' $ 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 Di�usion 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 � �
' $ 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 ' $ Overview � the p roject � neutron di�usion equations � eigenvalue p roblem � a va riant of inverse iteration � combination with iterative solvers � numerical results � � Overview ICIAM-95 2 � �
' $ 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 ' $ The Project Development of e�cient algo rithms fo r the nu- merical solution of 3D neutron transp o rt equa- tions in nuclea r reacto rs Co op 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 on � � Project ICIAM-95 3 � �
' $ 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 ' $ Nuclea r Reacto r Chain reaction of nuclea r �ssions induced b y free neutrons Critical reacto r Neutron p ro duction and neutron losses balanced: constant energy output Neutron sinks Neutron sources � �ssion � abso rption � di�usion � external neutron sources Simulation � Behaviour in a neutron equilib ri um ) stationa ry equations � Dynamical b ehaviour ) transient equations � � Reacto r ICIAM-95 4 � �
' $ 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 ' $ Balance fo r free neutrons neutron �ux , dep endent on direction of neutron motion, energy , space, time �( ~ r ; E ; � ; t ) = v � n ( ~ r ; E ; � ; t ) leads to transp o rt equations 1 @ � + � � r � + � � + : : : T v @ t Simplifying assumptions lead to di�usion theo ry � isotropical scattering � only w eak dep endence of the �ux on the di- rection of neutron motion � � Mathematical mo delling ICIAM-95 5 � �
' $ 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 ' $ Stationa ry t w o group di�usion equations �r � ( D r � ) + (� + � ) � 1 1 a 1 12 1 1 = � � (� � + � � ) 1 1 2 f 1 f 2 � �r � ( D r � ) + � � � � � 2 2 a 2 2 12 1 1 = � � (� � + � � ) 2 f 1 1 f 2 2 � o r " # " # �r � D r + � + � 0 � 1 a 1 12 1 � � �r � D r + � � 12 2 a 2 2 " # " # � � � � � � 1 1 f 1 f 2 1 = � � � � � � 2 f 1 2 f 2 2 o r ( L � �F ) � x = 0 1 T where � = and x = [ � ; � ] 1 2 � � � Di�usion equations ICIAM-95 6 � �
' $ 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 ' $ Inverse iteration Given: Initial vecto r x , Estimates � ~ 0 i F o r i = 0 ; 1 ; 2 ; : : : � 1 x = ( L � ~ � F ) F x ; i +1 i i x = x = k x k ; i +1 i +1 i +1 < x ; Lx > i +1 i +1 � = i +1 < x ; F x > i +1 i +1 Strategy to chose � ~ (dep ending on � ?), when i i iterative solvers must b e used. Inverse co rrection iteration (W. Schmid, 94; cf. Hackbusch 85) Given: Initial vecto r x , estimates � ~ 0 i initial eigenvalue app ro ximation � 0 F o r i = 0 ; 1 ; 2 ; : : : d = ( L � � F ) x i i i v = I T E R [( L � � ~ F ) ; d ; 0] i i i x = x � v i +1 i i < x ; Lx > i +1 i +1 � = i +1 < x ; F x > i +1 i +1 � � Inverse Iteration ICIAM-95 7 � �
' $ 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 ' $ Analysis of inverse co rrection � Assume x = e and � a re an eigenvecto r/ eigen- i value pair, i.e. � Le � � F e = 0 Then � d = ( L � � F ) e = (1 � � =� ) Le i i i ( � � � ~ ) i i � 1 x = x + ( L � � ~ F ) Le = e: i +1 i i � ( � � � ~ ) i | {z } = � � � � � = � (1 + � ), � ~ = � (1 + ~ � ) p erturbations of � : i i i i � � ( ~ � � � ) ( ~ � � � ) i i i i � = = : � � ~ � � ~ � i i lim � = 1 ; indep endent of ~ � 6 = 0 i � ! 0 i lim � = 0 ; indep endent of � 6 = 0 i ~ � ! � i i � � Analysis ICIAM-95 8 � �
' $ 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 ' $ F urther Analyis Consider ~ � = � � fo r constant � : i i � ( � � 1) � � 1 i lim � = lim = : � � � i � ! 0 � ! 0 i i Thus � � 1 = ) � � 1. General case (fo r symmetric L; F ): n X x = � e ; ( e ; � eigenvecto rs/ values) i j j j j j =0 n X x = � � e j j j i +1 j =0 where � ( � � � ~ ) � ( � � ~ � ) i i i i � = = : j � ( � � � ~ ) ( � � � (1 + ~ � )) j i j i Convergence if � � Both � and � ~ tend to � . i i � Relative di�erence � � ~ � � i � = � 1 � � � � i � � F urther analysis ICIAM-95 9 � �
' $ 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 ' $ T ypical 2D mo del geometry T ypical adaptive mesh Phi = 0 Computed b y KASKADE Region 1 (Reflector) dPhi Region 2 = 0 Phi = 0 dn Region 3 dPhi = 0 dn � � 2D numerical example ICIAM-95 10 � �
' $ 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 ' $ Numerical example 1 � 1D with 129 gridp oints � 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 ostsmo othing Jacobi smo other on 6 levels. � � ~ = 0 : 995 � . i i � � Numerical example 1 ICIAM-95 11 � �
' $ 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 ' $ 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 � 4 � 7 � Stop CGS, when residual < 10 and < 10 fo r inverse co rrection and inverse iteration, resp. � � ~ = 0 : 95 � i i 1 Inv. Iter. Inv. Corr. � � Numerical example 2 ICIAM-95 12 � � rel. Fehler 0.1 0.01 20 40 60 80 100 CGS-Iterationen
' $ 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 ' $ Conclusions � further analysis of inverse co rrection � in combination with multigrid � compa rison with Davidson metho d � compa rison with Cai/Mandel/McCo rmick � multigrid fo r eigenvalues (Hackbusch/Brandt) � multigrid in 3D � adaptivit y (fo r eigenvalue computation?) � transient calculations � � Conclusions ICIAM-95 13 � �
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