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Bridging-Scale Methods and A Brief Introduction to Finite Element Methods

Wing Kam Liu

Northwestern University

Department of Mechanical Engineering


Multiple Scale References

• F.F. Abraham, J. Broughton, N. Bernstein and E. Kaxiras, Europhysics Letters1998; 44:783-787

• E. Tadmor, M. Ortiz and R. Phillips, Philosophical Magazine A 1996; 73:1529-1563

• L. Shilkrot, W.A. Curtin and R.E. Miller, Journal of the Mechanics and Physics of Solids 2002; 50:2085-2106

• G.J. Wagner and W.K. Liu, Journal of Computational Physics 2003; 190:249-274• W.K. Liu, E.G. Karpov, S. Zhang and H.S. Park. “An Introduction to

Computational Nano Mechanics and Materials.” Computer Methods in Applied Mechanics and Engineering 2004; 193: 1529-1578.

• H.S. Park and W.K. Liu, Computer Methods in Applied Mechanics and Engineering 2004; 193:1733-1772

• H.S. Park, E.G. Karpov, P.A. Klein and W.K. Liu, submitted to Philosophical Magazine, 2003

• D. Qian. G.J. Wagner and W.K. Liu, Computer Methods in Applied Mechanics and Engineering 2004; 193:1603-1632

• S.P. Xiao and T. Belytschko, Computer Methods in Applied Mechanics and Engineering 2004; 193:1645-1669


MD Boundary Condition References

• S.A. Adelman and J.D. Doll, Journal of Chemical Physics 1976; 64:2375-2388

• W. Cai, M. DeKoning, V.V. Bulatov and S. Yip, Physical Review Letters 2000; 85:3213-3216

• W.E. and Z.Y. Huang, Journal of Computational Physics 2002; 182:234-261

• G.J. Wagner, E.G. Karpov and W.K. Liu, Computer Methods in Applied Mechanics and Engineering 2004; 193:1579-1601

• E.G. Karpov, G.J. Wagner and W.K. Liu, submitted to Computational Materials Science 2003

• E.G. Karpov, H. Yu, H.S. Park, W.K. Liu, J. Wang and D. Qian, submitted to Physical Review B 2003


Overview

• Motivation for multiple scale methods– Summary of previous concurrent methods

• Bridging scale concurrent method– Molecular dynamics (MD) boundary condition

• Numerical examples– 1D wave propagation

– 2D wave propagation

– 2D dynamic crack propagation

– 3D dynamic crack propagation

– Extension of MD boundary condition to non-nearest neighbor interactions

• Conclusions and future research


Multiscale Phenomena in Solids - Fracture/Failure

17:1

250:1250:1

200 µm

S.Li, W.K. Liu, A.J. Rosakis, T. Belytschko, and W. Hao , International Journal of Solids and Structures, Vol. 39, pp. 1213-1240.


Motivation for Multi-Scale Methods

• Bottlenecks in industrial design process– Experiments are expensive and time consuming

• Solution of mechanics problems which have been resistant to single-scale solution methods– Fracture in solids

• Need to incorporate small scale behavior into larger scale models– Predictive capabilities needed, particularly for material failure

• Solution to above problems first requires a fully developed multiple scale method


Hierarchical vs. Concurrent Multiscale

• Hierarchical Multiscale– Use known information at one scale to generate model for larger scale

– Information passing typically through some sort of averaging process

– Example: Young’s modulus in elasticity

• Concurrent Multiscale– Perform simulations at different length and time scales simultaneously

– Dynamic sharing of information between disparate simulations

• Major issues to consider:– Coupling between length scales - how do you do it?

– Handling interface where small and large scales intersect


Concurrent Multiscale: FE/MD/TB

• Finite elements (FE), molecular dynamics (MD), and tight binding (TB) all used in a single calculation (MAAD)

• MAAD = macroscopic, atomistic, ab initio dynamics

• Demonstrated in brittle fracture of Silicon

• Developed by Abraham, Broughton and co-workers

From Nakano et al, Comput. In Sci. and Eng., 3(4) (2001).


Concurrent Multiscale: FE/MD/TB

• Scales are coupled in “handshake” regions

• Finite element mesh graded down to atomic lattice in overlap region

• Total Hamiltonian is energy in each domain, plus overlap regions

( ) ( )( ) ( )

( )rr

rrrr

rruuuu

&

&&

&&&

,

,,

,,,,

/

/

TB

TBMDMD

MDFEFETot

H

HH

HHH

++++=

From Nakano et al, Comput. In Sci. and Eng., 3(4) (2001).


Concurrent Multiscale: Quasicontinuum (QC) Method

From Tadmor and Phillips, Langmuir 12, 1996

• Adaptive finite element method• Unique point: stress derived directly from

atomistic interatomic potential via Cauchy-Born rule

• Validated on quasistatic nanoindentation, fracture, grain boundary interaction problems

• Originally developed by Tadmor, Ortiz and Phillips


(Quasicontinuum method)

Cauchy-Born rule: able to derive a continuum stress tensor and tangent stiffness directly from the interatomic potential

gradient of deformation

( ) ( )

( )X

uX

xF

∂∂

+=

∂∂

=

tX

tXtX

,1

,,

gradient of deformation is constant for each element if linear shape function is used

E. Tadmor et al. “Quasicontinuum analysis of defects in solids” Philosphical Magazine A 1996; 73:1529-1563

Cauchy Born Rule


FP

∂∂

≡w

2

2

FC

∂∂

≡w

Lagrangian tangent stiffness tensor

1st Piola-Kirchhoff stress

If the potential energy for the atomic volume in the reference configuration is

( )∑∑>

=i ij

ijrWW

Strain energy density ( )0V

Ww ≡F

0V

Total strain energy ( ) ( ) WdwWtot =Ω≡ ∫Ω0

0FF


FEM vs. MD

( ) ( ) ( )∫

∑∑

Ω

Ω≈

∆∆

==

0

0dNN

VNNV

mNNmM

JI

ii

iJ

iI

i

i

i

iJ

iIiIJ

XXXρ

The atom with mass can be assigned a volumeim iV∆i

i atomic number I nodal number

The mass matrix in FEM can be obtained from the atomic mass

Shape function ( )iIiI NN X= density ( )

i

ii V

m

∆=Xρ

im


FEM vs. MD

( )i

I

i

i i

ii

i I

i

II

MDI V

wV

wW∆

∂∂

∂∂

=∆∂∂

=∂

∂=≡ ∑∑ x

F

Fxx

xff int

( )i

ii

w

FXP

∂∂

= ( ) ( )∑ ∂∂

=∂

∂≡

II

iIii

Nx

X

X

X

XxF

because

( ) ( ) ( ) ( )∫∑Ω

Ω∂

∂≈∆

∂∂

=0

0int d

NV

N Iii

i

iII XP

X

XXP

X

Xf

The internal force of FEM can be obtained from MD potential

Strain energy densityiw

Total Strain energyW ( ) ii

i VwW ∆= ∑X


( ) ( )0

0

Ω∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−= ∫

Ω

dwN

MT

III F

F

X

Xd&&

( )i

iiW

mx

xx

∂∂

−=&&

FEM

MD

Discrete equations

.....1=i number of atoms

It is too expensive to solve the above equation. Cauchy Born rule is used so that the MD equations of motion is approximated by a FEM approach.

