spanning time-scales in simulations of irradiated materials · 2009-07-15 · testing of candidate...
TRANSCRIPT
1
Lawrence Livermore National Laboratory
Spanning time-scales in simulations
of irradiated materials
Vasily Bulatov
Lawrence Livermore National Laboratory
Livermore, California
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Lawrence Livermore National Laboratory
Outline
Materials in nuclear reactors
Physics of material damage under irradiation: time-scales
Modeling and simulations in nuclear materials R&D
Spectrum of simulation approaches Atomistic simulations Lattice Monte Carlo Object Monte Carlo Mean-field Rate Theory (Cluster Dynamics)
Summary
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Lawrence Livermore National Laboratory
Outline
Materials in nuclear reactors
Physics of material damage under irradiation: time-scales
Modeling and simulations in nuclear materials R&D
Spectrum of simulation approaches Atomistic simulations Lattice Monte Carlo Object Monte Carlo Mean-field Rate Theory (Cluster Dynamics)
Summary
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Lawrence Livermore National Laboratory
Uses of materials in nuclear power generation
Materials: fuel, reactor core, pressure vessel, control rods,
coolant, moderator, waste forms, …
Extreme conditions: intense irradiation, high temperature,
high pressure, mechanical stress, aggressive
chemistry, long replacement cycle
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Lawrence Livermore National Laboratory
Material properties place constraints on reactor design
High radiation tolerance - aggressive (streamlined) design
High operation temperature - efficient energy conversion
High mechanical strength - lower cost
High corrosion resistance - longer replacement cycle
…
Important: need to reduce and/or quantify variations in materials properties
Improved materials reduce cost and environmental impact of nuclear energy generation
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Lawrence Livermore National Laboratory
Materials challenge for future nuclear energy
Fusion energy at LLNL
• NIF inspires research in ICF and new materials • LIFE-engine concept for fusion/fission energy
• Applications for dozens of new nuclear plants in NRC • ANES initiatives: Gen-IV, GNEP, ITER
Critical need for new materials to resist very large doses of irradiation, high temperature and aggressive chemistry
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Testing of candidate materials
To be useful material testing must be accelerated
IFMIF - International Fusion Materials Irradiation Facility
Accelerator-based source of high-energy neutrons
Estimated construction cost ~ €1 billion (Japan, EU, US, Russia)
Estimated completion ~ 2017
Host country - Japan
Catch-22 in materials R&D for nuclear energy
The only fully reliable means to assess a candidate material is …
to build the reactor and run it over the material lifetime
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Lawrence Livermore National Laboratory
Ion accelerators for accelerated materials testing
Test Reactors • ATR ~ 10 dpa/year • HIFR ~ 20 dpa/year
Ion Beam Systems • IBA ~ 50 dpa/day • CAMS ~ 50 dpa/day
JANNUS facility at CEA France
Three accelerators (1) heavy ions (2) helium (3) hydrogen
Advantages
Estimated construction cost ~ €10-30M
High testing throughput
Materials are not activated (safe to handle)
Big question: Can material performance be extrapolated from 6 hours of violent irradiation to 10-50 years in the reactor?
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Lawrence Livermore National Laboratory
Computer simulations can play an important role
10-12 10-6 1 106 1012 seconds
Accelerated irradiation tests
Material lifetime in a reactor
Cooling off of hot collision cascades
To bridge the time scale gap simulations must be efficient
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Lawrence Livermore National Laboratory
To be useful material simulations should be accurate
(model accuracy) x (computational efficiency) = simulation utility
(acc
urac
y) =
(err
or) -
1
(efficiency) = (computational cost) -1
Often accuracy can be traded for efficiency
Method 1
Method 2
Method 3
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Lawrence Livermore National Laboratory
Outline
Materials in nuclear reactors
Physics of material damage under irradiation: time-scales
Modeling and simulations in nuclear materials R&D
Spectrum of simulation approaches Atomistic simulations Lattice Monte Carlo Object Monte Carlo Mean-field Rate Theory (Cluster Dynamics)
Summary
Radiation damage 101
PRA - T PRA - T’ PRA - T’’
SRA - T’ SRA - T’’
TRA - T’’
etc.
PRA - T γ, α (Ηε), p (H) n E’
n - E
Neutrons propagate over 1-10 cm
between scattering and reaction events
n-scattering and reactions → primary recoil atoms (PRA)
PRA average T ≈ 10 keV fission
PRA average T ≈ 50 keV fusion
E E’
E’’ E’’’ n
Displacement cascades
5 nm
Primary damage in�iron at 100K
~ 50 keV PKA
10 keV PKA
* Stoller,JNM 276 (2000) 22.
