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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008

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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation

Part II - lecture 7

Atomistic and molecular methods

2

Content overviewLectures 2-10February/March

Lectures 11-19March/April

Lectures 20-27April/May

I. Continuum methods1. Discrete modeling of simple physical systems:

Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,

Variational formulations3. Linear elasticity: Review of basic equations,

Weak formulation: the principle of virtual work, Numerical discretization: the finite element method

II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and

how to model them

III. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?

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Overview: Material covered Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics)

Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions)

Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/EAM & applications, neighbor lists, periodic BCs, how to apply BCs)

Lecture 5: Interatomic potential and force field (cont’d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NPT), application: mechanical properties –basic introduction)

Lecture 6: Application to mechanics of materials-brittle materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms in brittle fracture, theory, case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model), modeling approaches: brittle-pair potential/ReaxF, applied to silicon

Lecture 7: Application to mechanics of materials-ductile materials (dislocation mechanisms), case study: failure of copper nanocrystal, supercomputing/parallelization

Lecture 8: Review session

Lecture 9: QUIZ

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II. Atomistic and molecular methods

Lecture 7: Application to mechanics of materials-ductile materials

Outline:1. Review: brittle fracture mechanics & atomistic modeling of brittle

fracture1.1 Brittle fracture – supersonic and subsonic cracking

2. Multi-paradigm modeling of brittle fracture in silicon 3. Basic deformation mechanism in ductile materials: dislocations

3.1 Atomistic modeling studies of dislocation plasticity4. Supercomputing and parallelization

Goal of today’s lecture: Review main concepts of brittle fracture, application to chemically complex material – silicon Discuss advanced molecular dynamics simulation methods (multi-paradigm modeling)Learn basics in mechanics of ductile materials, focused on dislocations, review atomistic modeling approaches for ductile materials

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1. Review: Ductile versus brittle materials

6

Tensile test of a wire: 2 types of material responses

Brittle Ductile

Strain

Stre

ss

BrittleDuctile

Necking

Figures by MIT OpenCourseWare.

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Ductile versus brittle materials

Difficultto deform,breaks easily

Easy to deformhard to break

BRITTLE DUCTILE

Glass Polymers Ice...

Shear load

Copper, Gold

Figure by MIT OpenCourseWare.

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Deformation of materials:Nothing is perfect, and flaws or cracks matter

Any material has imperfections – here shown as elliptical inclusions

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Significance of material flaws

Image removed due to copyright restrictions. Please see Fig. 1.3 in: Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

Stress concentrators: local stress >> global stress

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“Macro, global”

“Micro (nano), local”

)(rσ r

Deformation of materials:Nothing is perfect, and flaws or cracks matter

Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections

Failure of materials initiates at cracks

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Crack extension: brittle response

Large stresses lead to rupture of chemicalbonds between atoms

Thus, crack extends

Image removed due to copyright restrictions. Please see: Fig. 1.4 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.


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Lattice shearing: ductile response

Instead of crack extension, induce shearing of atomic latticeDue to large shear stresses at crack tipLecture 7



τ

τ

τ

τ

13

1.1 Brittle fracture – supersonic and subsonic cracking

Focus on two aspects:1. Basic physical aspects of fracture initiation

(dissipation of elastic energy)2. Crack limiting speed, once the crack begins to

propagate

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Basic fracture process: dissipation of elastic energy

Undeformed Stretching=store elastic energy Release elastic energydissipated into breaking chemical bonds


15

Limiting speeds of cracks

• Under presence of hyperelastic effects, cracks can exceed the conventional barrier given by the wave speeds• This is a “local” effect due to enhancement of energy flux• Subsonic fracture due to local softening, that is, reduction of energy flux

Images removed due to copyright restrictions. Please see: Fig. 6.56 and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

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How to build a “simple” atomistic model of fracture

Must be able to capture the process of storing elastic energy

Must be able to capture the process of breaking bonds

The simple harmonic potential & harmonic bond snapping potential are the simplest representations of such a model

Undeformed Stretching=store elastic energy Release elastic energydissipated into breaking chemical bonds

weak layer=fracture plane


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A “simple” atomistic model: geometry2