.....1=I number of nodes << number of atoms


Recently Developed Methods

• Coupled Atomistic/Discrete Dislocation method (CADD)– Developed by Shilkrot, Curtin and Miller

– Couples atomistics with discrete dislocation continuum

– Allows the passing of defects (in two-dimensions) from atomistic to continuum region

– Currently valid for quasi-static problems

• Bridging Domain method– Developed by Xiao and Belytschko

– Dynamic, concurrent coupling of atomistics and finite elements

– Dissipation of high frequency waves using spatial filtering

– Demonstrated on two-dimensional problems


Summary of Multiple Scale Approaches

• Pros:– QC allows full atomistic resolution around defects, crucial atomic regions

– MAAD successfully applied to brittle fracture of silicon

– CADD combines atomistic and discrete dislocation methods, allowing defects to propagate from atomistic to continuum regions

• Cons:– FE region meshed down to atomic scale in MAAD

– No explicit treatment of small scale waves in continuum region in MAAD

– QC and CADD currently limited to quasistatic problems

• Remaining issues:– Need for dynamic, finite temperature multiple scale method

– True coarse scale representation

– Mathematically consistent treatment of high frequency waves emitted from MD region in continuum


Spurious Wave Reflection in MD


Concurrent Multiple Scales: Goals

• Development of method for coupling molecular dynamics to finite element or meshfree computations in concurrent simulations

• Simulation of time dependent, finite temperature problems

• True “coarse scale” continuum discretization -- no meshing down to atomic scale

• Multiple time-stepping algorithms to take advantage of multiple time scales

– don’t want to be limited to nano time scale everywhere in the domain

• Re-use of existing MD and continuum codes

• Easily parallelizable algorithms


Introduction to Bridging Scale Concurrent Method

• 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

• Mathematically consistent treatment for MD boundary condition/high frequency waves

• 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


Boundary Conditions

• We want to avoid grading the coarse mesh down to the atomic lattice scale

– expensive

– too much information

– limits coarse scale time step

• Information passes from a fine MD lattice directly into a coarse scale mesh

• Main problem: this leads to internal reflection of small-scale waves– small-scale energy can’t be represented on the coarse scale, has nowhere

else to go


Boundary Conditions: the Wrong Way

• Most obvious approach is to set MD velocity equal to coarse scale velocity on the boundary:

( )∑=I

IIN dxq && αα


numerical solution of coarse scale calculation at a nodal point I

Wagner and Liu, “Coupling of Atomistic and Continuum simulations using a Bridging scale decomposition”, Journal of Computational Physics 2003

2x

Id

u

ω

qα

Coarse scale calculation Fine scale calculation

dq numerical solution of fine scale calculation at an atomic position α

u total solution at an atomic position α

Bridging scale concurrent method

u Nd Qq= +Express total solution with the sum of two solution of differentresolution level. N and Q are defined later.


We represent the total solution by the sum of coarse scale solution and the enriched part.

u

uuu ′+=

Bridging scale decomposition

Note: interpolation by finite element shape function NI(x)

1d2d

3d 4d

x

u1

2

1 2( ) ( ) ( ) ( )c

c

n

n

d

du x N x N x N x

d

α α α α

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤= ×⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

LM u (xα )

xα:number of nodes in coarse scalecn

is the interpolated coarse scale solution at atomic positionsu xα


1-D Interpolation

• In FEM, the nodal displacements are unknown variables, and the displacement field is obtained by using finite element approximation.

( , ) ( ) ( )I I

INu X t X u t=∑

where are nodal displacements

and are shape functions or interpolation functions

( )Iu t( )IN X

Note: Here, we only consider Lagrange interpolation functions. Therefore, the interpolation functions are functions of material coordinates X.


Linear Interpolation Functions

• Linear interpolation is done between two adjacent nodes

1 1 2 2( , ) ( ) ( ) ( ) ( )u X t N X u t N X u t= +

1 1 1 1 1 2 1 2( ) ( , ) ( ) ( ) ( ) ( )u t u X t N X u t N X u t= = +

2 2 1 2 1 2 2 2( ) ( , ) ( ) ( ) ( ) ( )u t u X t N X u t N X u t= = +

2 1( ) 0N X =

1 1( ) 1N X = 1 2( ) 0N X =

2 2( ) 1N X =


Shape functions in 1-D

(1) Linear shape function2

11 2

( )X X

N XX X

−=

−1

21 2

( )X X

N XX X

−=

−

1 32

2 1 2 3

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

(2) Quadratic shape function

2 31

1 2 1 3

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

1 23

3 1 3 2

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

1( )N X2 ( )N X

1X 2X

1( )N X3( )N X

2 ( )N X

1X 3X2X


1-D Lagrange Polynomials

11

1

( )

( )( )

nen

b

bb anen

a nen

a b

bb a

l

ξ ξ

ξξ ξ

=≠−

=≠

−

=−

∏

∏

order of polynomial: nen-1

the following is the general function:


2 node 1-D Lagrange Polynomial

• nen=2; order=nen-1=1

2111

1 2

( ) ( 1) 1( ) (1 )

( ) ( 1 1) 2l N

ξ ξ ξξ ξξ ξ

− −= = = − =

− − −

1122

2 1

( ) ( 1) 1( ) (1 )

( ) (1 1) 2l N

ξ ξ ξξ ξξ ξ

− += = = + =

− +

21

1

( ) ( )a aa

X l Xξ ξ=

= ∑




2 3211

1 2 1 3

( )*( ) 1( ) ( 1)

( )*( ) 2l N

ξ ξ ξ ξξ ξ ξξ ξ ξ ξ

− −= = − =

− −

1 3222

2 1 2 3

( )*( ) 1( ) (1 )

( )*( ) 2l N


− −= = + =

− −

32

1

( ) ( )a aa

X l Xξ ξ=

= ∑

2 23 ( ) 1l ξ ξ= −


Finite Element Interpolation in 2-D

For each element, we define

44332211 uNuNuNuNu +++=

44332211 vNvNvNvNv +++=

∑=

= ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧ 4

1

NEN

I I

II v

uN

v

u

Shape functions are defined in such a way thatIN

IJJI xN δ=)(Bi-linear interpolation

xyayaxaau 3210 +++=xybybxbbv 3210 +++=



1 1 2 2 3 3 4 4( , ) ( , ) ( , ) ( , ) ( , )X N X N X N X N Xξ η ξ η ξ η ξ η ξ η= + + +

remember that

( , )I J J IJN ξ η δ=

1 1 1

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = − −

2 2 1

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = + −

3 2 2

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = + +

4 1 2

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = − +


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

u x( )= u x( )+ ′ u x( )

u x( )= Pq x( )= NI x( )dI

I

∑


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:

′ u x( )= q x( )− Pq x( )

= +

( )xu ( )xu ( )xu′

Slide courtesy Dr. Greg Wagner


The coarse scale part

With this matrix, we can evaluate interpolated coarse scale solution at each atomic position.

