Post on 17-Jul-2020
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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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.
2/30
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”.
3/30
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]).
4/30
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.
5/30
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.
6/30
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
7/30
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).
8/30
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 .
10/30
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).
11/30
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
4ε
]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
4ε
]dx ≥ Hn−1(Ju).
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∫Ω
[ε|Dv |2 +
(1− v)2
4ε
]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)
16/30
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.
17/30
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
19/30
Elliptic approximation of I :
Iε(uε, vε,wε) :=
∫Ω
(v2ε + ηε)W (Duε) dx
+
∫Ω
[ε|Dvε|2
2+
(1− vε)2
2ε
]dx
+ 6
∫Q
[ε|Dwε|2
2+
w2ε (1− wε)2
2ε
]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).
21/30
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).
22/30
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.
23/30
Numerical experiments
For the moment,
Iε(uε, vε) =
∫Ω
(v2ε + ηε)W (Duε)dx+
∫Ω
[ε|Dvε|2
2+
(1− vε)2
2ε
]dx.
Alternate minimization: uε and vε.
Gradient flow for uε, stabilized Crouzeix-Raviart.
Ill-conditioned linear equation for vε.
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A ball with one micro-hole, λ = 1.5: void expansion
u(x) = λx, x ∈ ∂Ω.
Nearly incompressible:
W (F) =|F|p
p+ (detF− 1)2 +
2p/2−1
detF,
p = 1.5, ε = 0.01, η = 10−7.25/30
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A ball with two micro-holes, λ = 1.5: void coalescence26/30
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A slab with three micro-holes, λ = 4: void coalescence and rupture27/30
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A flawless ball, λ = 1.665: void and crack nucleation28/30
Gent & Lindley’s experiments: cavitation in rubber
(A. Gent & P. Lindley ’59)
(Aıt Hocinea, Hamdib, Naıt Abdelazizc, Heuilletb, Zaıric ’11)
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