Information Technology Reference
In-Depth Information
These results are consistent with those obtained in Chapter 3, Example 3.13. Substitution
of the boundary values of (3) into Eq. (9.9) leads to an expression for the deflection along
the beam. For example, at
ξ
=
L
/
2
,
00234375
p
0
L
4
EI
w(
L
/
2
)
=
0
.
(4)
which is the exact solution.
9.2
Poisson's and Laplace's Equations
In the previous section, the basic relationship (Eq. 9.8) for the boundary element formu-
lation is the expression of the displacement at any point along the beam in terms of the
displacement, slope, moment, and shear force at the boundaries of the beam. In the case of
two- and three-dimensional problems, the relationship for the boundary element formula-
tion is similar to the one for beams in that the unknown variables are expressed in terms
of the unknown variables and other quantities on the boundary. This leads to boundary
integral equations.
In this section, an integral equation formulation and boundary element solution for field
theory problems represented by the Poisson's and Laplace's equations will be derived.
9.2.1 Direct Formulation
Poisson's equation has the form
2
u
∇
=
b
inside the domain
V
and
u
=
u
on
S
u
(9.13)
q
on
S
q
where
b
is the nonhomogeneous term of the Poisson's equation,
S
u
and
S
q
are the parts of the
boundary of
V
on which
u
and
q
q
=
=
∂
/∂
n,
the derivative of
u
with respect to the outer nor-
mal of the boundary, are prescribed, respectively. Also
u
3
∂
2
2
∇
=
for three-dimensional problems
x
i
∂
i
=
1
and
2
2
∂
2
∇
=
for two-dimensional problems
x
i
∂
i
=
1
By definition,
S
Use the extended Galerkin
method as the basis for a boundary element formulation. Write Eq. (9.13) in the equivalent
integral form
∗
=
S
u
+
S
q
.
For Laplace's equation, set
b
=
0
.
S
q
δ
S
u
δ
∂
u
2
u
V
δ
u
(
∇
−
b
)
dV
+
n
(
u
−
u
)
dS
−
u
(
q
−
q
)
dS
=
0
(9.14)
∂
∗
Refer to the procedures of Chapter 2, Section 2.2 for establishing global integral equations from the local, differ-
ential equations of static admissibility.
Search WWH ::
Custom Search