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New Developments in Computational Modelingof Cryosphere SystemsFracture of an ice floe: modelisation and simulation
Dimitri Bălăs,oiuJuly 19, 2019
Université Grenoble Alpes, Jean Kuntzmann Laboratory
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Context of the presentation
We already have a granular model for the Marginal Ice Zone. Itdeals with:
• real-time simulation• 1 million floes• floes scale from several meters to several kilometers• variety of shapes• simulation window of 100 kilometers
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Context of the presentation: video
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Outline of the presentation
1. A brief overview of brittle fracture models
2. Our brittle fracture model : theoretical results
3. Numerical methods and results
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A brief overview of brittle fracturemodels
Griffith’s criterion for brittle fracture
Assume that the crack path is known. Denote by l(t) the crackat time t, and by k the toughness of the material.Energy relase rate:
G(t, l(t)) = − limh→0
W(t, l(t) + h)−W(t, l(t)).
Then G(t, l(t)) follows the evolution law:G(t, l(t)) ≤ kl(t) increasingdldt
(t) 6= 0 ⇒ G(t, l(t)) = k
Limitations:
• it does not account for fracture initiation• it does no predict fracture path
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A variational model for brittle fracture
Denote by Ω a floe, apply a displacement UD on ∂ΩD.Competition between:
• elastic energy : 12∫Ω\σ Ae(u) : e(u)dx,
• fracture energy : kH1(σ),
with A the stiffness tensor, e(u) = 12
(∇u+∇ut
)the symetric
gradient and H1(σ) the fracture’s lengh.
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A variational model for brittle fracture
Theorem (Dal Maso and Toader (02), Chambolle (03))The functional :
E(u, σ) = 1
2
∫Ω\σ
Ae(u) : e(u)dx + kH1(σ)
has a minimum over
(u, σ) |σ ⊂ Ω, σ closed ,
u ∈ L2(Ω,R2),u = UD on ∂ΩD, e(u) ∈ L2(Ω, S2(R))
The solution is a minimum of the floe’s energy.
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A variational model for brittle fracture
Theorem from image segmentation.
Theorem (De Giorgi, Carriero, Leacci (89))The Mumford-Shah functional:
F(u, K) = 1
2
∫Ω\K
|∇u|2 dx +∫Ω|u− g|2 + αH1(K)
has a minimum over(u, K)
∣∣ K closed ,u ∈ C1(Ω \ K).
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The classical numerical approach : Ambrosio-Tortorelli / phasefield approximation
We replace the unknown set σ by a function v :
v = 0 on the fracture, v = 1 on Ω \ Vε(σ).
ε
1v
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The classical numerical approach : Ambrosio-Tortorelli / phasefield approximation
We replace the unknown set σ by a function v :
v = 0 on the fracture, v = 1 on Ω \ Vε(σ).
Theorem (Ambrosio and Tortorelli (90), Bourdin (98))The sequence of functionals
Fε(u, v) =1
2
∫Ωv2Ae(u):e(u)dx+ε
∫Ω‖∇v‖2 dx+ 1
4ε
∫Ω(1−v)2 dx
Γ-converges to the functional F(u, σ)
Major inconvenient in our case: it needs tight meshs.
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Our brittle fracture model :theoretical results
Our take on the problem
We make restrictive hypothesis on the fracture: they are lines.
We study the minimum problem:
infσ∈Σ
infu∈Vσ
∫Ω\σ
Ae(u) : e(u)dx +H1(σ)
with the variational spaces
Σ =[a ,b] ⊂ R2
∣∣a ∈ ∂Ω,b ∈ Ω, ]a ,b[ ⊂ Ω
Vσ = u ∈ H1(Ω \ σ,R2) | u = UD on ∂ΩD \ σ,(u+ − u−) · ν ≥ 0 on σ
Σ is endowed with the Hausdorff distance dH.
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An existence result
Theorem (Labbé, Lattes, Weiss, Balasoiu)The minimun problem:
infσ∈Σ
infu∈Vσ
∫Ω\σ
Ae(u) : e(u)dx +H1(σ)
has a solution over⋃
σ∈Σ σ × Vσ .
ProofSuppose (σn)n∈N ⊂ Σ such that:
infu∈Vσn
E(u, σn) → infσ∈Σ
infu∈Vσ
E(u, σ).
There exists un ∈ Vσn such that: E(un, σn) = infu∈Vσn E(u, σn).There exists σ ∈ Σ such that dH(σn, σ) → 0.
