h questions history, current state of the art, and open researc · 2008. 10. 8. · rs for pn,...
TRANSCRIPT
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Fas
t Rea
ctor
Sim
ulat
ion
His
tory
, cur
rent
sta
te o
f the
art
, and
ope
n re
sear
ch q
uest
ions
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Sod
ium
coo
led
fast
rea
ctor
(S
FR
) Lo
op D
esig
n
~40
0C
~55
0C
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Key
poi
nt o
f fas
t vs.
ther
mal
rea
ctor
s
T
herm
al r
eact
ors
(e.g
. LW
Rs)
–N
eutr
ons
mod
erat
ed to
ther
mal
ene
rgie
s (u
sual
ly u
sing
wat
er)
–H
ighe
r pr
obab
ility
of f
issi
on ->
rel
ativ
ely
low
U-2
35 e
nric
hmen
t (~3%
)
–A
lso
high
pro
babi
lity
of c
aptu
re b
y U
-238
->
bui
ldup
of t
rans
uran
ics
–A
ll ex
istin
g co
mm
erci
al re
acto
rs a
re L
WR
s–
Maj
or s
pent
fuel
bur
den
+ in
effic
ient
reso
urce
usa
ge
F
ast r
eact
ors
(e.g
. LM
FB
Rs)
–N
eutr
on m
oder
atio
n m
inim
ized
–Lo
wer
-pro
babi
lity
of fi
ssio
n ->
hig
her
enric
hmen
t nee
ded
–Lo
w p
roba
bilit
y of
cap
ture
and
abi
lity
to fi
ssio
n tr
ansu
rani
cs/b
reed
plu
toni
um–
Key
to c
losi
ng fu
el c
ycle
+ lo
ng-t
erm
res
ourc
e m
anag
emen
t–
Mos
t bui
lt to
dat
e ha
ve b
een
rese
arch
or
prot
otyp
e
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Fas
t rea
ctor
s to
dat
e
A
num
ber
of fa
st r
eact
ors
have
bee
n de
sign
ed/o
pera
ted
over
the
last
50
year
s–
Mos
t hav
e be
en r
esea
rch
or p
roto
type
rea
ctor
s–
Yet
to b
e su
cces
sful
ly c
omm
erci
aliz
ed
M
ajor
bot
tlene
cks
–C
apita
l cos
t–
Dem
onst
ratio
n of
saf
ety
LW
R p
erfo
rman
ce h
as b
enef
ited
trem
endo
usly
from
de
cade
s of
ope
ratio
nal e
xper
ienc
e
W
ant t
o us
e si
mul
atio
n to
gre
atly
acc
eler
ate
for
LMF
BR
s
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P5
Det
ails
on
core
geo
met
ry
1/6
AB
TR
cor
e•
7k v
olum
es (c
ore,
ctr
l, re
flect
, shi
eld)
•43
k-5m
hex
ele
men
ts•~
6 G
B to
gen
erat
e us
ing
CU
BIT
217-
pin
fuel
ass
'y•
Con
form
al h
ex m
esh
•15
20 v
ols
•M
ultip
le h
omog
eniz
atio
n op
tions
, e.g
. pin
s re
solv
ed
Wire
-Wra
pped
Fue
l Pin
Ass
embl
y(N
ear-
) E
xact
Geo
met
ry
•N
on-c
onfo
rmal
hex
mes
h•
Use
s “c
ompo
site
sur
face
s” fo
r si
des
of e
ach
unit
cell
•S
pace
-fill
ing
geom
etry
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Clo
se-in
of w
ire-w
rapp
ed fu
el a
ssem
bly
-W
ire w
rap
used
to s
pace
pin
s-
Has
sig
nific
ant i
mpa
ct o
n pr
essu
re d
rop,
m
ixin
g, c
ross
flow
H
Fuel Pin
and Wire
Corner
Subchannel
Edge
Subchannel
Interior
Subchannel
Duct
Wall Fuel
Pin
D
P
Wire
Wrap
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Cur
rent
sta
te o
f LM
FB
R m
odel
ing
T
wo
broa
d cl
asse
s of
pro
blem
s --
safe
ty a
nd d
esig
n
H
uge
rang
e of
pro
blem
s to
be
addr
esse
d w
ithin
the
se–
Mix
ing,
shi
eldi
ng, p
ower
gen
erat
ion,
str
uctu
ral f
eedb
ack,
fuel
dep
letio
n,
clad
ding
failu
re, t
rans
ient
ove
rpow
er, t
rans
ient
und
erco
olin
g, fi
ssio
n pr
oduc
t rel
ease
, sod
ium
boi
ling,
etc
etc
A
ll in
volv
e on
e or
sev
eral
of a
han
dful
of p
heno
men
a–
Com
plex
geo
met
ries
–N
eutr
on tr
ansp
ort
–C
onju
gate
hea
t tra
nsfe
r (lo
w P
r fo
r LM
FB
R, m
ostly
sin
gle
phas
e)–
Str
uctu
ral d
efor
mat
ion
–F
uel p
rope
rtie
s/be
havi
or (U
nal t
alk)
–Lo
ts o
f dat
a --
cros
s se
ctio
ns, d
iffus
iviti
es, e
tc.