1 1 1 1

1

1

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

N

c

c

f f c f

I n

I a n

n I n n n

N x N x N x

N x N x N x

N x N x N x

α α

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥≡⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L L

M O M O M

L L

M O M O M

L L

We define a matrix N

number of nodes in coarse scalenumber of atoms in fine scalefn

cn

u ≡ Nd


We define as the difference between fine scale solution qand its projection onto the coarse scale Pq.

Minimizing above

qMNMw AT1−=

αm Atomic mass of an atom α

,NMNM AT=

u′

The fine scale part: projection operator

P can be obtained by finding a coarse scale nodal displacement w which minimize the mass weighted square error.

where

′u ≡ q − Pq

Error ≡ mα q − NIα w

II

∑2

α∑

1

2

3

0 0

0 0for 1-D case

0 0M A

m

m

m

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

L

L

L

M M M O


With this w, we can write in terms of q

AT MNNMP 1−=

The operator for the projection onto the coarse scale basis

PPP =

Total displacement

The fine scale part: projection operator ( cont.)

u′

′u = q − Nw = (I − NM−1NT MA)q

u = u + ′u = Nd + q − Pq = Nd +Qq

Q = I − P


Recall u can be represented by coarse degrees of freedomd and fine scale degrees of freedom q.

The Lagrangian L is the kinetic energy minus the potential energy if no external forces are considered

Multiscale Lagrangian

Kinetic energy is partitioned into two parts.

( ) ( ) ( ) ( )1

2u,u u u u M u uT

AL K V U= − = −& & & &

1 1 1

2 2 2T T T

A = +u M u d Md q q& & && & M

T TA A= =Q M Q Q MM


u

uuuf

∂∂

−≡′+)(

)(U

Multiscale equations of motion

d0

d

d0

d

L L

t

L L

t

⎧ ∂ ∂⎛ ⎞ − =⎜ ⎟⎪ ∂ ∂⎝ ⎠⎪⎨

⎛ ⎞∂ ∂⎪ − =⎜ ⎟⎪ ∂ ∂⎝ ⎠⎩

d d

q q

&

&

We can obtain equations of motion by

( )

( )

( ) ( )

( ) ( )

T T

T T T

U U

U U

⎧ ∂ ∂ ∂ ′= − = − = + = +⎪ ∂ ∂ ∂⎪⎨ ∂ ∂ ∂⎪ ′= − = − = + = +⎪ ∂ ∂ ∂⎩

d,q uΜd N f u u N f Nd Qq

d u dd,q u

q Q N f u u Q f Nd Qqq u q

&&

&&M

Substituting the expression for the multiscale Lagrangian into the above equations gives

where

( ) ( ) ( )1 1,

2 2u u d,d,q,q d Μd q q d,qT TL L U= = + −& & && & && M


If external forces are considered

( )extT fQqNdfNdΜ ++= )(&&

( )extT fQqNdfQq ++= )(&&

( )extTA

T fQqNdfQqMQ ++= )(&&

extA fQqNdfqM ++= )(&&

Since Q is singular, there are many candidates for q

We can pick one, which satisfies

Multiscale equations of motion (cont.)

Look at the fine scale equation of motion

Hence, the set of coupled coarse and fine scale equations are:

( ) QqqfqNdfNdΜ =′+′+= ,)( extT&&

extA fqNdfqM +′+= )(&&

Coarse scale

Fine scale


Coupled Multiscale Equations of Motion for LJ Potential

• Coupled equations of motion are:

• First equation is MD equation of motion

• Second equation is FE equation of motion with internal force obtained from MD forces

• FE equation of motion is redundant if MD and FE exist everywhere

intAM q f=&&

intTMd N f=&& f int = −∂φ r( )

∂r

φLJ r( )= 4εσr

⎛⎝⎜

⎞⎠⎟

12

−σr

⎛⎝⎜

⎞⎠⎟

6⎛

⎝⎜

⎞

⎠⎟


Recap what we have done so far

We separate the total solution as u = u + q’ , q’= Qq such that:

We can interpret this equation in the frequency domain. Fourier tranform of the equation:

total frequency spectrum of u

= FEM + molecular dynamic - “coarse scale projector, Pq”

Hence, in the limit Pq=Nd.

Substitute into the Lagrangian equation gives the coupled multiscale equations:

( )u Nd Qq Nd qcoarsescale finescale I P FEM MD overlap= + = + = + − = + −

( ) QqqfqNdfNdΜ =′+′+= ,)( extT&&ext

A fqNdfqM +′+= )(&&Coarse scale

Fine scale


1. Outside of the region of interest,

• we only perform the coarse-scale calculation, because we do not need detailed information about this region. It can be solved quickly.

2. In the region of interest,

• We perform both the fine-scale and the coarse-scale calculation.

• The coarse scale can describe the homogeneous part of the deformation using just a few degrees of freedom, and can be solved quickly; the solution of fine scale equations provides a refinement to the coarse scale solution

• Retaining the coarse scale equation in this region provides a seamless transition from the MD region to surrounding coarse scale.

How to reduce the computational cost?

MD calculation of entire domain is too expensive.


Bridging Scale Schematic

+ =

MD FEM

M q fA =&& M d N fT=&&

MD + FEM

Reduced MD & Impedance Force + FEM

M d N fT=&&

( ),M q f f d qim pA = +&&

M d N fT=&&

M q fA =&&


Since nf DOFs are too many to compute, we intend to write the solution of most of the fine scale atoms in terms of solution of coarse scale nodes and a few of fine scale atoms.

Id qα

u

ω

atom -1

atom 0 atom 1

We need to take cumulative effect of eliminated atoms into account

2x

Handling MD Boundaries

bqaq

DOFs to compute DOFs to eliminateRegion a Region b


Handling of MD boundaries DOFs (affordable)an

DOFs (Too many)bn1

2

( , , )

( , , )a a bAa

b a bAb

′ ′⎡ ⎤ ⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥ ′ ′⎩ ⎭ ⎩ ⎭⎣ ⎦

q f d q qM 0

q f d q q0 M

&&

&&

Proposed ApproachSplit the domain and solve for .

1

1

( , , )

( , , )a Aa a a b

b Ab b a b

−

−

′ ′⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬′ ′⎩ ⎭ ⎩ ⎭

q M f d q q

q M f d q q

&&

&&

Solve the bottom part for b′q

Substitute it into the top part

b′qSolution Techniques

•Assume periodic structure.

