Challenges For
Multiscale Modeling ofFusion Materials
N.M. Ghoniem (UCLA) and B. Wirth (LLNL)
Fusion Simulation Project Workshop
September 17-18, 2002
Hyatt Regency Islandia
San Diego, CA

9/16/20021

Outline Approach and Materials Environment; Surface and Bulk Phenomena;
 Plasma Physics – Materials ScienceAnalogies;
 Multi-scale Modeling Strategy;
 Fundamental Equations and Algorithms;
 Modeling Challenges and Limitations.

9/16/2002

2

Approach and Materials
Environment

Environment Heat Flux: FW ~1 MW/m2; Divertor ~5 –15
MW/m2
 Predictive; Neutron Flux: ~ 3 – 5 MW/m2

Approach Physics-Based; Particle Flux:
Divertor ~1021-1022 m-2s-1  Computational Design of Materials;
Mechanical Loads: Pressure ~ 2-5 MPa
 Experimentally-verifiable at Scale Interfaces.
9/16/2002

3Surface Phenomena
Sourceatom

High Heat Flux/ Particle Flux result in:(0.01eV)

Short timescale phenomena (e.g. 10-12 – 10-9 s):
Sputtering;[011]

Implantation of helium and tritium;Re-deposition and tritium co-deposition;
Near-surface damage (collision cascades).

[111]

[211]

Uni-axialTension

Thermostatlayers
Fixedlayers

Long timescale phenomena (e.g. 10-3 – 106 s):
Atomic transport (e.g diffusion, trapping,adsorption, recombination and desorption);Surface roughening and re-structuring;Microstructure and phase evolution (e.g.
voids, bubbles, dislocations, grains & newphases).
Surface Re-structuring after re-deposition.
H. Huang, RPI
9/16/20024

Bulk PhenomenaHigh Heat Flux/ Neutron flux/ Mechanical
Loads result in:
Short timescale phenomena (e.g. 10-12 – 10-9 s):
Atomic Displacements;
Fast Transport;

Lifetime(Yrs)Lattice Defects (Vacancies and Interstitials).

Ductile-toBrittle
Transition(DBTT)

Dimensional
Instability(SwellingCreep)

HeliumEmbrittlement

Long timescale phenomena (e.g. 10-3 – 106 s):
Microstructure Evolution (Voids, Bubbles,Dislocations, Phases);

Materials Design
Window

Dimensional Instabilities (Swelling andCreep);Shear Bands (Localized plasticity);

Temperature (OC)Helium Embrittlement.

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5

Material-Plasma Interfacing

Material
Bulk

Material
Surface

PlasmaEdge

PlasmaCore

VFTRIM;
REDEPHEIGHTSBPHI-3D
UEDGE-2D

Transport;Turbulence;
MHD;
Confinement;
Islands,Stability &Oscillations.

CracksAb initio;
MD
KMC
DD
Rate Theory
FEM
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6

Correspondence & Analogy
Phenomenon

PlasmaMaterialDensity & Degrees ofFreedom per cm3

1014 - 1016

1023

ForcesLong-range: Coulomb,
Electromagnetic

Short-range: Atomic > Pair,
Many-body
Long-range: Elastic

Particle Methods

Particle-Particle (P-P);Particle-Field (PIC);
KMC

Particle-Particle (MD);Particle-Field (DD-FEM);
KMC, Lattice MC, Event MC.

Transport &ContinuumCollisions & Fokker-Planck;Fluid, MHD
Reaction Cross-sections;Turbulence Microstructure Evolution &Fokker-Planck*;Elasticity;Rate Theory;
Plasticity

Space: Islands, Coherent
Structures;
Time: Oscillations, Disruptions

Space: Self-organization,segregation;
Time: shear bands, cracks.

Instabilities*H. Huang and N.M. Ghoniem, "Formulation of a Moment Method for n-dimensional Fokker-Planck Equations",
Phys. Rev. E, 51, 6: 5251-5260, 1995.
9/16/2002

7

Correspondence & Analogy
Bio-Savart

Electromagnetics

Dislocation Dynamics

Magnetic intensity

HiStrain

ε ijMagnetic induction

BiStressσ ij

Current density

Ji

Incompatibility tensor

ηijPermeabilityµElastic constants

E, νVector potential

Ai

Stress function

χ ij

Current

IBurgers vectorbiMaxwell’s Equation:∈ijk Hk , j = Ji

Incompatibility − ∈ ∈ εikl jmn nl ,km = ηij
Equation:
SdS

(a )
(b)

9/16/2002(c )Contour Gdl

8

Multi-scale Modeling StrategymContinuum
Mechanics
Statistical
Mechanics

mm

Dislocation
Dynamics

Monte
Carloµm

Classical
MDX

4000

3000

Quantum
Mechanics

2000

1000

01000
2000

psy(aS3

9/16/2002

01000

Sfs

Y

5000z(a)nmZσ = 165 Mpabns)

2000
3000

300040003)
x(a4000
5000 5000µsTimemssks9

Atomistic Simulations*
Molecular dynamics
First-principles(<200 atoms,<10ps)

 Start from Schroedinger’sEquation;

 Empirical Potentials;
 Initial defect distribution;
 Verlet or predictor-corrector;
 time-step ~ 1 fs;
 Short-range forces; Parallelization by spatial decomposition withMPI.

