*DECK SNTT2D SUBROUTINE SNTT2D (IGE,IMPX,LX,LY,SIDE,IELEM,NLF,NPQ,NSCT,IQUAD, 1 NCODE,ZCODE,MAT,XXX,YYY,VOL,IDL,DU,DE,W,MRM,MRMY,DB,DA,DAL,PL, 2 LL4,NUN,EELEM,WX,WE,CST,IBFP,ISCHM,ESCHM,IGLK,MN,DN,IL,IM,ISCAT) * *----------------------------------------------------------------------- * *Purpose: * Numbering corresponding to a 2-D Cartesian or R-Z geometry with * discrete ordinates approximation of the flux. * *Copyright: * Copyright (C) 2005 Ecole Polytechnique de Montreal * This library is free software; you can redistribute it and/or * modify it under the terms of the GNU Lesser General Public * License as published by the Free Software Foundation; either * version 2.1 of the License, or (at your option) any later version * *Author(s): A. Hebert and C. Bienvenue * *Parameters: input * IGE type of 2D geometry (=0 Cartesian; =1 R-Z; =2 Hexagonal). * IMPX print parameter. * LX number of elements along the X axis. * LY number of elements along the Y axis. * SIDE side of an hexagon. * IELEM measure of order of the spatial approximation polynomial: * =1 constant - only for HODD, classical diamond scheme * (default for HODD); * =2 linear - default for DG; * =3 parabolic; * =4 cubic - only for DG. * NLF SN order for the flux (even number). * NPQ number of SN directions in four octants (including zero-weight * directions). * NSCT maximum number of spherical harmonics moments of the flux. * IQUAD type of SN quadrature (1 Level symmetric, type IQUAD; * 4 Legendre-Chebyshev; 5 symmetric Legendre-Chebyshev; * 6 quadruple range). * NCODE type of boundary condition applied on each side * (i=1 X-; i=2 X+; i=3 Y-; i=4 Y+): * =1: VOID; =2: REFL; =4: TRAN. * ZCODE ZCODE(I) is the albedo corresponding to boundary condition * 'VOID' on each side (ZCODE(I)=0.0 by default). * MAT mixture index assigned to each element. * XXX Cartesian coordinates along the X axis. * YYY Cartesian coordinates along the Y axis. * EELEM measure of order of the energy approximation polynomial: * =1 constant - default for HODD; * =2 linear - default for DG; * >3 higher orders. * IBFP type of energy proparation relation: * =0 no Fokker-Planck term; * =1 Galerkin type; * =2 heuristic Przybylski and Ligou type. * ISCHM method of spatial discretisation: * =1 High-Order Diamond Differencing (HODD) - default; * =2 Discontinuous Galerkin finite element method (DG); * =3 Adaptive weighted method (AWD). * ESCHM method of energy discretisation: * =1 High-Order Diamond Differencing (HODD) - default; * =2 Discontinuous Galerkin finite element method (DG); * =3 Adaptive weighted method (AWD). * IGLK angular interpolation type: * =0 classical SN method. * =1 Galerkin quadrature