  • Q

dist2(u, ZN) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy Functional

Eε(u) =

  • Q
  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y)) dx dy+1 ε

  • Q

dist2(u, ZN) dx A multi-well potential functional with a non local, singular and anisotropic perturbation

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy Functional

Eε(u) =

  • Q
  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y)) dx dy+1 ε

  • Q

dist2(u, ZN) dx A multi-well potential functional with a non local, singular and anisotropic perturbation Crystallographic Slip ⇐ ⇒ Phase Transition

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Phase Fields

dist2(u(x), ZN) = min

ξ(x)∈ZN |u(x) − ξ(x)|2

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Phase Fields

dist2(u(x), ZN) = min

ξ(x)∈ZN |u(x) − ξ(x)|2

Phase Field ξ = (ξ1, ..., ξN) : S → ZN

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Phase Fields

dist2(u(x), ZN) = min

ξ(x)∈ZN |u(x) − ξ(x)|2

Phase Field ξ = (ξ1, ..., ξN) : S → ZN Dislocations are identified with the jump set of ξ

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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The Energy Functional

Eε(u) =

  • Q
  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y)) dx dy+1 ε

  • Q

dist2(u, ZN) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence

Let (X, d) be a metric space and Fε, F : X → [0, +∞].

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence

Let (X, d) be a metric space and Fε, F : X → [0, +∞]. Fε Γ(d)-converges to F if

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence

Let (X, d) be a metric space and Fε, F : X → [0, +∞]. Fε Γ(d)-converges to F if For every u ∈ X and every sequence {uε} ⊆ X such that uε

d

→ u it follows that lim inf

ε→0 Fε(uε) ≥ F(u) .

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence

Let (X, d) be a metric space and Fε, F : X → [0, +∞]. Fε Γ(d)-converges to F if For every u ∈ X and every sequence {uε} ⊆ X such that uε

d

→ u it follows that lim inf

ε→0 Fε(uε) ≥ F(u) .

For every u ∈ X there exists {uε} ⊆ X such that uε

d

→ u and lim

ε→0 Fε(uε) = F(u) .

uε is called an optimal sequence or a recovery sequence

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Rescaling of the Energy

How much a phase transition ”costs”?

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Rescaling of the Energy

How much a phase transition ”costs”?

= u = u

s ε ε s

A B

0, s ∈ ZN

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Rescaling of the Energy

How much a phase transition ”costs”?

= u = u

s ε ε s

A B

0, s ∈ ZN Eε(uε) ∼ C

  • Q
  • Q

|uε(x) − uε(y)|2 |x − y|3 dx dy + l.o.t. = C

  • A
  • B

|uε(x) − uε(y)|2 |x − y|3 dx dy + l.o.t. = C| log ε| + l.o.t.

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence of the Energy Functional

Fε(u) = 1 | log ε|

  • Q
  • Q

(u(x)−u(y))T J(x−y)(u(x)−u(y)) dx dy+ 1 ε| log ε|

  • Q

dist2 (u, ZN) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Γ-convergence of the Energy Functional

Fε(u) = 1 | log ε|

  • Q
  • Q

(u(x)−u(y))T J(x−y)(u(x)−u(y)) dx dy+ 1 ε| log ε|

  • Q

dist2 (u, ZN) dx

Theorem (C.-Garroni) Compactness If Fε(uε) ≤ M, then ∃ aε ∈ ZN and u ∈ BV (Q, ZN) such that (up to sub-sequences) uε − aε → u in L1(Q) Γ-convergence ∃ a sub-sequence εk → 0 and a function ϕ : ZN × S1 → R such that Fεk(u) Γ(L1)-converges to F(u) =

  • Su

ϕ([u], nu) dH1 Su = jump set of u [u] = jump of u nu = unit normal to Su

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 52

Problems of the Abstract Approach

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Problems of the Abstract Approach

The functional F depends on the sub-sequence εk

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 54

Problems of the Abstract Approach

The functional F depends on the sub-sequence εk ϕ(s, n) =?

