Coupling deterministic and stochastic simulations: an ...cermics.enpc.fr/~terrierp/Terrier_MMM_2016.pdfP. Terrier, M. Athènes, T. Jourdan, G. Adjanor, G. Stoltz (Universities of Somewhere

Coupling deterministic and stochastic simulations: anapplication to cluster dynamics in materials

P. Terrier1 M. Athènes2 T. Jourdan2 G. Adjanor3 G. Stoltz1

1CermicsEcole des Ponts Paristech

2CEA/DEN/DANS/DMN/SRMP

3EDF R&D dpt. MMC

MMM, 2016

P. Terrier, M. Athènes, T. Jourdan, G. Adjanor, G. Stoltz (Universities of Somewhere and Elsewhere)Coupling deterministic and stochastic simulations: an application to cluster dynamics in materialsMMM, 2016 1 / 22

Outline

1 Background and model

2 Main ideaAssumptions and linearityAlgorithm

3 Simulating the evolution of large size clustersJump process approachFokker-Planck approach

4 Results


Background and model

Outline




4 Results



Cluster dynamics (Motivations)

X Simulate the microstructure evolution of materials under irradiation orthermal ageing

X Cluster dynamics (CD): mean-field approach, large set of rateequations → curse of dimensionality.→ deterministic approaches: Fokker-Plank approach1

→ stochastic approaches: SCD2

→ hybrid approaches 3

X We propose a hybrid deterministic/stochastic algorithm takingadvantage of both methods.

1[Wolfer et al., Tech. rep., Wisconsin Univ., 1977]2[Marian et al., J. Nucl. Mater., 2011]3[Surh et al., J. Nucl. Mater., 2004]



Cluster dynamics (Model)

Rate equations (for clusters of size n)

dCn

dt= βn−1Cn−1Cvac − (βnCvac + αn)Cn + αn+1Cn+1, n ≥ 2 (REn)

dCvac

dt= −β1C

2vac −

∑

n≥2

βnCnCvac +∑

n≥2

αnCn + α2C2 (REvac)

Conservation law

d

dt

Cvac +

∑

n≥2

nCn

= 0 (1)

Initial and boundary conditions

Cvac(0) = Cinit, Cn(0) = 0 ∀n ≥ 2 (2)




n − 1 n n + 1 n + 2n − 2

βn−1Cvac βnCvac

αn αn+1Figure: Emission and absorption rates in size space




Rate equations (for clusters of size n)

dCn

dt= βn−1Cn−1Cvac − (βnCvac + αn)Cn + αn+1Cn+1, n ≥ 2 (REn)

dCvac

dt= −β1C

2vac −

∑

n≥2

βnCnCvac +∑

n≥2

αnCn + α2C2 (REvac)

Conservation law

d

dt

Cvac +

∑

n≥2

nCn

= 0 (3)

Initial and boundary conditions

Cvac(0) = Cinit, Cn(0) = 0 ∀n ≥ 2 (4)


Main idea

Outline




4 Results


Main idea Assumptions and linearity

The problem we solve (1/2)

X Large set of non-linear ODEs→ Splitting into one non-linear ODE (Cvac) and a large set of now linear

ODEs (C2,C3, · · · )→ Evolve (C2,C3, · · · ) with fixed Cvac, then evolve Cvac with fixed

(C2,C3, · · · ).

t = t0 t0 + Kδt = tft0 + δt t0 + 2δt t0 + (K − 1)δt

Evolve C1

and C0vac

Set C0

and fix C1vac

Evolve

Evolve C2

and fix Ck−1vac

Evolve

Evolve Ck

(REn) is now linear → decomposition between small and large size clusters,with appropriate methods for each class.


Main idea Assumptions and linearity

The problem we solve (2/2)

X Large set of linear ODEs→ Decomposition and solving with dedicated methods

0 100 200 300 400 500 600 700 800Cluster size (n)

0.0

0.5

1.0

1.5

2.0

C(n)

×10−12

Boundary: Nfront

Buffer zone:[Nfront −Nbuff , Nfront +Nbuff ]

Small size clustersLarge size clustersStarting distribution

(a) Initial distributions

0 100 200 300 400 500 600 700 800Cluster size (n)

0.0

0.5

1.0

1.5

2.0

C(n)

×10−12

Boundary: Nfront

Buffer zone:[Nfront −Nbuff , Nfront +Nbuff ]

Small size clustersLarge size clustersTotal distributionStarting distribution

(b) Propagated distributions

Figure: Main idea of the algorithm: initial distribution decomposed into twodistributions, both distributions are propagated independently.