Each PKA inflicts further damage
First 10 picoseconds: hot stage of damage evolution
7 keV Cascade in Ni (fcc)
• V and V-clusters
• I- and I-clusters
• Replaced atoms or ballistic jumps
self-interstitial atom
vacancy
solute atom (e.g., Cr, Cu,
Mn, Ni)
Diffusion-reaction processes
Vacancy clustering Collapse into vacancy loops
Interstitial clustering Form interstitial loops
Annihilation
10 picoseconds to years
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• Primary damage creates enormous SIA and vacancy super-saturations
• Some vacancies and SIA recombine
• Annihilation at sinks (dislocations, grain boundaries,…)
• Sink bias driven clustering of vacancies (cavities) and SIA (dislocation loops) as well as dislocation climb (by excess SIA flux)
Long range defect migration and interactions
Ei
Dislocations actively remove interstitials
I
V Ev
I
V
Dislocation bias
Co-evolution of irradiation defects and material microstructure
Ji Jv
growing SIA loop
Jv
Ji
growing void absorbs I
jog moves right
absorbs V
jog moves right
extra half-plane of atoms
jogs dislocation
Dislocation climb
Helium accumulation and storage
PRA - T γ, α (Ηε), p (H) n E’
n - E
He
n
n
i v
n
v
B
L
FMS Dislocation
GB
SIA vacancy
voids
GB He bubbles
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Complex microstructural evolution in multicomponent-multiphase alloys
Ballistic mixing and amorphization
Vacancy/SIA supersaturations cause radiation enhanced diffusion (RED)
Coupled solute-defect fluxes to sinks cause radiation induced
segregation (RIS) and precipitation (RIP)
Destabilization of Fe-Cr-Ni austenite
Kinetically modified phase boundaries
Grain boundary segregation
…
Radiation assisted (induced, driven) alloy kinetics
GB solute segregation in SS
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Exposure to neutrons degrades mechanical properties: - Volumetric swelling (0.3 - 0.6 Tm) - Irradiation hardening and embrittlement/decreased uniform elongation (< 0.4 Tm) - Irradiation (<0.45 Tm) and thermal (>~0.45 Tm) creep - High temperature He embrittlement (> 0.5 Tm)
Irradiation effects on materials properties
0.01
0.1
1
10
100
1000
104
0.1 1 10 100 1000 104
Creep Rupture Life of 20% Cold-workedType 316 Stainless Steel at 550˚C, 310 MPa
Cre
ep
ru
ptu
re life
(h
)
Helium concentration (appm)
(2.0 dpa)
(3.4 dpa)
(43 dpa)
0
Severe embrittlement due to He bubbles
Austenitic steels irradiated infast breeder reactors
Before After
Radiation-inducedswelling Void formation
Complex synergistic interactions with He and H
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Lawrence Livermore National Laboratory
Atomistic Monte Carlo simulations of collision cascades
Brute force (honest) atomistic simulations are limited to the first 10-1000 ps: further cascade annealing by defect diffusion takes place on longer time-scales
85 keV displacement cascade in U-235
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Lawrence Livermore National Laboratory
How to extend the time horizon of atomistic simulations
The notorious rare-event problem
To access the more interesting infrequent events taking place over longer time scales, the atomistic simulation must be somehow accelerated
A
B
S E
x
kT
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Lawrence Livermore National Laboratory
The notorious rare-event problem
To access the more interesting infrequent events taking place over Longer time-scales, atomistic simulations must be somehow accelerated
A
B
S E
x
kT
How to extend the time horizon of atomistic simulations
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Lawrence Livermore National Laboratory
Atomistic Monte Carlo simulations of cascade annealing
Further cascade annealing by defect diffusion takes place on time scales not accessible to brute force atomistic simulations
The notorious rare-event problem
To access the more interesting infrequent events taking place over longer time scales, the atomistic simulation must be somehow accelerated
A
B
S E
x
kT
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Lawrence Livermore National Laboratory
Atomistic simulations of infrequent events is a vast and striving area of current research Distort the system’s dynamics in such a way as to enhance
the probability of sampling infrequent events of interest
An ideal accelerated method should
• sample rare events much more efficiently than the brute-force (honest) atomistic simulation
• be unbiased, i.e. should sample the same rare events and with the same probabilities as the brute-force simulations
• supply an estimate of sampling errors
• contain information on the true rates of sampled rare events
Slow dynamics of cascade annealing presents a relevant and challenging case for method development
No existing method satisfies all the above requirements
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Building an optimal bias potential on the fly
• The evolution of the atomistic system is guided via a (external) bias potential defined on a low dimensional subspace of steering coordinates. • The choice of the steering subspace is not unique but a good choice will make simulations more efficient.