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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Problems of the Abstract Approach

The functional F depends on the sub-sequence εk ϕ(s, n) =? ∀ (s, n) ∈ ZN × S1 ϕ(s, n) = F(un

s , Qn)

un

s (x) = s χ{x·n>0} =

n

u

n

Qn s s

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 56

Problems of the Abstract Approach

The functional F depends on the sub-sequence εk ϕ(s, n) =? ∀ (s, n) ∈ ZN × S1 ϕ(s, n) = F(un

s , Qn)

un

s (x) = s χ{x·n>0} =

n

u

n

Qn s s

We do not know optimal sequences: uε → un

s

lim

ε→0 Fε(uε, Qn) = F(un s , Qn)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 57

Problems of the Abstract Approach

The functional F depends on the sub-sequence εk ϕ(s, n) =? ∀ (s, n) ∈ ZN × S1 ϕ(s, n) = F(un

s , Qn)

un

s (x) = s χ{x·n>0} =

n

u

n

Qn s s

We do not know optimal sequences: uε → un

s

lim

ε→0 Fε(uε, Qn) = F(un s , Qn)

Completely solved in a scalar case (Garroni-M¨ uller)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 58

A Scalar Case

Only one slip system active Slip Field = ub where u : Q → R is a scalar function and b is a given Burgers vector.

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 59

A Scalar Case

Only one slip system active Slip Field = ub where u : Q → R is a scalar function and b is a given Burgers vector. The functional reduces to Fε(u, Q) = 1 | log ε|

  • Q
  • Q

J(x−y)|u(x)−u(y)|2 dx dy+ 1 ε| log ε|

  • Q

dist2 (u, Z) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 60

A Scalar Case

Only one slip system active Slip Field = ub where u : Q → R is a scalar function and b is a given Burgers vector. The functional reduces to Fε(u, Q) = 1 | log ε|

  • Q
  • Q

J(x−y)|u(x)−u(y)|2 dx dy+ 1 ε| log ε|

  • Q

dist2 (u, Z) dx Theorem (Garroni-M¨ uller) Fε(u, Q) Γ(L1)-converges to F(u, Q) =

  • Su∩Q

γ(nu)|[u]| dH1 where γ(n) = 2

  • x·n=1

J(x) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 61

A Scalar Case

Optimal sequence: it sufficies to consider any mollification of un = χ{x·n>0}, i.e. uε = un ∗ φε

uε

1

Qn

n

= =

n

u ε

n

1

Qn

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 62

A Scalar Case

Optimal sequence: it sufficies to consider any mollification of un = χ{x·n>0}, i.e. uε = un ∗ φε

uε

1

Qn

n

= =

n

u ε

n

1

Qn

lim

ε→0 Fε(uε, Qn) = lim ε→0

1 | log ε|

  • Qn
  • Qn J(x−y)|u(x)−u(y)|2 dx dy = γ(n) = F(un, Qn)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 63

A Scalar Case

Optimal sequence: it sufficies to consider any mollification of un = χ{x·n>0}, i.e. uε = un ∗ φε

uε

1

Qn

n

= =

n

u ε

n

1

Qn

lim

ε→0 Fε(uε, Qn) = lim ε→0

1 | log ε|

  • Qn
  • Qn J(x−y)|u(x)−u(y)|2 dx dy = γ(n) = F(un, Qn)

”One-Dimensional Profile”

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 64

A Vector Case: the Cubic Lattice

Two slip systems active Slip Field = u1b1 + u2b2 where u = (u1, u2) : Q → R2 and b1, b2 are two Burgers vectors parallel to the versors of the canonical basis of R2.

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 65

A Vector Case: the Cubic Lattice

Two slip systems active Slip Field = u1b1 + u2b2 where u = (u1, u2) : Q → R2 and b1, b2 are two Burgers vectors parallel to the versors of the canonical basis of R2. The functional reduces to

Fε(u, Q) = 1 | log ε|

  • Q
  • Q

(u(x)−u(y))T J(x−y)(u(x)−u(y)) dx dy+ 1 ε| log ε|

  • Q

dist2 (u, Z2) dx

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 66

A Vector Case: the Cubic Lattice

Two slip systems active Slip Field = u1b1 + u2b2 where u = (u1, u2) : Q → R2 and b1, b2 are two Burgers vectors parallel to the versors of the canonical basis of R2. The functional reduces to

