Elliptic approximation of free-discontinuity problems ... · Approximation of free discontinuity...

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Elliptic approximation of free-discontinuity problems: cavitation and fracture Carlos Mora Corral Universidad Aut´ onoma de Madrid (Joint with D. Henao and X. Xu) 1/30

Transcript of Elliptic approximation of free-discontinuity problems ... · Approximation of free discontinuity...

Page 1: Elliptic approximation of free-discontinuity problems ... · Approximation of free discontinuity problems Direct approach to minimization (e.g., nite elements) is numerically unfeasible.

Elliptic approximation of free-discontinuityproblems: cavitation and fracture

Carlos Mora CorralUniversidad Autonoma de Madrid

(Joint with D. Henao and X. Xu)

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Page 2: Elliptic approximation of free-discontinuity problems ... · Approximation of free discontinuity problems Direct approach to minimization (e.g., nite elements) is numerically unfeasible.

Free discontinuity problems (E. De Giorgi ’91)

A free discontinuity problem is the Calculus of Variations analogue

of a free boundary problem in PDEs.

min I (u), where Ω ⊂ Rn, u : Ω→ Rm,

I = volume energy︸ ︷︷ ︸n dimensional

+ surface energy︸ ︷︷ ︸n−1 dimensional

=

∫Ω

(· · · ) dx +

∫S

(· · · ) dHn−1(x).

The surface energy involves a surface S , which is an unknown of

the problem.

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A useful functional space: SBV (E. De Giorgi & L. Ambrosio ’88)

W 1,p is the set of u ∈ Lp such that Du ∈ Lp.

BV is the set of u ∈ L1 such that Du is a measure.

Du can be decomposed as

Du = g dHn−1 + Cu +∇u dx.

Support of Cu has fractal dimension n − 1 < α < n.

SBV is the set of u ∈ BV such that Cu = 0.

Ju = set of jumps of u = support of g dHn−1.

Intuitively, SBV can be thought of “piecewise W 1,1”.

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Free discontinuity problems. Example 1: Mumford-Shah

(D. Mumford & J. Shah ’89, E. De Giorgi, M. Carriero & A. Leaci ’89)

Image segmentation.

min

∫Ω|∇u|2 dx +

∫Ω

(u − f )2 dx +Hn−1(Ju)

subject to u ∈ SBV (Ω, [0, 1]).

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Free discontinuity problems. Example 2: Perimeter

min PerA

subject to A ⊂ Ω, Ln(A) = λ.

Two immiscible liquids “0” and “1” in container Ω.

χA: place where liquid 1 is.

PerA: surface tension between liquid 0 and liquid 1.

λ: quantity of liquid 1.

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Free discontinuity problems. Example 3: Fracture

(A. Griffith ’21, L. Ambrosio & A. Braides ’95,

G. Francfort & J-J. Marigo ’98)

min

∫ΩW (∇u) dx +Hn−1(Ju)

subject to u ∈ SBV (Ω,Rn).

Ω: body in reference configuration.

u : Ω→ Rn: deformation of the body.∫Ω W (∇u)dx: elastic energy of deformation.

Hn−1(Ju): fracture energy.

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Free discontinuity problems. Example 4: Cavitation

(A. Gent & P. Lindley ’59, S. Muller & S. Spector ’95)

min

∫ΩW (Du) dx + Per u(Ω)

subject to u ∈W 1,p(Ω,Rn).

Ω: body in reference configuration.

u : Ω→ Rn: deformation of the body.∫Ω W (Du)dx: elastic energy of deformation.

Per u(Ω): cavitation energy, formation of voids.

u

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Existence in free discontinuity problems

Minimize I (u) =

∫ΩW (x , u,∇u) dx +

∫Ju

ϕ(x , u+, u−, νu)dHn−1

Direct method of Calculus of Variations. Minimizing sequence:

I (uj)→ inf I .

Compactness in SBV (L. Ambrosio ’89): coercivity of W and ϕ

implies uj u in SBV .

Lower semicontinuity in SBV (G. Bouchitte & G. Butazzo ’90,

L. Ambrosio & A. Braides ’90): “Convexity” of W and ϕ implies

I (u) ≤ lim infj→∞

I (uj).