 (1-100 million atoms, < 100 ns)

 Approximate: DFT;
 Accurateenergetics ofpoint defects and defectclusters

KMC (<µm,<ms) Freeze atomic degrees of Freedom; Track defects only; Microstructure evolution of defects
 Spatial inhomogeneity.

* Srolovitz and Carr - Princeton
9/16/200210

Interatomic Potentials and MD Simulations
Stoller, ORNL

 Born-Oppenheimer: Adiabatically eliminate
nuclear degrees of freedom. Solve only forelectrons.
 Kohn-Sham-Hohenberg: Density Functional
Theory (DFT) reduces to the single electron
quantum problem, with effective potentials.
Exchange-Correlation potentials areapproximated with the Local Density
Approximation (LDA).
 Using DFT-LDA material properties havebeen calculated without input.

Quantum MD
9/16/2002

Bond-order
Potentials

Classical MD with Empirical Potentials
11

Dislocation-Microstructure Interaction:
KMC Modeling of Pinning and Rafting

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12

Mesoscopic Simulations: Dislocation
Dynamics
Covariant Vectors

S( a1 = e , a2 = t , a3 = b)

e=R
R

t=

ΤT

T=dldwdS

Contravariant Vectors
a1 =1( a 2 × a3 )
2πV

a2 =

1
( a3 × a1 )
2πV

a3 =

1
( a1 × a2 )
2πV

P(0)

bROx3T ( w)Curved dislocation segment
Represented by a spline

QP

w=1x2x1dl

Contour GT(0)V = ( a1 × a2 ) ⋅ a3
9/16/2002

w=0

P(1)
T(1)
13

Differential Forms of DD are analogous toElectromagnetics, but of higher dimensionality
Bio-Savart

Displacement
du T  ( s × a1 ) ⋅ a2V1 1
12
ν
aaa1 =+−+() 3
dw 4 R  π (1 + s ⋅ a1 )1 −ν2π Stress
dσ µVT  11112
2=( a ⊗ a1 + a1 ⊗ a ) + ( a ⊗ a2 + a2 ⊗ a ) − 2π (1 − ν ) ( a1 ⊗ a1 + I ) 
dw 2 R 2 1 − νInteraction Energy
1 I IIIIIIIII⋅⋅−⋅⋅
aaaaaaaa()()
()()
23
2
32233
dEintµT T 1 −ν
= − I II 
14π R  2νdwI dwII
IIIIIIIIII
II+a2 ⋅ a3 )( a2 ⋅ a3 ) −a2 ⋅ a 2 )( a3 ⋅ a1 )( a3 ⋅ a1 )
((1 −ν 1 − ν

9/16/2002

14

Weak Variational Form for DD
Equations of Motion
Equations of Motion∫ ( f − BV ) δ r ds = 0ΓDefine:tkk

V =krr =af* =*drdt

fµaf = fP−K + fself + fothers

t* =µtBFinal Equation of Motion

KdQ=F*dt

Q = [ P1 , T1 , P2 , T2 ]Q=Nodal coordinate vector
F=Nodal Forces
K=Mobility Matrix
9/16/200215

Mesoscopic Simulations ofPlasticity

One micron

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16

Parallel Dislocation Dynamics Code
Algorithm Description
 Similar to the N-body problem in plasma & astrophysics;
 Hierarchical tree representation can be utilized in the DD code.
 The Major difference is that dislocation lines are irregular filaments &not particles.
 Far-field interaction is approximated by multipole expansions.
 Hierarchical tree-based methods reduce the computational complexity
from O(N*N) to O(N log(N)) or even O(N) .
 Special techniques are used to maintain load balancing and reduce
communications between computer cluster nodes.

9/16/2002

1.

Jaswinder Pal Singh, et al. J. Parallel and Distributed Computing. 1995; 27:118.

2.

Ananth Y. Grama, et al. SIAM Conference on Parallel Processing, San Francisco, 1994.

3.

H.Y. Yang, and R. LeSar. Philosophical Magazine A, 1995; 71(1):149.
17

DD & KMC Parallel Algorithms work
well for a small number of processors
DD Algorithm

KMC Algorithm

9/16/2002# of processors

12

3

4

8

16

Time (s)

897.6

449.9

327.3

237.8

135.1

78.1

18

Continuum Modeling of Microstructure
Instabilities and Self-Organization
Dose(isotropic)

Dose(anisotropic)

Simulations &experimentson self-organizationof irradiated Cu
microstructure.

Continuum Rate Equations for concentrations
Ginzburg-Landau Dynamics Give Amplitude
Equation for Patterns; bc= critical bifurcation
Parameter

9/16/2002

19