00 )(21 rrk −=φ

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<−=

break2

0break0

break2

00

)(21

)(21

rrrrk

rrrrkφ

9165.012246.10

==

ρrEQ distance & density

X

YV

31/2rora

Iy

Ix

Weak fracture layer

bond can

y (store energy) 2D triangular (hexagonal) lattice

break

elasticit

stable configuration of pair potential


18

How to calculate elastic properties and wave speeds – triangular lattice, pair potential φ

12246.10 =r

EQ distance & density (same in pset 2)

''φ=k 0r

**2****rbreak0 /][/][][],,[ TLcLEkELrrr ==== φ

3**3** /][/][ LMLEE == ρ :::

*

*

*

MEL reference length (1 Å)

reference energy (347 kJ/mole)

reference mass (12 amu)

Example

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Harmonic and harmonic bond snapping potential

200 )(

21 rrk −=φ

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<−=

break2

0break0

break2

00

)(21

)(21

rrrrk

rrrrkφ


20

Hyperelasticity or nonlinear elasticity


21

MD model development: biharmonic potential

Polymers, rubber

Metals

• Stiffness change under deformation, with different strength

• Atomic bonds break at critical atomic separation

• Want: simple set of parameters that control these properties (as few as possible, to gain generic insight)

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Biharmonic potential – control parameters

breakε

0k

1k

01ratio / kkk =

breakronr r 0

0

rrr −

=ε

'φstiffening

softening

23

Physical basis for subsonic/supersonic fracture

Changes in energy flow at the crack tip due to changes in local wave speed (energy flux higher in materials with higher wave speed


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Supersonic fracture: mode II (shear)

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Experimental verificationIntersonic mode I fracture

Recent experiment has verified existence of intersonic mode I cracks in stiffening material (rubber)Possible explanation: Hyperelasticity or nonlinear elasticity

Theory/MD experiment

Image removed due to copyright restrictions.Please see Fig. 2 in Petersan, Paul J., Robert D. Deegan, M. Marder, and Harry L. Swinney. "Cracks in Rubber under Tension Exceed the Shear Wave Speed." Phys Rev Lett 93 (2004): 015504.

Image removed due to copyright restrictions. Please see: Fig. 9 in Buehler, Markus, and Huajian Gao. "Modeling Dynamic Fracture Using Large-Scale Atomistic Simulations." Chapter 1 in Shukla, Arun. Dynamic Fracture Mechanics. Hackensack, NJ: World Scientific, 2006.

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2. Multi-paradigm modeling of brittle fracture in silicon

Focus: model particular fracture properties of silicon (chemically complex material)

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Fracture of silicon: problem statement

Pair potential insufficientto describe bond breaking(chemical complexity)

Image courtesy of NASA.

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Concept: concurrent multi-paradigm simulations

ReaxFF

FE (c

ontin

uum

)

Organic phase

Inorganic phase

nonreactiveatomistic

nonreactiveatomistic

• Multi-paradigm approach: combine different computational methods (different resolution, accuracy, different numerical requirements..) in a single computational domain

• Decomposition of domainbased on suitability of different approaches

• Example: concurrent FE-atomistic-ReaxFF scheme in a crack problem (crack tip treated by ReaxFF) and an interface problem (interface treated by ReaxFF)

Interfaces (oxidation, grain boundaries,..)Sites of chemical reactions

Crack tips, defects (dislocations)

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Concurrent multi-paradigm simulations:link nanoscale to macroscale

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Multi-paradigm concept for fracture

Need method“good” for elastic properties

Need method“good” for describing rupture of chemical bonds



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Simulation geometry: fracture of silicon

• Consider a crack in a silicon crystal under mode I loading.

• Periodic boundary conditions in the z direction (corresponding to a plane strain case).

Buehler et al., Phys. Rev. Lett., 2006, 2007

Tersoff potential: Good for modeling elasticity of silicon

ReaxFF potential: Good for modeling break of bonds

Image removed due to copyright restrictions.

