05-Introduction to Multiple Scale Modeling

7/28/2019 05-Introduction to Multiple Scale Modeling

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Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

by

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


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5. Introduction to Multiple Scale Modeling

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


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Role of Computational Methods

Nano- and micro-structure

Electronic structure

Molecular mechanics

Continuum

mechanics

Potentials

Const.

laws

Plasticity

Mul tiscale methods

Manufacturing

platform

Function

Performance

Reliability

Computations

and design

Prediction

Validation

Structural and material design

Optimization

Prediction and validation


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Examples of Multi-Scale Phenomena in Solids

17:1

250:1250:1

200 m

Shear bands Mechanics of carbon nanotubes

Figures: D. Qian, E. Karpov, NU

Shaofan Li, UC-Berkeley

Movie: Michael Griebel, Universitt Bonn


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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


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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 scalessimultaneously

Relationships between length scales are dynamic

Classic example:heat bath techniques


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Macroscopic, Atomistic, Ab Initio Dynamics (MAAD)

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).


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MAAD: Concurrent Coupling of Length Scales

/

/

, , , ,

, ,

,

Tot FE FE MD

MD MD TB

TB

H H H

H H

H

u u u u r r

r r r r

r r

Scales are coupled in

handshake regions

Finite element meshgraded down to atomiclattice in the overlap region

Total Hamiltonian isenergy in each domain, plusoverlap regions

Nakano et al,

Comput. In Sci. and

Eng., 3(4) (2001).

Broughton, et al,PRB60(4) (1999).

Handshake

at MD/FE

interface


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Quasicontinuum Method

Tadmor and Phillips,Langmuir12, 1996

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 usingshape functions and summation rules

Adaptivity criteria used to reselect representative latticepoints in regions of high deformation

Applications to dislocations, grain boundary interactions,nanoindentation, and fracture (quasistatic modeling)

Cauchy-Born rule assumes 1) continuum energy densitycan be derived from the atomic potential; 2) deformationgradientFdescribes deformation at both continuum andatomic scales, and therefore serves as the link. Thus, atomicdeformation has to be homogeneous

Issues: non-local interaction, long dislocations/illconditioning, separation of scales, finite temperatures,universal scenarios

Later improved by Arroyo, Belytschko, 2004, in applicationto CNT:PRB69, article 115415

T

W

P F


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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- & nano-

level 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


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Typical Issues

1. True coarse scale discretization and coupling between the scales

2. Handling interfaces where small and large scales intersect; handshake isexpensive and non-physical; spurious wave reflection

3. Double counting of the strain energy

4. Implementation: usage of existing MD and continuumcodes is hard; parallel computing

5. Dynamic mesh refinement/enrichment

6. Finite temperatures

7. Multiple time scales and dynamics of infrequent events

- BSM has resolved issues 1-2, and partially 3-6.

- The alternative: MSBC method, where issues 1-4 DO NOT ARISE

Typical interface model


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The Bridging Scale Method

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

x x x

x x

x x x x x

u u u

u Pu

u u Pu I P u Qu

= +

Two most important components:

- bridging scale projection

- impedance boundary conditions appliedMD/FE interface in the form of a time-history integral

Assumes asingle solution u(x) for the entire domain.This solution is decomposed into the fine and coarse

scale fields:

( )xu ( )xu ( )xu

BS projection

T

1 T

A

A

M N M N

Q I NM N M


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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

Lagrangian formulation gives coupled,

coarse and fine scale, equations of motion

' u u u Nd Qq

T

T

( )

( )A

U

Md N f uf

uM q Q f u

Bridging-Scale Equations of Motion

MD, q FEM, d MD + FE, (q, d)

T T T1 1( , , , ) ( , )2 2

AL U d d q q d Md q Q M q d q


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Impedance Boundary Conditions / MD Domain Reduction

MD degrees of freedom outside the

localized domain are solved implicitly

+

Due to atomistic nature of the model, the structural impedance is evaluated computed at

the atomic scale.

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 forceapplied at the formal MD/continuum interface:

FE + Reduced MD +

Impedance BC

MD FE

T

0

( )

( ) ( ) ( ) ( )t

A t d

Md N f u

M q f u q u


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1D Illustration: Non-Reflecting MD/FE InterfaceImpedance 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 1:10Direct 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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Why is Multiscale Modeling Difficult?

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 MDFEM FEM


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The phase velocity of progressive

waves is given by

Dependence on the wave number:

Value v0 is the phase velocity of the

longest waves (at p 0).

-1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p

0/ ontinuum

vp

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

1

/p

continuum

0/v v

0 0

12sin sin

2 2

p v p

v p

Incompatible Dispersion Properties of Lattices and Continua

lattice structure

lattice structure


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Issues in Multiscale Modeling

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

Need for a dynamic, finite temperature multiple scale method

True coarse scale representation

- No meshing FEM down to MD lattice spacing

- Different time steps for MD and FEM simulations

Mathematically sound and physically motivated treatment of highfrequency waves emitted from MD region at MD/FE interface


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Selected References

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 AppliedMechanics and Engineering2004; 193:1645-1669

Review articles:

W.A. Curtin and R.E. Miller, Modelling and Simulation in Materials

Science and Engineering2003; 11:R33-R68

W.K. Liu, E.G. Karpov. S. Zhang and H.S. Park, Computer Methodsin Applied Mechanics and Engineering2004; 193:1529-1578

Documents

05-Introduction to Multiple Scale Modeling