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An existence result: proof
Proof.Embeding: ∀σ ∈ Σ, Vσ ⊂ V = L2(Ω,R2)2 × L2(Ω,R4)2.Mosco convergence: Vσn → Vσ
• ∀ϕ ∈ Vσ,∃ϕn ∈ Vσn , ϕn → ϕ strongly• ∀ϕn ∈ Vσn bounded in V , ∃ϕ ∈ Vσ, ϕn → ϕ weakly
There exists u ∈ Vσ , such that un → u weakly.Let v ∈ Vσ , there exists vn ∈ Vσn such that vn → v strongly.Use variational formulation for fixed n ∈ N:∫
Ω\σnAe(un) : e(vn − un)dx ≥ 0.
Go to the limit with strong convergence of vn and weakconvergence of un: u verifies the variational formulation.
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A quasistatic framework: mitigation of the strong geometric hy-pothesis
Traction on ∂ΩD : UD(t) = tUDLet 0 = t0 < · · · < ti < · · · < tp = 1 be a subdivision of [0 , 1].A discrete evolution satisfies:
1. u0 = 0 IdΩ and σ0 = ∅,2. ∀i ∈ 1, . . . ,p , σi ∈ Σσi−1
,
3. ∀i ∈ 1, . . . ,p , ∀σ ∈ Σσi−1,∀u ∈ Vti,σ, E(ui, σi) ≤ E(u, σ).
with the variational spaces:
Σσ =σ ∪ [a ,b] ⊂ R2
∣∣a = end(σ),b ∈ Ω, ]a ,b[ ⊂ Ω \ σ
Vt,σ = u ∈ H1(Ω \ σ,R2) | u = UD(t) on ∂ΩD \ σ,(u+ − u−) · ν ≥ 0 on σ
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A quasistatic framework: convergence result
Theorem (Dal Maso and Toader, Chambolle)The discrete evolution converges to the continuous evolution:
1. σ(0) = ∅,2. ∀t ∈ [0 , 1], σ(t) is a rectifiable curve,3. ∀t1 < t2, σ(t1) ⊂ σ(t2),4. for all rectifiable curves τ ⊃
⋃s<t σ(s):
infu∈Vt,σ(t)
E(u, σ(t)) ≤ infu∈Vt,τ
E(u, τ).
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Numerical methods and results
Numerical scheme: requirements
The number of mesh in a floe does not deppend of the floe’ssize.On large floes : large cells.We need the fracture’s location to be independent of themesh’s precision.
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An ad-hod finite element method
Set:
Tτ,σ = ω ∈ τ2 |ω ∩ σ 6= ∅ντ,σ = p ∈ τ0 |p ∈ σ∪
p ∈ τ0 | ∃ω ∈ Tτ,σ,p ∈ ω
For each finite element ep based on a node p, define:
e+τ,p = eτ,p∑
ω∈Tτ,σ
1ω+ , e−τ,p = eτ,p∑
ω∈Tτ,σ
1ω− ,
Finite element space:
Wτ,σ = spanp∈τ0\ντ,σ
(eτ,pux, eτ,puy)+ spanp∈ντ,σ
(e+τ,pux, e−τ,pux, e+τ,puy, e−τ,puy).
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A convergence result
Set:Σn ⊂ Σ a discretization of Σ
andV =
⋃σ∈Σ
Vσ × σ , Vn =⋃
σ∈Σn
Wτn,σ × σ .
Define:
En : V → R
(u, σ) 7→E(u, σ) if (u, σ) ∈ Vn+∞ otherwise
Theorem (Labbé, Lattes, Weiss, Balasoiu)The sequence of functionals (En)n∈N are equi-coercive, andΓ-converges to E (for the topology of V).
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Numerical solution: initiation in finite time
Classic elastic behavior (smalltraction) With fracture (huge traction)
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Numerical solution: influence of fracture boundary precision
Small precision High precision
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Numerical solution: influence of fracture angular precision
Small precision High precision
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Numerical solution: path prediction
Rigid circular inclusion with k1 >> k.
Coarse mesh Tight mesh
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References
Dimitri Balasoiu, Stéphane Labbé, Philippe Lattes, andJérome Weiss. «An Efficient Variational Model for BrittleFracture». In : (To appear).
Matthias Rabatel, Stéphane Labbé, and Jérôme Weiss.«Dynamics of an Assembly of Rigid Ice Floes». In : Journalof Geophysical Research: Oceans 120.9 (2015),pp. 5887–5909.
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