>
100
0 pe
rson
-yea
rs o
f cod
es d
evel
oped
and
dep
loye
d in
70s
-80s
to
desi
gn e
arly
LM
FB
Rs
–M
any
code
s/m
odel
s ex
ist s
ince
mos
tly o
ne c
ode/
mod
el p
er p
heno
men
on
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Rea
lly b
oilin
g it
dow
n
M
uch
of th
ese
phen
omen
a ad
dres
s tw
o ov
erar
chin
g pr
oble
ms
–D
emon
stra
te in
crea
se o
f lin
ear
pow
er to
mel
ting
–D
emon
stra
te u
npro
tect
ed
(pas
sive
) sa
fety
fea
ture
s
T
wo
appr
oach
es–
Adv
ance
d si
mul
atio
n le
ads
to
low
er r
ule-
of-t
hum
b de
sign
m
argi
ns fo
r ex
istin
g de
sign
s
–A
dvan
ced
sim
ulat
ion
lead
s to
de
sign
inno
vatio
ns w
ith m
uch
bette
r ec
onom
ics/
safe
ty
Impr
oved
Sim
ulat
ion
Val
idat
ion
and
Ope
ratin
g E
xper
ienc
e
Impr
oved
Des
ign
and
Sim
ulat
ion
Exp
erim
enta
l Unc
erta
inty
Ope
ratio
nal M
argi
n
Pre
dict
ion
Unc
erta
inty
Tem
pera
ture
Li
mit
Nom
inal
Pea
k Te
mpe
ratu
re
Ave
rage
Tem
pera
ture
Ope
ratin
g lim
it
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
SH
AR
P -
-S
imul
atio
n-ba
sed
Hig
h-ef
ficie
ncy
Adv
ance
d R
eact
or P
roto
typi
ng
Exi
stin
g m
odel
s/fr
amew
ork
SH
AR
P o
vera
ll f
ram
ewor
k
In
fast
rea
ctor
s, s
afet
y de
pend
s on
SM
-neu
tron
ics
feed
back
In
ther
mal
rea
ctor
s, T
H-n
eutr
onic
s fe
edba
ck s
igni
fican
t
Wan
t to
take
adv
anta
ge o
f too
ls d
evel
oped
els
ewhe
re–
Min
imiz
e re
quire
men
ts fo
r be
ing
used
in fr
amew
ork
& c
alcu
latio
ns•
Min
imal
cha
nges
to g
uts
of p
hysi
cs c
ode
–lib
rary
/driv
er•
Eac
h ph
ysic
s ca
n ch
oose
its
own
mes
h–
Cou
plin
g re
quire
s kn
owle
dge
of b
oth
mes
hes
& fi
elds
on
them
neu
tron
tra
nsp
ort
fuel
the
rmo
hydr
aul
ics
Str
uctu
ral
me
cha
nics
bala
nce
of
pla
nt
Cou
plin
g
Visu
aliz
atio
n
Mes
h ge
nera
tion
Hig
h-pe
rfor
man
ce i/
o
Ult
ra-s
cala
ble
solv
ers
•MC
•MO
L•D
ire
ct
Unc
erta
inty
Geo
met
ry
Ena
blin
g te
chno
logi
es
Nek
5000
& In
itial
The
rmal
Hyd
raul
ics
Sim
ulat
ion
Eff
ort
A
NL
code
for
fluid
s / h
eat t
rans
fer
(Fis
cher
, Lot
tes,
Tho
mas
)–
DN
S a
nd L
ES
–H
igh-
orde
r spe
ctra
l ele
men
t bas
ed c
ode
–S
cale
s to
P >
10,
000
proc
esso
rs–
Sta
te o
f the
art
mul
tigrid
sol
vers
–2
deca
des
of d
evel
opm
ent /
ver
ifica
tion
/ val
idat
ion
–S
uppo
rts
conj
ugat
e he
at tr
ansf
er, v
aria
ble
prop
ertie
s, M
HD
, ALE
, UR
AN
S
E
xten
sive
rea
ctor
TH
exp
erie
nce:
(Fan
ning
, Poi
nter
, Yan
g)–
RA
NS
mod
elin
g –
Sta
r C
D–
Sub
chan
nel c
odes
(S
AS
)
UN
IC n
eutr
onic
s co
de
P
N2N
D
–S
cale
d to
64K
pro
cs o
n B
G/P
(m
ore
deta
il la
ter)
–E
ven-
parit
y se
cond
-ord
er f
orm
ulat
ion
–U
se s
pher
ical
har
mon
ics
to r
epre
sent
ang
ular
dis
trib
utio
n
–U
se fi
nite
ele
men
ts fo
r sp
atia
l app
roxi
mat
ion
–S
uper
ior
solu
tion
of h
omog
eneo
us p
robl
ems
S
N2N
D
–S
cale
d to
64K
prc
s on
BG
/P (
mor
e de
tail
late
r)
–E
ven-
parit
y se
cond
-ord
er f
orm
ulat
ion
–U
se d
iscr
ete
ordi
nate
s cu
batu
re t
o re
pres
ent
the
angu
lar
dist
ribut
ion
–U
se fi
nite
ele
men
ts fo
r sp
atia
l app
roxi
mat
ion
–P
rovi
des
chea
p m
emor
y m
atrix
met
hod
M
OC
FE
–Li
mite
d sc
alab
ility
in 3
d (1
00s
of p
rocs
)
–F
irst o
rder
inte
gral
tra
nspo
rt f
orm
ulat
ion
–U
se d
iscr
ete
ordi
nate
s cu
batu
re t
o re
pres
ent
the
angu
lar
dist
ribut
ion
–U
se fi
nite
ele
men
ts fo
r sp
atia
l app
roxi
mat
ion
–C
an h
andl
e th
ousa
nds
of g
roup
s du
e to
mat
rix fr
ee fo
rmul
atio
n
–S
houl
d be
sca
labl
e to
thou
sand
s of
pro
cess
ors
The
rmo
mec
hani
cs m
odel
ing
Ju
st b
egin
ning
sim
ple
calc
ulat
ions
with
LLN
L co
de D
iabl
o
Li
ttle
fund
ing
so fa
r fo
r th
is c
ompo
nent
but
will
bec
ome
criti
cal i
n ne
ar
futu
re.
Add
ition
al to
ols/
tech
nolo
gies
use
d so
far
in S
HA
RP
M
eshi
ng -
-C
ubit
V
isua
lizat
ion
--V
isit
S
olve
rs -
-P
etsc
Cou
plin
g m
esh
repr
esen
tatio
n --
MO
AB
La
ngua
ge(s
): C
/FO
RT
RA
N m
ixed
with
MP
I
Pro
ject
goa
ls
W
hat w
e ar
e ju
dged
on
–C
arry
ing
out s
imul
atio
ns th
at p
redi
ct g
ener
al tr
ends
, elu
cida
te
fund
amen
tal p
heno
men
a, o
r m
ore
accu
rate
ly c
alcu
late
key
phy
sica
l qu
antit
ies
for
mor
e op
timiz
ed d
esig
n an
d lic
ensi
ng o
f SF
Rs.
V
ery
little
rew
ard
for
the
tool
itse
lf so
far
S
ome
rem
aini
ng s
kept
icis
m in
com
mun
ity (t
houg
h w
anin
g ra
pidl
y) th
at
adva
nced
sim
ulat
ion
is c
orre
ct a
ppro
ach
–Li
ttle
time
to r
espo
nsib
ly la
y gr
ound
wor
k fo
r a
long
er-t
erm
effo
rt if
not
un
iform
bel
ief i
n its
effe
ctiv
enes
s
B
ette
r ba
lanc
e m
ust b
e st
ruck
--
not a
ll of
f-th
e-sh
elf t
echn
olog
y ne
arly
go
od e
noug
h. M
ust p
ush
forw
ard
mat
h an
d cs
res
earc
h
Gen
eral
tool
s/te
chno
logi
es th
at w
e ar
e no
t fun
ded
to
purs
ue b
ut w
ould
hav
e bi
g im
pact
on
proj
ect
La
ngua
ges/
com
pile
rs w
ith m
ore
tran
spar
ency
in u
se o
f mem
ory
hier
arch
y
Q
uick
er to
ols
for
algo
rithm
ic p
roto
typi
ng/e
xplo
ratio
n be
yond
sin
gle
proc
M
orph
ing
of M
atla
b/V
isit
type
cap
abili
ties
for
anal
ysis
Q
uant
ifyin
g un
cert
aint
y fo
r m
ultip
hysi
cs c
oupl
ed s
imul
atio
ns
R
educ
tion
of c
ompl
exity
of c
reat
ing
deta
iled
mes
hes,
incl
udin
g cu
t-ce
ll te
chni
ques
, be
tter
mes
hing
tool
s/al
gorit
hms
A
goo
d op
en s
ourc
e pa
ralle
l cou
pler
that
doe
s w
hat
I nee
d it
to
M
ore
adva
nced
par
alle
l deb
uggi
ng to
ols
with
adv
ance
d m
emor
y sn
oopi
ng, e
tc.