•Use Fourier/LaplaceTransform in space/time and solve for the entire domain.

•Obtain the relation of displacements between the atoms inside and outside of the boundary.

•Applicable to multi-DOF case


Periodic structure (1-DOF )

A is a circulant matrix of infinite size.

um

k node 0 node 1

qAq ′−=′&&

A−1= −k / m, A

0= 2k / m, A−1

= −k / mLet

Fine part2

1

0

1

2

2

2

2

2

2

0 0

0 0

0 0

0 0

0 0

k k k

m m mk k k

m m mk k k

m m mk k k

m m mk k k

m

q

q

q

m

q

q

m

−

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎧ ⎫⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪

′ ⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪′ ⎢ ⎥⎪ ⎪⎪ ⎪ ⎢ ⎥′ =⎨ ⎬⎢ ⎥⎪ ⎪′ ⎢⎪ ⎪⎢′⎪ ⎪⎢⎪ ⎪⎢⎪ ⎪⎢⎪ ⎪⎩ ⎭ ⎢⎢⎢⎣ ⎦

−

−

−

−

−

L L L L L L

L L L L L L

ML L L L

M

&& L L L L&&

&& L L L L

&&

L L L L&&

ML L L L

M

L L L L L O

L L L L

O O O

O O O

O O

O

O

OO OL

2

1

0

1

2

q

q

q

q

q

−

−

⎧ ⎫⎪ ⎪⎪ ⎪

′⎪ ⎪⎪ ⎪′⎪ ⎪⎪ ⎪′⎨ ⎬⎪ ⎪′⎥ ⎪ ⎪

⎥ ′⎪ ⎪⎥ ⎪ ⎪⎥ ⎪ ⎪⎥ ⎪ ⎪⎩ ⎭⎥⎥⎥

M

M

M

M


Splitting the domain (1DOF)

Instead, we express ′q1 in terms of ′q

0

2 2

1

0

1

2

1 0 1

1 0 1

1 0 1

1 0 1

1 0 1

0 0

0 0

0 0

0 0

0 0

A A A

A A A

A A A

A A A

A A

q q

q q

q

q

q A

−

−

−

−

− −

−

−

⎧ ⎫ ⎡ ⎤⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎢ ⎥

′ ′⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎢ ⎥′ ′⎪ ⎪ ⎢ ⎥⎪ ⎪′ ⎢ ⎥= −⎨ ⎬

⎢ ⎥⎪ ⎪′ ⎢ ⎥⎪ ⎪⎢ ⎥′⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎩ ⎭ ⎣ ⎦

M L L L L L L M

M L L L L L L M

&& L L L L

&& L L L L

&& L L L L

&& L L L L

&& L L L L

M L L L L L O

M L L L

O O O

O O O

O O

O O OL L O

O

1

0

1

2

q

q

q

−

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪′⎨ ⎬⎪ ⎪′⎪ ⎪

′⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

M

M

1 1 0 1 1n n n nq A q A q A q+ − − +′ ′ ′ ′= − − −&&

DOFs to compute

DOFs to eliminate

For arbitrary row we can write

From node -∞ to node -1, no problem for computation.

For node 0 0 1 1 0 0 1 1q A q A q A q+ − −′ ′ ′ ′= − − −&& We don’t compute 1q′analytically.

aq′

bq′


Assume a localized force (1-DOF)

We assume an external point force acting at only node 0.

We can write the equation of motion for each node as follows.

ext0

1 1 0 1 1n

n n n n

fq A q A q A q

m

δ+ − − +′ ′ ′ ′= − − − +&&

fnext = δ

0nf ext

Note: Discrete Fourier Transform (DFT) of this external force is a constant in wave number space.

δ

0nf

0ext = e− inpδ

0nf ext

n=−∞

∞

∑ = e0 f ext = f ext


Procedure (1-DOF)

1. Take the DFT in space to obtain the equation for

2. Take the LT in time to obtain the equation for

3. Solve for

4. Take the inverse DFT to obtainand derive the relation between and

5. Take the numerical inverse LT to obtain the relationbetween

ext0

1 1 0 1 1( ) ( ) ( ) ( ) nn n n n

fq t A q t A q t A q t

m

δ+ − − +′ ′ ′ ′= − − − +&&

ˆ ( , )q t p′

ˆ ( , )Q s p

ˆ ( , )Q s p′

′Qn(s)

′Q0(s) ′Q

1(s)

′q0(t) and ′q

1(t)

Equation of motion


Step 1: Take DFT

ext

ˆ ( , ) ( ) ( , )f

q p t A p q p tm

′ ′= − +&&

1. Taking the DFT of the above equation gives

2 2ˆ( ) (1 cos( ))ip ipn

k k k kA A p e e p

m m m m−= = − + − = −F

DFT of An is a sum of only 3 terms(because of tri-diagonal shape of A)

Note

ext0

1 1 0 1 1( ) ( ) ( ) ( ) nn n n n

fq t A q t A q t A q t

m

δ+ − − +′ ′ ′ ′= − − − +&&

( ) ˆ ˆ( ) ( )n m mm

A q A p q p∞

−=−∞

⎧ ⎫=⎨ ⎬

⎩ ⎭∑F

We use the characteristics of the DFT of convolution

Note

Fine scale equation of motion.


Step 2 and 3: Take the LT

( ) ( )ext

2 ( )ˆˆ ˆ ˆ( , ) 0, 0, ( ) ( , )F s

s Q s p sQ p Q p A p Q s pm

′ ′ ′ ′− − = − +&

( ) ( ) ( )ˆˆ ˆ ˆ( , ) ( , ) 0, 0,extF s

Q s p G s p sQ p Q pm

⎛ ⎞′ ′ ′= + +⎜ ⎟

⎝ ⎠&

12ˆ ( , ) ( )G s p s A p−

⎡ ⎤= +⎣ ⎦It is easy to obtain the inverse

2. Taking the LT in time gives (here we pick up the initial conditions)

3. Solve for the displacement

1DOF/node: scalar2DOFs/node: 2x2 matrix

ˆ ( , )Q s p′

where

ext

ˆ ( , ) ( ) ( , )f

q p t A p q p tm

′ ′= − +&&

Fine scale equation of motion in wave number space.