Fε(u, Q) = 1 | log ε|

  • Q
  • Q

(u(x)−u(y))T J(x−y)(u(x)−u(y)) dx dy+ 1 ε| log ε|

  • Q

dist2 (u, Z2) dx

Strategy of the scalar case: un

s = sχ{x·n>0} with s ∈ Z2, n ∈ S1 and

uε = un

s ∗ φε =

n

u uε ε

n

Qn Qn

n

= s s s

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 67

A Vector Case: the Cubic Lattice

lim

ε→0 Fε(uε, Qn) = sTγ(n)s = γ11(n)s2 1+γ22(n)s2 2+2γ12(n)s1 s2 =: Fflat(un s , Qn)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 68

A Vector Case: the Cubic Lattice

lim

ε→0 Fε(uε, Qn) = sTγ(n)s = γ11(n)s2 1+γ22(n)s2 2+2γ12(n)s1 s2 =: Fflat(un s , Qn)

n

θ γ(n) = 2

  • x·n=1

J(x) dx = γ(θ) = Cν   2 − 2ν sin2 θ ν sin 2θ ν sin 2θ 2 − 2ν cos2 θ  

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 69

A Vector Case: the Cubic Lattice

lim

ε→0 Fε(uε, Qn) = sTγ(n)s = γ11(n)s2 1+γ22(n)s2 2+2γ12(n)s1 s2 =: Fflat(un s , Qn)

n

θ γ(n) = 2

  • x·n=1

J(x) dx = γ(θ) = Cν   2 − 2ν sin2 θ ν sin 2θ ν sin 2θ 2 − 2ν cos2 θ   γ(n) is positive defined, but the second diagonal may change sign

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 70

A Vector Case: the Cubic Lattice

In some directions it is better to split the jumps:

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 71

A Vector Case: the Cubic Lattice

In some directions it is better to split the jumps: θ = π

4 ,

s = (1, 1) u = (0,0) (1,1) vε δε (1,0) (0,0) (1,1)

>> ε

=

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 72

A Vector Case: the Cubic Lattice

In some directions it is better to split the jumps: θ = π

4 ,

s = (1, 1) u = (0,0) (1,1) vε δε (1,0) (0,0) (1,1)

>> ε

=

lim

ε→0 Fε(u∗φε, Qn) = γ11(θ)+γ22(θ)+2γ12(θ) > γ11(θ)+γ22(θ) = lim ε→0 Fε(vε ∗φε, Qn)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 73

A Vector Case: the Cubic Lattice

In some directions it is better to pile-up the jumps:

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 74

A Vector Case: the Cubic Lattice

In some directions it is better to pile-up the jumps: θ = − π

4 ,

s = (1, 1) vε δ ε ε >> u = (0,0) (1,1) = (0,0) (1,1) (1,0)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 75

A Vector Case: the Cubic Lattice

In some directions it is better to pile-up the jumps: θ = − π

4 ,

s = (1, 1) vε δ ε ε >> u = (0,0) (1,1) = (0,0) (1,1) (1,0)

lim

ε→0 Fε(u∗φε, Qn) = γ11(θ)+γ22(θ)+2γ12(θ) < γ11(θ)+γ22(θ) = lim ε→0 Fε(vε ∗φε, Qn)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 76

A Vector Case: the 1D Profile is Not Optimal

= u0

(0,0) (1,1)

Fflat(u0, Q) = lim

ε→0 Fε(u0∗φε, Q) = γ11(e1)+γ22(e1)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 77

A Vector Case: the 1D Profile is Not Optimal

= u0

(0,0) (1,1)

Fflat(u0, Q) = lim

ε→0 Fε(u0∗φε, Q) = γ11(e1)+γ22(e1)

Consider

vδ

δ (0,0) (1,1)

= =

v

(1,1) (0,0) (1,0)

lim

δ→0 vδ = u0

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 78

A Vector Case: the 1D Profile is Not Optimal

= u0

(0,0) (1,1)

Fflat(u0, Q) = lim

ε→0 Fε(u0∗φε, Q) = γ11(e1)+γ22(e1)

Consider

vδ

δ (0,0) (1,1)

