Peridynamic Modeling of Structural Damage and FailureDepartment of Energy’s National Nuclear...

Computational Physics Department

1 of 32/home/sasilli/emu/salishan1/salishan1.frm

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United StatesDepartment of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Peridynamic Modeling of Structural Damageand Failure

Stewart SillingSandia National Laboratories

[email protected]

Simon KahanCray Inc.

[email protected]

April 21, 2004

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Ou

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•W

hy

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her

met

hod

?•

Th

eory

•E

xam

ples

•P

aral

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on a

nd

perf

orm

ance

•E

MU

on

th

e C

ray

MT

A-2

Con

trib

uto

rs•

Abe

Ask

ari,

Boe

ing

Ph

anto

m W

orks

•T

ony

Cri

spin

o, C

entu

ry D

ynam

ics,

In

c.•

Pau

l Dem

mie

, San

dia

•B

ob C

ole,

San

dia

•P

rof.

Flo

rin

Bob

aru

, Un

iver

sity

of

Neb

rask

a

Sp

onso

rs•

Dep

artm

ent

of E

ner

gy•

Join

t D

OE

/DoD

Mu

nit

ion

s T

ech

nol

ogy

Pro

gram

•T

he

Boe

ing

Com

pan

y•

US

Nu

clea

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egu

lato

ry C

omm

issi

on

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Nee

d f

or a

new

th

eory

:W

hy

is f

ract

ure

a h

ard

pro

ble

m?

•C

lass

ical

con

tin

uu

m m

ech

anic

s u

ses

part

ial d

iffe

ren

tial

equ

atio

ns.

•B

ut

the

part

ial d

eriv

ativ

es d

o n

ot e

xist

alo

ng

crac

ks a

nd

oth

erdi

scon

tin

uit

ies.

•S

peci

al t

ech

niq

ues

of

frac

ture

mec

han

ics

are

cum

bers

ome.

Goa

l

Dev

elop

a m

odel

in w

hic

h e

xact

ly t

he

sam

e eq

uat

ion

s h

old

ever

ywh

ere,

rega

rdle

ss o

f an

y di

scon

tin

uit

ies.

♦T

o do

th

is, g

et r

id o

f sp

atia

l der

ivat

ives

.

u ˜+

u ˜-



Basic idea of theperidynamic theory

• Equation of motion:

where is a functional.• A useful special case:

.

where is any point in the reference configuration, and is a vector-valued function.

More concisely:

.

• is the pairwise forcefunction. It contains allconstitutive information.

• It is convenient to assume that vanishes outside some

horizon .

ρu˜˙̇ L

˜ u b˜

+=

L˜ u

L˜ u x

˜t,( ) f

˜u˜

x'˜

t,( ) u˜

x t,( )– x'˜

x˜

–,( ) V x'˜

dR∫=

x˜

f˜

L˜

f˜

u'˜

u˜

– x'˜

x˜

–,( ) V 'dR∫=

x˜

x'˜

f˜

R

δ

f˜

f˜ δ



Microelastic materials

• Simplest class of constitutive models: microelastic:

♦ There exists a scalar-valued function , called themicropotential, such that

♦ Interaction between particles is equivalent to an elasticspring.

♦ The spring properties can depend on the referenceseparation vector.

♦ Work done by external forces is stored in arecoverable form

w

f˜

η˜

ξ˜

,( ) η˜

∂∂w η

˜ξ˜

,( )=

x˜

x'˜

u˜

u'˜

ξ˜

η˜

+

ξ˜



Some useful material models

• Microelastic♦ Each pair of particles is connected by a spring.

• Linear microelastic♦ The springs are all linear.

• Microviscoelastic♦ Springs + dashpots

• Microelastic-plastic♦ Springs have a yield point

• Ideal brittle microelastic: springs break at somecritical stretch ε.

Spring stretch

Springforce

Yielded

Elastic

Failed

Yielded

Unload

ε

need to storebond data!