Step 4: Take the inverse DFT

( ) ( ) ( ) ( )11 1

0 0

( ) ( ) ( ) ( ) 0 0j j k k k k k kk k

Q s G s G s Q s s G s q G s q−′ ′ ′ ′− −

′ ′> >

′ ′ ′ ′= + +∑ ∑ &

( ) ( ) ( ) ( ) ( )( ) ( ) 0 0ext

n n n n n n n nn n

F sQ s G s s G s q G s q

m′ ′ ′ ′− −

′ ′

′ ′ ′= + +∑ ∑ &

4. Take Inverse DFT

We can write down the equation above for n=j and n=k( )( ) ( ) ( ) ( ) ( )1 1

0 0

( ) ( ) /

( ) ( ) / 0 0

extj j

extk k k k k k

k k

Q s G s F s m

Q s G s F s m s G s q G s q′ ′ ′ ′− −′ ′> >

′⎧ =⎪⎨ ′ ′ ′= + +⎪⎩

∑ ∑ &

Eliminate external force and solve for ′Qj(s)

Expression of in terms of in Laplace space. ′Qj(s) ′Q

k(s)


Step 5: Take the numerical inverse LT

′qj(t) = θ(t − τ ) ′q

k(τ )dτ

0

t

∫ + R t( )

5. Taking Numerical Inverse LT gives the expression in real time space.

where θ(t) = −1 G

j(s)G

k−1(s)

Since we only need the relation between ′q0(t) and ′q

1(t)

′q1(t) = θ(t − τ ) ′q

0(τ )dτ

0

t

∫ + R t( ) where θ(t) = −1 G

1(s)G

0−1(s)

( ) ( ) ( ) ( ) ( )1 10 0

0 0n n n nn n

R t g t q g t q′ ′ ′ ′− −′ ′> >

′ ′= +∑ ∑& &The forcing term due to initial conditions


Step 6: Recombine Coarse and Fine Scales

Substitute into the top part of the split discrete equation

( )

1

0 1

1

2 2

1 1

0 0 00

0 1

1 0 1 1

0000

0 ( ) ( )t

q q

q q

q

A

A A

A A A

A A A Aq t q d R tθ τ τ τ

−

−− −

− −−

− −

⎧ ⎫⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪⎪ ⎪ ⎢ ⎥ ⎪ ⎪′ ′⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥= − −⎨ ⎬ ⎨ ⎬ ⎨ ⎬′ ′⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪′ ′ ′− +⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭∫

MM M M

&& &&L

&& &&L

&

O

&& &L

Add to the coarse scale part

( )

1

0 1

1 0 1

1

21

1

0 0010

1

1

000

( ) 0

0 ( ) ( )

( , , )

q q M f d

M f d q q 0

a a Aa a

t

Aa a a b

A

A A

A A A

q

q

q t qA tA A Ad Rθ τ τ τ

−

−

−

− −

−−

−

−

⎧ ⎫⎡ ⎤ ⎧ ⎫⎪ ⎪⎢ ⎥ ⎪ ⎪′⎪ ⎪ ⎪ ⎪⎢ ⎥′+ = − −⎨ ⎬ ⎨ ⎬′⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪′ ′− +⎣ ⎦ ⎩ ⎭ ⎩ ⎭

′ ′ =∫

MM M

&&L&& &&&&L

&&L14444444244444443

O

1 ( )q M f da Aa a−=&&


Final Form

( )1

1 0 10

( , , )( ) ( )

0q M f d q q 0 ta Aa a a b

A t q d A R tθ τ τ τ−

− −

⎧ ⎫⎪ ⎪′ ′= = − ⎨ ⎬′− +⎪ ⎪⎩ ⎭∫&&

Time history kernelAdded only to the interface nodeobtained from the history of node 0

Obtained from fine scalecalculation by setting ′q

b= 0

Random/Stochastic ForceThermally motivated forces that continuum exerts on reduced MD system


Reduced Fine Scale Equation

• Define damping kernel:

• Integrate by parts:

• This is known as a generalized Langevin equation, which includes a dissipation term and a random force term R(t)

– R(t) is used to describe chaotic thermal motion

( )11 0 10

( , , ) ( ) ( )q M f d q q 0 qt

a Aa a a b A t d A R tθ τ τ τ−− −′ ′ ′= = − − +∫&&

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )11 1 1 1 1 10

0q M f u q u q ut

At t t t d R tβ β τ τ−= + − − − − +∫ &&& &%

( ) ( ) ( ) ( )ttdtt

Θ−=β⇒ττΘ=β ∫∞ &

'q q u= −


Remarks

• The total forcing term consists of four major parts:– The standard force computed in MD simulation by assuming

displacements of all atoms just outside the boundary are given by the coarse scale

– A modified stiffness at the boundary (due to slight difference between total scale and coarse scale)

– Time history-dependent dissipation at the boundary

– Random forcing term at the boundary

– this term can be related to the temperature of the solid:

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )tRdttttt

A 10 111111

1 0~ +τ−τ−β−−β+= ∫− uququfMq &&&&

( ) ( ) ( ) TkttRtR Bijji βδ−=


1D Bridging Scale Wave Propagation Example

• 111 atoms in bridging scale MD system• 30 finite elements• 10 atoms per finite element• ∆tfe = 55∆tmd

• Lennard-Jones 6-12 potential

• Initial MD & FEM displacements

• H.S. Park, W.K. Liu. An Introduction and Tutorial on Multiple Scale Analysis in Solids. Computer Methods in Applied Mechanics and Engineering (193), 1733-1772, 2004


1D Wave Propagation Example (Movies)

• Compare with Full MD simulation• With impedance force


1D Wave Propagation - Energy Transfer

• 99.97% of total energy transferred from MD domain• Only 9.4% of total energy transferred without impedance force


Statistical Mechanics and Energy

• Coarse scales are those that can be observed directly in experiments

• Averaged effects of fine scales are important

• Ensemble averaging: average over the atomic positions and momenta qand p while holding the coarse scales fixed

• Example: averaged kinetic energy gives coarse scale kinetic energy plus internal energy proportional to temperature:

• Internal energy = 0 when Natoms = nnodes

∫ ∫∫ ∫

+−

+−

=pq

pq

dde

ddAeA

TkUK

TkUK

BE

BE

/)(

/)(

( ) ( ) TkmnNMK BaJI

JIIJE nodesatoms, 2

3

2

1−+⋅= ∑ dd &&


Energy Equation

• Temperature can be related to average of fine scale velocities:

• In the MD region, MD provides exact rhs for coarse scale equation

• Outside MD region, treat fine scale as Gaussian random variable

( ) ( )

( ) ( ) ( )( )αααααα

α

ααα

ααα

α

−′=′′≈

′′≈′′=

ffuPk

uuPk

mxT

uuPk

muu

k

mxT

BB

BB

&&&&&

&&&&

22

( ) ( )∑α

αααα −′= ffuxNk

TM IB

JIJ && 2


Temperature Field from Small Scales


1D Example: Heat Propogation


2D Bridging Scale Problems

• Lennard-Jones (LJ) 6-12 potential

• Potential parameters σ=ε=1

• Nearest neighbor interactions

• Hexagonal lattice structure ([111] plane of FCC lattice)

• Impedance force derived numericallyfor hexagonal lattice, LJ potential

• Hexagonal lattice with nearest neighbors


2D Wave Propagation

• MD region given initial displacements with both high and low frequencies similar to 1D example

• 30000 bridging scale atoms, 90000 full MD atoms

• 1920 finite elements (600 in coupled MD/FE region)

