Civil Engineering Reference
In-Depth Information
Schematic showing different parts of pseudo steady-state of weld and their significance with respect
to the calculation of temperature fields using geometric constraints.
FIGURE 6.1
that of an elliptic equation where the state of the weld is represented by the Laplace equation (or Poisson
equation in general) over a closed domain.
Given a representation of the pseudo-steady-state temperature field via a parabolic equation, specifi-
cation of the two-dimension temperature distribution on the upstream boundary and top surface bound-
ary (over the downstream region) is equivalent to specification of the three-dimensional distribution for
deposition of energy into the material resulting from an energy source located upstream (above the
upstream boundary). This equivalence is described schematically in
Fig. 6.1
.
Referring to
Fig. 6.1
,
it
follows from the conservation of energy that the heat input per unit distance along the direction of
relative motion between the energy source and workpiece provides a global constraint on the three-
dimension temperature distribution extending over the downstream portion of the pseudo steady state
of the weld. It is to be noted, however, that this constraint cannot be adopted as a strong condition on
the temperature field in cases where the efficiency for the coupling of energy into the workpiece is not
known with sufficient accuracy. In any case, however, this constraint does provide a bound condition on
the system. The significance of this bound condition in terms of the theory of constrained optimization
is discussed further below.
The weighted finite-sum defined by Eq. (6.4) provides a means for adopting a more general approach
for inclusion of information concerning a given weld process. We have designated this approach the
method of geometric constraints.
This method follows from an extension of the definition of the
quantities and (defined in Eq. (6.4)) and from the mathematical properties of a closed and
bounded solution domain over which the pseudo steady state temperature field of a weld is represented
by an elliptic equation. Accordingly, by adopting an interpretation based on constrained optimization
theory, the quantities and defined in Eq. (6.4) are assumed to be contained in a set of quantities
such that each provides for the input of a specific type of information. That is to say,
1,2
T
i
()
T
p
()
T
i
n
(
)
T
p
()
T
i
()
and
T
p
()
{
TT
B
T
C
T
E
T
,
,
,
,
T
S
}
(6.8)
where
is represents those discrete temperature-field quantities determined according to the procedure
defined by Eq. (6.4) whose values vary with each iteration until they are optimized subject to the constrained
or specified values of the discrete temperature-field quantities
T
T
,
T
,
T
, and
T
. The quantity
T
B
C
E
S
B