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Approximation of free discontinuity problems

Direct approach to minimization (e.g., finite elements) is

numerically unfeasible. Main difficulties: mixture of dimensions,

discontinuity set is unknown.

Need of converting (approximating) a surface integral into a

volume integral. Best way, Γ-convergence.

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Γ-convergence (E. De Giorgi & T. Franzoni ’75)

A sequence of functionals Ij : X → [−∞,∞] Γ-converges to

I : X → [−∞,∞] if

o) (Compactness): If Ij(uj) is bounded then, for a subsequence,

uj → u ∈ X .

i) (Lower bound): If uj → u then I (u) ≤ lim infj→∞

Ij(uj).

ii) (Upper bound): For each u there exists uj such that

I (u) = limj→∞

Ij(uj).

We write IjΓ−→ I .

Main property:

arg min Ij → arg min I and min Ij → min I .

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Approximation of free discontinuity problems

Aim: define a sequence Iε of functionals easy to handle numerically

(e.g., elliptic) and regular (e.g., defined in W 1,p) such that

IεΓ−→ I .

The functional space changes (from SBV to W 1,p): it is more

regular (sort of singular perturbation problem).

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Modica-Mortola approximation for phase transitions

(S. Allen & J. Cahn 72–79, L. Modica & S. Mortola ’77, L. Modica ’87)

ε

∫Ω|Dvε|2 dx +

1

ε

∫Ωv2ε (1− vε)2 dx

Γ−→ 1

3Perv = 1.

among vε ∈W 1,2(Ω, [0, 1]) with∫

Ω vε dx = λ.

vε ' 0 first fluid, vε ' 1 second fluid.

λ = volume of second fluid.

Surface tension codified in transitions from 0 to 1: detected by

|Dvε|2.

vε → v ∈ SBV (Ω, 0, 1).

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Ambrosio-Tortorelli approximation of fracture

(L. Ambrosio & V. Tortorelli ’90, ’92, B. Bourdin, G. Francfort &

J-J. Marigo ’00, A. Braides, A. Chambolle & M. Solci ’07)

∫Ω

(v2ε + ηε)W (Duε) dx +

∫Ω

[ε|Dvε|2 +

1

4ε(1− vε)2

]dx

Γ−→∫

ΩW (∇u) dx +Hn−1(Ju).

among uε ∈W 1,p(Ω,Rn), vε ∈W 1,2(Ω, [0, 1]).

vε ' 0 damaged material, vε ' 1 healthy material.

uε → u ∈ SBV (Ω,Rn), vε → 1 as ηε ε→ 0.

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Main idea (Modica & Mortola):∫Ω

[ε|Dv |2 +

(1− v)2

]dx

C-S≥

∫Ω|Dv | (1− v)dx

coarea=

∫ 1

0(1− s)Hn−1 (x : v(x) = s)ds.

But Hn−1 (v = s) ' constant ' 2Hn−1(Ju)

v = 0.1

v ' 1

v ' 1 v = 0.1

Juv = 0.01

v = 0.01

so

∫Ω

[ε|Dv |2 +

(1− v)2

]dx ≥ Hn−1(Ju).

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∫Ω

[ε|Dv |2 +

(1− v)2

]dx ≥ Hn−1(Ju).

Moreover, “=” holds iff (Cauchy-Schwarz)

|Dv | =1

2ε(1− v).

Solving the O.D.E.

σ′ε =1

2ε(1− σε), σε(0) = 0,

the optimal vε is

vε(x) = σε(dist(x, Ju)).

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Fracture by void coalescence

(N. Petrinic, J. L. Curiel Sosa, C. R. Siviour, B. C. F. Elliot ’06)

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A free discontinuity model for cavitation and fracture in

nonlinear elasticity (Henao & M.-C. ’10–11, Henao, M.-C. & Xu ’13)

∫ΩW (∇u) dx︸ ︷︷ ︸

elastic

+Hn−1(Ju)︸ ︷︷ ︸fracture

+ Per u(Ω)︸ ︷︷ ︸new surface in

deformed configuration

u ∈ SBV (Ω,Rn), u one-to-one a.e., det∇u > 0 a.e.