Please see: Fig. 2 in Buehler, Markus J., et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field."Physical Review Letters 96 (2006): 095505. orFig. 6.95 in Buehler, Markus J. Atomistic Modeling of Materials Failure.New York, NY: Springer, 2008.

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Hybrid Hamiltonians

TersoffReaxFFTersoffReaxFF −++= UUUUtot

transition region

xxUF tot

∂∂

−=)(

Weights = describe how mucha particular FF counts (assigned to each atom)

To obtain forces:

Approach: handshaking via mixed Hamiltonians

Image removed due to copyright restrictions. Please see: Fig. 1 in Buehler, Markus J., et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field."Physical Review Letters 96 (2006): 095505.

.

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Assigning weights to atoms

Percentage ReaxFFPercentage Tersoff(relative contribution to total energy)

100% … 100% 70% 30% 0% … 0%

0% … 0% 30% 70% 100% … 100%


.

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TersoffReaxFFTersoffReaxFF −++= UUUUtot

TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( UwUxwU −+=−

( ) ( )⎥⎦⎤

⎢⎣⎡ −

∂∂

−−+−=− TersoffReaxFFReaxFF

TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( UUx

wFwFxwF

wReaxFF is the weight of the reactive force field in the handshaking region.

Hybrid Hamiltonians – force calculation

D. Sen and M. Buehler, Int. J. Multiscale Comput. Engrg., 2007

xU

F∂

∂−=Recall:

Need potential energy for force calculation )(rUtot )(xUtot xxUF tot

∂∂

−=)(

Image removed due to copyright restrictions. Please see

:

Fig. 1 in Buehler, Markus J., et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field."Physical Review Letters 96 (2006): 095505.

xxwxw ∀=+ 1)()( TersoffReaxFF

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( ) ( )⎥⎦⎤

⎢⎣⎡ −

∂∂

−−+−=− TersoffReaxFFReaxFF

TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( UUx

wFwFxwF

Slowly varying weights (wide transition region):

If (i.e., both force fields have similar energy landscape)

0/ReaxFF ≈∂∂ xw

0TersoffReaxFF ≈−UU

Hybrid Hamiltonians – force calculation

( )TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( FwFxwF −+=− xxwxw ∀=+ 1)()( TersoffReaxFF

≈0

D. Sen and M. Buehler, Int. J. Multiscale Comput. Engrg., 2007

≈0

Simplified result: can interpolate forces from one end to the other

36

Energy landscape of two force fields

• Schematic showing the coupling of reactive and nonreactive potentials

• At small deviations, energy landscape is identical in nonreactive and reactive modelsU

x

0EAMReaxFF ≈−UU

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Summary: mixed Hamiltonian approach

( )TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( FwFxwF −+=−

xxwxw ∀=+ 1)()( TersoffReaxFF


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Use multi-paradigm scheme that combines the Tersoff potential and ReaxFF

Reactive region (red) is moving with crack tip

Fracture of silicon single crystals


Please see: Fig. 4a in Buehler, Markus J., et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field."Physical Review Letters 96 (2006): 095505.or

Fig. 6.97 in Buehler, Markus J. Atomistic Modeling of Materials Failure.New York, NY: Springer, 2008.

39

Cracking in Silicon: Hybrid model versusTersoff based model

Illustrates: Pure Tersoff can not describe correct crack dynamics


Please see: Fig. 3a,b in Buehler, Markus J., et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field."Physical Review Letters 96 (2006): 095505.

or

Fig. 6.96 in Buehler, Markus J. Atomistic Modeling of Materials Failure .New York, NY: Springer, 2008.

40

Quantitative comparison w/ experiment

Image removed due to copyright restrictions. Please see: Fig. 1c inBuehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): 165502.orFig. 6.99b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

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Crack dynamics

Image removed due to copyright restrictions. Please see: Fig. 2 inBuehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): 165502.orFig. 6.102 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

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Atomistic fracture mechanism

M.J. Buehler et al., Phys. Rev. Lett., 2007

Image removed due to copyright restrictions. Please see: Fig. 3 inBuehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): 165502.

orFig. 6.104 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

43

Fracture initiation and instabilities: movie

44

Shear (mode II) loading:Crack branching

Tensile (mode I) loading:Straight cracking

Fracture mechanism: tensile vs. shear loading


45

Shear (mode II) loading: Crack branchingTensile (mode I) loading: Straight cracking

Fracture mechanism: tensile vs. shear loading

Mode I Mode II

Images removed due to copyright restrictions.

