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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