Civil Engineering Reference
In-Depth Information
FIGURE 6.12
Schematic representation of a pseudo steady state temperature field illustrating the equivalence
between a given finite distribution of subdomains defined by constrained surface values of temperature and an energy
source defined by a given power and spatial distribution.
is an equal weight of the summed quantities by all the boundaries defining the subdomain. Discretization
errors resulting from stiffness over regions near the upstream boundaries are reduced because of a
propagation of information concerning the gradient from downstream boundaries. Second, a partitioning
of the solution domain via a given set of distributed subdomains is equivalent to specifying an energy
source of a given power and spatial distribution. This equivalence is illustrated in
Fig. 6.12
.
Referring to
this figure, we consider a partitioning of the region defined by the pseudo steady state of the weld into
two regions, one which is upstream and defined mostly by the spatial distribution of the energy source
and another which is downstream and defined mostly by the total volume of the weld but excluding that
part occupied by the energy source. Shown in the downstream section of
Fig. 6.12
is a representation of
a closed and finite sized subregion which is defined by an upstream boundary and a set of isothermal
surfaces located downstream. The temperature field within this subregion is assumed to have been
generated by two point sources of energy located on the surface of the workpiece. The isothermal surfaces
partition the solution domain into three subdomains. The equivalence between the energy sources and
domain partitioning is illustrated by noting that for a given domain partitioning defined by an upstream
boundary upon which there is a given temperature distribution and two isothermal surfaces of given
temperatures and relative spatial extent, one can find two point energy sources whose relative separation
and magnitude are such that they generate the given set of subdomains. By superposition, it follows that
this equivalence can be extended to the more general case of volumetric deposition of energy and of
temperature dependent material properties. A general restriction on domain partitioning, however,
follows from this equivalence. It is possible to adopt a specific partitioning of the solution domain that
does not correspond to a physically realistic energy source. Considering this, it then follows that in the
absence of any constraints based on experimental information concerning the three-dimensional char-
acter of the temperature field (e.g., the case considered here) the generating function
T
E
(
x
) must be
sufficiently accurate for generating a partitioning of the solution domain which is physically realistic. It
should be noted that this does not represent an inconvenient restriction in practice on the construction
of
T
E
(
x
).
