Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

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Nano Mechanics and MaterialsTheory, Multiscale
Methods and Applications

• by
• Wing Kam Liu, Eduard G. Karpov, Harold S. Park

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6. Introduction to Bridging Scale

• Molecular dynamics to be used near crack/shear
  band tip, inside shear band, at area of large
  deformation, etc.
• Finite element/meshless coarse scale defined
  everywhere in domain
• Two-way coupled MD boundary condition accounts
  for high frequency wavelengths
• G.J. Wagner and W.K. Liu, Coupling of atomistic
  and continuum simulations using a bridging scale
  decomposition, Journal of Computational Physics
  190 (2003), 249-274

Slide courtesy of Dr. Greg Wagner, formerly
Research Assistant Professor at Northwestern,
currently at Sandia National Laboratories
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6.1 Bridging Scale Fundamentals

• Based on coarse/fine decomposition of
  displacement field u(x)
• Coarse scale defined to be projection of MD
  displacements q(x) onto FEM shape functions NI
• P minimizes least square error between MD
  displacements q(x) and FEM displacements dI

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Bridging Scale Fundamentals

• Fine scale defined to be that part of MD
  displacements q(x) that FEM shape functions
  cannot capture
• Example of coarse/fine decomposition of
  displacement field

Slide courtesy Dr. Greg Wagner
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Multiscale Lagrangian

• Total displacement written as sum of coarse and
  fine scales
• Write multiscale Lagrangian as difference between
  system kinetic and potential energies
• Multiscale equations of motion obtained via

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Coupled Multiscale Equations of Motion

• First equation is MD equation of motion
• Second equation is FE equation of motion with
  internal force obtained from MD forces
• Kinetic energies (and thus mass matrices) of
  coarse/fine scales decoupled due to bridging
  scale term Pq
• FE equation of motion is redundant if MD and FE
  exist everywhere

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Bridging Scale Schematic
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MD Boundary Condition Approaches

• Generalized Langevin Equation (GLE)
• S.A. Adelman and J.D. Doll, Journal of Chemical
  Physics 64, 1976.
• Limited to one-dimensional cases
• Minimizing boundary reflections
• W. Cai, M. de Koning, V.V. Bulatov and S. Yip,
  Physical Review Letters 85, 2000.
• Size of time history kernel related to number of
  boundary atoms
• Matching conditions
• W.E., B. Engquist and Z. Huang, Physical Review B
  67, 2003.
• Geometry of lattice must be explicitly modeled
• Still lacking consistently derived MD boundary
  condition that is valid for arbitrary lattice
  structures, interatomic potentials

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MD Boundary Condition Assumptions

• Utilize inherently periodic/repetitive structure
  of crystalline lattices
• Difficult to apply to fluids, amorphous solids
  (polymers)
• Eliminate all MD DOFs which are assumed to
  behave harmonically/linear elastically away from
  nonlinear physics of interest (crack/defects)
• Work needed to mathematically define where
  linear/nonlinear transition actually occurs in
  practice
• Similar to approach by Wagner, Karpov and Liu
  (2004), Karpov, Wagner and Liu (2004)

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Transformation of Effective Information into Heat
Due to reflective boundaries, the wave
packages/signals gradually transforms into heat
(chaotic motion) Important information about
physics of the process can be lost. It is
required that wave packages propagate to the
coarse scale without reflection at the
fine/coarse interface. The successive tracking
of wave packages is unnecessary.
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Multiscale Boundary Conditions
Spurious wave reflection occurs at the
atomistic/continuum interface. For periodic
crystal lattices, the response of the coarse can
be computed at the atomistic level, without
involving the continuum model.
Multiscale BC
(atomistic solution is not sought on the coarse
grain)
The solution for atom 0 can be found without
solving the entire domain, if one knows the
dependence
(multiscale boundary condition)
is a known coarse scale displacement
For this 1D problem (quasistatic case)
The single equation to solve
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Dynamic Multiscale Boundary Conditions with a
Damping Kernel
Domain of interest (fine grain)
Bulk domain (coarse grain)
-2 -1 0 1
2 3 4
2. Force boundary conditions (currently used in
bridging scale)
1. Displacement boundary conditions
u1(t) and all other DoF ngt1 are eliminated. Their
effect is described by an external force term,
introduced into the MD equations
Displacements of the first atom on the coarse
scale u1(t) are considered as dynamic boundary
conditions for MD simulation
In both cases, the knowledge of time history
kernel Q(t) is important
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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 110 Direct coupling with continuum

Impedance BC are involved
Over 90 of the kinetic wave energy is reflected
back to the fine grain.
Less than 1 of the energy is reflected.
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Several Degrees of Freedom in One Cell
In case of multiple degrees if freedom per unit
cell, the equation of motion is still identical
for all repetitive cells n, though it takes a
matrix form
General definition of K-matrices
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Several Degrees of Freedom in One Cell
Response function
Time history kernel
Multiscale boundary conditions
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Further Explanation on Assumption of Linearity

