Civil Engineering Reference
In-Depth Information
FIGURE 6.10
Cross-section of temperature field showing an example of partitioning of solution domain into
subdomains for the purpose of embedding information concerning a weld.
of this particular calculation that should be noted. First, for the region of the solution domain close to
the upstream boundary and greater than
T
M
, the calculated temperature field does not represent any type
of reasonable estimate, or for that matter, any physically realistic temperature distribution. For the
calculations shown in
Fig. 6.9
,
the primary condition on the function
T
E
(
x
) is that of generating an
isothermal surface at
T
M
that is consistent with
Fig. 6.7
.
Second, for this calculation we have adopted a
generating function which is symmetric with respect to the midplane boundary and which consists of
a sufficiently large number of terms in each of the Rosenthal-type solutions comprising
T
E
(
x
) that a relatively
high level of accuracy is maintained. In addition, we have used
T
E
(
x
) to generate field values over the
entire solution domain. As a result, the values generated using
T
E
(
x
) should represent a realistic estimate
of the temperature field for regions of the solution domain that are not close to the upstream boundary.
This level of accuracy, however, is not necessary at this stage of the procedure. The level of computation
associated with generating the potentially constrained surfaces can be reduced by using a generating
function which is not necessarily symmetric with respect to the midplane of the weld, does not necessarily
represent a fully converged Rosenthal-type solution and which is only used to generate temperature
values in regions of the solution domain that are either on or close to those surfaces to be constrained.
It is significant to note that any approximations or unrealistic distributions of temperature that are
generated at this stage (within the solution domain) are removed by optimization of the objective function
Eq. (6.32) for a given distribution of temperatures on the upstream boundary. And thirdly, the region
corresponding to the solution domain does not include the entire pseudo steady state of the weld.
Therefore we are not, for the present case, considering a full optimization of the objective function with
respect to input energy. The relative significance of achieving a relatively full optimization of
Z
with
respect to
q
0
/
V
B
is discussed below and in the appendix.
The second stage of our analysis consists of partitioning the solution domain into subdomains. It is
this stage of our procedure which provides the basis for embedding known information about the weld
into the solution domain. An example of this partitioning is shown in
Fig. 6.10
,
where the solution
domain is partitioned into four subregions, each defined by a given set of upstream and downstream
boundaries. Subdomain 1 in
Fig. 6.10
is defined by a closed boundary consisting of a section of the
upstream boundary of the solution domain, a section of the midplane boundary and the isothermal
surface at
T
M
. Subdomain 2 in
Fig. 6.10
is defined by a closed boundary consisting of a section of the