F
aste
r co
mpu
ters
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
App
licat
ion-
spec
ific
prob
lem
s w
here
gre
at a
pplie
d m
athe
mat
icia
ns a
re n
eede
d fo
r br
eakt
hrou
ghs
H
ighl
y sc
alab
le t
rans
port
met
hods
--
impr
oved
pre
cond
ition
ers
for
PN
, sca
laba
le r
ay tr
acin
g al
gorit
hms
for
deco
mpo
sed
geom
etrie
s, h
ybrid
met
hods
, sc
alab
le M
C e
tc. t
owar
ds a
sin
gle
reac
tor
anal
ysis
tool
for
VH
TR
, SF
R, L
WR
, etc
.
M
ulti-
scal
e ap
proa
ch f
or h
eat
tran
sfer
, tr
ansp
ort,
brid
ging
ab
initi
o to
eng
inee
ring
scal
e m
odel
ing
for
fuel
s, li
nkin
g D
NS
, RA
NS
, LE
S, a
nd s
ubch
anne
l, et
c.
A
dapt
ive
tran
spor
t m
etho
ds (
in e
nerg
y, a
ngle
, spa
ce,
and
poly
nom
ial o
rder
)
N
ew id
eas
beyo
nd m
ultig
roup
for
hand
ling
of e
nerg
y va
riabl
e
A
ccur
ate
coup
ling
tech
niqu
es f
or fa
st tr
ansi
ents
in r
eact
or a
ccid
ent
scen
ario
s
S
truc
ture
d C
FD
on
com
plex
geo
met
ries
C
ompo
nent
arc
hite
ctur
es f
or ti
ght/l
oose
cou
plin
g
D
eriv
ing
subg
rid fl
uid
mod
els
for
low
-PR
flo
ws,
sodi
um b
oilin
g fr
om D
NS
S
truc
tura
l mod
elin
g fo
r ro
d bo
win
g, v
esse
l exp
ansi
on,
etc.
Wor
k co
nduc
ted
byA
NL
for t
he G
NE
P
Neu
tron
tran
spor
t alg
orith
ms
in d
epth
Hom
ogen
izat
ion
at v
ario
us le
vels
Hom
ogen
ized
as
sem
bly
Hom
ogen
ized
as
sem
bly
inte
rnal
s
Hom
ogen
ized
pi
n ce
llsF
ully
exp
licit
asse
mbl
y
Neu
tron
Tra
nspo
rt: T
wo
Bro
ad C
ateg
orie
s
S
toch
astic
(Mon
te C
arlo
)
Det
erm
inis
tic–
Eve
n P
arity
For
mul
atio
n (P
n-F
E)
–D
iscr
ete
Ord
inat
es (
Sn)
•S
atis
fy th
e tr
ansp
ort e
quat
ion
only
alo
ng s
ome
pred
eter
min
ed a
ngul
ar d
irect
ions
–In
tegr
al F
orm
(M
etho
d of
Cha
ract
eris
tics)
•R
ay tr
acin
g
•W
orks
wel
l in
low
den
sity
reg
ions
Mon
te C
arlo
Met
hods
F
ollo
w in
divi
dual
par
ticle
his
torie
s fr
om b
irth
(som
e so
urce
) to
dea
th
(abs
orpt
ion
and
leak
age)
N
ucle
ar d
ata
is e
xpre
ssed
in c
ontin
uum
of e
nerg
y
Abi
lity
to h
andl
e co
mpl
ex g
eom
etrie
s
Cal
cula
te in
tegr
al q
uant
ities
ove
r th
e w
hole
dom
ain
C
ons
–V
ery
time
cons
umin
g fo
r sc
atte
ring
dom
inan
t tra
nspo
rt (
e.g.
opt
ical
ly
thic
k m
ediu
m)
–N
ot a
ppro
pria
te w
hen
a de
taile
d so
lutio
n of
neu
tron
den
sity
th
roug
hout
the
geom
etry
is n
eede
d
Det
erm
inis
tic M
etho
ds
E
ffici
ent i
n sc
atte
ring
dom
inan
t situ
atio
ns
Now
can
han
dle
com
plex
geo
met
ries
(FE
M)
and
hete
roge
neou
s m
ater
ials
G
ive
deta
iled
solu
tions
thro
ugho
ut th
e do
mai
n
Con
s–
Mem
ory
inte
nsiv
e (s
even
dim
ensi
onal
pha
se s
pace
)–
Ene
rgy
varia
ble
hand
led
less
acc
urat
ely
(tha
n M
onte
Car
lo m
etho
ds)
–Lo
w d
ensi
ty r
egio
ns a
re p
robl
emat
ic fo
r se
cond
ord
er m
etho
ds
Som
e N
otes
on
Cro
ss S
ectio
ns
V
ital s
ourc
e of
inac
cura
cy in
det
erm
inis
tic c
odes
V
ery
invo
lved
pro
cess
to g
et c
ross
sec
tions
hom
ogen
ized
ove
r an
ene
rgy
grou
p–
All
the
phys
ics
know
ledg
e go
es h
ere
La
rge
num
ber
of e
nerg
y gr
oups
(~
10,0
00)
need
ed fo
r fa
st r
eact
ors
to
impr
ove
accu
racy
and
red
uce
unce
rtai
ntie
s
Nee
d to
rep
eat t
he c
ross
sec
tion
gene
ratio
n w
ith th
erm
al h
ydra
ulic
s fe
edba
ck (t
empe
ratu
re a
nd d
ensi
ty c
hang
es)
The
Ste
ady
Sta
te T
rans
port
Equ
atio
n (p
46 in
Lew
is)
(,
)(
,)
(,
)f
rE
rE
rE
χν
Σr
rr
ˆˆ
ˆˆ
ˆˆ
(,
,)
(,
)(
,,
)(
,'
,'
)(
,',
')'
'
1ˆ
(,
)(
,')
(,
')(
,',
')'
'
ˆ(
,,
)
ts
f
rE
rE
rE
rE
Er
Ed
dE
rE
rE
rE
rE
ddE
k Sr
E
ψψ
ψ
χν
ψ
Ω⋅∇
Ω+
ΣΩ
=Σ
Ω→
Ω→
ΩΩ
+Σ
ΩΩ
+Ω
∫∫
∫∫
rr
rr
rr
rr
rr
r
ˆ(
,,
)r
Eψ
Ωr (
,)
tr
EΣ
r ˆˆ
(,
',
')
sr
EE
ddE
ΣΩ
→Ω
→Ω
r
ˆ(
,,
)S
rE
Ω
The
neu
tron
flux
(ne
utro
n de
nsity
mul
tiplie
d by
spe
ed)
The
tota
l pro
babi