Assumptions: W polyconvex,

W (F) ≥ c |F|p + h1(| adjF|) + h2(detF),

p ≥ n − 1, h1, h2 superlinear at ∞.

Key of the proof: if uj u in W 1,p then det∇uj det∇u in L1.

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Geometric interpretation of Per u(Ω).

Per u(Ω) = Hn−1(∂u(Ω)), ∂u(Ω) = u(∂Ω)︸ ︷︷ ︸stretching of ∂Ω

∪ ∂u(Ω) \ u(∂Ω)︸ ︷︷ ︸new surface

Γ1

u(∂Ω)

Ju

u

Ω

Γ2

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Aim: Approximation of∫ΩW (∇u) dx +Hn−1(Ju) + Per u(Ω).

Term Hn−1(Ju) like Ambrosio-Tortorelli.

Term Per u(Ω) like Modica-Mortola.

Two phase-field functions:

v ' 0 in Ju, v ' 1 in Ω \ Ju (Ambrosio-Tortorelli)

w ' 1 in u(Ω), w ' 0 in Q \ u(Ω) (Modica-Mortola)

v ' w u.

Q

u

v ' 0

v ' 1

Ω

Ju

w ' 0

w ' 1

w ' 0w ' 0

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Elliptic approximation of I :

Iε(uε, vε,wε) :=

∫Ω

(v2ε + ηε)W (Duε) dx

+

∫Ω

[ε|Dvε|2

2+

(1− vε)2

]dx

+ 6

∫Q

[ε|Dwε|2

2+

w2ε (1− wε)2

]dy.

I uε ∈W 1,p(Ω,Rn), uε = u0 on ∂DΩ, one-to-one a.e.

I uε does not create surface: ∂∗uε(Ω) = uε(∂Ω).

I vε ∈W 1,2(Ω), wε ∈W 1,2(Q), Q ⊃ uε(Ω).

I vε|∂DΩ = 1, wε|Q\uε(Ω) = 0.

I∫

Ω |vε − wε uε| ≤ bε → 0.

I ηε ε→ 0.

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Towards a Γ-convergence result

Would like to prove IεΓ−→ I .

Theorem (existence, compactness and lower bound)

I Existence of minimizers for Iε.

I If supε Iε(uε, vε,wε) <∞, then, for a subsequence, uε → u

a.e. for some u ∈ SBV (Ω,Rn), one-to-one a.e., det∇u > 0

a.e. Moreover, vε → 1 a.e., wε → χu(Ω) a.e.

I

∫ΩW (∇u) dx +Hn−1(Ju) + Per u(Ω)

≤ lim infε→0

Iε(uε, vε,wε).

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Particular cases of upper bound

For u piecewise regular (in particular, for u creating one cavity or

one crack), there exists (uε, vε,wε) admissible such that uε → u,

vε → 1, wε → χu(Ω) a.e., and∫ΩW (∇u) dx +Hn−1(Ju) + Per u(Ω) = lim

ε→0Iε(uε, vε,wε).

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Construction of recovery sequence

Ω

Juu

I Regularize u around the singularity. uε = u outside the

singularity.

I vε and wε follow optimal profile given by the O.D.E., from 0

to 1. Interface of width ε.

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Numerical experiments

For the moment,

Iε(uε, vε) =

∫Ω

(v2ε + ηε)W (Duε)dx+

∫Ω

[ε|Dvε|2

2+

(1− vε)2

]dx.

Alternate minimization: uε and vε.

Gradient flow for uε, stabilized Crouzeix-Raviart.

Ill-conditioned linear equation for vε.

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u(x) = λx, x ∈ ∂Ω.

Nearly incompressible:

W (F) =|F|p

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p = 1.5, ε = 0.01, η = 10−7.25/30

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A slab with three micro-holes, λ = 4: void coalescence and rupture27/30

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Gent & Lindley’s experiments: cavitation in rubber

(A. Gent & P. Lindley ’59)

(Aıt Hocinea, Hamdib, Naıt Abdelazizc, Heuilletb, Zaıric ’11)

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