= =

v

(1,1) (0,0) (1,0)

lim

δ→0 vδ = u0

lim

ε→0 Fε(vδε ∗ φε, Q) < Fflat(u0, Q)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 79

Numerical Approximation: A First Step

Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 80

Numerical Approximation: A First Step

Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher Fε(u) =     

1 | log ε|

  • 1

2

  • I
  • I

|u(x)−u(y)|2 |x−y|2

dx dy + 1

ε

  • I W (u) dx
  • if u ∈ H

1 2 (I)

+∞

  • therwise in L1(I)

with I = (0, 1), u : I → R, W : R → [0, +∞), {W = 0} = Z

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 81

Numerical Approximation: A First Step

Model Problem: a One-Dimensional Scalar Isotropic Functional by Alberti-Bouchitt´ e-Seppecher Fε(u) =     

1 | log ε|

  • 1

2

  • I
  • I

|u(x)−u(y)|2 |x−y|2

dx dy + 1

ε

  • I W (u) dx
  • if u ∈ H

1 2 (I)

+∞

  • therwise in L1(I)

with I = (0, 1), u : I → R, W : R → [0, +∞), {W = 0} = Z Fε Γ(L1)-converges to F(u) =   

  • Su |[u]| dH0

if u ∈ BV (I, Z) +∞

  • therwise in L1(I) .

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 82

Numerical Approximation: Finite Elements Method

For every h > 0 we denote by Ih the partition of I = (0, 1) in intervals of width h and consider the Linear Finite Elements Space Vh(I) =

  • u : I → R : u ∈ C(I), u|I ∈ P1(I)

∀I ∈ Ih

  • .

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 83

Numerical Approximation: Finite Elements Method

For every h > 0 we denote by Ih the partition of I = (0, 1) in intervals of width h and consider the Linear Finite Elements Space Vh(I) =

  • u : I → R : u ∈ C(I), u|I ∈ P1(I)

∀I ∈ Ih

  • .

Let {φi} be a basis for Vh. Every u ∈ Vh(I) writes as u(x) =

N(h)

  • i=1

uiφi(x) u ← → U := {u1, ..., uN} , where N(h) is the (finite) dimension of Vh(I).

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 84

Numerical Approximation: Finite Elements Method

For every h > 0 we denote by Ih the partition of I = (0, 1) in intervals of width h and consider the Linear Finite Elements Space Vh(I) =

  • u : I → R : u ∈ C(I), u|I ∈ P1(I)

∀I ∈ Ih

  • .

Let {φi} be a basis for Vh. Every u ∈ Vh(I) writes as u(x) =

N(h)

  • i=1

uiφi(x) u ← → U := {u1, ..., uN} , where N(h) is the (finite) dimension of Vh(I). We define the bilinear form on Vh(I): A(u, v) :=

  • I
  • I
  • u(x) − u(y)
  • v(x) − v(y)
  • |x − y|2

dx dy .

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 85

Numerical Approximation: Finite Elements Method

It follows that

  • I
  • I

|u(x) − u(y)|2 |x − y|2 dx dy = A(u, u) = UTAhU with Ah = (Aij

h) = (A(φi, φj))

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 86

Numerical Approximation: Finite Elements Method

We set Wh(U) =

  • I

πh

  • W (u)
  • dx

where πh : C(I) → Vh(I) denotes the Lagrange interpolation operator.

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 87

Numerical Approximation: Finite Elements Method

We set Wh(U) =

  • I

πh

  • W (u)
  • dx

where πh : C(I) → Vh(I) denotes the Lagrange interpolation operator. We then define the following discrete approximation of the functional Fε: Fε,h(u) = 1 | log ε| 1 2UTAhU + 1 εWh(U)

  • Simone Cacace - Adriana Garroni

A multi-phase transition model with interfacial microstructure

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SLIDE 88

Numerical Approximation: Finite Elements Method

We set Wh(U) =

  • I

πh

  • W (u)
  • dx

where πh : C(I) → Vh(I) denotes the Lagrange interpolation operator. We then define the following discrete approximation of the functional Fε: Fε,h(u) = 1 | log ε| 1 2UTAhU + 1 εWh(U)

  • If h = h(ε) = o(ε) we expect that Fε,h Γ-converges to the same limit

functional F: Fε,h(u)

