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' $ r r r r INSTITUT F UR INF ORMA TIK r r r r r r r r r r r r r Ulrich R ude r r r ' $ Multilevel Eigenvalue Solvers fo r the Multigroup Diusion Equations U. R ude and W. Schmid T
SLIDE 1' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Multilevel Eigenvalue Solvers fo r the Multigroup Diusion Equations U. R ude and W. Schmid T echnische Universit
at
M unchen ICIAM 95 Hamburg, July 3{7, 1995
Title
page ICIAM-95 1
SLIDE 2' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Overview
the
p roject
neutron
diusion equations
eigenvalue
p roblem
a
va riant
f
inverse iteration
combination
with iterative solvers
numerical
results
Overview
ICIAM-95 2
SLIDE 3' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r The Project Development
f
ecient algo rithms fo r the nu- merical solution
f
3D neutron transp
rt
equa- tions in nuclea r reacto rs Co
p
eration b et w een
Siemens
KWU
Siemens
ZFE
Universit
at
Augsburg (R.H.W. Hopp e, W. Schmid, F. W agner)
T
ech. Univ. M unchen (U. R ude) Goals
adaptive
discretization in space b y (noncon- fo rming) nite elements
adaptive
timestepping
fast
multilevel solution techniques
pa
rallelizat i
n
Project
ICIAM-95 3
SLIDE 4' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Nuclea r Reacto r Chain reaction
f
nuclea r ssions induced b y free neutrons Critical reacto r Neutron p ro duction and neutron losses balanced: constant energy
utput
Neutron sinks
abso
rption
diusion
Neutron sources
ssion
external
neutron sources Simulation
Behaviour
in a neutron equilib ri um ) stationa ry equations
Dynamical
b ehaviour ) transient equations
Reacto
r ICIAM-95 4
SLIDE 5' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Balance fo r free neutrons neutron ux , dep endent
n
direction
f
neutron motion, energy , space, time ( ~ r ; E ; ; t) = v
n(
~ r ; E ; ; t) leads to transp
rt
equations 1 v @
@
t +
r
+
T
+
: : : Simplifying assumptions lead to diusion theo ry
isotropical
scattering
nly
w eak dep endence
f
the ux
n
the di- rection
f
neutron motion
Mathematical
mo delling ICIAM-95 5
SLIDE 6' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Stationa ry t w
group
diusion equations r
(D
1 r 1 ) + ( a1 +
12
)
1
= 1
1
( f 1
1
+
f
2
2
) r
(D
2 r 2 ) +
a2
2
12
1
= 1
2
( f 1
1
+
f
2
2
)
r
" r
D
1 r +
a1
+
12
12
r
D
2 r +
a2
# "
1
2
# =
"
1
f
1
1
f
2
2
f
1
2
f
2 # "
1
2
#
r
(L
F
)
x
= where
=
1
and
x = [ 1 ;
2
] T
Diusion
equations ICIAM-95 6
SLIDE 7' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Inverse iteration Given: Initial vecto r x , Estimates ~
i
F
r
i = 0; 1; 2; : : : x i+1 = (L
~
i
F ) 1 F x i ; x i+1 = x i+1 =kx i+1 k;
i+1
= < x i+1 ; Lx i+1 > < x i+1 ; F x i+1 > Strategy to chose ~
i
(dep ending
n
i
?), when iterative solvers must b e used. Inverse co rrection iteration (W. Schmid, 94; cf. Hackbusch 85) Given: Initial vecto r x , estimates ~
i
initial eigenvalue app ro ximation
F
r
i = 0; 1; 2; : : : d i = (L
i
F )x i v i = I T E R [(L
~
i
F ); d i ; 0] x i+1 = x i
v
i
i+1
= < x i+1 ; Lx i+1 > < x i+1 ; F x i+1 >
Inverse
Iteration ICIAM-95 7
SLIDE 8' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Analysis
f
inverse co rrection Assume x i = e and
a
re an eigenvecto r/ eigen- value pair, i.e. Le
F
e = Then d i = (L
i
F )e = (1
i
=
)Le
x i+1 = x i + (L
~
i
F ) 1 Le = ( i
~
i
) (
~
i
) | {z } =
e:
i
=
(1
+
i
), ~
i
=
(1
+ ~
i
) p erturbations
f
:
=
(
~
i
i
)
~
i
= ( ~
i
i
) ~
i
: lim
i
!0
=
1; indep endent
f
~
i
6= lim ~
i
! i
=
0; indep endent
f
i
6=
Analysis
ICIAM-95 8
SLIDE 9' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r F urther Analyis Consider ~
i
=
i
fo r constant
:
lim
i
!0
=
lim
i
!0
i
(
1)
i
=
1
:
Thus
1
= )
1.
General case (fo r symmetric L; F ): x i = n X j =0
j
e j ; (e j ;
j
eigenvecto rs/ values) x i+1 = n X j =0
j
j
e j where
j
= ( i
~
i
) ( j
~
i
) =
(
i
~
i
) ( j
(1
+ ~
i
)) : Convergence if
Both
i
and ~
i
tend to
.
Relative
dierence
=
~
i
i
1
F
urther analysis ICIAM-95 9
SLIDE 10' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r T ypical 2D mo del geometry
dPhi = 0 dn dPhi dn Phi = 0 Phi = 0 Region 1 Region 2 Region 3 (Reflector) = 0
T ypical adaptive mesh Computed b y KASKADE
2D
numerical example ICIAM-95 10
SLIDE 11' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Numerical example 1
1D
with 129 gridp
ints
dominating
eigenvalues: 0:9501; 0:9287; 0:8950; : : :
Inverse
iteration vs. Inverse Co rrection when combined with multigrid V-cycles
2
p re- and 1 p
stsmo
thing
Jacobi smo
ther
n
6 levels.
~
i
= 0:995 i .
Numerical
example 1 ICIAM-95 11
SLIDE 12' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Numerical example 2
3D
with nonconfo rming Rannacher/T urek
nite
elements
numb
er unkno wns = 6538
Blo
ck-ILU p reconditioned CGS as solver
Stop
CGS, when residual < 10 4 and < 10 7 fo r inverse co rrection and inverse iteration, resp.
~
i
= 0:95 i
0.01 0.1 1 20 40 60 80 100
rel. Fehler
CGS-Iterationen
Inv. Iter.
Inv. Corr.
Numerical
example 2 ICIAM-95 12
SLIDE 13' ' $ $ INSTITUT F
UR
INF ORMA TIK Ulrich R ude r r r r r r r r r r r r r r r r r r r r Conclusions
