7.4 Finite Element Procedure

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Similar to the discretization procedure for the displacements in Section 2.1, the temperatures

are interpolated within an element between the nodal values by shape functions

T (ξ,η, ζ, t) =

N

_

i=1

ϕi(ξ, η, ζ)Ti (t ). (7.29)

The time derivative yields

˙ T (ξ,η, ζ, t) =

N

_

i=1

ϕi(ξ, η, ζ) ˙ Ti (t) (7.30)

and similarly

δT (ξ, η, ζ, t) =

N

_

i=1

ϕi(ξ, η, ζ )δTi (t ). (7.31)

Substituting these expressions into Equation (7.27) and breaking down the volume integration

on the element level yields

_

e

N

_

i=1

N

_

j=1

__

Ve

ρc(T )ϕjϕi dve

_ ˙ TjδTi +_

e

N

_

i=1

N

_

j=1

__

Ve

κkl(T )ϕj,lϕi,k dve

_

TjδTi

= −_

e

N

_

i=1

__

Ae

h(T )(T − Te)ϕi dae

_

δTi −_

e

N

_

i=1

__

Ae

A(T )(θ 4 − θ4

e )ϕi dae

_

δTi

−_

e

N

_

i=1

__

Ae

qϕi dae

_

δTi +_

e

N

_

i=1

__

Ve

_ρh + σ : ˙_p + qi : ˙α − β : ˙_eθ_ ϕi dve

_

δTi .

(7.32)

Defining for each element a vector containing the nodal temperatures

_T _e :=



 T1

T2

...

TN





(7.33)

Equation (7.32) can be written as

_

e

δ _T _T

e _C_e

D

Dt

_T _e

+_

e

δ _T _T

e _K_e _T _e

=_

e

δ _T _T

e _Q_e (7.34)

where

_C_eij

= _

Ve

ρc(T )ϕiϕj dve (7.35)

_K_eij

= _

Ve

κkl(T )ϕi,kϕj,l dve (7.36)

310 HEAT TRANSFER

_Q_ei

= −_

Ae

h(T )(T − Te)ϕi dae − _

Ae

A(T )(θ 4 − θ4

e )ϕi dae − _

Ae

qϕi dae

+ _

Ve

_ρh + σ : ˙_ p + qi : ˙α − β : ˙_ eθ_ ϕidve. (7.37)

_C_e is the element capacity matrix and _K_e is the element conduction matrix. Both are

symmetric matrices (κkl is a symmetric tensor). Defining the localization matrix _L_e that

localizes element “e” within the structure by

_T _e

= _L_e _T _ (7.38)

where _T _ contains the temperatures of all nodes, Equation (7.34) now reads

δ _T _T _C_

D

Dt

_T _ + δ _T _T _K_ _T _ = δ _T _T _Q_ (7.39)

where

_C_ =_

e

_L_T

e _C_e _L_e (7.40)

_K_ =_

e

_L_T

e _K_e _L_e (7.41)

_Q_ =_

e

_L_T

e _Q_e . (7.42)

Since Equation (7.39) must apply for any δ _T _T, one finally arrives at the following

governing set of finite element equations:

_C_

D

Dt

_T _ + _K_ _T _ = _Q_ . (7.43)

Although Equation (7.43) looks linear in the temperature, it is not linear at all. Indeed,

both _C_ and _K_ are a function of the temperature, since the capacity and conduction

coefficients are temperature-dependent. Furthermore, the driving flux _Q_ (units of power)

is highly nonlinear because of the radiation terms.

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