Please see figures in Buehler, M.J., A. Cohen, and D. Sen. “Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading.” Journal

of Algorithms and Computational Technology 2 (2008): 203-221.


46

Summary: main concept of this section

Can combine different force fields in a single computational domain = multi-paradigm modeling

Enables one to combine the strengths of different force fields

Simple approach by interpolating force contributions from individual force fields, use of weights (sum of weights = 1 at all points)

ReaxFF based models quite successful, e.g. for describing fracture in silicon, quantitative agreement with experimental results

47

3. Basic deformation mechanism in ductile materials: dislocations

48

Significance of material flaws

Image removed due to copyright restrictions. Please see Fig. 1.3 in: Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

Stress concentrators: local stress >> global stress

49

Crack extension: brittle response

Large stresses lead to rupture of chemicalbonds between atoms

Thus, crack extends



50

Lattice shearing: ductile response

Instead of crack extension, induce shearing of atomic latticeDue to large shear stresses at crack tipLecture 7



τ

τ

τ

τ

51

Driving forces for ductile failure: shear stresses

Images removed due to copyright restrictions. Please see: Fig. 6.30 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

52

Hoop or opening stress

Maximum principal stress

1.5

1

1σ

0.5

00 50 100 150

θ

v/cr = 0

0.2σ xx

0.4

0.6v/cr = 0

0

0 50θ

100 150

Continuum theoryMD modeling

σ XY

v/cr = 0

0-0.2

-0.1

0.1

0.2

0

50θ

100 150

0.2σ yy

0.4

0.6

v/cr = 0

0

0 50θ

100 150

σ θ

0.2

0.4

0.6

0

0 50 100 150θ

v/cr = 0

Figures by MIT OpenCourseWare.

Angular distribution

Perfect crystal: deformation must be cooperative movement of all atoms; the critical shear stress for this mechanism was calculated by Frenkel(1926), 1..2 GPa (μ = 50 GPa for copper):

However: shear strength of crystals measured in experiment is much lower, 100..200 MPa:

Difference explained by existence of dislocations by Orowan, Polanyi and Taylor in 1934

Confirmed by experiments with whiskers (defect free crystals that are sheared cooperatively)

μτ th ≈30


μτ exp =100...1000

Theoretical shear strength and concept of dislocations


τ

τ

Concept of dislocations

Localization of shear rather than homogeneous shear that requires cooperative motion

“size of single dislocation” = b, Burgers vector 54

additional half plane


τ

τ

τ

τ

55

Animation: Dislocation motion

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/a5_1_1.html

Courtesy of Dr. Helmut Foell. Used with permission.

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/a5_1_1.html

56

3.1 Atomistic modeling studies of dislocation plasticity

Typically use LJ potentialor EAM potential, parameterized for metals

Yields ductile material response

57

Bending a paper clip: deforming copper

58

A closer look: surface cracks

http://www2.ijs.si/~goran/sd96/e6sem1y.gif

Courtesy of Goran Drazic. Used with permission.

http://www2.ijs.si/~goran/sd96/e6sem1y.gif

59

Case study: Crack in a copper crystal…

Do not observe crack extension! Rather, shear of latticeCreation of dislocations

Copper


60

Atomistic details of dislocation nucleation

• Dislocation nucleation from a traction-free grain boundary in an ultra thin copper film

• Atomistic results depict mechanism of nucleation of partial dislocation

[111]

[112]

stretch 10x

Figure removed for copyright reasons.

See figure 16 in Buehler, Markus J., John Balk, Eduard Arzt, and Huajian Gao. "Constrained Grain Boundary Diffusion in Thin Copper Films." Chapter 13 in Handbook of Theoretical and Computational Nanotechnology. Edited by Michael Rieth and Wolfram

Schommers. Stevenson Ranch, CA: American Scientific Publishers, 2006.or

Fig. 8.27 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

61

Dislocations in TEM

Balk, Dehm, Arzt, 2003 Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.

http://www.sciencedirect.com

62

Case study: plasticity in a micrometer crystal of copper

Image removed for copyright reasons.