lity
of in
tera
ctio
n in
the
dom
ain
The
sca
tterin
g tr
ansf
er k
erne
l
The
ste
ady
stat
e m
ultip
licat
ive
fissi
on s
ourc
e
If a
fixed
sou
rce
is p
rese
nt th
en k
= 1
kT
he m
ultip
licat
ion
eige
nval
ue
Mul
tigro
up F
orm
alis
m
T
o da
te th
e fo
llow
ing
has
been
inef
fect
ive
–P
olyn
omia
l exp
ansi
ons
of th
e ne
utro
n flu
x de
pend
ence
–F
inite
ele
men
t exp
ansi
ons
of th
e ne
utro
n flu
x de
pend
ence
In
gen
eral
this
is m
easu
red
by th
e ga
in in
acc
urac
y re
lativ
e to
the
cost
as
soci
ated
with
the
solu
tion
algo
rithm
T
here
fore
, we
chos
e to
con
tinue
usi
ng th
e m
ultig
roup
form
alis
m
,,
''
'1
',
''
'1
ˆˆ
ˆˆ
ˆˆ
(,
)(
)(
,)
(,
')
(,
')'
1ˆ
()
()
()
(,
')'
ˆ(
,)
G
gt
gg
sg
gg
g
G
gg
fg
gg
g
rr
rr
rd
rr
rr
dk S
r
ψψ
ψ
χν
ψ
→=
=
Ω⋅∇
Ω+
ΣΩ
=Σ
Ω→
ΩΩ
Ω
+Σ
ΩΩ
+Ω
∑∫
∑∫
rr
rr
rr
rr
rr
r
Sol
ving
the
Eig
enva
lue
Pro
blem
1A
TF
x
kψλ
−= = =
12
ˆˆ
ˆ(
,)
(,
)(
,)
T
Gr
rr
ψψ
ψψ
=Ω
ΩΩ
rr
rL
Ax
xλ
=
,,
''
'1
ˆˆ
ˆˆ
ˆˆ
(,
)(
)(
,)
(,
')
(,
')'
G
gt
gg
sg
gg
g
Tr
rr
rr
dψ
ψψ
ψ→
=
=Ω
⋅∇Ω
+Σ
Ω−
ΣΩ
→Ω
ΩΩ
∑∫
rr
rr
rr
',
''
'1
ˆ(
)(
)(
)(
,')
'G
gg
fg
gg
Fr
rr
rd
ψχ
νψ
=
=Σ
ΩΩ
∑∫
rr
rr
1T
Fk
ψψ
=
Ax
xλ
=
Sta
ndar
d ei
genv
alue
not
atio
n:
Cas
t the
tran
spor
t equ
atio
n as
a p
seud
o m
atrix
-vec
tor
oper
atio
n
T =
str
eam
ing/
colli
sion
/sca
tterin
g F
= fi
ssio
n
Eig
enva
lue
Sol
utio
n P
roce
ss (
Inve
rse
Pow
er M
etho
d)
1T
Fk F
ψψ
θψ
= =
()
1(
1)1
(1)
(1)
(1)
()
()
(1)
(1)
11
, ,
ii
ii
i
i
ii
i
FT
FT
kk
wk
kw
θθ
θ
θ θ−−
−−
−−
−−
==
=
Fis
sion
sou
rce
Last
equ
atio
n ha
s th
e ite
rativ
e so
lutio
n:
1
1 1
TF
k
Tk
kF
T
ψψ
ψθ
θθ
−
= = =
The
w is
an
arbi
trar
y ve
ctor
whi
ch w
e ch
oose
to b
e 1
such
that
we
get t
he to
tal f
issi
on s
ourc
e.
Neu
tron
ics
Cod
e S
olut
ion
Pro
cess
In
vers
ion
of T
is g
ener
ally
ver
y ex
pens
ive
due
to s
catte
ring
coup
ling.
R
ecas
t usi
ng th
e w
ithin
gro
up n
otat
ion
and
swee
p th
e gr
oups
–Lo
gica
l app
roac
h gi
ven
that
iter
atio
n on
fiss
ion
is a
lread
y ne
cess
ary
–“D
owns
catte
r” o
nly
regi
on h
as s
trai
ghtfo
rwar
d so
lutio
n (d
own
swee
p)–
“Ups
catte
r” re
gion
req
uire
s ad
ditio
nal i
tera
tion
(mor
e do
wn
swee
ps)
()
()
()
()
,,
ˆˆ
ˆˆ
ˆˆ
ˆ(
,)
()
(,
)(
,'
)(
,')
'(
,)
ii
ii
gt
gg
sg
gg
gr
rr
rr
dQ
rψ
ψψ
→Ω
⋅∇Ω
+Σ
Ω=
ΣΩ
→Ω
ΩΩ
+Ω
∫r
rr
rr
rr
()
()
,'
''
1,'
(1)
',
''
(1)
'1
ˆˆ
ˆˆ
(,
)(
,'
)(
,')
'
1ˆ
()
()
()
(,
')'
ˆ(
,)
ii
gs
gg
gg
Gg
g
Gi
gg
fg
gi
g
g
Qr
rr
d
rr
rr
dk S
r
ψ
χν
ψ
→∈ ≠
−−
=
Ω=
ΣΩ
→Ω
ΩΩ
+Σ
ΩΩ
+Ω
∑∫
∑∫
rr
r
rr
rr
r
Pic
king
a M
etho
d
Mon
te C
arlo
–U
ses
part
icle
“wal
king
” w
ith r
ando
m n
umbe
r de
cisi
ons
to d
ecid
e pa
th–
Has
bee
n im
plem
ente
d in
mul
ti-gr
oup
and
cont
inuo
us e
nerg
y
Firs
t ord
er d
iscr
ete
ordi
nate
s (m
atrix
free
)–
Dom
ain
is s
wep
t in
spec
ific
dire
ctio
ns, e
lem
ent b
y el
emen
t–
Ele
men
t siz
ed A
mat
rix is
form
ed a
nd in
vert
ed d
urin
g sw
eep
F
irst o
rder
met
hod
of c
hara
cter
istic
s (m
atrix
free
cp)
–S
erie
s of
par
alle
l tra
ject
orie
s ar
e us
ed to
pie
rce
the
dom
ain
–S
olut
ion
alon
g th
e ra
y is
che
ap a
nd s
ourc
e by
ele
men
t is
sim
ple
sum
S
econ
d or
der
disc
rete
ord
inat
es (g
ener
ally
a m
atrix
met
hod)
–B
lock
dia
gona
l mat
rix m
akes
met
hod
reas
onab
le to
sto
re
Sec
ond
orde
r sp
heric
al h
arm
onic
s (g
ener
ally
a m
atrix
met
hod)
–In
clud
es a
ll di
ffusi
on th
eory
met
hods
–S
tora
ge o
f mat
rices
is q
uite
exp
ensi
ve
Sec
ond
orde
r an
gula
r fin
ite e
lem
ents
–R
are
usag
e du
e to
com
plic
atio
n of
ani
sotr
opic
sca
tterin
g
Met
hods
Em
ploy
ed in
UN
IC
P
N2N
D–
Eve
n-pa
rity
seco
nd-o
rder
form
ulat
ion
–U
se s
pher
ical
har
mon
ics
to r
epre
sent
ang
ular
dis
trib
utio
n–
Use
fini
te e
lem
ents
for
spat
ial a
ppro