• 50 atoms per finite element

• Initial MD displacements


2D Wave Propagation

• Snapshots of wave propagation


2D Wave Propagation

Energy Transfer Rates:• No BC: 35.47%• ncrit = 0: 90.94%• ncrit = 4: 95.27%• Full MD: 100%

• ncrit = 0: 0 neighbors• ncrit = 1: 3 neighbors• ncrit = 2: 5 neighbors

n n+1 n+2n-1n-2


A 15-walled NT with over 3 million atoms. Outer most layer is (140,140) tube with interlayer distance of 3.34 Å

=Multi-ScaleSimulation

90 nm

18.98nm

MD region: Two 15-walled NT with the total 340,200 atomsand interlayer distance of 3.34 Å

+

Fine scale represented byMolecular Dynamics

6.6 nm

6.6 nm

18.98 nm

Meshfree region: 15 layers of continuumShell structure approximated by meshfree method with 27,450 particles

Coarse scale represented byMeshfree approximation+

18.98 nm

90 nm

Nanoscale Material : Bending of a 15-Walled CNT(Professor Dong Qian, U. of Cincinnati)


Multi-Scale Simulation Coupled MD Simulation

Nanoscale material : Bending of a 15-Walled CNT

Movie


MWCNT embedded in thin film made by polymer material. Thermo-loading causes distortion of the polymer and MWCNT is then loaded.

Multi-Scale Simulation

Nanoscale material : Bending of a 15-Walled CNT


Example : 2D Dynamic Crack Propagation

Problem Description:• LJ 6-12 potential• Nearest neighbor interactions• 90000 atoms, 1800 finite elements (900 in coupled region)• 100 atoms per finite element• 40 MD time steps per FEM time step• Ramp velocity BC on FEM• Full MD = 180,000 atoms

• H.S. Park, E.G. Karpov, P.A. Klein and W.K. Liu, The Bridging Scale for Two-Dimensional Atomistic/Continuum Coupling, submitted to Philosophical Magazine A, 2003


2D Dynamic Crack Propagation (movies)

Entire domain for coupled crackpropagation example

results

Ref: Harold Park

Finite elements

MD domain


2D Dynamic Crack Propagation

• Full MD = 601x601

• Full domain = 601 atoms• Multiscale 1 = 301 atoms • Multiscale 2 = 201 atoms• Multiscale 3 = 101 atoms


Zoom in of Fracture Surface (movies)

•Zoom in of fracture surface •100 atoms in each finite element•Note extreme deformation of edge elements

Fine grain(coupled MD/FE region)


Zoom in of Cracked Edge


3D Dynamic Crack Propagation

• 3D FCC lattice• Lennard Jones 6-12 potential• Each FEM = 200 atoms• 1000 FEM, 117000 atoms

Time

Velocity

t1

Vmax

FEM

FEM

MD+FEM

Pre-crack

V(t)

V(t)

• H.S. Park, P.A. Klein, E.G. Karpov and W.K. Liu, Three-Dimensional Multiple Scale Analysis of DynamicFracture, in preparation 2004

(Harold Park PhD thesis)


Initial Configuration

• Velocity BC applied out of plane(z-direction)• All non-equilibrium atoms shown


Onset of Crack Branching

• Full MD • Bridging Scale


Final Configuration - 3D Crack Propagation



MD/Bridging Scale Comparison



MD/Bridging Scale Comparison



Movie of MD region around crack initiation region


Movie of FEM/MD regions around crack


Summary: Bridging Scale (Coupled MD/FEM)

• Bridging scale projection provides a unique decomposition of total solution for separation into coarse and fine scales

• Decomposition allows concurrent simulation of fine scale using MD and coarse scale using FEM

• Coarse scale mesh need not correspond to atomic lattice for coupling

• Coarse scale equations and boundary conditions follow directly from the multi-scale formulation

• Projection allows statistical description of fine scale, leading to a definition for the internal energy (temperature)

• Solid mechanics applications:– nano-scale devices

– fracture and failure

– friction and wear


• Self Study Materials


Final Coupled Equations of Motion

• θ(t-τ) called “time history kernel”, and acts to dissipate fine scale energy from MD to surrounding continuum, and results in non-reflecting MD boundary conditions

• Impedance and random forces only at MD boundary atoms; standard MD equation of motion elsewhere

• Stochastic thermal effects captured through random force R(t)

TMd N f=&&

( ) ( ) ( ) ( )( ) ( )0 0

0

t

mq t f t q d d R tθ τ τ τ τ= + − − +∫&&

Standard MD Impedance Force Random Force


is used to represent time history quantities that are needed for integration.

let ( )h,u,up,q,fqM &&& =A

h

nnnnn p,q,u,u,u ΓΓΓ&&& nsand are known

Γ

The superscripts are used to denote the time step, with the bracket notation [j] as a shorthand for the fractional timeStep n +j/m, and the sub-cycle time step will be denoted

. The subscript on coarse scale quantitiesindicates that those quantities are only needed close to the MD boundary.

mttm /∆=∆

Multiple time steps algorithm (fine scale)


nm

jm

jj tt uuuu &&& 2][][]1[

2

1∆+∆+=+

nm

jj t uuu &&&& ∆+=+ ][]1[

q[ j +1] = q[ j ] + ∆tmp[ j ] +1

2∆tm

2 s[ j ]

%p[ j +1] = p[ j ] + ∆tms[ j ]

( )][]1[]1[]1[]1[1]1[ ,~, jjjjjA

j h,u,upqfMs ++++−+ = &

p[ j +1] = p[ j ] +1

2∆tm s[ j +1] + s[ j ]( )

is a predicted atomic velocity, is the FE displacementis the FE velocity, is the MD displacement and is the

MD acceleration.

p~ uu& q s

Multiple time steps algorithm (fine scale) cont.

1. update boundary displacement

2. update boundary velocity

3. update MD displacement

4. predict MD velocity

5. update MD acceleration

6. update MD velocity


nnnn tt avdd 21

2

1∆+∆+=+

( )1111 ++−+ += nnTn QqNdfNMa

( )nnnn t aavv +∆+= ++ 11

2

1

Once the MD quantities are obtained using the fine scale time step algorithm at time n+1, the FE displacement d, velocitiesv and accelerations a are updated from time n to n+1

Multiple time steps algorithm (coarse scale)


2D Wave Propagation

• Above picture is without MD boundary condition

• Above picture is with MD boundary condition


Extension to Non-Nearest Neighbor Interactions

• Atomic interactions inherently non-local

• Some potentials have angular dependence

• Usage of nearest neighbor potentials leads to different physics being displayed (Holian, Physical Review A 1991), particularly at large deformations/high strain rates

• Two formulations:– Impedance force

– Semi-analytical displacement control of ghost atoms - alleviates necessity of controlling ghost atom displacements with finite elements