Γ

− → F(u) =

  • Su

|[u]| dH0

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 89

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 90

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

u0

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 91

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

uε Fε,h(uε) = 1.0173

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 92

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

u Fε,h(uε) = 1.0173 ∼ 1 = F(u)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 93

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u0

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 94

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

uε Fε,h(uε) = 1.9593

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 95

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u Fε,h(uε) = 1.9593 ∼ 2 = F(u)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 96

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3

u0

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 97

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3

uε Fε,h(uε) = 2.9919

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 98

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3

u Fε,h(uε) = 2.9919 ∼ 3 = F(u)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 99

Numerical Tests

We look for Local Minimizers of Fε,h by means of the Gradient Descent Method ε = 10−2 h = 10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3

u The logarithmic rescaling slows down the convergence of the energy Fε,h

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 100

To Do ...

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 101

To Do ...

Extension to Dimension 2 Fε(u) = 1 | log ε|

Q

  • Q

|u(x) − u(y)|2 |x − y|3 dx dy + 1 ε

  • Q

dist2(u, Z) dx

  • Simone Cacace - Adriana Garroni

A multi-phase transition model with interfacial microstructure

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SLIDE 102

To Do ...

Extension to Dimension 2 Fε(u) = 1 | log ε|

Q

  • Q

|u(x) − u(y)|2 |x − y|3 dx dy + 1 ε

  • Q

dist2(u, Z) dx

  • Parallelize the Computation of the Interaction Matrix Ah (Quadruple

Integrals)

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 103

To Do ...

Extension to Dimension 2 Fε(u) = 1 | log ε|

Q

  • Q

|u(x) − u(y)|2 |x − y|3 dx dy + 1 ε

  • Q

dist2(u, Z) dx

  • Parallelize the Computation of the Interaction Matrix Ah (Quadruple

Integrals) Extension to the Anisotropic Case Fε(u) = 1 | log ε|

Q

  • Q

J(x−y)|u(x)−u(y)|2 dx dy+1 ε

  • Q

dist2(u, Z) dx

  • Simone Cacace - Adriana Garroni

A multi-phase transition model with interfacial microstructure

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SLIDE 104

To Do ...

Extension to Dimension 2 Fε(u) = 1 | log ε|

Q

  • Q

|u(x) − u(y)|2 |x − y|3 dx dy + 1 ε

  • Q

dist2(u, Z) dx

  • Parallelize the Computation of the Interaction Matrix Ah (Quadruple

Integrals) Extension to the Anisotropic Case Fε(u) = 1 | log ε|

Q

  • Q

J(x−y)|u(x)−u(y)|2 dx dy+1 ε

  • Q

dist2(u, Z) dx

  • Extension to the Vector Case

Fε(u) = 1 | log ε|

Q

  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y))dxdy+1 ε

  • Q

dist2(u, Z2)dx

  • Simone Cacace - Adriana Garroni

A multi-phase transition model with interfacial microstructure

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SLIDE 105

To Do ...

Extension to Dimension 2 Fε(u) = 1 | log ε|

Q

  • Q

|u(x) − u(y)|2 |x − y|3 dx dy + 1 ε

  • Q

dist2(u, Z) dx

  • Parallelize the Computation of the Interaction Matrix Ah (Quadruple

Integrals) Extension to the Anisotropic Case Fε(u) = 1 | log ε|

Q

  • Q

J(x−y)|u(x)−u(y)|2 dx dy+1 ε

  • Q

dist2(u, Z) dx

  • Extension to the Vector Case

Fε(u) = 1 | log ε|

Q

  • Q

(u(x)−u(y))TJ(x−y)(u(x)−u(y))dxdy+1 ε

  • Q

dist2(u, Z2)dx

  • Simulate Interfacial Microstructures: ”Zig-Zag Profiles”

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure

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SLIDE 106

A multi-phase transition model for dislocations with interfacial microstructure

Simone Cacace Adriana Garroni

Mathematics Department ”G.Castelnuovo” University of Rome ”La Sapienza”

CANUM 29 May 2008

Simone Cacace - Adriana Garroni A multi-phase transition model with interfacial microstructure