xim
atio
n–
Sup
erio
r sol
utio
n of
hom
ogen
eous
pro
blem
s
SN
2ND
–E
ven-
parit
y se
cond
-ord
er fo
rmul
atio
n–
Use
dis
cret
e or
dina
tes
cuba
ture
to r
epre
sent
the
angu
lar
dist
ribut
ion
–U
se fi
nite
ele
men
ts fo
r sp
atia
l app
roxi
mat
ion
–P
rovi
des
chea
p m
emor
y m
atrix
met
hod
M
OC
FE
–F
irst o
rder
inte
gral
tran
spor
t for
mul
atio
n–
Use
dis
cret
e or
dina
tes
cuba
ture
to r
epre
sent
the
angu
lar
dist
ribut
ion
–U
se fi
nite
ele
men
ts fo
r sp
atia
l app
roxi
mat
ion
–C
an h
andl
e th
ousa
nds
of g
roup
s du
e to
mat
rix fr
ee fo
rmul
atio
n–
Sho
uld
be s
cala
ble
to th
ousa
nds
of p
roce
ssor
s
Sec
ond
Ord
er M
etho
ds; W
ithin
Gro
up F
orm
ulat
ion
Ang
ular
flux
is w
ritte
n in
term
s of
eve
n-an
d od
d-pa
rity
com
pone
nts
ˆˆ
ˆ(
,)
(,
)(
,)
gg
gr
rr
ψψ
ψ+
−Ω
=Ω
+Ω
rr
r
,ˆ
ˆˆ
ˆˆ
(,
)(
)(
,)
(,
)(
,)
gt
gg
gg
rr
rW
rQ
rψ
ψ−
++
+Ω
⋅∇Ω
+Σ
Ω=
Ω+
Ωr
rr
rr
r
,ˆ
ˆˆ
ˆˆ
(,
)(
)(
,)
(,
)(
,)
gt
gg
gg
rr
rW
rQ
rψ
ψ+
−−
−Ω
⋅∇Ω
+Σ
Ω=
Ω+
Ωr
rr
rr
r
Eve
n-P
arity
Odd
-Par
ity
Eve
n-an
d od
d-pa
rity
form
s of
the
tran
spor
t equ
atio
n ar
e ob
tain
ed
11
22
ˆˆ
ˆ(
,)
(,
)(
,)
gg
gQ
rQ
rQ
r±
Ω=
Ω±
−Ωr
rr
,
,,
ˆˆ
ˆˆ
ˆ(
,)
(,
')(
,')
'
ˆˆ
ˆˆ
()
(')
(,
')'
gs
gg
g
sg
gm
mg
m
Wr
rr
d
rP
rd
ψ
ψ
±±
±→ ±
±±
→
Ω=
ΣΩ
⋅ΩΩ
Ω
=Σ
Ω⋅Ω
ΩΩ
∫ ∑ ∫
rr
r
rr
With
in g
roup
Sca
tterin
g
Ext
erna
l Sou
rce
PN
2ND
,,
,0
,2
ˆˆ
(,
)(
)(
)(
)(
)L
lT
gl
mg
lm
gl
ml
rY
rr
ψψ
ψ+
++
+=
=−
Ω=
Ω=
Ω∑∑
rr
rY
,,
,1,
3
ˆˆ
(,
)(
)(
)(
)(
)L
lT
gl
mg
lm
gl
ml
rY
rr
ψψ
ψ−
−−
−=
=−
Ω=
Ω=
Ω∑∑
rr
rY
Sph
eric
al h
arm
onic
s ap
prox
imat
ion
Fin
ite e
lem
ent a
ppro
xim
atio
n ˆ(
,)
()
()
TT
gg
rL
rψ
ψ±
±±
Ω=
Ω⊗
rr
Y
SN
2ND
,,0
,0ˆ (
)l
mn
nl
mn
Yw
δδ
Ω=
∑
,
,,
ˆˆ
(,
)(
)(
)
ˆ(
)
ˆ(
)
TT
gg
Tg
ng
nn
gn
SNP
NP
N
nn
nrL
r
wM N
ψψ
ψϕ
ϕ
++
+
++
++
+
++
++
+
Ω=
Ω⊗
=Ω
=
=Ω
=
∑
rr
Y
Y Y
,(
,)
(,
)(
)T
gg
ng
nr
rL
rψ
ϕϕ
++
+Ω
⇒Ω
=r
rr
Dis
cret
e or
dina
tes
appr
oxim
atio
n
ˆ(
,)
()
()
TT
gg
rL
rψ
ψ−
−−
Ω=
Ω⊗
rr
Y
PN
2ND
& S
N2N
D G
over
ning
Equ
atio
ns ()1
,,
,g
mt
sg
gm
−−
→Ξ
=Σ
−Σ
'1,
:'
gG
RU
LE
gg
∈ ≠P
Ng
gg
gA
ψ+
+−
=+
()
()
()
()
()
()
(1)
,,
'',
,,
',',
,(
1)'
1
()
()
()
,,
,,
,,
,,
,,
()
()
,,
,'
,',
,,
,,
1G
ii
ig
es
gg
eg
eg
ef
ge
eg
ee
ge
iR
UL
Eg
iT
Ti
ig
em
gm
KL
eK
Lg
eg
mg
em
mK
L
ii
Tg
eg
ms
gg
mg
em
gm
Ke
KL
gm
RU
LE
QF
JF
IF
Sk
VV
PQ
QV
US
ψχ
νψ
ξψ
ξ
++
++
+−
++
→−
=
−+
−
−−
−→
⊗⊗
⊗
⊗
⊗
=Σ
+Σ
+
=Ξ
+Ξ
=Ξ
Σ+
Ξ
∑∑
∑ ∑(
),
ie
−
PN
2ND
SN
2ND
()1
,g
mt
−Ξ
=Σ
:'
1,R
UL
Eg
G∈
Gro
up s
ourc
e eq
uatio
ns
()
SN
gg
gg
gg
AN
M
ϕ
ψϕ
++
−+
++
+=+
=
Gro
up S
wee
ping
Alg
orith
m
()
1
''
&'
1,
gg
gg
gg
Solv
eA
Upd
ate
gG
ψ+
−+
−
+−
=+
∈
()
()
12
11
22
()
1
()
()
()
1(
1)
''
&'
1,
ig
gg
gg
ii
ii
gg
gS
gg
gg
Solv
eM
MA
Solv
eF
Upd
ate
gG
ψϕ
ψψ
αψ
ψ
−+
+−
+−
++
−−
++
−+
+−
+−
==
+
=+
−
∈
()
1
''
&'
1,
gg
gg
gg
Solv
eA
Upd
ate
gG
ψ+
−+
−
+−
=+
∈
()
()
1 2
11
22
()
1
()
()
()
1(
1)
''
&'
1,
ig
gg
gg
ii
ii
gg
gS
gg
gg
Solv
eM
MA
Solv
eF
Upd
ate
gG
ψϕ
ψψ
αψ
ψ
−+
+−
+−
++
−−
++
−+
+−
+−
==
+
=+
−
∈
Dow
nsca
tter
only
reg
ime
Ups
catte
r reg
ime
Cnv
g or
Max
Iter
No
Yes
Sta
rt
End
PN
2ND
Ups
catte
r reg
ime
Cnv
g or
Max
Iter
No
Yes
End
Dow
nsca
tter
only
reg
ime
Sta
rtS
N2N
D
Fea
ture
s of
Sec
ond
Ord
er F
orm
Sol
utio
ns in
UN
IC
P
N2N
D a
nd S
N2N
D s
olve
rs h
ave
been
dev
elop
ed to
sol
ve th
e st
eady
-sta
te, s
econ
d-or
der,
eve
n-pa
rity
neut
ron
tran
spor
t equ
atio
n–
PN
2ND
: Sph
eric
al h
arm
onic
met
hod
in 1
D, 2
D a
nd 3
D g
eom
etrie
s w
ith F
E m
ixed
m
esh
capa
bilit
ies
–S
N2N
D: D
iscr
ete
ordi
nate
s in
2D
and
3D
geo
met
ries
with
FE
mix
ed m
esh
capa
bilit
ies
T
hese
sec
ond
orde
r m
etho
ds h
ave
been
impl
emen
ted
on la
rge
scal
e pa
ralle
l mac
hine
s–
Line
ar te
trah
edra
l and
qua
drat
ic h
exah
edra
