Nano Mechanics and Materials:
Theory, Multiscale Methods and Applications
by
Wing Kam Liu, Eduard G. Karpov, Harold S. Park
5. Introduction to Multiple Scale Modeling
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Motivation for multiple scale methods
 Coupling of length scales
Bridging scale concurrent method
 Molecular dynamics (MD) boundary condition
Numerical examples
 1D wave propagation
 2D dynamic crack propagation
 3D dynamic crack propagation
Discussion/Areas of improvement for bridging scale
Conclusions and future research
Role of Computational Methods
•
Structural and material design
•
Optimization
•
Prediction and validation
Nano- and micro-structure
Electronic structure
Molecular mechanics
Continuum
mechanics
Potentials
Const.
laws
Plasticity
Multiscale methods
Computations
and design
Manufacturing
platform
Function
Performance
Reliability
Prediction
Validation
Examples of Multi-Scale Phenomena in Solids
Shear bands
Mechanics of carbon nanotubes
17:1
250:1
250:1
200 m
Figures: D. Qian, E. Karpov, NU
Shaofan Li, UC-Berkeley
Movie: Michael Griebel, Universität Bonn
Why Multiscale Methods?
Limitations of industrial simulations today:
a)
Continuum models are good, but not always adequate
•
Problems in fracture and failure of solids require improved constitutive models
to describe material behavior
•
Macroscopic material properties of new materials and composites are not readily
available, while they are needed in simulation-based design
•
Detailed atomistic information is required in regions of high deformation or
discontinuity
b)
Molecular dynamics simulations
•
Limited to small domains (~106-108 atoms) and small time frames (~nanoseconds)
•
Experiments, even on nano-systems, involve much larger systems over longer times
Opportunities: 1) Obtain material properties by subscale (multiscale) simulation
2) Enrich information about material/structural performance across scales
via concurrent multiscale methodologies
Hierarchical vs. Concurrent
• Hierarchical approach
– Use known information at one scale to generate model for larger scale
– Information passing typically through some sort of averaging process
– Example: bonding models/potentials, constitutive laws
• Concurrent approach
– Perform simulations at different length scales simultaneously
– Relationships between length scales are dynamic
– Classic example: “heat bath” techniques
Macroscopic, Atomistic, Ab Initio Dynamics (MAAD)
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Finite elements (FE), molecular
dynamics (MD), and tight binding (TB)
all used in a single calculation
(MAAD)
MAAD = macroscopic, atomistic, ab
initio dynamics
Atomistics used to resolve features of
interest (crack)
Continuum used to extend size of
domain
Developed by Abraham (IBM),
Broughton (NRL), and co-workers
From Nakano et al, Comput. In Sci. and Eng., 3(4) (2001).
MAAD: Concurrent Coupling of Length Scales
• Scales are coupled in
“handshake” regions
• Finite element mesh
graded down to atomic
lattice in the overlap region
• Total Hamiltonian is
energy in each domain, plus
overlap regions
H Tot  H FE  u, u   H FE / MD  u, u, r, r 
Broughton, et al, PRB 60(4) (1999).
Handshake
at MD/FE
interface
 H MD  r, r   H MD / TB  r, r 
 H TB  r, r 
Nakano et al,
Comput. In Sci. and
Eng., 3(4) (2001).
Quasicontinuum Method
• Developed by Ortiz, Phillips and coworkers in 1996.
• Deformation is represented on a triangulation of a subset
of lattice points; points in between are interpolated using
shape functions and summation rules
P
W
F T
• Adaptivity criteria used to reselect representative lattice
points in regions of high deformation
• Applications to dislocations, grain boundary interactions,
nanoindentation, and fracture (quasistatic modeling)
• Cauchy-Born rule assumes 1) continuum energy density
can be derived from the atomic potential; 2) deformation
gradient F describes deformation at both continuum and
atomic scales, and therefore serves as the link. Thus, atomic
deformation has to be homogeneous
Issues: non-local interaction, long dislocations/ill
conditioning, separation of scales, finite temperatures,
universal scenarios
Later improved by Arroyo, Belytschko, 2004, in application
to CNT: PRB 69, article 115415
Tadmor and Phillips, Langmuir 12, 1996
Challenges
• Large number of degrees of freedom at the atomic scale
• Interfaces: mismatch of dynamic properties, and other issues
• Consistent and accurate representation of meso-, micro- & nanolevel behavior within continuum models
• Multiple time scales
• Potentials
• Interdisciplinary nature of multiscale methods
- continuum mechanics
- classical particle dynamics (MD), and lattice mechanics
- quantum mechanics and quantum chemistry
- thermodynamics and statistical physics
• Atomic scale plasticity: lattice dislocations
• Finite temperatures
• Entropic elasticity, soft materials
• Dynamics of infrequent events: diffusion, protein dynamics
• Algorithmic issues in large scale coupled simulations
Typical Issues
1.
True coarse scale discretization and coupling between the scales
2.
Handling interfaces where small and large scales intersect; handshake is
expensive and non-physical; spurious wave reflection
3.
Double counting of the strain energy
4.
Implementation: usage of existing MD and continuum
codes is hard; parallel computing
5.
Dynamic mesh refinement/enrichment
6.
Finite temperatures
7.
Multiple time scales and dynamics of infrequent events
Typical interface model
- BSM has resolved issues 1-2, and partially 3-6.
- The alternative: MSBC method, where issues 1-4 DO NOT ARISE
The Bridging Scale Method
• Two most important components:
- bridging scale projection
- impedance boundary conditions applied
MD/FE interface in the form of a time-history integral
• Assumes a single solution u(x) for the entire domain.
This solution is decomposed into the fine and coarse
scale fields:
u( x)  u( x)  u( x)
BS projection
u( x)  Pu( x)
u( x)  u( x)  Pu( x)   I  P  u( x)  Qu( x)
u( x)
M  NT M AN
Q  I  NM 1NT M A
u( x)
u( x)
=
+
Bridging-Scale Equations of Motion
Within the bridging scale method, the MD and FE formulation exist
simultaneously over the entire computational domain:
+
=
The total displacement is a combination of
the FE and MD solutions:
Multiscale Lagrangian
MD + FE, (q, d)
FEM, d
MD, q
u  u  u '  Nd  Qq
1
1
L  (d, d, q, q)  d T Md  q T  Q T M A  q  U (d, q)
2
2
Lagrangian formulation gives coupled,
coarse and fine scale, equations of motion
Md  N T f (u)
M Aq  QT f (u)
f
U
u
Impedance Boundary Conditions / MD Domain Reduction
The MD domain is too large to solve, so that we eliminate the MD degrees of freedom
outside the localized domain of interest.
Collective atomic behavior of in the bulk material is represented by an impedance force
applied at the formal MD/continuum interface:
Md  N Tf (u)
M Aq  f (u)   Θ(t   )  q( )  u( )  d