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Dyn

amic

fra

ctu

re i

n a

PM

MA

pla

te

•P

late

is s

tret

ched

ver

tica

lly.

•C

ode

pred

icts

sta

ble-

un

stab

le t

ran

siti

on.

*J. F

ineb

erg

& M

. Mar

der,

Ph

ysic

s R

epor

ts31

3 (1

999)

1-1

08

Cal

cula

ted

dam

age

con

tou

rs

Cra

ck g

row

th d

irec

tion

Exp

erim

ent*

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Dyn

amic

fra

ctu

re i

n a

tou

gh s

teel

:m

ode

tran

siti

on

Not

ches

Mar

agin

g st

eel p

late

•C

ode

pred

icts

cor

rect

cra

ck a

ngl

es*.

•C

rack

vel

ocit

y ~

900

m/s

.

*J. F

. Kal

thof

f &

S. W

inkl

er, i

nIm

pact

Loa

din

g an

d D

ynam

ic B

ehav

ior

of M

ater

ials

, C. Y

. Ch

iem

, ed.

(19

88)



Peridynamic fracture modelis “autonomous”

• Cracks grow when and where it is energeticallyfavorable for them to do so.

• Path, growth rate, arrest, branching, mutualinteraction are predicted by the constitutive modeland equation of motion (alone).

• No need for anyexternallysupplied relationcontrolling thesethings.

• Any number of crackscan occur and interact.

•Interfaces betweenmaterials have theirown bond properties.

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•P

eak

forc

e oc

curs

at

abou

t 0.

4ms

(en

d of

dri

llin

g ph

ase)

:1

1. N

ot a

ll of

the

targ

et is

sho

wn.

1.02

ms

0.68

ms

0.38

ms

2.05

ms

1.70

ms

1.36

ms

Exa

mp

le:

Per

fora

tion

of

thin

du

ctil

e ta

rget

s

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Exa

mp

le:

Com

pos

ite

mat

eria

l fr

actu

re

•C

rack

pat

h, g

row

th, a

nd

stab

ilit

y de

pen

d on

ly o

n m

ater

ial p

rope

rtie

s.•

No

nee

d fo

r se

para

te la

ws

gove

rnin

g cr

ack

grow

th.

Init

ial c

ondi

tion

Wea

k in

terf

ace

Wea

k m

atri

xW

eak

fibe

r

DC

B t

est

ona

com

posi

te

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ics

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art

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Exa

mp

le:

Dyn

amic

fra

ctu

re i

n a

bal

loon

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art

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Exa

mp

le:

Pen

etra

tion

in

to r

ein

forc

ed c

oncr

ete

Pu

llou

t da

mag

eE

xit

debr

is

Dam

age

on s

urf

ace

Exi

t cr

ater

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pu

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hys

ics

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art

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t

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Exa

mp

le:

Mem

bra

ne

frac

ture

•E

last

ic s

hee

t is

hel

d fi

xed

on 3

sid

es. P

art

of t

he

4th

sid

e is

pu

lled

upw

ard.

•C

rack

s in

tera

ct w

ith

eac

h o

ther

an

d ev

entu

ally

join

up.

“Exp

erim

ent”

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Exa

mp

les:

Mec

han

ics

of fi

ber

s

•F

iber

sca

nin

tera

ctth

rou

ghlo

ng-

ran

ge(e

.g.v

ande

rW

aals

)or

con

tact

.f

Sel

f-sh

apin

g of

a fi

ber

due

to in

tera

ctio

ns

betw

een

dif

fere

nt

part

s

Str

etch

ing

of a

net

wor

k of

fibe

rs(c

ourt

esy

of P

rof.

