3次元FDM用CG法(C)プログラム Y-Z-対称NO.2
2003/08/26 日立製作所 & 早稲田大学 後 保範 ( Ushiro Yasunori )
--------------------------------------------------------------
1. 概要
3次元差分法で離散化した疎行列を係数とする連立一次方程式Ax=bに対してCG法で
反復解xを求める。
α=(r,r)/(p,Ap),β=new(r,r)/old(r,r)を使用した、演算量の少ないバージョン
係数行列Aと右辺ベクトルbが解析領域に対してy及びzの中心で対称なら、計算解xも
対称になる工夫をしたプログラムである。Cコンパイラーが同一式中の加減算の順序を
変更する場合も、対称性を保証する。但し、ループ結合まで行う最適化の場合は結果の
対称性は保証されない。
2. プログラム
#include <stdio.h>
#include <math.h>
// Global Define
#define NDX 51
#define NDY 51
#define NDZ 51
#define A(i,j,k,l) A[l-1][k][j][i]
#define B(i,j,k) B[k][j][i]
#define X(i,j,k) X[k][j][i]
#define R(i,j,k) R[k][j][i]
#define P(i,j,k) P[k][j][i]
#define Q(i,j,k) Q[k][j][i]
extern double A[4][NDZ][NDY][NDX], B[NDZ][NDY][NDX], X[NDZ][NDY][NDX] ;
extern double R[NDZ][NDY][NDX], P[NDZ][NDY][NDX], Q[NDZ][NDY][NDX] ;
extern FILE *FT1 ;
//=================================================================C
int CG3DM(int NX, int NY, int NZ, double EPS, int *ITER, double *ERR)
//=================================================================C
// Solve Ax=b by CG NO.1 with 3 dimensional FDM C
// Alpha=(P,R)/(P,AP), Beta=-(R,AP)/(P,AP) C
// for symmetric solution on Y-axis C
// Modified version of computing Q=A*P for symmetric solution C
//-----------------------------------------------------------------C
// NX I*4, In, Grid Numbers on X-axis C
// NY I*4, In, Grid Numbers on Y-axis C
// NZ I*4, In, Grid Numbers on Y-axis C
// EPS R*8, In, if ||r||/||b|| <= EPS --> return C
// ITER I*4, I/O, Number of Iteration C
// ERR R*8, Out, ERR=||r||/||b|| C
// return I*4, Out, IERR=0, Normal Return C
// =1, No Convergent C
//-----------------------------------------------------------------C
// Written by Yasunori Ushiro, 2003/08/25 C
// ( Hitachi Ltd. and Waseda University ) C
//=================================================================C
{ int i, j, k, l, m, kk, NY2, NZ2, MNY, MNZ, IERR ;
double BN, C0, C1, Alpha, Beta ;
// Even or Odd of NY,NZ
MNY = NY % 2 ;
if(MNY == 0) { NY2 = NY/2 - 1 ; }
else { NY2 = (NY - 1)/2 ; }
MNZ = NZ % 2 ;
if(MNZ == 0) { NZ2 = NZ/2 - 1 ; }
else { NZ2 = (NZ - 1)/2 ; }
// P=R=B-A*X
IERR = 0 ;
BN = 0.0 ;
C0 = 0.0 ;
// main points
for (k=1; k<=NZ2; k++) {
m = NZ - k ;
for (j=1; j<=NY2; j++) {
l = NY - j ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
BN = BN + B(i,j,k)*B(i,j,k) + B(i,j,m)*B(i,j,m)
+ B(i,l,k)*B(i,l,k) + B(i,l,m)*B(i,l,m) ;
R(i,j,k) = B(i,j,k) - A(i,j,k,1)*X(i,j,k-1) ;
R(i,j,m) = B(i,j,m) - A(i,j,m+1,1)*X(i,j,m+1) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,1)*X(i,l,k-1) ;
R(i,l,m) = B(i,l,m) - A(i,l,m+1,1)*X(i,l,m+1) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,2)*X(i,j-1,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,2)*X(i,j-1,m) ;
R(i,l,k) = B(i,l,k) - A(i,l,k+1,2)*X(i,l+1,k) ;