l ele
men
ts–
Fix
ed s
ourc
e an
d ei
genv
alue
pro
blem
s–
Arb
itrar
ily o
rient
ed r
efle
ctiv
e an
d va
cuum
bou
ndar
y co
nditi
ons
–P
ET
Sc
solv
ers
are
utili
zed
to s
olve
with
in-g
roup
equ
atio
ns•
Con
juga
te g
radi
ent m
etho
d w
ith S
SO
R a
nd IC
C p
reco
nditi
oner
s•
Oth
er s
olut
ion
met
hods
and
pre
cond
ition
ers
will
be
inve
stig
ated
–S
ynth
etic
diff
usio
n ac
cele
ratio
n fo
r w
ithin
-gro
up s
catte
ring
itera
tion
–P
ower
iter
atio
n m
etho
d fo
r ei
genv
alue
pro
blem
•V
ario
us a
ccel
erat
ion
sche
mes
are
bei
ng in
vest
igat
ed–
MeT
iS is
em
ploy
ed fo
r m
esh
part
ition
ing
U
nstr
uctu
red
finite
ele
men
t mes
hes
are
empl
oyed
–C
UB
IT p
acka
ge is
the
prim
ary
mes
h ge
nera
tion
tool
–he
xahe
dral
and
tetr
ahed
ral e
lem
ents
(lin
ear
and
quad
ratic
)–
Fur
ther
res
earc
h is
req
uire
d fo
r re
duci
ng m
esh
gene
ratio
n ef
fort
s an
d ro
bust
mer
ging
of t
he m
eshe
s of
indi
vidu
al g
eom
etric
al c
ompo
nent
s
Gen
eral
Geo
met
ry C
apab
ility
in U
NIC
–A
BT
R A
ssem
bly
Key
Fea
ture
s of
the
Impl
emen
tatio
n S
trat
egy
F
ollo
w th
e “o
wne
r co
mpu
tes”
rule
und
er th
e du
al c
onst
rain
ts o
f min
imiz
ing
the
num
ber
of
mes
sage
s a
nd o
verla
ppin
g co
mm
unic
atio
n w
ith c
ompu
tatio
n
Eac
h pr
oces
sor “
ghos
ts” i
ts s
tenc
il de
pend
ence
s in
its
neig
hbor
s
Gho
st n
odes
ord
ered
afte
r con
tiguo
us o
wne
d no
des
D
omai
n m
appe
d fr
om (u
ser)
glo
bal o
rder
ing
into
loca
l ord
erin
gs
Sca
tter/
gath
er o
pera
tions
cre
ated
bet
wee
n lo
cal s
eque
ntia
lve
ctor
s an
d gl
obal
di
strib
uted
vect
ors,
bas
ed o
n ru
ntim
e co
nnec
tivity
pat
tern
s
Kry
lov-
Sch
war
z op
erat
ions
tran
slat
ed in
to lo
cal t
asks
and
com
mun
icat
ion
task
s
Pro
filin
g us
ed to
hel
p el
imin
ate
perf
orm
ance
bot
tlene
cks
in c
omm
unic
atio
n an
d m
emor
y hi
erar
chy
Impr
oved
Par
alle
lism
and
Acc
eler
atio
n
R
ecen
tly a
dded
cap
abili
ty to
par
alle
lize
by s
pace
, ang
le, a
nd e
nerg
y in
UN
IC–
Pre
viou
s im
plem
enta
tion
only
con
side
red
spat
ial p
aral
leliz
atio
n–
Mem
ory
limita
tions
pre
vent
ed la
rge
prob
lem
s fr
om b
eing
exe
cute
d
Im
plem
ente
d m
atrix
-fre
e C
G s
olve
rs fo
r P
N2N
D a
nd S
N2N
D–
New
ver
sion
use
s si
ngle
ste
ncile
d pr
econ
ditio
ner
mat
rix fo
r ea
ch a
ngle
-gr
oup
part
ition
(fa
r le
ss m
emor
y)–
Par
titio
ns c
an b
e so
lved
sim
ulta
neou
sly
–C
urre
nt fi
ssio
n so
urce
iter
atio
n sc
hem
e st
ill u
ses
Gau
ss S
eide
l in
ener
gy
Add
ed fi
ssio
n so
urce
acc
eler
atio
n–
Tche
bych
ev–
Sel
f-ad
just
ing
with
in-g
roup
flux
err
or
tole
ranc
e to
min
imiz
e pr
econ
ditio
ner
effo
rt–
Tim
e to
sol
utio
n re
duce
d by
10+
fact
or
Sam
ple
Per
form
ance
of A
ccel
erat
ion
Sch
emes
-T
aked
a B
ench
mar
k 4
1.E
-08
1.E
-07
1.E
-06
1.E
-05
1.E
-04
1.E
-03
1.E
-02
1.E
-01
1.E
+00
02
46
810
1214
1618
2022
24O
uter
Iter
atio
n
Error
Eig
enva
lue
Fis
sion
Flu
xG
roup
1G
roup
2G
roup
3G
roup
4
1.E
-08
1.E
-07
1.E
-06
1.E
-05
1.E
-04
1.E
-03
1.E
-02
1.E
-01
1.E
+00
02
46
810
1214
1618
2022
24O
uter
Iter
atio
n
Error
Eig
enva
lue
Fis
sion
Flu
xG
roup
1G
roup
2G
roup
3G
roup
4
`
With
out T
cheb
yche
v ac
cele
ratio
n an
d a
fixed
tole
ranc
e of
with
in-g
roup
flu
x er
ror
With
Tch
ebyc
hev
acce
lera
tion
and
self-
adju
stin
g w
ithin
-gro
up f
lux
erro
r to
lera
nce
Ene
rgy
Gro
up P
aral
lelis
m Is
sues
Sod
ium
Fas
t Rea
ctor
Fue
l A
ssem
bly
(230
Gro
ups)
PW
R F
uel A
ssem
bly
(172
G
roup
s)
Sca
tterin
g C
ross
Sec
tion
Ste
ncili
ng
k-E
igen
valu
e P
ower
Iter
atio
n
S
tead
yst
ate
mul
tigro
upei
genv
alue
and/
orfix
edso
urce
itera
tions
for
one,
two,
and
thre
edi
men
sion
alun
stru
ctur
edfin
iteel
emen
tm
esh
geom
etrie
s.