Begin the simulation with zero bias potential
Find the number of times the simulation trajectory passed through each state in the steering subspace
Update the bias potential to steer away from the frequently visited states in the steering subspace
Continue the simulation with the new bias potential
A puppet master
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Lawrence Livermore National Laboratory
Two important potential advantages of the low dimensional steering method
Adaptively constructed on-the-fly, the bias potential is provably optimal, steering the atomistic
model most efficiently towards the unexplored states and providing the best statistics on the
weights of the previously explored states.
The method is firmly rooted in the rigorous mathematical formalism of Importance Function
that contains information on the rates at which the observed transitions would have taken
place in an honest simulation as well as the variance between the “smart” and the “honest”
atomistic simulations.
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Lawrence Livermore National Laboratory
Kinetic Monte Carlo simulations of damage accumulation kinetics
KMC uses inputs from MD simulations and/or experiment to model kinetics of damage accumulation over longer time scales
Detailed analysis of defect mobility
Cu atoms
Vacancies
MD simulations Individual defects
MD simulations Displacement cascade
Point defect production
10-11sec 10-8m
Kinetic Monte Carlo Damage accumulation
Diffusion, annihilation, clustering
? sec 10-7m Damage source
Adapted from N. Soneda
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2. Lattice Monte Carlo simulations
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Lawrence Livermore National Laboratory
B B
B B
B B
B B
B B
B B
B B
B
B B
B
B B
B
B B
B B
B B
B B
B B
B B
B B
B
B B
B
B B
B
The lattice gas models of alloys
*V = vacancy
A sequence of vacancy-atom exchanges
B B
B B
B B
B B
B B
B B
B B
B
B B
B
B B
B
A sequence of vacancy-atom exchanges
B B
B B
B B
B B
B B
B B
B B
B
B B
B
B B
B
A sequence of vacancy-atom exchanges
B B
B B
B B
B B
B B
B B
B B
B
B B
B
B B
B
Has been extended to binary alloys under irradiation
The ABV model describes kinetics of alloy
microstructure evolution
• phase nucleation, growth and coarsening
• spinodal decomposition
A simple and accurate model of binary alloys, e.g.
Fe-Cu and Fe-Cr.
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Lawrence Livermore National Laboratory
Two sources of inefficiency in lattice KMC simulations
1. Low density of mobile species (vacancies, interstitials)
Remedy: use spatial protection and asynchronous propagation as in FPKMC
2. Trapping of mobile species
Trapping of mobile species
Vacancy bound to a B7 cluster
B B
B B
B B
B B
B
B B
B
B
B
B
B
B B
B B
B
B
B
This ubiquitous problem is known as stiffness
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Lawrence Livermore National Laboratory
Local traps: clusters of micro-states connected by fast transitions
Potential for efficiency • Exact and fast first-passage exploration of traps - no repeat transitions! • Size of the growing web can be effectively controlled by pruning • Locally adaptive self-tuning algorithm - the web grows larger in stronger traps • Efficient even when there is no clear time scale separation between fast and slow transitions
First-passage propagation on Markov webs
This method is likely to increase computational efficiency of lattice Monte Carlo simulations by orders of magnitude
System’s micro-states
Current state
Any other state
Transition between two states
Possible next states
Previously visited states
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Lawrence Livermore National Laboratory
Direct coarse-graining of the microscopic Master Equation Master Equation:
€
˙ P (x, t) = P(x',t) ⋅ wx'∑ (x'→ x) − P(x, t) ⋅ w(x → x')
• Microstate x = (σ1, σ2, …, σN) • P(x,t) is the probability of microstate x at time t
• w(x→x’) is the rate of transitions from microstate x to microstate x’ 8x8 cells
Microscopic lattice sites
Define coarse variables:
€
Si = σ jcell∑
€
P(X,t) = P(x, t)x /X∑Coarse state X = (S1, S2, …, SM) and its probability
€
˙ P (X,t) = P(X ',t)w(X '→ X)(t) − P(X,t) w(X → X ')(t)X '∑
X '∑Projected ME:
€
w(X → X ')(t) =def P(x, t)
P(X,t)x /X∑ w(x→ x') =
x ' /X '∑ P(x | X)(t) w(x→ x')
x ' /X '∑
x /X∑where:
Adiabatic elimination closure:
€
P(x | X)(t)≈ Peq (x | X)
time-independent coarse rates
€
w(X → X ')
Solve the coarse-grained Master Equation by KMC on the cells
AB model
+ Coarse-graining
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Lawrence Livermore National Laboratory
Multi-resolution adaptive Monte Carlo
Nested coarse-graining can be repeated as many times as necessary
Local refinement: initialize microscopic simulations where and when needed
(to resolve interesting sub-scale events) Adaptivity: On the finest scale reduces to the (exact) microscopic Monte Carlo
On the coarsest scale treats the largest elements of material microstructure
The Master Equation has the same structure on all scales
AMR software manages refinement and coarsening
AMR research issues Criteria for refinement and coarsening in a fully stochastic simulation
Kolmogorov distance, other measures?