R(i,l,m) = B(i,l,m) - A(i,l+1,m,2)*X(i,l+1,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,3)*X(i-1,j,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,3)*X(i-1,j,m) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,3)*X(i-1,l,k) ;
R(i,l,m) = B(i,l,m) - A(i,l,m,3)*X(i-1,l,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,4)*X(i,j,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,4)*X(i,j,m) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,4)*X(i,l,k) ;
R(i,l,m) = B(i,l,m) - A(i,l,m,4)*X(i,l,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i+1,j,k,3)*X(i+1,j,k) ;
R(i,j,m) = B(i,j,m) - A(i+1,j,m,3)*X(i+1,j,m) ;
R(i,l,k) = B(i,l,k) - A(i+1,l,k,3)*X(i+1,l,k) ;
R(i,l,m) = B(i,l,m) - A(i+1,l,m,3)*X(i+1,l,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j+1,k,2)*X(i,j+1,k) ;
R(i,j,m) = B(i,j,m) - A(i,j+1,m,2)*X(i,j+1,m) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,2)*X(i,l-1,k) ;
R(i,l,m) = B(i,l,m) - A(i,l,m,2)*X(i,l-1,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k+1,1)*X(i,j,k+1) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,1)*X(i,j,m-1) ;
R(i,l,k) = B(i,l,k) - A(i,l,k+1,1)*X(i,l,k+1) ;
R(i,l,m) = B(i,l,m) - A(i,l,m,1)*X(i,l,m-1) ;
C0 = C0 + R(i,j,k)*R(i,j,k) + R(i,j,m)*R(i,j,m)
+ R(i,l,k)*R(i,l,k) + R(i,l,m)*R(i,l,m) ;
P(i,j,k) = R(i,j,k) ;
P(i,j,m) = R(i,j,m) ;
P(i,l,k) = R(i,l,k) ;
P(i,l,m) = R(i,l,m) ; }
} }
// Y center line
if(MNY == 0) {
for (k=1; k<=NZ2; k++) {
m = NZ - k ;
j = NY/2 ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
BN = BN + B(i,j,k)*B(i,j,k) + B(i,j,m)*B(i,j,m) ;
R(i,j,k) = B(i,j,k) - A(i,j,k,1)*X(i,j,k-1) ;
R(i,j,m) = B(i,j,m) - A(i,j,m+1,1)*X(i,j,m+1) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,2)*X(i,j-1,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,2)*X(i,j-1,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,3)*X(i-1,j,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,3)*X(i-1,j,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,4)*X(i,j,k) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,4)*X(i,j,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i+1,j,k,3)*X(i+1,j,k) ;
R(i,j,m) = B(i,j,m) - A(i+1,j,m,3)*X(i+1,j,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j+1,k,2)*X(i,j+1,k) ;
R(i,j,m) = B(i,j,m) - A(i,j+1,m,2)*X(i,j+1,m) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k+1,1)*X(i,j,k+1) ;
R(i,j,m) = B(i,j,m) - A(i,j,m,1)*X(i,j,m-1) ;
C0 = C0 + R(i,j,k)*R(i,j,k) + R(i,j,m)*R(i,j,m) ;
P(i,j,k) = R(i,j,k) ;
P(i,j,m) = R(i,j,m) ; }
} }
// Z center line
if(MNZ == 0 ) {
k = NZ/2 ;
for (j=1; j<=NY2; j++) {
l = NY - j ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
BN = BN + B(i,j,k)*B(i,j,k) + B(i,l,k)*B(i,l,k) ;
R(i,j,k) = B(i,j,k) - A(i,j,k,1)*X(i,j,k-1) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,1)*X(i,l,k-1) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,2)*X(i,j-1,k) ;