MD Impedance Boundary Condition

• There are 4 θ’s for second-nearest neighbor interactions instead of 1 for nearest neighbor interactions

f 2→0 = θ1 t − τ( ) q0 τ( )− u0 τ( )( )0

t

∫ dτ + θ2 t − τ( ) q1 τ( )− u1 τ( )( )0

t

∫ dτ

f 2,3→1 = θ3 t − τ( ) q0 τ( )− u0 τ( )( )0

t

∫ dτ + θ4 t − τ( ) q1 τ( )− u1 τ( )( )0

t

∫ dτ

0 1 2 3

Eliminated AtomsBoundary atoms


1D Bridging Scale Wave Propagation

Problem Description:• Non-nearest neighbor harmonic potential of form:

• Initial MD displacements like previous 1D examples• 10 atoms per finite element

• MD energy transfer if MD impedance force applied correctly

Φ r( )=1

2k r − r0( )2


A SHORT COURSE ONFINITE ELEMENT METHODS

WING KAM LIUProfessor of Mechanical Engineering

Northwestern University2145 Sheridan Road, Evanston, IL 60208

Tech A326Voice: 847-491-7094Fax: 847-491-3915

Email: [email protected] Site: http://www.tam.nwu.edu/wkl/liu.html


References

• S.P.Timoshenko, “Theory of Elasticity”, 3rd edition, McGraw-Hill Publishing Company, 1987

• T.J.R.Hughes, “The Finite Element Method”, Prentice-Hall International Inc., 1987

• O.C.Zienkiewicz and R.L.Taylor, “The Finite Element Method”, 4th edition, Vol.1,McGraw-Hill International Editions, 1989

• R.D.Cook, D.S.Malkus and M.E.Plesha, “Concepts and Applications of Finite Element Analysis”, 3rd edition, John Wiley & Sons, 1989

• R.H.Gallagher, “Finite Element Analysis: Fundamentals”, Prentice-Hall Inc.,1975

•T. Belytschko, W. K. Liu and B. Moran, “Nonlinear Finite Elements for Continua and Structures”, John Wiley and Sons, 2000.


OUTLINE

Section I. Nature of Finite Element Method

Section II. 1-D Linear Elasticity

Section III. 1-D Nonlinear Elasticity

Section IV. Hamiltonian in Molecular Dynamics and

Comparison of FEM with MD


Section 1

NATURE OF F.E.M.

Step 1: Discretion of a continuous system

Step 2: Determination of finite element matrices from the

geometry, material, and loading data

Step 3: Assembly of the finite element matrices

Step 4: Displacement and traction boundry conditions

Step 5: Solve the resulting equation system and interpretation of

the results (post-processing)


Type of Elements


Example of Finite Element ModelsBoeing 747 Aircraft

a. Boeing 747 Aircraft. (Cross-hatched area indicatesportion of the airframe analyzed by finite element method.)

b. Substructures for finite element analysisof cross-hatched region.


Ship Structure


Section II

LINEAR ELASTICITY IN 1-D

• Matrix Method

1) Stiffness matrix

2) Assembly

• Finite Element Method

1) Governing equations

2) Strong form and weak form

3) Principle of virtual work

4) Finite element approximation


Matrix Method of a Bar Element

• Equilibrium

• Strain-Displacement Relation

•Stress-Strain Law (Hooke’s Law)

xJ Af σ= 0=+ JI ff

xx εσ Ε=

eee d

J

I

Kf

J

I

IJJI

IJxxJ

IJx

d

d

L

AE

f

fL

ddAEff

L

ddAEAAf

L

dd

L

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−−+

=⎭⎬⎫

⎩⎨⎧

+−=−=

−=Ε==

−==

44 344 2111

11

εσ

δε


Stiffness Matrix Assembly

Element 1

Element 2

Element 3

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

)1(

)1(

)1(22

)1(21

)1(12

)1(11

int

)1(

)1(

)1(

J

I

K

J

I

d

d

kk

kk

f

f

local

434211

2

==

J

I

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

)2(

)2(

)2(22

)2(21

)2(12

)2(11

int

)2(

)2(

)2(

J

I

K

J

I

d

d

kk

kk

f

f

local

434213

1

==

J

I

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

)3(

)3(

)3(22

)3(21

)3(12

)3(11

int

)3(

)3(

)3(

J

I

K

J

I

d

d

kk

kk

f

f

local

434214

3

==

J

I

1f 3f

02 =d 04 =d


Augmented matrices

)1(

4

3

2

1

)1(11

)1(12

)1(21

)1(22

int)1(

4

3

2

1

0000

0000

00

00

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

d

d

d

d

kk

kk

f

f

f

f

)2(

4

3

2

1

)2(22

)2(21

)2(12

)2(11

int)2(

4

3

2

1

0000

00

0000

00

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

d

d

d

d

kk

kk

f

f

f

f

)3(

4

3

2

1

)3(22

)3(21

)3(12

)3(11

int)3(

4

3

2

1

00

00

0000

0000

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

d

d

d

d

kk

kk

f

f

f

f


Equilibrium intFFext =int)3(

4

3

2

1

int)2(

4

3

2

1

int)1(

4

3

2

1

int

4

3

2

1

4

3

2

1

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

fext

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

0000

0000

00

00

()1(

11)1(

12

)1(21

)1(22

4

3

2

1

kk

kk

f

f

f

fext

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

0000

00

0000

00

)2(22

)2(21

)2(12

)2(11

kk

kk

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

4

3

2

1

)3(22

)3(21

)3(12

)3(11

)

00

00

0000

0000

d

d

d

d

kk

kk


Assembled stiffness matrix

ext

f

f

f

f

d

d

d

d

kk

kkkk

kk

kkkk

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

+

4

3

2

1

4

3

2

1

)3(22

)3(21

)3(12

)3(11

)2(22

)2(21

)1(11

)1(12

)2(12

)1(21

)2(11

)1(22

00

0

00

0

Impose boundary conditions

042 ==dd

ext

f

f

d

d

d

d

kkk

kkk

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

+

0

0

1000

00

0010

00

3

1

4

3

2

1

)3(11

)2(22

)2(21

)2(12

)2(11

)1(22

Alternative (reduced) final equations

;3

1

3

1

)3(11

)2(22

)2(21

)2(12

)2(11

)1(22

ext

f

f

d

d

kkk

kkk

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

++

1f 3f

02 =d 04 =d

( ) extRextR fKdfdK1−

=⇒=


Flowchart of FEM Program


1-D Interpolation

• In FEM, the nodal displacements are unknown variables, and the displacement field is obtained by using finite element approximation.

( , ) ( ) ( )I I

INu X t X u t=∑

where are nodal displacements

and are shape functions or interpolation functions

( )Iu t( )IN X

Note: Here, we only consider Lagrange interpolation functions. Therefore, the interpolation functions are functions of material coordinates X.


