PROGRAM ELEVEN C======================================================================= C SOLUTION OF D2U/DXDX + ALPHA*U = 0 USING WEIGHTED RESIDUAL METHOD C WITH AN APPROXIMATING FUNCTION OF U(X)=F0(X)+A1*F1(X) C AND BOUNDARY CONDITIONS OF U(0)=1. & DU/DX(L)=S; C -------------- VARIABLE DEFNITION ----------- 12/2/2004 EIJI FUKUMORI C XST & XEN: INTEGRATION LIMITS; NSEG: NUMBER OF SEGMENTS IN LIMITS; C UNKNOWN COEFFICENT (A1) IN THE APPROXIMATING FUNCTION IS EVALUATED C BY THE FOLLOWING EQUATION: B1 * A1 + C1 = -S C======================================================================= IMPLICIT REAL * 8 ( A-H , O-Z ) PARAMETER ( N = 3, ALPHA=1., XST=0., XEN=0.5, NSEG=100, MULTI=10 ) DIMENSION SAI(N) , W(N) COMMON / DEL / DELTAX /DOMAIN / RL, S, U0 EXTERNAL F0, F1 C======================================================================= C THREE-SAMPLING-POINT GAUSS INTEGRATION METHOD C N: NUMBER OF SAMPLING POINTS IN EACH SEGMENET C SAI(I) & W(I): NON-DIMENSIONALIZED COORDINATE & WEIGHTING FACTOR DATA SAI/-0.7745966692415D0,0.0000000000000D0, 0.7745966692415D0/ DATA W / 0.5555555555555D0,0.8888888888888D0, 0.5555555555555D0/ C======================================================================= RL = XEN - XST U0 = 1.D0 WRITE (*,240) 240 FORMAT( 'Type in S = ' $ ) READ(*,*) S C======================================================================= OPEN ( 1, FILE='ELEVEN.FEM',STATUS='UNKNOWN' ) WRITE(1,*)' APPROXIMATING FUNCTION: F0(X) + A1*F1(X)' C======================================================================= C DELTAX: SPACIAL DEFERENTIAL LENGTH FOR DERIVATIVE EVALUATION. DELTAX = ( XEN - XST ) / ( MULTI * NSEG ) C======================================================================= C COMPUTATION OF H(F0,F1) AND H(F1,F1) CALL INTE ( ALPHA, XST, XEN, NSEG, N, SAI, W, F0, F1, C1 ) CALL INTE ( ALPHA, XST, XEN, NSEG, N, SAI, W, F1, F1, B1 ) C======================================================================= C EVALUATION OF UNKNOWN A1 IN THE APPROXIMATING FUNCTION U(X) A1 = (-S - C1) / B1 C======================================================================= C PRINTING RESULTS WRITE (1,100) B1, C1, -S WRITE (1,110) A1 100 FORMAT ( 1X, F20.10, 1X, '* A1 +', F20.10, ' =', F20.10 ) 110 FORMAT ( 2X, 'U(X) = F0(X) + ',F15.10, ' * F1(X)' ) X = 0.5 UOFX = F0(X) + A1*F1(X) WRITE (1,*) ' X=',X, ' U(X)=',UOFX CALL INTE1 ( ALPHA,XST, XEN, NSEG, N, SAI, W, A1,SQERROR ) WRITE (1,*) ' SQUARE ERROR =', SQERROR CLOSE (1) STOP END C C SUBROUTINE INTE ( ALPHA,XST,XEN,NSEG, N,SAI,W, G1,G2, TOTAL ) IMPLICIT REAL * 8 ( A-H , O-Z ) DIMENSION SAI(N) , W(N) EXTERNAL G1, G2 TOTAL = 0. DX = ( XEN - XST ) / NSEG DO I = 1 , NSEG X1 = DX*(I-1) X2 = X1 + DX SUM = 0. SH = ( X2 - X1 ) / 2. AVE = ( X1 + X2 ) / 2. DO J = 1 , N X = SH * SAI(J) + AVE SUM = SUM + (-DERIV(G1,X)*DERIV(G2,X)+ALPHA*G1(X)*G2(X)) * W(J) END DO TOTAL = TOTAL + SH * SUM END DO RETURN END C C FUNCTION F0(X) IMPLICIT REAL * 8 ( A-H , O-Z ) COMMON / DOMAIN / RL, S, U0 F0 = U0 - S*RL* ( 1 - X/RL )* ( X/RL ) RETURN END C C FUNCTION F1(X) IMPLICIT REAL * 8 ( A-H , O-Z ) COMMON / DOMAIN / RL, S, U0 F1 = (1-X/RL )* (X/RL ) + (X/RL ) RETURN END C C FUNCTION DERIV(F,X) IMPLICIT REAL * 8 ( A-H , O-Z ) COMMON / DEL / DELTAX EXTERNAL F DERIV = ( F(X+DELTAX) - F(X-DELTAX) ) / ( 2.*DELTAX ) RETURN END C C SUBROUTINE INTE1 ( ALPHA,XST,XEN,NSEG, N,SAI,W,A1,TOTAL ) IMPLICIT REAL * 8 ( A-H , O-Z ) COMMON /DOMAIN / RL, S, U0 DIMENSION SAI(N) , W(N) TOTAL = 0. DX = ( XEN - XST ) / NSEG C=DSQRT(ALPHA) A = ( S+U0*C*DSIN(C*RL) ) / (C*DCOS(C*RL)) B = U0 DO I = 1 , NSEG X1 = DX*(I-1) X2 = X1 + DX SUM = 0. SH = ( X2 - X1 ) / 2. AVE = ( X1 + X2 ) / 2. DO J = 1 , N X = SH * SAI(J) + AVE EXACT = A*DSIN(C*X) + B*DCOS(C*X) APPRO = F0(X) + A1*F1(X) SUM = SUM + W(J)*(APPRO-EXACT)**2 END DO TOTAL = TOTAL + SH * SUM END DO X = 0. N = 10 DX = ( XEN - XST ) / N WRITE(1,*) ' X APPROXIMATION EXACT' WRITE(1,*) X, U0, U0 DO I = 1 , N X = X + DX EXACT = A*DSIN(C*X) + B*DCOS(C*X) APPRO = F0(X) + A1*F1(X) WRITE (1,*) X, APPRO , EXACT END DO RETURN END