Civil Engineering Reference
In-Depth Information
is satisfied because constrained values of the field quantity
T
E
represent an implicit inclusion of infor-
mation concerning the temperature dependence of the material properties and of the motion of the
workpiece relative to the energy source. The representation defined by Eq. (6.35) is the discrete analog
of the Laplace equation and implies that within each subdomain the temperature field is determined via
Eq. (6.33).
Referring to
Fig. 6.20
,
it follows that by linear interpolation in three-dimensions, the discrete temper-
ature field
T
p
is given by
6
T
p
W
Ei
T
Ei
,
(6.36)
,
,
i
1
where
6
1
1
1
W
E
,
i
----------
------------
(6.37)
l
E
,
i
l
Em
m
1
,
and the discrete quantities
6
W
E
,
i
1
and
l
E
,
i
l
(6.38)
i
1
where the has been partitioned into a set of discrete volume elements of volume (
l
)
3
. The quantities
1, …, 6) are the distances between a given discrete location within a subdomain (indexed by
the integer variable
p)
and its six discrete nearest neighbor locations (indexed by the integer variable
i
)
which are on the closed surface of the subdomain.
Equation (6.36) represents a weighted average of the temperature field
T
E
over the closed surface of a
subdomain. It is significant to note that the general form of the sum defined by Eq. (6.36) implies that
Eq. (6.35) is an appropriate representation of the discrete temperature field within a closed subdomain.
A semi-quantitative proof showing that the existence of the sum defined by Eq. (6.36) implies that the
equality represented by Eq. (6.35) is satisfied within a closed subdomain is given in Appendix D. Our
proof, which is based on Green's theorem,
8
is in terms of discrete analogs of mathematical representations
based on continuous functions and operators. A qualitative and intuitive understanding of the equality
expressed by Eq. (6.35) follows from the observation that the right side of this equality, applied iteratively
over a closed domain, is simply an interpolation among the surface values
T
E
that bound that domain.
l
E
,
i
(
i
6.8
Implicit Representation of Coupling Between Heat
Transfer and Fluid Convection
In this section we discuss a specific property of constraints which provides a means for including
information concerning the coupling between heat transfer and fluid convection. This property is related
to that examined in the previous section and provides a basis for posing the weld thermal-history problem
in terms of the solution of an elliptic equation (i.e., the Laplace equation Eq. (6.33)) defined over a closed
domain. The mathematical foundation for posing the problem as a solution of Eq. (6.33) over a distrib-
uted set of subdomains, and consistently representing the coupling between heat transfer and fluid
convection, follows from the formal structure of the discretization procedure employed by the SIMPLE
algorithm.
9
With respect to the structure of this discretization procedure, we first consider the process
of calculating a discrete temperature field
T
p
over an infinite solution domain for a given set of upstream
boundary conditions and a specified flow field U representing the flow of liquid in the weld meltpool.
