Page 165 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

TRANSIENT HEAT CONDUCTION ANALYSIS
If the second- and higher-order terms in the above equation are neglected, then
∂T n T n+1 − T n 157
≈ + O( t) (6.33)
∂t t
which is first-order accurate in time. If we now introduce a parameter θ such that
T n+θ = θT n+1 + (1 − θ)T n (6.34)
into Equation 6.16 then, along with Equation 6.33, we have
n+1 n
T − T n+θ n+θ
[C] + [K]{T} ={f} (6.35)
t
or
n+1 n
T − T n+1 n n+1 n
[C] + [K] θT + (1 − θ)T = θ{f} + (1 − θ){f} (6.36)
t
The above equation can be rearranged as follows:

n+1 n n+1 n
([C] + θ t[K]) {T} = ([C] − (1 − θ) t[K]) {T} + t θ{f} + (1 − θ){f}
(6.37)
Equation 6.37 gives the nodal values of temperature at the n + 1 time level. These
temperature values are calculated using the n time level values. However, both the n + 1
and n time level values of the forcing vector {f} must be known. By varying the parameter
θ, different transient schemes can be constructed, which are shown in Table 6.1 for varying
values of θ.
In the following numerical example, we demonstrate how the Crank–Nicolson time-
stepping scheme can be used to solve a one-dimensional transient problem.

Example 6.4.1 In Example 3.5.1, let us assume that the initial temperature of the fin is
◦
equal to the atmospheric temperature, 25 C. If the base temperature is suddenly raised to a
temperature of 100 C, and maintained at that value, determine the temperature distribution
◦
6
in the fin with respect to time. Assume a heat capacity of 2.42 × 10 W/m C.
3◦
Let us assume that the problem is to be solved using the Crank–Nicolson method, in
which θ is equal to 0.5. Assume a time step, t, of 0.1 s. Equation 6.37 can be rewritten
with the given value for θ and t as
n+1 n
([C] + 0.5 × 0.1[K]){T} = ([C] − 0.5 × 0.1[K]){T} + 0.1{f} (6.38)

Table 6.1 Different time-stepping schemes

θ Name of the scheme Comments
0.0 Fully explicit scheme Forward difference method
1.0 Fully implicit scheme Backward difference method
0.5 Semi-implicit scheme Crank–Nicolson method