See fig. 4 at http://tinyurl.com/nbhev6

(1 1 0)

[1 1 0]

[1 1 0] Crack faces

Mode 1 tensile loading

Y

Z

X

Figure by MIT OpenCourseWare. After Buehler et al., 2005.

http://tinyurl.com/nbhev6

63

Simulation details

- 1,000,000,000 atoms

- Lennard-Jones ductile material, for copper

- Visualization using energy filtering method (only show high energy atoms)

φ

r

Genericfeatures of atomicbonding: „repulsion vs. attraction“

Case study: plasticity in a micrometer crystal of copper

(1 1 0)

[1 1 0]

[1 1 0] Crack faces

Mode 1 tensile loading

Y

Z

X

Figures by MIT OpenCourseWare. After Buehler et al., 2005.

[001]

Crack Direction [1 1 0]

y = [100]

z = [001]x = [110]

[010]

64

A simulation with 1,000,000,000 particles

Image removed due to copyright restrictions.Please see: Fig. 1 in Abraham, Farid F., et al. "Simulating Materials Failure by Using up to One BillionAtoms and the World's Fastest Computer: Work-hardening." PNAS 99 (April 30, 2002): 5783-5787.

65

Analysis of a one-billion atom simulation of work-hardening

Image removed due to copyright restrictions.Please see: Fig. 1c in Buehler, Markus J., et al. "The Dynamical Complexity of Work-hardening: ALarge-scale Molecular Dynamics Simulation." Acta Mechanica Sinica 21 (2005): 103-111.

66

Lattice around dislocation

Atoms with higher energy than bulk are highlighted

[121]

[111]

b

Geometry of dislocation in atomic lattice

67

Hardening mechanismscreation of sessile structure – makes material brittle

(Hull & Bacon)

ImPl "

age removed due to copyright restrictions. ease see "Precipitation Hardening,

http://www.bss.phy.cam.ac.uk/~amd3/teaching/A_Donald/Crystalline_solids_2.htm Image removed due to copyright restrictions. Please see:

Fig. 4 in Buehler, M., et al. "The Dynamical Complexity of Work-Hardening: A Large-Scale Molecular Dynamics Simulation."

Acta Mechanica Sinica 21 (2005): 103-111.

http://www.bss.phy.cam.ac.uk/%7Eamd3/teaching/A_Donald/Crystalline_solids_2.htm

http://www.bss.phy.cam.ac.uk/%7Eamd3/teaching/A_Donald/Crystalline_solids_2.htm

68

Cross-slip

Activation of secondary slip systems by cross-slip and Frank-Read mechanisms: At later stages of the simulation• After activation of secondary slip systems: More dislocation reactions (e.g. cutting processes).

• Fleischer’s mechanism of cross-slip of partials that was proposed 1959

b xz

(111)

(111)[101]

w a

b

c

d

B

D

C

A

y


Image removed due to copyright restrictions.Please see: Fig. 5b in Buehler, Markus J., et al. "The DynamicalComplexity of Work-hardening: A Large-scale Molecular DynamicsSimulation." Acta Mechanica Sinica 21 (2005): 103-111.


69

• At critical dislocation density, secondary slip systems are activated

• This enables for additional plasticity to occur, but also further contributes to work-hardening as the dislocation density increases making it more difficult for dislocations to move


Images removed due to copyright restrictions.Please see: Fig. 5c,d in Buehler, Markus J., et al. "The DynamicalComplexity of Work-hardening: A Large-scale Molecular DynamicsSimulation." Acta Mechanica Sinica 21 (2005): 103-111.

70

Formation of Lomer-Cottrell locks

• Formation of sessile Lomer-Cottrell locks, with its typical shape of a straight sessile arm connected to two partial dislocations

• Sessile junctions provide a severe burden for further dislocation glide


Images removed due to copyright restrictions.Please see: Fig. 5e,f in Buehler, Markus J., et al. "The DynamicalComplexity of Work-hardening: A Large-scale Molecular DynamicsSimulation." Acta Mechanica Sinica 21 (2005): 103-111.