• Most interatomic potentials function of distance
  r (LJ 6-12)
• Stiffness for a potential can be evaluated as
• Thus, stiffnesses K are function of position r as
  well
• But, if K evaluated about equilibrium separation
  req2(1/6)?
• Linearized MD internal force, i.e. fint Ku
• Key result from assumption of linearity
  constant K
• Leads to repetitive expression for MD internal
  force

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Theoretical Developments in 1D

• 1D Lagrangian for linearized lattice
• Equation of motion
• Note equation of motion valid for every atom n
  (repetitive structure)!

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Stiffness (K) Matrices (Nearest Neighbors)

• Harmonic potential
• Potential energy per unit cell
• K constants

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Tie to Finite Elements

• Force on atom n becomes
• Equation of motion for three atoms
• The conclusion, if FE nodes MD atoms

Repetitive, and results from constant K assumption
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One Final Comparison

• Re-writing the MD equations of motion
• Equations of motion for ngt0 atoms no longer
  necessary effects implicitly included in time
  history kernel ?(t)

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Final Coupled Equations of Motion

• ?(t-?) called time history kernel, and acts to
  dissipate fine scale energy from MD to
  surrounding continuum assumptions of linearity
  only contained within ?(t-?)
• Impedance and random forces act only on MD
  boundary atoms standard MD equation of motion
  elsewhere
• Stochastic thermal effects captured through
  random force R(t)

Standard MD
Impedance Force
Random Force
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Features of MD Boundary Condition

• MD equation of motion is two-way coupled with
  coarse scale
• If information begins in the continuum, can be
  transferred naturally to MD as boundary condition
• ? has dimensions of minimum number of degrees of
  freedom in each unit cell, and is re-used for
  every boundary atom
• Size of ? remains constant as size of structure
  grows - leads to computational scalability for
  any lattice structure
• Automated numerical procedure to calculate time
  history kernel ? for a given multi-dimensional
  lattice structure and potential
• Standard numerical Laplace and Fourier transform
  techniques
• ? derived consistently using lattice dynamics
  principles
• No ad hoc damping used to eliminate high
  frequency waves
• Ease of implementation
• Only additional external force required for MD
  boundary atoms

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2-D Lattices
The general idea of MS boundary conditions for
N-D structures is similar to the 1D case.
Response of the outer (bulk) material is modeled
by additional external forces applied at the
MD/continuum interface.
Reduced MD Domain Multiscale BC
MD Domain
Update for the equation of motion 1D
lattice
2D lattice
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2-D Formulation
Equation of motion
Response function
Mixed real space/Fourier domain function
Time history kernel - depends on a spatial
parameter m
Multiscale boundary conditions
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Numerical Transform Inversion
Numerical inverse Laplace transform
Week (J Assoc Comp Machinery 13, 1966, p.419)
Papoulis (Quart Appl Math 14, 1956, p.405)
Laguerre polynomials,
coefficients to be computed using F(s)
Inverse discrete Fourier transform
Fast Fourier transform reduce computational cost
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Performance Study Problem Statement
Initial conditions
K-matrices and mass matrix
Time history kernel
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Performance Study Size Effect
Reflection coefficient
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Performance Study Method Parameters
Temporal and spatial truncation
Time steps management
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Application Bridging Scale Simulation of Crack
Growth
Problem statement
The impedance boundary conditions were used along
the interface between the reduced fine scale
domain and the coarse scale domain in dynamic
crack propagation problems (H.S. Park, E.G.
Karpov, W.K. Liu, 2003). The Lennard-Jones
potential is utilized. The 2D time history kernel
represents the effect of eliminated fine scale
degrees of freedom.
Model description
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Application Bridging Scale Simulation of Crack
Growth
Results of the simulations, compared with
benchmark (full atomistic solution)
Fine grain (coupled MD/FE region)
Full atomistic domain
Crack propagation speeds are virtually identical
in the benchmark and multiscale simulations
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Removing Fine Scale Degrees of Freedom in Coarse
Scale Region
Equation of motion is identical for all
repetitive cells n
Introduce the stiffness operator K
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Periodic Structure Response Function
Dynamic response function Gn(t) is a basic
structural characteristic. G describes
lattice motion due to an external, unit momentum,
pulse
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Response Function Example
Assume first neighbor interaction only
Displacements
Velocities
Illustration (transfer of a unit pulse due to
collision)
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Time History Kernel (THK)
The time history kernel shows the dependence of
dynamics in two distinct cells. Any time history
kernel is related to the response function.
f(t)
-2 -1 0 1
2
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Elimination of Degrees of Freedom
Equations for atoms n gt 0 are no longer required