Civil Engineering Reference
In-Depth Information
—
An important aspect of the weighted finite-sum representation, e.g.,
Eq. (6.17), is that the discrete field quantities
T
i
and
Duality between
T
and
i
have an equivalent status formally. That is to
say, minimization of the objective function can be effected with respect to the
field as well as the
T
field.
This property provides the basis for further generalization of the method of geometric constraints. First,
there can be cases where information concerning the thermal diffusivity
(
T
) may only be known
partially. In these cases, for those regions of the solution domain where information concerning
(
T
) is
i
can also be adopted as optimization
parameters. The requirement is that there exists a sufficient amount of geometric constraint information
for assuring a unique minimum of the objective function with respect to both
T
i
, spanning the entire
solution domain, and the set of quantities
either unavailable or approximate, the discrete field quantities
i
whose values are not known
a priori
. Second, there can be
cases where the available geometric constraint information may be such that the problem is over con-
strained. That is to say, the number of equality constraints of the type Eq. (6.21) and Eq. (6.22) exceeds
the number of discrete field quantities whose values are to be optimized. In these cases, an increase in
the number of discrete field quantities can be effected by adopting the quantities
i
as well as
T
i
for
optimization. It follows then that the duality between
T
and
with respect to the weighted finite-sum
representation provides a means for increasing the amount of information which can be put into a model
system via constraints.
Extending the range of system parameters—
An objective function such as Eq. (6.18) can be easily
adapted to include additional information concerning a weld which may be associated with regions that
are outside a specified solution domain. This would include detailed information concerning the heat
source and would in principle represent constraints on the upstream boundary values of the solution
domain. In this context, it is significant to compare minimization of the objective function defined by
Eq. (6.14) to the method of distributed heat sources. Stated formally in terms of a parameter optimization
problem, the method of distributed heat sources represents minimization of the objective function
N
Z
q
o
q
i
(6.26)
i
1
subject to the type of constraint that is defined by Eq. (6.19) and the assumption that the value of
q
o
is known. The quantities
q
o
and
q
i
(
i
1 ,…,
N
) are the total power input and individual power inputs
associated with a finite distribution of sources, respectively. It follows that inclusion of any information
concerning details of the energy source, i.e., its power and distribution, can be accomplished by adopting
constraints on the distribution of temperatures above and including the upstream boundary of the
solution domain. It is to be noted that optimization of the objective function Eq. (6.18) with the
condition
Z
BD
0 is equivalent to the iterative solution of Eq. (6.4) for a given set of upstream boundary
conditions. For the prototype analyses presented below, a restricted form of Eq. (6.18) is adopted which
does not adopt in terms of constraints detailed characteristics of the energy source.
A reformulation of the problem (a strong condition for uniqueness)
—Adopting the conditions for
a pseudo steady state, i.e., Eq. (6.16) and Eq. (6.24), the weighted finite-sum Eq. (6.4) can be expressed
in the form
6
1
T
P
----------------------------------
i
T
i
(
V
B
l
)
T
p
1
.
(6.27
)
(
6
P
V
B
l
)
i
1
Note that Eq. (6.27) can also be deduced from a Rosenthal-type solution of Eq. (6.1) (see Appendix B).
Next, we adopt the substitution
i
i
,
(6.28)