method (M = inv(D)) * =2 Galerkin quadrature method (D = inv(M)) * ISCAT maximum number of spherical harmonics moments of the flux. * *Parameters: output * VOL volume of each element. * IDL isotropic flux indices. * DU first direction cosines ($\\mu$). * DE second direction cosines ($\\eta$). * W weights. * MRM quadrature index. * MRMY quadrature index. * DB diamond-scheme parameter. * DA diamond-scheme parameter. * DAL diamond-scheme angular redistribution parameter. * PL discrete values of the spherical harmonics corresponding * to the 2D SN quadrature. * LL4 number of unknowns being solved for, over the domain. This * includes the various moments of the isotropic (and if present, * anisotropic) flux. * NUN total number of unknowns stored in the FLUX vector per group. * This includes LL4 (see above) as well as any surface boundary * fluxes, if present. * WX spatial closure relation weighting factors. * WE energy closure relation weighting factors. * CST constants for the polynomial approximations. * MN moment-to-discrete matrix. * DN discrete-to-moment matrix. * IL indexes (l) of each spherical harmonics in the * interpolation basis. * IM indexes (m) of each spherical harmonics in the * interpolation basis. * *----------------------------------------------------------------------- * *---- * SUBROUTINE ARGUMENTS *---- INTEGER IGE,IMPX,LX,LY,IELEM,NLF,NPQ,NSCT,IQUAD,NCODE(4), 1 MAT(LX,LY),IDL(LX*LY),MRM(NPQ),MRMY(NPQ),LL4,NUN,EELEM,IBFP, 2 ISCHM,ESCHM,IL(NSCT),IM(NSCT),ISCAT,IGLK REAL ZCODE(4),VOL(LX,LY),XXX(LX+1),YYY(LY+1),DU(NPQ),DE(NPQ), 1 W(NPQ),DB(LX,NPQ),DA(LX,LY,NPQ),DAL(LX,LY,NPQ),PL(NSCT,NPQ), 2 WX(IELEM+1),WE(EELEM+1),CST(MAX(IELEM,EELEM)),MN(NPQ,NSCT), 3 DN(NSCT,NPQ) *---- * LOCAL VARIABLES *---- CHARACTER HSMG*131 LOGICAL L1,L2,L3,L4 PARAMETER(RLOG=1.0E-8,PI=3.141592654) REAL PX,PE DOUBLE PRECISION NORM,IPROD INTEGER, ALLOCATABLE, DIMENSION(:) :: JOP REAL, ALLOCATABLE, DIMENSION(:) :: XX,YY,UU,WW,TPQ,UPQ,VPQ,WPQ DOUBLE PRECISION, ALLOCATABLE, DIMENSION(:) :: V,V2 DOUBLE PRECISION, ALLOCATABLE, DIMENSION(:,:) :: U,MND DOUBLE PRECISION, ALLOCATABLE, DIMENSION(:,:,:) :: RLM *---- * SCRATCH STORAGE ALLOCATION *---- ALLOCATE(XX(LX),YY(LY)) *---- * UNFOLD FOUR-OCTANT QUADRATURES. *---- IF(MOD(NLF,2).EQ.1) CALL XABORT('SNTT2D: EVEN NLF EXPECTED.') IF(IQUAD.EQ.10) THEN NPQ0=NLF**2/4 ELSE NPQ0=NLF*(NLF/2+1)/4 ENDIF ALLOCATE(JOP(NLF/2),UU(NLF/2),WW(NLF/2),TPQ(NPQ0),UPQ(NPQ0), 1 VPQ(NPQ0),WPQ(NPQ0)) IF(IQUAD.EQ.1) THEN CALL SNQU01(NLF,JOP,UU,WW,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.2) THEN CALL SNQU02(NLF,JOP,UU,WW,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.3) THEN CALL