R(i,l,k) = B(i,l,k) - A(i,l+1,k,2)*X(i,l+1,k) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,3)*X(i-1,j,k) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,3)*X(i-1,l,k) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k,4)*X(i,j,k) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,4)*X(i,l,k) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i+1,j,k,3)*X(i+1,j,k) ;
R(i,l,k) = B(i,l,k) - A(i+1,l,k,3)*X(i+1,l,k) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j+1,k,2)*X(i,j+1,k) ;
R(i,l,k) = B(i,l,k) - A(i,l,k,2)*X(i,l-1,k) ; }
for (i=1; i<=NX-1; i++) {
R(i,j,k) = B(i,j,k) - A(i,j,k+1,1)*X(i,j,k+1) ;
R(i,l,k) = B(i,l,k) - A(i,l,k+1,1)*X(i,l,k+1) ;
C0 = C0 + R(i,j,k)*R(i,j,k) + R(i,l,k)*R(i,l,k) ;
P(i,j,k) = R(i,j,k) ;
P(i,l,k) = R(i,l,k) ; }
} }
// Y and Z center line
if(MNY == 0 & MNZ ==0) {
k = NZ/2 ;
j = NY/2 ;
for (i=1; i<=NX-1; i++) {
BN = BN + B(i,j,k)*B(i,j,k) ;
R(i,j,k) = B(i,j,k) - A(i,j,k,1)*X(i,j,k-1)
- A(i,j,k,2)*X(i,j-1,k) - A(i,j,k,3)*X(i-1,j,k)
- A(i,j,k,4)*X(i,j,k) - A(i+1,j,k,3)*X(i+1,j,k)
- A(i,j+1,k,2)*X(i,j+1,k) - A(i,j,k+1,1)*X(i,j,k+1) ;
C0 = C0 + R(i,j,k)*R(i,j,k) ;
P(i,j,k) = R(i,j,k) ; }
}
// Main Loop
for (kk=1; kk<=*ITER; kk++) {
// Q=A*P, C=(P,Q), Alpha=C0/(P,Q)
Alpha = 0.0 ;
for (k=1; k<=NZ2; k++) {
m = NZ - k ;
for (j=1; j<=NY2; j++) {
l = NY - j ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = A(i,j,k,1)*P(i,j,k-1) + A(i,j,k,2)*P(i,j-1,k) ;
Q(i,j,m) = A(i,j,m+1,1)*P(i,j,m+1) + A(i,j,k,2)*P(i,j-1,m) ;
Q(i,l,k) = A(i,l,k,1)*P(i,l,k-1) + A(i,l+1,k,2)*P(i,l+1,k) ;
Q(i,l,m) = A(i,l,m+1,1)*P(i,l,m+1) + A(i,l+1,m,2)*P(i,l+1,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,3)*P(i-1,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,m,3)*P(i-1,j,m) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,3)*P(i-1,l,k) ;
Q(i,l,m) = Q(i,l,m) + A(i,l,m,3)*P(i-1,l,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,4)*P(i,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,k,4)*P(i,j,m) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,4)*P(i,l,k) ;
Q(i,l,m) = Q(i,l,m) + A(i,l,m,4)*P(i,l,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i+1,j,k,3)*P(i+1,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i+1,j,m,3)*P(i+1,j,m) ;
Q(i,l,k) = Q(i,l,k) + A(i+1,l,k,3)*P(i+1,l,k) ;
Q(i,l,m) = Q(i,l,m) + A(i+1,l,m,3)*P(i+1,l,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j+1,k,2)*P(i,j+1,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j+1,k,2)*P(i,j+1,m) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,2)*P(i,l-1,k) ;
Q(i,l,m) = Q(i,l,m) + A(i,l,m,2)*P(i,l-1,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k+1,1)*P(i,j,k+1) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,m,1)*P(i,j,m-1) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k+1,1)*P(i,l,k+1) ;
Q(i,l,m) = Q(i,l,m) + A(i,l,m,1)*P(i,l,m-1) ;
Alpha = Alpha + P(i,j,k)*Q(i,j,k) + P(i,j,m)*Q(i,j,m)
+ P(i,l,k)*Q(i,l,k) + P(i,l,m)*Q(i,l,m) ; }
} }
// Y center line
if(MNY == 0) {
for (k=1; k<=NZ2; k++) {
m = NZ - k ;
j = NY/2 ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = A(i,j,k,1)*P(i,j,k-1) + A(i,j,k,2)*P(i,j-1,k) ;
Q(i,j,m) = A(i,j,m+1,1)*P(i,j,m+1) + A(i,j,k,2)*P(i,j-1,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,3)*P(i-1,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,m,3)*P(i-1,j,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,4)*P(i,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,k,4)*P(i,j,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i+1,j,k,3)*P(i+1,j,k) ;
Q(i,j,m) = Q(i,j,m) + A(i+1,j,m,3)*P(i+1,j,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j+1,k,2)*P(i,j+1,k) ;
Q(i,j,m) = Q(i,j,m) + A(i,j+1,k,2)*P(i,j+1,m) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k+1,1)*P(i,j,k+1) ;
Q(i,j,m) = Q(i,j,m) + A(i,j,m,1)*P(i,j,m-1) ;
Alpha = Alpha + P(i,j,k)*Q(i,j,k) + P(i,j,m)*Q(i,j,m) ; }
} }
// Z center line
if(MNZ == 0) {
k = NZ/2 ;
for (j=1; j<=NY2; j++) {
l = NY - j ;
// Divided do loop for symmetric solution
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = A(i,j,k,1)*P(i,j,k-1) + A(i,j,k,2)*P(i,j-1,k) ;
Q(i,l,k) = A(i,l,k,1)*P(i,l,k-1) + A(i,l+1,k,2)*P(i,l+1,k) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,3)*P(i-1,j,k) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,3)*P(i-1,l,k) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k,4)*P(i,j,k) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,4)*P(i,l,k) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i+1,j,k,3)*P(i+1,j,k) ;
Q(i,l,k) = Q(i,l,k) + A(i+1,l,k,3)*P(i+1,l,k) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j+1,k,2)*P(i,j+1,k) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k,2)*P(i,l-1,k) ; }
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = Q(i,j,k) + A(i,j,k+1,1)*P(i,j,k+1) ;
Q(i,l,k) = Q(i,l,k) + A(i,l,k+1,1)*P(i,l,k+1) ;
Alpha = Alpha + P(i,j,k)*Q(i,j,k) + P(i,l,k)*Q(i,l,k) ; }
for (i=1; i<=NX-1; i++) {
}
for (i=1; i<=NX-1; i++) {
}
} }
// Y-Z center line
if(MNY == 0 & MNZ == 0) {
k = NZ/2 ;
j = NY/2 ;
for (i=1; i<=NX-1; i++) {
Q(i,j,k) = A(i,j,k,1)*P(i,j,k-1) + A(i,j,k,2)*P(i,j-1,k)
+ A(i,j,k,3)*P(i-1,j,k) + A(i,j,k,4)*P(i,j,k)
+ A(i+1,j,k,3)*P(i+1,j,k) + A(i,j+1,k,2)*P(i,j+1,k)
+ A(i,j,k+1,1)*P(i,j,k+1) ;
Alpha = Alpha + P(i,j,k)*Q(i,j,k) ; }
}
Alpha = C0/Alpha ;
// X=X+Alpha*P, R=R-Alpha*Q
C1 = 0.0 ;
for (k=1; k<=NZ-1; k++) {
for (j=1; j<=NY-1; j++) {
for (i=1; i<=NX-1; i++) {
X(i,j,k) = X(i,j,k) + Alpha*P(i,j,k) ;
R(i,j,k) = R(i,j,k) - Alpha*Q(i,j,k) ;
C1 = C1 + R(i,j,k)*R(i,j,k) ; }
} }
// if(ERR <= EPS) Convergent, Beta=C1/C0
*ERR = sqrt(C1/BN) ;
if(*ERR <= EPS) goto M100 ;
Beta = C1/C0 ;
C0 = C1 ;
// P=R+Beta*P
for (k=1; k<=NZ-1; k++) {
for (j=1; j<=NY-1; j++) {
for (i=1; i<=NX-1; i++) {
P(i,j,k) = R(i,j,k) + Beta*P(i,j,k) ; }
} }
}
IERR = 1 ;
//
M100: *ITER = kk ;
return (IERR) ; }