Main idea Algorithm


Main idea Algorithm

Summary


Main idea Algorithm

Summary


Simulating the evolution of large size clusters

Outline




4 Results


Simulating the evolution of large size clusters Jump process approach

Jump process approach (JP)

We see cluster dynamics as a jump process in size space:

n − 1 n n + 1 n + 2n − 2

βn−1Cvac βnCvac

αn αn+1Figure: Emission and absorption rates in size space

Characteristic jump time for a cluster of size n: ν−1(n) = (βnCvac +αn)−1.

Propagate independent particles → Highly parallelizable


Simulating the evolution of large size clusters Fokker-Planck approach

Fokker-Planck approach (FP)

Assuming that Cvac fixed, then for n� 1, the rate equations (REn) arewell described by the Fokker-Planck equation (FPE)

∂C

∂t= −∂FC

∂x+

12∂2DC

∂x2 , (5)

withCn(t) ' C (t, n), (6)

where F (x) = β(x)Cvac − α(x) and D(x) = β(x)Cvac + α(x). Thisequation is linked to the stochastic Langevin process in size space

dXt = F (Xt)dt +√D(Xt)dWt , (7)

where p(t, x) the law of (Xt)t≥0 is solution of the FPE.Propagate independent particles → Highly parallelizable


Results

Outline




4 Results


Results

Preliminary results

We present the results using a model of vacancy clustering during thermalageing4. Here, for a full ODE simulation:

0 500 1000 1500 2000Cluster size (n)

0

5

10

15

20

25

30

35

40

Jum

pti

me( ν−

1)

Buffer zone

t = 103 st = 104 st = 105 s

(a) Characteristic jump time

0 2000 4000 6000 8000 10000Time (s)

−1.6

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

dCvac

dt

×10−13

dCvac

dt

1

2

3

4

5

6

Cvac

×10−10

Cvac

(b) Cvac(t) and dCvacdt

4[Ovcharenko et al., Comput. Phys. Commun., 2003]


Results

Main result

Here with both coupling methods (Nsim = 3.107, Ncores = 30):

0 500 1000 1500 2000 2500Cluster size (n)

0.0

0.5

1.0

1.5

2.0

C(n)

×10−13

Boundary: Nfront

Buffer zone:

[Nfront −Nbuff , Nfront +Nbuff ]

FP approachJP approachReference solution

Good accuracy for both methods (FP and JP)


Conclusion

Conclusion

SummaryX Splitting of the dynamics, decomposition of the concentrationsX Versatile coupling algorithm, highly parallelizableX Simple resolution for small size cluster rate equationsX Dedicated stochastic methods for large size clustersX Avoiding high frequency events

Perspectives→ Introduce mobile clusters of size greater than one→ Study complex materials with different types of defects (higher

dimensions)See more on: Terrier et al., arXiv:1610.02949


Conclusion

Thank you

Thank you for your attention


Conclusion

Jump process approach (JP)

Assuming that Cvac is fixed, consider a particle of size N(t) that evolveswithin a small time step δτ with the following probabilities:

P [N(t + δτ) = n + 1|N(t) = n] = βnCvacδτ + o(δτ),

P [N(t + δτ) = n − 1|N(t) = n] = αnδτ + o(δτ),

P [N(t + δτ) = n|N(t) = n] = 1− (βnCvac + αn)δτ + o(δτ).

(8)

Let pn(t) = P(N(t) = n), then the forward-Kolmogorov equation of thisMarkov process is given by

dpndt

= βn−1pn−1Cvac − (βnCvac + αn) pn + αn+1pn+1, ∀n ≥ 2. (9)

Characteristic jump time for a cluster of size n: (βnCvac + αn)−1.

Propagate independent particles → highly parallelizable


Documents

Coupling deterministic and stochastic simulations: an ...cermics.enpc.fr/~terrierp/Terrier_MMM_2016.pdfP. Terrier, M. Athènes, T. Jourdan, G. Adjanor, G. Stoltz (Universities of Somewhere