SNQU03(NLF,JOP,UU,WW,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.4) THEN CALL SNQU04(NLF,JOP,UU,WW,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.5) THEN UU(:NLF/2)=0.0 CALL SNQU05(NLF,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.6) THEN UU(:NLF/2)=0.0 CALL SNQU06(NLF,TPQ,UPQ,VPQ,WPQ) ELSE IF(IQUAD.EQ.10) THEN CALL SNQU10(NLF,JOP,UU,WW,TPQ,UPQ,VPQ,WPQ) ELSE CALL XABORT('SNTT2D: UNKNOWN QUADRATURE TYPE.') ENDIF N=0 IOF=0 DO 30 I=1,NLF/2 IF(IGLK.NE.0) THEN JOF = NLF-2*I+2 KOF = (NLF+4)*NLF/4 ELSE IOF=IOF+1 JOF=IOF+NLF-2*I+2 KOF=IOF+(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=-SQRT(1.0-UU(I)*UU(I)) DE(IOF)=-UU(I) W(IOF)=0.0 ENDIF DO 10 J=0,NLF/2-I IOF=IOF+1 KOF=IOF+(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=-UPQ(N+J+1) DE(IOF)=-VPQ(N+J+1) W(IOF)=WPQ(N+J+1) JOF=JOF-1 10 CONTINUE DO 20 J=NLF/2-I,0,-1 IOF=IOF+1 KOF=IOF+(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=UPQ(N+J+1) DE(IOF)=-VPQ(N+J+1) W(IOF)=WPQ(N+J+1) JOF=JOF-1 20 CONTINUE N=N+NLF/2-I+1 30 CONTINUE N=0 DO 60 I=1,NLF/2 IF(IGLK.NE.0) THEN JOF=NLF-2*I+2 KOF=-(NLF+4)*NLF/4 ELSE IOF=IOF+1 JOF=IOF+NLF-2*I+2 KOF=IOF-(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=-SQRT(1.0-UU(I)*UU(I)) DE(IOF)=UU(I) W(IOF)=0.0 ENDIF DO 40 J=0,NLF/2-I IOF=IOF+1 KOF=IOF-(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=-UPQ(N+J+1) DE(IOF)=VPQ(N+J+1) W(IOF)=WPQ(N+J+1) JOF=JOF-1 40 CONTINUE DO 50 J=NLF/2-I,0,-1 IOF=IOF+1 KOF=IOF-(NLF+4)*NLF/4 MRM(IOF)=JOF MRMY(IOF)=KOF DU(IOF)=UPQ(N+J+1) DE(IOF)=VPQ(N+J+1) W(IOF)=WPQ(N+J+1) JOF=JOF-1 50 CONTINUE N=N+NLF/2-I+1 60 CONTINUE DEALLOCATE(WPQ,VPQ,UPQ,TPQ,WW,UU,JOP) IF(IMPX.GE.4) THEN WRITE(6,'(/41H SNTT2D: FOUR-OCTANT ANGULAR QUADRATURES:/26X, 1 2HMU,9X,3HETA,10X,2HXI,6X,6HWEIGHT)') SUM=0.0 DO 70 N=1,NPQ SUM=SUM+W(N) ZI=SQRT(ABS(1.0-DU(N)**2-DE(N)**2)) IF(ZI.LT.1.0E-3) ZI=0.0 WRITE(6,'(1X,3I5,1P,4E12.4)') N,MRM(N),MRMY(N),DU(N),DE(N),ZI, 1 W(N) 70 CONTINUE WRITE(6,'(54X,10(1H-)/52X,1P,E12.4)') SUM ENDIF *---- * IDENTIFICATION OF THE GEOMETRY. *---- IF(IGE.EQ.0) THEN * ---------- * 2D CARTESIAN * ---------- DO 82 N=1,NPQ VU=DU(N) VE=DE(N) DO 81 I=1,LX XX(I)=XXX(I+1)-XXX(I) DB(I,N)=VE*XX(I) DO 80 J=1,LY YY(J)=YYY(J+1)-YYY(J) DA(I,J,N)=VU*YY(J) DAL(I,J,N)=0.0 80 CONTINUE 81 CONTINUE 82 CONTINUE DO 91 I=1,LX DO 90 J=1,LY VOL(I,J)=XX(I)*YY(J) 90 CONTINUE 91 CONTINUE ELSEIF(IGE.EQ.1) THEN * ---------- * 2D TUBE * ---------- DO 95 J=1,LY YY(J)=YYY(J+1)-YYY(J) 95 CONTINUE DO 102 N=1,NPQ VU=DU(N)*PI DO 101 I=1,LX XX(I)=XXX(I+1)-XXX(I) VE=(XXX(I)+XXX(I+1))*VU DO 100 J=1,LY DA(I,J,N)=VE*YY(J) 100 CONTINUE 101 CONTINUE 102 CONTINUE DB(:LX,:NPQ)=0.0 DAL(:LX,:LY,:NPQ)=0.0 DO 135 J=1,LY DO 111 I=1,LX VE=2.0*PI*(XXX(I+1)-XXX(I))*YY(J) DO 