F. B

obar

u, U

niv

ersi

ty o

f N

ebra

ska)

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Air

craf

t im

pac

t on

to s

tru

ctu

res

•F

4 in

to a

3.6

m t

hic

k co

ncr

ete

bloc

k*

*sim

ula

tion

of

full

-sca

le e

xper

imen

tal d

ata

in o

pen

lite

ratu

re (

Su

gan

o et

al.,

Nu

clea

rE

ngi

nee

rin

g an

d D

esig

n14

0 37

3-38

5 (1

993)

.



Numerical solution methodfor dynamic problems

• Theory lends itself to mesh-free numerical methods.• No elements.• Changing connectivity.

• Brute-force integration in space.

• Typical (macroscale) model:♦ If long-range forces are important, could need

much larger .

δi

j

Δx

ρu˜˙̇

if˜

u˜

ju˜

i– x

˜j, x

i

˜–( ) Δx( )3

x˜

jx

i

˜– δ<∑=

δ 3Δx≈

δ



Parallelization

• Each processor is assigned a fixed rectangular regionof space.

♦ Regions are assigned so that each slice (in eachdirection (x,y,z) contains an equal number ofnodes.

♦ Easy to implement.♦ OK if the grid is more or less rectangular.♦ Static.♦ Some processors may do nothing!

Proc 0 Proc 1 Proc 2 Proc 3






Parallelization, ctd.

• Material nodes can migrate between processors.

• Could improve load balancing by changing the regionowned by each processor as the calculation progresses.







Parallelization, ctd.Communication requirements

• Exchange of data must take place for nodes withinof any other processor’s region.

• The cost of this depends strongly on !

δ

δ

Proc BProc A

δ

Com

mu

nic

ate

to B

Com

mu

nic

ate

to A

δ



Parallelization, ctd.Timings

• Performance depends on material model used.

♦ Microelastic model requires only node data.♦ Microplastic model requires bond data.

♦ For each interacting pair of nodes.♦ Each node interacts with ~200 neighbors.♦ Results in much heavier communication

requirements.

Cplant



Parallelization, ctd.Issue with current approach

• In the limit of one node per processor, each processormust communicate with a large number of others:

• Results in different limiting speedup propertiesthan for a typical hydrocode.

δ



Parallelization, ctd.Issue with current approach

• In nanoscale modeling, may be large because oflong-range forces.

♦ Greatly increases communication requirements.

δΔx------



Parallelization, ctd.Discussion

• What would an ideal architecture for this algorithmlook like?

• Shared memory seems near ideal.♦ Avoids communication issues.♦ Avoids need to assign fixed regions of space to each

processor.

• Multi-threaded architecture (MTA) may have bigadvantages.

♦ Any processor can do any node without regard toits location.

♦ No need to write MPI calls.♦ No need for load balancing.

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Wh

at i

s th

e C

ray

MT

A-2

?

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A-2

Pro

cess

or

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Con

cep

tual

Cra

y M

TA

-2 S

yste

m

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EM

U o

n t

he

MT

A-2

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EM

U o

n t

he

MT

A-2

:A

n e

xam

ple

Con

stan

t vel

ocity

mot

ion

Load

bal

anci

ng a

s de

scrib

edpr

evio

usly

doe

s no

t wor

k w

ell!

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EM

U o

n t

he

MT

A-2

:T

imin

g re

sult

s fr

om t

he

exam

ple

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Com

men

ts o

n M

TA

per

form

ance



Conclusions

• EMU, because it is based on integral equations,performs differently from traditional codes.

• Each node interacts with many neighbors.• This influences communication requirements on

distributed memory systems.• Experience with EMU on the MTA-2:

• What is required by the programmer...♦ Ensure loops are parallel -- period.♦ Larger problems should scale to larger MTA.♦ Shared scalars accessed by too many threads may

lead to “hot spots” that require mitigation.

• Evolution of EMU, e.g., to adaptive grid, is likelyto be straightforward on the MTA.

Peridynamic Modeling of Structural Damage and FailureDepartment of Energy’s National Nuclear...

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