Mai
nlo
op(P
ower
itera
tion)
Beg
inO
uter
Itera
tion
Beg
inLo
opov
eren
ergy
grou
psO
btai
ngr
oup
scat
terin
g+fis
sion
+fix
edso
urce
sS
olve
asy
mm
etric
posi
tive
defin
itelin
ear
syst
emfo
rflu
x(p
reco
nditi
oned
conj
ugat
egr
adie
nt)
End
Loop
over
ener
gygr
oups
Com
pute
tota
lfis
sion
sour
ceC
heck
for
conv
erge
nce
inei
genv
alue
,ang
ular
flux,
and
sour
ces
End
Out
erIte
ratio
n
Res
ourc
e R
equi
rem
ents
for
UN
ICS
tora
ge (
PN
2ND
Sol
ver)
M
assi
ve li
near
sys
tem
s
N e
nerg
y gr
oups
, m m
esh
poin
ts, q
ang
ular
term
s, a
nd p
non
zero
es p
er
row
, we
need
m∗q
∗p ∗8
∗1.
5 by
tes
are
requ
ired
to s
tore
the
mat
rix in
co
mpr
esse
d ro
w fo
rmat
.
Sin
ce th
is is
a s
ymm
etric
mat
rix, w
e ne
ed h
alf o
f thi
s.
As
an e
stim
ate,
for
4000
non
zero
es in
eac
h ro
w (
p) fo
r 10
8ro
ws
(m =
106
and
q =
100
):–
2.4
tera
byte
s (T
B)
of m
emor
y pe
r gr
oup
for
the
mat
rix a
lone
(80
0 M
B fo
r on
e ve
ctor
) an
d 24
pet
abyt
es (
PB
) fo
r 10
,000
gro
ups.
Res
ourc
e R
equi
rem
ents
for
UN
ICP
erfo
rman
ce –
A C
rude
Ana
lysi
s
T
he c
onju
gate
gra
dien
t (C
G)
met
hod
is O
[n ∗
i] w
here
n
= n
umbe
r of
non
zer
oes
in th
e m
atrix
(p ∗m
∗q)
. For
a w
ell c
ondi
tione
d sy
stem
, the
num
ber
of it
erat
ions
(i)
need
ed to
con
verg
e is
sm
all.
M
atrix
vec
tor
prod
uct d
omin
ates
the
exec
utio
n tim
e–
Tw
o m
atrix
-vec
tor
prod
ucts
per
CG
iter
atio
n–
2 ∗n
floa
ting
poin
t ope
ratio
ns (f
lops
) ne
eded
for
each
mat
rix v
ecto
r pr
oduc
t–
Neg
lect
the
vect
or o
pera
tions
(dot
pro
duct
s an
d sc
ale)
for
seria
l cas
e si
nce
(n >
> m∗q
)
Res
ourc
e R
equi
rem
ents
for
UN
ICP
erfo
rman
ce (c
ontd
.)
F
or i
itera
tions
of C
G, O
out
er it
erat
ions
, and
N g
roup
s (a
ssum
ing
the
sam
e nu
mbe
r of i
nner
iter
atio
ns fo
r al
l the
gro
ups)
, the
tota
l flo
ps a
re 4
∗n∗
i ∗N
∗O
.
CG
per
form
ance
is m
emor
y ba
ndw
idth
lim
ited
–10
-20
% o
f mac
hine
pea
k is
pra
ctic
al
For
i =
100
, O =
50,
N =
10,
000,
and
n =
4∗1
011, t
he to
tal w
ork
is 8
x1019
flops
(80,
000
peta
flops
)–
At 1
00 T
Flo
p/s,
111
1 hr
s–
At 1
PF
lops
/s, 1
11 h
rs
Ben
chm
ark
Pro
blem
Usi
ng U
NIC
PN
2ND
–Z
PP
R-1
5 C
ritic
al E
xper
imen
tM
esh
with
76,
000
vert
ices
, P3
scat
terin
g ke
rnel
and
230
ene
rgy
grou
ps
Com
puta
tiona
l Mes
h an
d E
xam
ple
Flu
x S
olut
ions
of Z
PP
R-1
5 C
ritic
al E
xper
imen
t
Flu
x ex
pans
ion
orde
rS
catte
ring
ord
erE
igen
valu
e
P1
P1
0.99
258
P3
P3
0.99
640
P5
P3
0.99
651
Mon
te C
arlo
(VIM
)0.
9964
7±0.
0001
0
Pow
er D
istr
ibut
ion
for
AB
TR
Ful
l Cor
e B
ench
mar
k
Sam
ple
UN
IC s
olut
ion
for A
BT
R.
(Lef
t) M
eTiS
dec
ompo
sitio
n fo
r 51
2 pr
oces
sors
(cen
ter)
hom
ogen
ized
fuel
reg
ion
s ar
e ex
trac
ted
and
disp
lay
the
pow
er d
istr
ibut
ion
whe
re
the
low
er
rem
nant
dis
play
s th
e M
eTiS
dec
ompo
sitio
n an
d (r
ight
) ve
rtic
al
slic
e th
roug
h th
e m
odel
sho
win
g th
e po
wer
dis
trib
utio
n.
Gro
up 1
Flu
x
Pow
er D
istr
ibut
ion
Per
form
ance
of U
NIC
(P
N2N
D)
on C
ray
XT
4
P
aral
lel p
erfo
rman
ce fr
om 5
12 to
409
6 C
ray
XT
4 pr
oces
sors
–12
0ºpe
riodi
c A
BT
R c
ore
with
hom
ogen
ized
ass
embl
ies
–33
gro
up P
5ca
lcul
atio
n–
Mes
h co
ntai
ns 5
87,4
58 q
uadr
atic
tetr
ahed
ral e
lem
ents
and
793
,668
ver
tices
•A
bout
12
mill
ion
spac
e-an
gle
degr
ees
of fr
eedo
m p
er e
nerg
y gr
oup
–A
bout
900
GF
lop/
s (~
4.2%
of m
achi
ne p
eak)
on
4,09
6 pr
oces
sors
Pro
cess
ors
AggregateGFlop/s
10
00
20
00
30
00
40
00
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
0
10
00
Per
form
ance
bot
tlene
cks
V
ecto
r dot
pro
duct
s–
14%
-29%
–Lo
ad im
bala
nce
T
otal
line
ar it
erat
ion
coun
t–
Incr
ease
s as
we
grow
the
num
ber
of p
roce
ssor
s
Pro
cess
ors
%ofTimeonVectorDotProducts
1000
2000
3000
4000
1012141618202224262830
Pro
cess
ors
TotalLinearIterations
1000
2000
3000
4000
5000
6000
0
6500
0
7000
0
7500
0
8000
0
Ang
ular
Con
verg
ence
of E
igen
valu
e of
AB
TR
Cor
e U
sin
g U
NIC
on
Blu
eGen
e/P
S
N2N
D S
olve
r
–9
ener
gy g
roup
s
–M
esh
(ass
embl
y ho
mog
eniz
ed)
•18
7,56
0 he
xahe
dral
qua
drat
ic e
lem
ents
and
785
,801
ver
tices
•
Spr
ead
over
512
pro
cess
or c
ores
Pro
cess
orC
ores
Ang
ula
rR
esol
utio
nA
ngle
sin
2π
Spa
ce-A
ngle
DO
FE
igen
valu
e
2,04
81
43,
143,
204
1.00
633
4,60
82
97,
072,
209
1.00
823
8,19
23
1612
,572
,816
1.00
754
12,8
004
2519
,645
,025
1.00
776
18,4
325
3628
,288
,836
1.00
773
25,0
886
4938
,504
,249
1.00
781
32,7
687
6450
,291
,264
1.00
780
41,4
728
8163
,649
,881
1.00
782
51,2
009
100
78,5
80,1
001.
