As stated before, the variable T was allowed to vary in both space and time. This dependency is separated as:
(6–16) |
where:
T = T(x,y,z,t) = temperature |
{N} = {N(x,y,z)} = element shape functions |
{Te} = {Te(t)} = nodal temperature vector of element |
Thus, the time derivatives of Equation 6–16 may be written as:
(6–17) |
δT has the same form as T:
(6–18) |
The combination {L}T is written as
(6–19) |
where:
[B] = {L}{N}T |
Now, the variational statement of Equation 6–11 can be combined with Equation 6–16 thru Equation 6–19 to yield:
(6–20) |
Terms are defined in Heat Flow Fundamentals. ρ is assumed to remain constant
over the volume of the element. On the other hand, c and may vary over the element. Finally, {Te},
, and {δTe} are nodal quantities and do not vary
over the element, so that they also may be removed from the integral. Now, since all quantities
are seen to be premultiplied by the arbitrary vector {δTe}, this term
may be dropped from the resulting equation. Thus, Equation 6–20 may be
reduced to:
(6–21) |
Equation 6–21 may be rewritten as:
(6–22) |
where:
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Comments on and modifications of the above definitions:
is not symmetric.
is calculated
as defined above, for SOLID90 only. All
other elements use a diagonal matrix, with the diagonal terms defined
by the vector
.
is frequently
diagonalized, as described in Lumped Matrices.
If
exists and
has been diagonalized and also the analysis is a transient (Key =
ON on the TIMINT command),
has its terms adjusted so that they are proportioned
to the main diagonal terms of
.
, the heat generation rate
vector for Joule heating is treated similarly, if present. This adjustment
ensures that elements subjected to uniform heating will have a uniform
temperature rise. However, this adjustment also changes nonuniform
input of heat generation to an average value over the element.
For phase change problems,
is evaluated from the enthalpy curve (Tamma
and Namnuru([42])) if enthalpy is input
(input as ENTH on MP command). This option should
be used for phase change problems.