A posteriori error estimates (by concurrent simulations on sub-scales) A priori error bounds?
Further topics What can justify adiabatic elimination in the absence of naturally disparate time scales?
Non-Markov Monte Carlo?
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Lawrence Livermore National Laboratory
Computational challenge of low density
Great many diffusion hops necessary to bring particles to collisions at low density
Time to collision ~ (inter-particle spacing)3
KMC is a powerful and robust method but grossly inefficient when the density of diffusing particles is low
A kinetic Monte Carlo simulation
Diffusion-controlled annihilation A + A → 0
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Lawrence Livermore National Laboratory
0.81 0.43 0.19 0.01
1.21 0.03 0.58
τ1
τ2
τ4 τ3
τ5
τ6
τ7
The new method of first-passage Monte Carlo
For each walker randomly sample first passage time from PFP(t<τ) and order them in a time queue
τ3 < τ4 < τ1 < τ2 < τ7 < τ6 < τ5
Construct disjoint protective regions (cubes, spheres) centered on the particles at t = 0
new τ3
Repeat:
• Find the earliest time in the queue
• Propagate the particle to boundary
• Construct a new protective region
• Pick a new event time, insert into the queue
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Lawrence Livermore National Laboratory
The algorithm is exact for N-body diffusion-reaction models
Efficiency: on each step only one particle propagates over a long distance
Asynchronous algorithm: every particle lives on its own time clock
Diffusion-controlled annihilation A + A → 0
FPKMC simulation
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Lawrence Livermore National Laboratory
A test of algorithm efficiency
Kinetics of A + A → 0 annihilation reaction in 1d
Our exact method: 220 decades of time evolution simulated in 15 minutes Previous record: 8 orders of magnitude in time over a month of number-crunching
For a wide class of reaction-diffusion processes the new algorithm is exact and super-efficient
Age of the universe One particle in the universe
5.0−t
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Lawrence Livermore National Laboratory
FPKMC has been adapted to simulations of irradiated materials
Needed to introduce additional kinetic mechanisms
Particle insertion: Frenkel pairs, cascades
Clustering: An + Am = An+m
Annihilation: An + Bm = An-m
Emission: An = An-1 + A1
Others
We developed a general protocol that allows to define material models of arbitrary complexity
single particle events
two particle events
Others
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Lawrence Livermore National Laboratory
A model of α-Fe under electron irradiation
A. Barbu et al., Philosophical Magazine 85, 541-547 (2005)
Periodic in X and Y
0.287µ
e-
SIA V
• Only single vacancies and single SIA can diffuse.
• All other defects are immobile.
This is not the most realistic but a well-studied model
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Lawrence Livermore National Laboratory
FPKMC pushes the simulation limits: α-Fe under electron irradiation
T=200oC and dose rate G = 1.5 x 10-4 dpa/s
10 min: 0.018 dpa 10 days: 18 dpa
Exceeds the previous simulated dose limit by 1000 times
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Lawrence Livermore National Laboratory
JERK versus FPKMC : computational efficiency
Cube of α-Fe ( L = 1000 a0 ) 70 K, 300 K, 500 K
From T. Luypaert and C. Marinica
JERK
FPKMC
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Simulations of resistivity recovery
T
electron irradiation isochronal annealing
e- e- e- e- e-
T
Res
istiv
ity (ρ
) T
-dρ/
dT
recovery stages
recovery stages correspond to annihilation, clustering or dissociation of defects experiments allow to calibrate the rates of various recovery processes which mechanisms ?
