Civil Engineering Reference
In-Depth Information
§
·
w
u
w
U
w
T
¨
©
¸
¹
³
³
q
P
k
P
k
t
Q
P
,
Q
dS
Q
u
Q
P
,
Q
dS
Q
x
a
a
a
a
¨
¸
w
x
w
x
w
x
S
S
§
·
w
u
w
U
w
T
¨
©
³
³
¸
¹
q
P
k
P
k
t
Q
P
,
Q
dS
Q
u
Q
P
,
Q
dS
Q
y
a
a
a
a
w
y
w
y
w
y
(9.26)
S
S
§
·
w
u
w
U
w
T
¨
©
³
³
¸
¹
q
P
k
P
k
t
Q
P
,
Q
dS
Q
u
Q
P
,
Q
dS
Q
z
a
a
a
a
w
z
w
z
w
z
S
S
where derivatives of
U
have been presented previously and derivatives of
T
are for two-
dimensional problems
ª
º
w
T
w
w
U
w
U
cos
2
cos
2
T
n
n
r
« »
ww w w
x
y
,
x
2
x
x
x
y
S
r
¬
¼
(9.27)
w
T
w
ª
w
U
w
U
º
T
n
n
r
« »
ww w w
x
y
,
y
y
y
x
y
2
S
r
¬
¼
and for three-dimensional problems
ª
º
w
T
w
w
U
w
U
w
U
cos
4
cos
4
cos
4
T
n
n
n
r
« »
ww w w w
x
y
z
,
x
3
x
x
x
y
z
S
r
¬
¼
w
T
w
ª
w
U
w
U
w
U
º
T
n
n
n
r
« »
ww w w w
(9.28)
x
y
z
,
y
y
y
x
y
z
3
S
r
¬
¼
w
T
w
ª
w
U
w
U
w
U
º
T
n
n
n
r
« »
ww w w w
x
y
z
,
z
3
z
z
x
y
z
S
r
¬
¼
The derivatives of fundamental solutions
T
for three-dimensional space have a
singularity of
1/r
2
for 2-D and
1/r
3
for 3-D problems and
are therefore
hypersingular
. We
now extend the Laplace_lib to include the derivatives of the fundamental solution.
FUNCTION
dU(r,dxr,Cdim)
!-------------------------------
!
Derivatives of Fundamental solution for Potential problems
!
Temperature/Potential
!------------------------------
REAL,INTENT(IN)::
r !
Distance between source and field point
REAL,INTENT(IN)::
dxr(:)!
Distances in x,y directions div. by r
REAL ::
dU(UBOUND(dxr,1)) ! dU is array of same dim as dxr
INTEGER ,INTENT(IN)::
Cdim ! Cartesian dimension (2-D,3-D)
REAL ::
C
SELECT CASE
(CDIM)
CASE
(2) ! Two-dimensional solution
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