110 N=2,NPQ DB(I,N)=DB(I,N-1)-W(N)*DU(N)*VE 110 CONTINUE 111 CONTINUE DO 130 N=2,NPQ VE=W(N) IF(VE.LE.RLOG) GOTO 130 DO 120 I=1,LX DAL(I,J,N)=(DB(I,N)+DB(I,N-1))/VE 120 CONTINUE 130 CONTINUE 135 CONTINUE DO 155 I=1,LX VE=PI*XX(I)*(XXX(I+1)+XXX(I)) DO 140 N=1,NPQ DB(I,N)=VE*DE(N) 140 CONTINUE DO 150 J=1,LY VOL(I,J)=YY(J)*VE 150 CONTINUE 155 CONTINUE ELSEIF(IGE.EQ.2) THEN * ---------- * 2D HEXAGONAL * ---------- DET = SQRT(3.0)*(SIDE**2)/2.0 DO 162 N=1,NPQ VU=DU(N) VE=DE(N) DO 161 I=1,LX DB(I,N)=VE DO 160 J=1,LY DA(I,J,N)=VU VOL(I,J)=DET 160 CONTINUE 161 CONTINUE 162 CONTINUE ENDIF *---- * GENERATE SPHERICAL HARMONICS FOR SCATTERING SOURCE. *---- IOF=0 DO 211 L=0,ISCAT-1 DO 210 M=-L,L IF(MOD(L+M,2).EQ.1) GO TO 210 IOF=IOF+1 IF(IOF.GT.NSCT) GO TO 211 DO 200 N=1,NPQ ZI=SQRT(ABS(1.0-DU(N)**2-DE(N)**2)) IF(ZI.LT.1.0E-3) ZI=0.0 PL(IOF,N)=PNSH(L,M,ZI,DU(N),DE(N)) 200 CONTINUE 210 CONTINUE 211 CONTINUE *---- * GENERATE MAPPING MATRIX FOR GALERKIN QUADRATURE METHOD *---- MN(:NPQ,:NSCT)=0.0 DN(:NSCT,:NPQ)=0.0 IL(:NSCT)=0 IM(:NSCT)=0 IF(IGLK.NE.0) THEN ALLOCATE(U(NPQ,NPQ),RLM(NPQ,ISCAT,2*ISCAT-1),V(NPQ),V2(NPQ), 1 MND(NPQ,NPQ)) RLM(:NPQ,:ISCAT,:2*ISCAT-1)=0.0 DO L=0,ISCAT-1 DO M=-L,L DO N=1,NPQ ZI=SQRT(ABS(1.0-DU(N)**2-DE(N)**2)) IF(ZI.LT.1.0E-3) ZI=0.0 RLM(N,L+1,M+L+1)=PNSH(L,M,DU(N),DE(N),ZI) ENDDO ENDDO ENDDO ! GRAM-SCHMIDT PROCEDURE TO FIND INDEPENDANT SET ! OF SPHERICAL HARMONICS WITH ANY QUADRATURE U(:NPQ,:NPQ)=0.0D0 NORM=0.0D0 DO N=1,NPQ NORM=NORM+RLM(N,1,1)**2 ENDDO NORM=SQRT(NORM) DO N=1,NPQ IF(IGLK.EQ.1) THEN MND(1,N)=2.0D0*W(N)*RLM(N,1,1) ELSEIF(IGLK.EQ.2) THEN MND(N,1)=(2.0*L+1.0)/(4.0*PI)*RLM(N,1,1) ELSE CALL XABORT('UNKNOWN GALERKIN QUADRATURE METHOD.') ENDIF U(N,1)=RLM(N,1,1)/NORM ENDDO IND=1 ! ITERATE OVER THE SPHERICAL HARMONICS DO 212 L=1,ISCAT-1 DO 213 M=0,L V2(:NPQ)=0.0D0 DO N=1,IND IPROD=0.0D0 DO N2=1,NPQ IPROD=IPROD+U(N2,N)*RLM(N2,L+1,M+L+1) ENDDO DO N2=1,NPQ V2(N2)=V2(N2)+IPROD*U(N2,N) ENDDO ENDDO V(:NPQ)=0.0D0 DO N=1,NPQ V(N)=RLM(N,L+1,M+L+1)-V2(N) ENDDO NORM=0.0D0 DO N=1,NPQ NORM=NORM+V(N)**2 ENDDO NORM=SQRT(NORM) ! KEEP THE SPHERICAL HARMONICS IF IT IS INDEPENDANT IF(NORM.GE.1.0E-5) THEN IND=IND+1 DO N=1,NPQ U(N,IND)=V(N)/NORM IF(IGLK.EQ.1) THEN MND(IND,N)=2.0D0*W(N)*RLM(N,L+1,M+L+1) ELSEIF(IGLK.EQ.2) THEN MND(N,IND)=(2.0*L+1.0)/(4.0*PI)*RLM(N,L+1,M+L+1) ELSE CALL XABORT('UNKNOWN GALERKIN QUADRATURE METHOD.') ENDIF ENDDO IL(IND)=L IM(IND)=M ENDIF IF(IND.EQ.NPQ) GOTO 217 213 ENDDO 212 ENDDO CALL XABORT('SNTT2D: THE'// 1 ' GRAM-SCHMIDTH PROCEDURE TO FIND A SUITABLE INTERPOLATION'// 2 ' BASIS REQUIRE HIGHER LEGENDRE ORDER.') ! FIND INVERSE MATRIX 217 IF(IGLK.EQ.1) THEN DN=REAL(MND) CALL ALINVD(NPQ,MND,NPQ,IER) IF(IER.NE.0) CALL