Civil Engineering Reference
In-Depth Information
(c)
FIGURE 6.13
(continued.)
Prototype case study 2. Optimization of discrete temperature field based on multiple con-
straints
—For the present case study we consider a prototype analysis of the weld whose cross section is
shown in
Fig. 6.8
and whose process parameters are given in data set 2 of Appendix A. For this analysis
we adopt three separate measurements as constraints. These are the transverse cross section of the
solidification boundary, the solidification boundary at the top surface, and the transverse cross section
of the
Ac
1
isotherm. For this analysis we adopt the same optimization procedure applied in the case study
presented above.
The first stage of our analysis for this case entails the generation of a three-dimensional isothermal surface
at
T
M
whose transverse projection maps onto the experimentally measured transverse cross section of the
solidification boundary and whose intersection with the top surface of the workpiece assumes the shape of
the experimentally measured solidification boundary at the top surface. In addition, we generate a three-
dimensional isothermal surface whose transverse projection maps onto the experimentally measured
Ac
1
cross section.
Shown in
Fig. 6.13a
a
re two-dimensional cross sections of a temperature field, calculated via a gener-
ating function
T
E
(
x),
for which the isothermal surfaces at temperatures
T
M
and
Ac
1
are consistent with
Fig, 6.8
.
The assumed values of
T
M
and
Ac
1
are given in appendix A. A feature of this calculation that
should be noted is that we have adopted a generating function which does not represent any type of
reasonable estimate of the temperature field. For this calculation we have arbitrarily adopted a generating
function which is not symmetric with respect to the midplane boundary.
The next stage of this analysis consists of partitioning the solution domain into subdomains. A
temperature field obtained via a function
T
E
(
x
) which satisfies this partitioning is shown in
Fig. 6.13b
.
It is significant to note that for this calculation two of the constrained boundaries are used for embedding
known information about the weld, in contrast to the previous analysis, where only one boundary served
this purpose.
In the next stage of this analysis the objective function Eq. (6.32) is minimized with respect to adjustment
of the distribution of temperatures on the upstream boundary. Shown in
Fig. 6.14
are values of the
temperature field which have been calculated by minimizing the objection function Eq. (6.32) for a
specific distribution of temperatures on the upstream boundary and the subdomain partitioning defined
by the temperature field shown in
Figure 6.13
.
Again, as in the analysis presented in the previous section,