MD degrees of freedom outside the
localized domain are solved implicitly
t
0
MD
FE + Reduced MD +
Impedance BC
FE
+

Due to atomistic nature of the model, the structural impedance is evaluated computed at
the atomic scale.
1D Illustration: Non-Reflecting MD/FE Interface
Impedance boundary conditions allows non-reflecting coupling of the fine and coarse grain solutions
within the bridging scale method.
Example: Bridging scale simulation of a wave propagation process; ratio of the characteristic lengths
at fine and coarse scales is 1:10
Direct coupling with continuum
Over 90% of the kinetic wave energy
is reflected back to the fine grain.
Impedance BC are involved
Less than 1% of the energy is reflected.
Why is Multiscale Modeling Difficult?
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Wave reflection at MD/FE interface
Larger length scales (FE) cannot represent wave lengths typically found at
smaller length scales (MD)
Also due to energy conserving formulations for both MD and FEM
MD
FEM
MD
FEM
Incompatible Dispersion Properties of Lattices and Continua
continuum
The phase velocity of progressive
waves is given by
v

 / 0
2.5
2
1.5
lattice structure
1
p
p /
0.5
-1
-0.5
0.5
1

Dependence on the wave number:

p
v 1
p
 2sin 
 sin
0
2
v0 p
2
3
1
continuum
v / v0
0.8
0.6
Value v0 is the phase velocity of the
longest waves (at p  0).
lattice structure
0.4
0.2
-6
-4
-2
p /
2
4
6
Issues in Multiscale Modeling
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Preventing high frequency wave reflection
- Need reduced MD system to behave like full MD system
- Reflection of high frequency waves can lead to melting of the
atomistic system
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Need for a dynamic, finite temperature multiple scale method
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True coarse scale representation
- No meshing FEM down to MD lattice spacing
- Different time steps for MD and FEM simulations
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Mathematically sound and physically motivated treatment of high
frequency waves emitted from MD region at MD/FE interface
Selected References
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Quasicontinuum Method
 E. Tadmor, M. Ortiz and R. Phillips, Philosophical Magazine A 1996;
73:1529-1563
Coupled Atomistic/Discrete Dislocation method (CADD)
 L. Shilkrot, R.E. Miller and W.A. Curtin, Journal of the Mechanics
and Physics of Solids 2004; 52:755-787
Bridging Domain method
 S.P. Xiao and T. Belytschko, Computer Methods in Applied
Mechanics and Engineering 2004; 193:1645-1669
Review articles:
 W.A. Curtin and R.E. Miller, Modelling and Simulation in Materials
Science and Engineering 2003; 11:R33-R68
 W.K. Liu, E.G. Karpov. S. Zhang and H.S. Park, Computer Methods
in Applied Mechanics and Engineering 2004; 193:1529-1578