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Low dose: 2.10-6 dpa
JERK versus FPKMC : cross-validation
JERK: 540 CPU hours
FPKMC: 0.25 CPU hour
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High doses and low dose rates
Such simulations have not been done with any other KMC algorithm
Dose rate dpa/s
Total dose dpa
Simulated time
Performance s/cpus*
Performance dpa/cpuday**
1.5 x 10-4 18 33 hours 0.14 1.80
1.5 x 10-5 2.9 54 hours 1.3 1.67
1.5 x 10-6 4.1 31 days 12.7 1.63
1.5 x 10-7 1.6 125 days 154 1.98
1.5 x 10-8 10 21 years 2,080 2.67
1.5 x 10-9 8.3 175 years 23,200 2.96
*one second of CPU time **one day of CPU time
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Accelerated test (10-4dpa/s) versus reactor (10-8dpa/s) irradiations to 10 dpa
More sophisticated models are being developed for FPKMC simulations of a-Fe JANNUS experiments are planned to test the new models
The new method is efficient at high and low dose rates
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1. FPKMC + Rate Theory
Here we propose:
To extend the FPKMC method to include elastic interactions among the radiation-induced defects
To use FPKMC simulations to examine and improve the accuracy of the Rate Theory models
A. Donev (LLNL), C. Marinica, T. Luypaert (Saclay) and S. Le Bourdiec (EDF)
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Lawrence Livermore National Laboratory
Rate Theory
Rate Theory is a mean-field method - ignores spatial correlations
€
˙ V n = kv (n −1)Vn−1V1− ki(n)VnI1+ kdis(n +1)Vn +1 − kdis(n)Vn
€
˙ I n = kv (n −1)In−1I1− ki(n)InV1+ kdis(n +1)In +1 − kdis(n)In
€
˙ V 1 = Γirr− ˙ V 1(sinks) + kdis(n)Vnn= 2
∞
∑ − kv (n)Vnn=1
∞
∑ V1
€
˙ I 1 = Γirr− ˙ I 1(sinks) + kdis(n)Inn= 2
∞
∑ − ki(n)Inn=1
∞
∑ I1
Solve the set of ODE for In and Vn (n = 1, 2, …
For the same simple model of α-Fe under electron irradiation
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Lawrence Livermore National Laboratory
The RT method is very efficient but …
1. Mean-field assumption intrinsic to RT can be suspect
Remedy:
Use FPKMC to identify conditions where MF assumption breaks down
Go beyond MF approximation?
2. Difficult to treat complex defect populations, e.g. VnHem
n vacancies
VnHem cluster concentration
Too many equations to solve
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Lawrence Livermore National Laboratory
Eureka: Gillespie’s finite volume stochastic method
Solve the master equation directly by kinetic Monte Carlo in a finite volume Ω
Stochastic Rate Theory
Finite (and integer) number of clusters in the finite volume Ω
Monte Carlo rates taken directly from the RT equations
One defect reaction is selected per Monte Carlo cycle with correct probability
€
˙ N i = γ i −α iNi + kijNiN jj∑
Advantages
Extension to complex clusters, e.g. C-V-He, is very simple
Computational cost is controlled by the selection of volume Ω
Finite volume fluctuations are retained
Exact conservation of monomers and clusters species
Gillespie’s algorithm prevailed over ODE in bio-chemistry and cell-biology
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Lawrence Livermore National Laboratory
Summary
Accurate and efficient simulations of irradiation damage and its effects on material properties will be a critical component of materials development for future nuclear energy applications.
Novel method of First-Passage Kinetic Monte Carlo enables fully resolved simulations of damage accumulation on the reactor time scale.
Ongoing work on high-performance Monte Carlo methods focuses on three main approaches to microstructure simulations actively employed in the nuclear materials community:
(1) Stochastic Monte Carlo implementation of the Rate Theory (2) Markov-web acceleration for lattice kinetic Monte Carlo (3) Guided atomistic Monte Carlo simulations of cascade annealing
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Collaborators
LLNL A. Donev, J. Marian, B. Sadigh, A, Arsenlis, G. Gilmer
CEA Saclay M.-J. Marinica, T. Luypaert, M. Nastar, M. Athenes
KTH Stockholm T. Oppelstrup
EDF S. Le Bourdiec
Stanford W. Cai
U. Campinas, Brazil M. de Koning