peridynamic modeling of structural damage and failuredepartment of energy’s national nuclear...
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Computational Physics Department
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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
Simon KahanCray Inc.
April 21, 2004
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Ou
tlin
e
•W
hy
anot
her
met
hod
?•
Th
eory
•E
xam
ples
•P
aral
leli
zati
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
r R
egu
lato
ry C
omm
issi
on
Com
pu
tati
ona
l P
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ics
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art
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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 ˜-
Computational Physics Department
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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
Computational Physics Department
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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˜
Computational Physics Department
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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*
Com
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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)
Computational Physics Department
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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
Com
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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
Com
pu
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ona
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ics
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t
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Exa
mp
le:
Dyn
amic
fra
ctu
re i
n a
bal
loon
Com
pu
tati
ona
l P
hys
ics
Dep
art
men
t
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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
Com
pu
tati
ona
l P
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”
Com
pu
tati
ona
l P
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ics
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art
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t
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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)
Com
pu
tati
ona
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ics
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art
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t
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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)
.
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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
Computational Physics Department
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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
Proc 4 Proc 5 Proc 6 Proc 7
Proc 8 Proc 9 Proc 10 Proc 11
Proc 12 Proc 13 Proc 14 Proc 15
Computational Physics Department
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Parallelization, ctd.
• Material nodes can migrate between processors.
• Could improve load balancing by changing the regionowned by each processor as the calculation progresses.
Proc 0 Proc 1 Proc 2 Proc 3
Proc 4 Proc 5 Proc 6 Proc 7
Proc 8 Proc 9 Proc 10 Proc 11
Proc 12 Proc 13 Proc 14 Proc 15
Computational Physics Department
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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
Computational Physics Department
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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
Computational Physics Department
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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
Computational Physics Department
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Parallelization, ctd.Issue with current approach
• In nanoscale modeling, may be large because oflong-range forces.
♦ Greatly increases communication requirements.
δΔx------
Computational Physics Department
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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
Com
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Con
cep
tual
Cra
y M
TA
-2 S
yste
m
Com
pu
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EM
U o
n t
he
MT
A-2
Com
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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
Computational Physics Department
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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.