0078
1
61,9
5210
121
95,0
81,9
211.
0078
2
73,7
2811
144
113,
155,
344
1.00
782
Wea
k S
calin
g of
UN
IC o
n B
lueG
ene/
P
S
N2N
D S
olve
r
2,04
8 to
73,
728
Cor
es (
Virt
ual N
ode
Mod
e)
Wea
k S
calin
g of
UN
IC o
n B
lueG
ene/
P (
cont
d.)
Lo
ad im
bala
nce
in r
educ
e op
erat
ions
N
eed
to b
alan
ce b
ound
ary
vert
ices
Thr
ee F
unda
men
tal L
imiti
ng F
acto
rs to
Pea
k P
erfo
rma
nce
M
emor
y B
andw
idth
–P
roce
ssor
doe
s no
t get
dat
a at
the
rate
it r
equi
res
In
stru
ctio
n Is
sue
Rat
e–
If th
e lo
ops
are
load
/sto
re b
ound
, we
will
not
be
able
to d
o a
float
ing
poin
t ope
ratio
n in
eve
ry c
ycle
eve
n if
the
oper
ands
are
ava
ilabl
e in
pr
imar
y ca
che
–S
ever
al c
onst
rain
ts (l
ike
prim
ary
cach
e la
tenc
y, la
tenc
y of
floa
ting
poin
t uni
ts e
tc.)
are
to b
e ob
serv
ed w
hile
com
ing
up w
ith a
n op
timal
sc
hedu
le
Fra
ctio
n of
Flo
atin
g P
oint
Ope
ratio
ns–
Not
eve
ry in
stru
ctio
n is
a fl
oatin
g po
int i
nstr
uctio
n
Pat
h fo
rwar
d
Lo
ad b
alan
cing
to b
e ha
ndle
d th
roug
h be
tter
part
ition
ing
P
aral
leliz
e ac
ross
gro
ups
to in
crea
se th
e am
ount
of c
oncu
rren
cy
Exp
lore
the
hybr
id (
mix
ed M
PI/O
penM
P)
prog
ram
min
g m
odel
–B
ette
r al
gorit
hmic
con
verg
ence
rate
M
emor
y re
duci
ng a
lgor
ithm
s fo
r m
atrix
vec
tor
prod
ucts
–T
enso
r m
atrix
vec
tor
prod
uct i
mpl
emen
tatio
n
Mor
e ef
ficie
nt c
usto
m p
reco
nditi
oner
s th
at ta
ke a
dvan
tage
of t
he m
atrix
sp
arsi
ty p
atte
rn
Wire
-Wra
pped
Fue
l Pin
Ass
embl
yS
etup
Bot
tlene
cks
C
AD
-bas
ed g
eom
etry
def
initi
on s
urpr
isin
gly
diffi
cult
–“s
wee
p w
ith r
otat
ion”
not
sup
port
ed/r
obus
t
Var
ious
pie
ces
of th
e pr
oces
s br
oke,
forc
ing
mor
e de
com
posi
tion
–S
wep
t+ro
tate
d su
rfac
e, v
olum
e m
esh
N
o w
ay to
ana
lytic
ally
spe
cify
geo
met
ry o
r m
esh,
th
ough
bot
h co
ncep
tual
ly s
trai
ghtfo
rwar
d
MA
TLA
B-b
ased
sol
utio
n no
t sca
labl
e, p
orta
ble,
ex
tens
ible
–37
pin
, 217
pin
pro
blem
s ne
eded
too
R
eal g
eom
etry
wou
ld h
elp
with
vol
ume
frac
tion
calc
ulat
ion
for
wire
hom
ogen
izat
ion
App
roac
hes
to T
H a
naly
sis
of s
ubas
sem
blie
s
D
NS
–di
rect
num
eric
al s
imul
atio
n of
all
scal
es
pa
ram
eter
-fre
e
LE
S –
larg
e ed
dy s
imul
atio
n +
dis
sipa
tion
pa
ram
eter
-fre
e
R
AN
S –
Rey
nold
s-av
erag
ed N
avie
r-S
toke
stu
ning
requ
ired
S
ubch
anne
l mod
elin
gem
piric
al in
put
40
0 x
200
subc
hann
els
in th
e co
re:
–S
ubch
anne
l ana
lysi
s w
ill c
ontin
ue to
be
used
for
reac
tor
desi
gn.
–R
AN
S w
ill in
form
des
ign
proc
ess.
–LE
S c
an h
elp
to v
alid
ate
/ inf
orm
RA
NS
and
sub
chan
nel a
naly
sis.
H
iera
rchi
cal a
ppro
ach
yiel
ds in
depe
nden
t/red
unda
nt re
fere
nce
calc
ulat
ionsim
prac
tical
107
p. p
er c
hann
el
105
p. p
er c
hann
el–
stea
dy s
tate
100
p. p
er c
hann
el–
stea
dy s
tate
Oct
ober
4, 2
007
GN
EP
Ann
ual R
evie
w M
eetin
g
App
roac
hes
to T
H a
naly
sis
of s
ubas
sem
blie
s
D
NS
–di
rect
num
eric
al s
imul
atio
n of
all
scal
es
p
aram
eter
-fre
e
LE
S –
larg
e ed
dy s
imul
atio
n +
diss
ipat
ion
pa
ram
eter
-fre
e
R
AN
S –
Rey
nold
s-av
erag
ed N
avie
r-S
toke
stu
ning
req
uire
d
S
ubch
anne
l mod
elin
gem
piric
al in
put
40
0 x
200
subc
hann
els
in th
e co
re:
–S
ubch
anne
l ana
lysi
s w
ill c
ontin
ue t
o be
use
d fo
r re
acto
r de
sign
.
–R
AN
S w
ill in
form
des
ign
proc
ess.
–LE
S c
an h
elp
to v
alid
ate
/ inf
orm
RA
NS
and
sub
chan
nel a
naly
sis.
H
iera
rchi
cal a
ppro
ach
yiel
ds in
depe
nden
t/red
unda
nt
refe
renc
e ca
lcul
atio
ns
impr
actic
al
107
p. p
er c
hann
el
105
p. p
er c
hann
el–
stea
dy s
tate
100
p. p
er c
hann
el–
stea
dy s
tate
SH
AR
P fr
amew
ork
Set
up B
ottle
neck
s
In
tero
pera
bilit
y cr
ucia
l–
Inte
ract
ing
with
mul
tiple
mod
ules
–N
eed
varie
ty o
f too
ls to
ope
rate
on
mes
h &
fiel
d da
ta
Cou
plin
g re
quire
s m
eshe
s &
res
ults
in th
e sa
me
plac
e
Cen
tral
ized
par
alle
l mes
h in
fras
truc
ture
nee
ded,
to s
uppo
rt c
oupl
ing
whi
le
allo
win
g m
odul
es fr
eedo
m
Met
adat
a ne
eded
to c
oord
inat
e co
uplin
g (n
orm
aliz
atio
n/co
nser
vatio
n)