Civil Engineering Reference
In-Depth Information
upstream boundary of the solution domain, the isothermal surface at
T
M
, a section of the midplane
boundary and the isothermal surface at 650°C. Subdomain 3 in
Fig. 6.10
i
s defined by a closed boundary
consisting of a section of the upstream boundary of the solution domain, the isothermal surface at 650°C,
a section of the midplane boundary, the isothermal surface at 350°C, and a section of the downstream
boundary of the solution domain. Subdomain 4 in
Fig. 6.10
is defined by a closed boundary consisting
of a section of the upstream boundary, the isothermal surface at 350°C, and various portions of the three
downstream boundaries of the solution domain.
The third stage of this analysis entails the specification of a distribution of temperatures on the
upstream boundary of the solution domain. This procedure is undertaken by means of a generating
function
T
B
(
x
) which is defined, as for the function
T
E
(
x
), by Eq. (6.11). For a specific partitioning of
the solution domain into a given set of subdomains, the objective function Eq. (6.32) is then minimized
over the solution domain for a given distribution of temperatures on its upstream boundary. Shown in
Fig. 6.11
are two-dimensional slices of a temperature field whose discrete values minimize the function
Eq. (6.32) for the partitioning of the solution domain shown in
Fig. 6.10
and a given distribution values
on its upstream boundary.
There are features of the second and third stages of our analysis and of this particular calculation that
should be noted. First, we have restricted our minimization of Eq. (6.32) only with respect to the subdo-
mains 1, 2, and 3 in
Fig. 6.10
.
For this calculation the generating function
T
E
(
x
) used to generate the
boundary at
T
M
provides, in addition, a sufficiently good estimate of the temperature field at downstream
regions of the pseudo-steady state (i.e.,
T
below 350°C) that the values of this function may themselves
be adopted as the solution within subregion 4 of
Fig. 6.10
.
That is to say, for this calculation we have in
fact adopted a hybrid procedure in which the method of distributed heat sources is used to calculate the
temperature field for
T
above 350 °C and the method of constraints is used for temperatures below 350°C.
Second, by partitioning the solution domain into four subdomains, we have provided, in principle, a
means for the inclusion into our model system of information corresponding to experimental measure-
ments taken at three separate regions of the dynamic weld. An example of this type of information is
shown in
Fig. 6.8
,
where it is assumed that (in addition to information concerning the solidification
boundary) there is information related to the
Ac
1
and
Ac
3
boundaries. Another example of this type of
information is that which would be obtained from thermocouple measurements. For the calculated
temperature field shown in
Fig. 6.11
,
the subdomain boundaries at temperatures other than
T
M
would
then be considered as representing constraints corresponding to specific
a priori
information concerning
the distribution of temperatures in the weld. Lastly, the partitioning of the solution domain into a relatively
distributed set of subdomains represents an aspect of our procedure which has significance beyond that
of providing a means for the inclusion of
a priori
information about weld temperatures. This stage of our
procedure, which may appear somewhat artificial if used without the inclusion
a priori
information,
actually provides a means for enhancing the accuracy of the calculated temperature field by helping to
make the objective function better conditioned. The underlying mathematical properties providing the
basis for this improved conditioning of the objective function are relatively subtle and are discussed below.
The final stage of this analysis entails in principle optimization of the temperature field with respect
to energy input. This stage is accomplished by repetition of the third stage of our analysis (described
above) according to adjustments of the distribution of upstream boundary values on the solution domain.
It should be noted that without relatively detailed information concerning the energy source, a full
optimization with respect to energy input is not in general well posed. That is to say, adjustment of
boundary values such that the objective function Eq. (6.32) obtains its minimum value with respect to
energy input may not be relevant. It is reasonable to obtain, however, a partial optimization where a
discrete temperature field is sufficiently optimum that the value of the objective function is a reasonably
close estimate of the minimum with respect to the energy input for a given set of constraints. It should
be noted further, that a discrete temperature field which is optimized under a sufficiently large set of
constraints will tend implicitly toward being optimum with respect to the energy input.
Before proceeding, we examine those features of our method that are related to the partitioning the
solution domain into a distributed set of subdomains. First, a distributed partitioning of the solution