71

Final sessile structure – “brittle”

Consists of:Vacancy tubes, interstitials, partial dislocations, and sessile dislocations

The more defects, the more difficult for new dislocations to pass

Makes material brittle (increased likelihood of crack extension rather than dislocation nucleation)

Images removed due to copyright restrictions.Please see: Fig. 6a in Buehler, Markus J., et al. "The DynamicalComplexity of Work-hardening: A Large-scale Molecular DynamicsSimulation." Acta Mechanica Sinica 21 (2005): 103-111.

72


73


74

4. Supercomputing and parallelization

How to solve the numerical problem of integrating the equations of motions in MD efficiently

Use multiple computers (CPUs) to solve the problemIndividual computers can communicate results via

network

75

Parallel computing – “supercomputers”

Supercomputers consist of a very large number of individual computing units (e.g. Central Processing Units, CPUs)

Images removed due to copyright restrictions

Photos of supercomputers.

76

Domain decomposition

Each piece worked on by one of the computers in the supercomputer

77

• Each CPU is responsible for part of the problem

• Atoms can move into other CPUs (migration)

Parallel molecular dynamics: domain decomposition

(after Schiotz)

Concept:

Divide the workload

No (immediate) long range interaction

physical simulation domain


78

Implementation of parallelization

Shared memory systems (all CPUs “see” same memory)OpenMP (easy to implement, allows incremental parallelization)POSIX threads

Distributed memory systemsMPI (=Message Passing Interface)Most widely accepted and used, very portable, but need to parallelize whole code at once

Parallelization can be very tedious and time-consuming and may distract from solving the actual problem; debugging difficult

Challenges: Load balancing, different platforms, input/output, compilers and libraries, modifications and updates to codes, “think parallel” as manager

Strategy for your own code: Find similar code and implement your own problem

79

How to measure the speed of a supercomputer

Measured in “Floating point operations per second” FLOPS

Performance

MFLOPS 106, Million

109, Billion (reached in 1990s)

1012, Trillion (reached around 2000)

1015, Quadrillion (estimated to be reached in 2010)

Explanation of the different measures of computing power, and list when this power became available. In comparison a state-of-the art PC provides aperformance of 30 GFLOPS.

GFLOPS

TFLOPS

PFLOPS

Floating point operations per second (FLOPS)


80

The largest supercomputers

Figure removed due to copyright restrictions.

Chart showing what organizations possess the world’s largest supercomputers.

81

Performance history

2 5

6

8

9

/L ( ) ( )

( )

10 atoms 10 atoms 10 atoms

10 atoms

10 atoms

1011 atoms

"Teraflop"

1965 1975 1985 1995 2005 2012 Year

Computer power BlueGene USA 70 TFLOP NASA Ames USA 50 TFLOP Earth Simulator Japan 40 TFLOP LINUX Clusters

IBM Almaden Spark

"Gigaflop"

"Petaflop" computers


82

Bridging length scales by direct numerical simulation (DNS)

Understand the behavior of complex many-particle systems, without imposing constraints or boundary conditions

Discover new physical phenomena, e.g. collective events that involve a large number of particles

Caution!

Need to make sure that model produces useful results, i.e. includes new scientific content and discoveries

Pictures may be pretty, but what do we learn?

Why is large-scale modeling useful?

83

Visualization challenges

Huge amount of data: Single frame may have 100 GB data and moreLarge tiled display at USC, showing hypervelocity impact damage of a ceramic plate with impact velocity

Courtesy of Elsevier, Inc, http://www.sciencedirect.com. Used with permission.

http://www.sciencedirect.com

84

The need for “size”

Modeling of mechanical behavior of materials is highly demanding and requires models with millions and billions of atoms

200570,000,000,000

particles70 TFLOP computers

20107,000,000,000,000 particles

1,000 TFLOP computers

0.3 µm

1.2 µm

5 µm20001,000,000,000

particles10 TFLOP computers

3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling...

Documents

Transcript of 3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling...