河海工程系 3B
學號: B94520143
姓名:李家瑋
BEM 作業(4.5)
國立台灣海洋大學河海工程研究所 BEM 第 4.5 次作業
(2008)/NTOU-MSV
In the course, we separated the closed-form fundamental solution into separable form
for the fundamental solution of Laplace problem as shown below:
Closed-form:
U ( s, x) = ln r + ij , r = x - s , j = tan - 1 (
x2 - s2
), (ln r : source, j : screw dislocation)
x1 - s1
Degenerate form (separable form):
m
¥
ìï
ïï Re[U I ( s, x)] = ln R - å 1 r cos[m(q - f )], R ³ r
m
ï
m= 1 m R
ln r = Re[U ( s, x)] = ïí
¥
ïï
1 Rm
E
cos[m(q - f )], R < r
ïï Re[U ( s, x)] = ln r - å
m
ïî
m= 1 m r
m
¥
ìï
ïï Im[U I ( s, x)] = q + p + å 1 r sin[m(q - f )], R ³ r
m
ïï
m= 1 m R
j = Im[U ( s, x)] = í
¥
ïï
1 Rm
sin[m(q - f )], R < r
ïï Im[U E ( s, x)] = f - å
m
ïî
m= 1 m r
where x  (  ,  ) and s  ( R, ) .
Based on the degenerate kernel, U ( s, x) please plot the boundary densities of
u (q) and t (q) along the circular boundary.
u (q) or t (q)
B
c= 1
g(3, 4)
Ä
15
s = ( R, q )
10
5
R
Ui
0
-5
-10
-15
-15
x = (r , f )
Ue
x = (r , f )
-10
-5
0
5
10
15
j
ln r
1
河海工程系 3B
學號: B94520143
姓名:李家瑋
BEM 作業(4.5)
大學部
(1) Using the degenerate kernel of ln r to plot the boundary densities u (q) an
t (q) along the circular boundary B .
d
PROGRAM BEM HW 4 5
IMPLICIT NONE
INTEGER I,J,M,M1,M2,M3,M4,M5,N,L
REAL*8 u,u1,u2,u3,u4,u5,t,t1,t2,t3,t4,t5,PI,R,Theta,Rho,Phi,cu,ct
100 FORMAT (100F16.8)
OPEN(11,FILE='u.TXT')
OPEN(13,FILE='t.TXT')
OPEN(15,FILE='close(u).TXT')
OPEN(16,FILE='close(t).TXT')
OPEN(17,FILE='u series(m=1).TXT')
OPEN(18,FILE='u series(m=2).TXT')
OPEN(19,FILE='u series(m=3).TXT')
OPEN(20,FILE='u series(m=4).TXT')
OPEN(21,FILE='u series(m=5).TXT')
OPEN(22,FILE='t series(m=1).TXT')
OPEN(23,FILE='t series(m=2).TXT')
OPEN(24,FILE='t series(m=3).TXT')
OPEN(25,FILE='t series(m=4).TXT')
OPEN(26,FILE='t series(m=5).TXT')
M=10
M1=1
M2=2
M3=3
M4=4
M5=5
N=360
Rho=1.
PI=4.*ATAN(1.0)
R=5.
Theta=ATAN(4.0/3.0)+PI
DO I=1,N
Phi=2*PI*(I-1)/N
u=LOG(R)
u1=LOG(R)
u2=LOG(R)
u3=LOG(R)
u4=LOG(R)
u5=LOG(R)
DO J=1,M
u=u-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
2
河海工程系 3B
學號: B94520143
姓名:李家瑋
BEM 作業(4.5)
DO J=1,M1
u1=u1-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
DO J=1,M2
u2=u2-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
DO J=1,M3
u3=u3-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
DO J=1,M4
u4=u4-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
DO J=1,M5
u5=u5-(1./J)*((Rho/R)**J)*(COS(J*(Theta-Phi)))
END DO
t=0.
t1=0.
t2=0.
t3=0.
t4=0.
t5=0.
DO L=1,M
t=t-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
DO L=1,M1
t1=t1-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
DO L=1,M2
t2=t2-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
DO L=1,M3
t3=t3-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
DO L=1,M4
t4=t4-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
DO L=1,M5
t5=t5-((Rho**(L-1))/(R**L))*COS(L*(Theta-Phi))
END DO
cu=0.5*LOG((R**2+Rho**2)-2.*R*Rho*COS(Theta-Phi))
3
河海工程系 3B
學號: B94520143
姓名:李家瑋
BEM 作業(4.5)
&
ct=0.5*(2.*Rho-2.*R*COS(Theta-Phi))/((R**2+Rho**2)2.*R*Rho*COS(Theta-Phi))
WRITE(11,100) (Rho+u)*COS(Phi),(Rho+u)*SIN(Phi)
WRITE(12,100) Phi,u
WRITE(13,100) (Rho+t)*COS(Phi),(Rho+t)*SIN(Phi)
WRITE(14,100) Phi,t
WRITE(15,100) Phi,cu
WRITE(16,100) Phi,ct
WRITE(17,100) Phi,u1
WRITE(18,100) Phi,u2
WRITE(19,100) Phi,u3
WRITE(20,100) Phi,u4
WRITE(21,100) Phi,u5
WRITE(22,100) Phi,t1
WRITE(23,100) Phi,t2
WRITE(24,100) Phi,t3
WRITE(25,100) Phi,t4
WRITE(26,100) Phi,t5
END DO
STOP
END
4
u and t
circular boundary (c=1)
u (m=10)
t (m=10)
2
0
-2
-4
-4
-2
0
2
4
4
河海工程系 3B
學號: B94520143
姓名:李家瑋
BEM 作業(4.5)
1.9
u(phi)
closed form
series form (m=1)
series form (m=2)
series form (m=3)
series form (m=4)
series form (m=5)
1.8
1.7
1.6
1.5
1.4
1.3
0
2
4
6
8
t(phi)
closed form
series form (m=1)
series form (m=2)
series form (m=3)
series form (m=4)
series form (m=5)
0.2
0.1
0
-0.1
-0.2
-0.3
0
2
4
5
6
8