1d finite element ii
TRANSCRIPT
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Finite Element Method:Finite Element Method:OneOne--Dimensional HeatDimensional Heat
Conduction IIConduction II
Engineering 471Engineering 471Heat TransferHeat Transfer
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Boundary ConditionsBoundary Conditions
x=0
x=L
x
( )
( ) 0TThnix
Tk
T0T L
=+
=
or
( )
( ) 0TThnix
Tk
TLT R
=+
=
orin
n
Each end can have either a
specified temperature or
convection boundary condition.
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IntegrationIntegration
( )
( ) ( )( ) ( ) ( )( ) 0TL!hAL!T0!hA0!
dxT!!hPdx
x
!KA
x
!dxA!qdx
t
!!CA
L
0
L
0
L
0
gen
L
0
=+
+
Governing Integral Equation
Nodes
Elements"2
3i
j
"2
k
( ) ( )dxxFdxxF
elem
x
x
L
0
j
i
=
x
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Element GeometryElement Geometry
"2
3i
j
"
2
k
( ) ( )
( ) ( )dx#fdxxF
xx#
dxxFdxxF
elem0
elem
x
x
i
elem
x
x
L
0
j
i
j
i
=
=
=
x
ij
#
k
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Interpolation FunctionsInterpolation Functions( ) ( ) ( )
( )
( )
##N
#
"#N
!#N!#N#!
j
i
jjii
=
=
+=
( )#N
#
".0
( )#Ni ( )#Nj
( ) ( )
known.arepointsnode
at the!ofvaluestheif
#anyat!ofvaluethefindtousedbecanThey
functions.
ioninterpolatcalled
are#Nand#N ji
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Thermal Conductivity IntegralThermal Conductivity Integral(Continued)(Continued)
d#!!""
"
"
!!KAd##!
#!KA
j
i
0
ji
0
=
( ) ( ) ( )
( )
( )
##N
#"#N
!#N!#N#!
j
i
jjii
=
=
+=
=
+=
j
i
ji
!
!""
!"
!"
#
!
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Thermal Conductivity IntegralThermal Conductivity Integral(Continued)(Continued)
=
=
j
i
ji
j
i
022
22
ji
j
i
0
ji
!
!
""
""
!!KA
!
!""
""
!!KA
d#!
!""
"
"
!!KA
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Convection IntegralConvection Integral
( )
( ) ( )( ) ( ) ( )( ) 0TL!hAL!T0!hA0!
dxT!!hPdxx
!KAx
!dxA!qdxt
!!CA
L
0
L
0
L
0
gen
L
0
=+
+
( ) ( )
=
=
elem 0elem 0
elem 0
L
0
d#T!hP!d#!hP
d#T!!hPdxT!!hP
First Integral Second Integral
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Convection IntegralConvection Integral(First Integral)(First Integral)
( ) ( ) ( )
( )
( )
( )
=
=
=
+=
j
i
j
i
jjii
!
!##
"#!
#
#N
#"#N
!#N!#N#!
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Convection IntegralConvection Integral(First Integral)(First Integral)
d#
!
!
###"
##"
#"
!!hP
d#!!##-"
#
#-"
!!hP!d#!hP
j
i
2
2
0
ji
j
i
0
ji
0
=
=
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Convection IntegralConvection Integral(First Integral)(First Integral)
= ji
ji
0 !
!
36
63!!hP!d#!hP
Second Integral
=
2
2!!hPTd#T!hP ji0
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Heat Capacity IntegralHeat Capacity Integral
( )
( ) ( )( ) ( ) ( )( ) 0TL!hAL!T0!hA0!
dxT!!hPdx
x
!KA
x
!dxA!qdx
t
!!CA
L
0
L
0
L
0
gen
L
0
=+
+
=
j
i
elem
ji
L
0 !
!
36
63!!CAdxt!!CA
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Internal Heat Generation IntegralInternal Heat Generation Integral
( )
( ) ( )( ) ( ) ( )( ) 0TL!hAL!T0!hA0!
dxT!!hPdx
x
!KA
x
!dxA!qdx
t
!!CA
L
0
L
0
L
0
gen
L
0
=+
+
=elems
jigen
elems 0
gen
2
2!!Aqd#Aq!