XABORT('SNTT2D: SINGULAR MATRIX.') MN=REAL(MND) ELSEIF(IGLK.EQ.2) THEN MN=REAL(MND) CALL ALINVD(NPQ,MND,NPQ,IER) IF(IER.NE.0) CALL XABORT('SNTT2D: SINGULAR MATRIX.') DN=REAL(MND) ELSE CALL XABORT('UNKNOWN GALERKIN QUADRATURE METHOD.') ENDIF DEALLOCATE(U,RLM,V,V2,MND) ELSE IND=1 DO L=0,ISCAT-1 DO 218 M=-L,L IF(MOD(L+M,2).EQ.1) GO TO 218 IL(IND)=L IM(IND)=M DO N=1,NPQ ZI=SQRT(ABS(1.0-DU(N)**2-DE(N)**2)) IF(ZI.LT.1.0E-3) ZI=0.0 DN(IND,N)=2.0*W(N)*PNSH(L,M,ZI,DU(N),DE(N)) MN(N,IND)=(2.0*L+1.0)/(4.0*PI) 1 *PNSH(L,M,ZI,DU(N),DE(N)) ENDDO IND=IND+1 218 ENDDO ENDDO ENDIF *---- * GENERATE THE WEIGHTING PARAMETERS OF THE CLOSURE RELATION. *---- PX=1 PE=1 IF(ISCHM.EQ.1.OR.ISCHM.EQ.3) THEN PX=1 ELSEIF(ISCHM.EQ.2) THEN PX=0 ELSE CALL XABORT('SNTT2D: UNKNOWN TYPE OF SPATIAL CLOSURE RELATION.') ENDIF IF(MOD(IELEM,2).EQ.1) THEN WX(1)=-PX WX(2:IELEM+1:2)=1+PX IF(IELEM.GE.2) WX(3:IELEM+1:2)=1-PX ELSE WX(1)=PX WX(2:IELEM+1:2)=1-PX IF(IELEM.GE.2) WX(3:IELEM+1:2)=1+PX ENDIF IF(IBFP.NE.0) THEN IF(ESCHM.EQ.1.OR.ESCHM.EQ.3) THEN PE=1 ELSEIF(ESCHM.EQ.2) THEN PE=0 ELSE CALL XABORT('SNTT2D: UNKNOWN TYPE OF ENERGY CLOSURE RELATION.') ENDIF IF(MOD(EELEM,2).EQ.1) THEN WE(1)=-PE WE(2:EELEM+1:2)=1+PE IF(EELEM.GE.2) WE(3:EELEM+1:2)=1-PE ELSE WE(1)=PE WE(2:EELEM+1:2)=1-PE IF(EELEM.GE.2) WE(3:EELEM+1:2)=1+PE ENDIF ENDIF ! NORMALIZED LEGENDRE POLYNOMIAL CONSTANTS DO IEL=1,MAX(IELEM,EELEM) CST(IEL)=SQRT(2.0*IEL-1.0) ENDDO *---- * COMPUTE ISOTROPIC FLUX INDICES. *---- NM=IELEM*IELEM*EELEM NMX=IELEM*EELEM NMY=IELEM*EELEM NME=IELEM**2 LL4=LX*LY*NSCT*NM IF(IGE.LT.2) THEN NUN=LL4+(LX*NMY+LY*NMX)*NPQ DO I=1,LX*LY IDL(I)=(I-1)*NSCT*NM+1 ENDDO ELSEIF(IGE.EQ.2) THEN NUN=LL4 DO I=1,LX IDL(I)=(I-1)*NSCT*NM+1 ENDDO ELSE CALL XABORT('SNTT2D: CHECK SPATIAL SCHEME DISCRETISATION '// 1 'PARAMETER.') ENDIF *---- * SET BOUNDARY CONDITIONS. *---- DO 240 I=1,4 IF(NCODE(I).NE.1) ZCODE(I)=1.0 IF(NCODE(I).EQ.5) CALL XABORT('SNTT2D: SYME BC NOT ALLOWED.') IF(NCODE(I).EQ.7) CALL XABORT('SNTT2D: ZERO FLUX BC NOT ALLOWED.') 240 CONTINUE *---- * CHECK FOR INVALID VIRTUAL ELEMENTS. *---- DO 295 I=2,LX-1 DO 290 J=2,LY-1 IF(MAT(I,J).EQ.0) THEN L1=(NCODE(1).NE.1) DO 250 J1=1,J-1 L1=L1.OR.(MAT(I,J1).NE.0) 250 CONTINUE L2=(NCODE(2).NE.1) DO 260 J1=J+1,LY L2=L2.OR.(MAT(I,J1).NE.0) 260 CONTINUE L3=(NCODE(3).NE.1) DO 270 I1=1,I-1 L3=L3.OR.(MAT(I1,J).NE.0) 270 CONTINUE L4=(NCODE(4).NE.1) DO 280 I1=I+1,LX L4=L4.OR.(MAT(I1,J).NE.0) 280 CONTINUE IF(L1.AND.L2.AND.L3.AND.L4) THEN WRITE(HSMG,'(17HSNTT2D: ELEMENT (,I3,1H,,I3,11H) CANNOT BE, 1 9H VIRTUAL.)') I,J CALL XABORT(HSMG) ENDIF ENDIF 290 CONTINUE 295 CONTINUE *---- * SCRATCH STORAGE DEALLOCATION *---- DEALLOCATE(YY,XX) RETURN END