Linear Interpolation Functions

• Linear interpolation is done between two adjacent nodes

1 1 2 2( , ) ( ) ( ) ( ) ( )u X t N X u t N X u t= +

1 1 1 1 1 2 1 2( ) ( , ) ( ) ( ) ( ) ( )u t u X t N X u t N X u t= = +

2 2 1 2 1 2 2 2( ) ( , ) ( ) ( ) ( ) ( )u t u X t N X u t N X u t= = +

2 1( ) 0N X =

1 1( ) 1N X = 1 2( ) 0N X =

2 2( ) 1N X =


Shape functions in 1-D

(1) Linear shape function2

11 2

( )X X

N XX X

−=

−1

21 2

( )X X

N XX X

−=

−

1 32

2 1 2 3

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

(2) Quadratic shape function

2 31

1 2 1 3

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

1 23

3 1 3 2

( )( )( )

( )( )

X X X XN X

X X X X

− −=

− −

1( )N X2 ( )N X

1X 2X

1( )N X3( )N X

2 ( )N X

1X 3X2X



11

1

( )

( )( )

nen

b

bb anen

a nen

a b

bb a

l

ξ ξ

ξξ ξ

=≠−

=≠

−

=−

∏

∏

order of polynomial: nen-1

the following is the general function:




2111

1 2

( ) ( 1) 1( ) (1 )

( ) ( 1 1) 2l N

ξ ξ ξξ ξξ ξ

− −= = = − =

− − −

1122

2 1

( ) ( 1) 1( ) (1 )

( ) (1 1) 2l N

ξ ξ ξξ ξξ ξ

− += = = + =

− +

21

1

( ) ( )a aa

X l Xξ ξ=

= ∑




2 3211

1 2 1 3

( )*( ) 1( ) ( 1)

( )*( ) 2l N


− −= = − =

− −

1 3222

2 1 2 3

( )*( ) 1( ) (1 )

( )*( ) 2l N


− −= = + =

− −

32

1

( ) ( )a aa

X l Xξ ξ=

= ∑

2 23 ( ) 1l ξ ξ= −


Finite Element Interpolation in 2-D

For each element, we define

44332211 uNuNuNuNu +++=

44332211 vNvNvNvNv +++=

∑=

= ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧ 4

1

NEN

I I

II v

uN

v

u

Shape functions are defined in such a way thatIN

IJJI xN δ=)(Bi-linear interpolation

xyayaxaau 3210 +++=xybybxbbv 3210 +++=



1 1 2 2 3 3 4 4( , ) ( , ) ( , ) ( , ) ( , )X N X N X N X N Xξ η ξ η ξ η ξ η ξ η= + + +

remember that

( , )I J J IJN ξ η δ=

1 1 1

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = − −

2 2 1

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = + −

3 2 2

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = + +

4 1 2

1 1( , ) ( ) ( ) (1 ) (1 )

2 2N l lξ η ξ η ξ η= = − +


Section IIINONLINEAR ELASTICITY

Linear Statics

Nonlinear Statics

ext

F

FKd =int

known

extFF =int

where is a nonlinear function of d

For elastostatics,

Nonlinear algebraic equations to be solved

intF

)(int dNF =


Finite Element Approach

TJ FSFσ ⋅⋅= −1

XX

xF

∂∂

=

Strain energy

stress

( ) ( )FC www ==

the first Piola-Kirchhoff stress( )F

FP

∂∂

=w

( )C

CS

∂∂

=w

22nd Piola-Kirchhoff stress

Cauchy stress

Deformation gradientcurrent coordinatesreference coordinates

x

)det(F=JJacobian

FFC T=Cauchy-Green deformation tensor


Mooney-Rivlin material

Strain energy

( ) ( )33 2211 −+−= IcIcw

( ) ( ) iiCtraceI == CC1

FFC ⋅= T

21,cc are constant

21, II are first and second principal invariants

( ) ( )( ) ( )( ) ( ) ijjiii CCCtracetraceI −=−= 2222 2

1

2

1CCC

is right Cauchy-Green deformation tensor


Mooney-Rivlin material

Second Piola-Kirchhoff stress tensor

CCCS

∂∂

+∂∂

=∂∂

= 22

11 222

Ic

Ic

w

IC

=∂∂ 1I

TII

CIC

−=∂∂

12

( ) CIC

S 2121 2222 cIccw

−+=∂∂

=

( ) TTTT cIcc FFFFFSP ⋅⋅−+=⋅= 2121 222

first Piola-Kirchhoff stress tensor


Polynomial strain energy in 1D

Strain energy

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

63

122

1 F

a

F

aaw

It is Lennard Jones potential in molecular scale

1st Piola-kirchhoff stress

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 7

63

13

122

1 612F

a

F

aaP

321 ,, aaa

are constants

Internal force

( )0

int

0

PdlX

XNf

l

T

∫ ⎟⎠⎞

⎜⎝⎛

∂∂

=


Discrete Equations of Motion

Internal force

Discrete equations

( ) ( )Ω⎟

⎠⎞

⎜⎝⎛

∂∂

=Ω⎟⎠⎞

⎜⎝⎛

∂∂

= ∫∫ΩΩ

dN

dN

TT

σx

xP

X

Xf 0

int

0

If no external force considered

intint fffdM −=−= ext&&

Nodal displacements

Mass matrixM

d


Mass matrix

Consistent mass matrix

Lumped mass matrix

00

0

Ω= ∫Ω

dT NNM ρ

00

0

Ω= ∫Ω

dNNM JIIJ ρ

IJIIIJ MM δ=

∑ ∫ Ω=ΩJ

JIII dNNM 00

0

ρ


Section IV: Hamiltonian in Molecular Dynamics and Comparison with FEM

Hamiltonian is the summation of kinetic and potential energy of the molecules in an isolated system and it is constant in time

Current position of molecule Ix

( ) ( ) constanttWm

ttH II

II

II =+= ∑ )(2

1)(),( 2 xppx

Im Mass of molecule I

Momentum of molecule IIpTotal potential energyW

III m xp &=


Discrete Equations of Motion

Time derivative of Hamiltonian( )

011

=⋅∂

∂+⋅=+⋅= ∑∑∑

II

I

I

III

IIII

I

W

mdt

dW

mdt

dHx

x

xpppp &&&

( )0=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+∑ I

I I

II

Wx

x

xp &&

Because the velocities are independent, so that the above is satisfied only if

( ) ( )I

III

I

II

Wmor

W

x

xx

x

xp

∂∂

−==∂

∂+ &&& 0

III m xp &=(since )


Comparison of FEM with MD

Discrete equations (Assume )

( ) ( ) ( )

int

00

00

II

TT

M

dwN

dN

fd

F

F

X

XP

X

XdM

−=

Ω∂

∂⎟⎠⎞

⎜⎝⎛

∂∂

−=Ω⎟⎠⎞

⎜⎝⎛

∂∂

−= ∫∫ΩΩ

&&

&&

( ) intI

I

III

Wm f

x

xx −≡

∂∂

−=&&

Because , the above discrete equations have identical form

X

xF

∂∂

=

FEM

MD

0=extf

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