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A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT
DIMENSIONAL REDUCTION
Dubravka Mijuca, Bojan MedjoFaculty of Mathematics, Department of Mechanics
University of Belgradedmijuca@matf.bg.ac.yu
Seminar for Rheology, 15 Mart, 2005
Reference
• The Finite Element Method - Volume 1: The Basis; O.C. Zienkiewicz, R.L. Taylor
• Finite Element Procedures; K. J. Bathe
• On hexahedral finite element HC8/27 in elasticity, Mijuca D.
• Mijuca D, Žiberna A, Medjo B (2005) A new multifield finite element method in steady state heat analysis, Thermal Science, in press
• Cannarozzi AA, Ubertini F (2001) A mixed variational method for linear coupled thermoelastic analysis. International Journal of Solids and Structures. 38: 717-739
• LUSAS Theory Manual 1, Version 13
• STRAUS 7 Verification Manual
• ANSYS Verification Manual
1st Law of Thermodynamics
Tc f
t
q
00tT T
Initial condition:
Boundary conditions:
4 4
( ) on
( ) on
T
h h
c c a c
r r a r
T h c
T T na
q h na
q h T T
q h A T T
q n
q n
q n
Heat Transfer Modes
• Conduction
• Convection
• Radiation
Conduction
T q k
( , ( , ))T t r rk k
Fourrier’s Law (1822.)
k - Thermal Conductivity
Thermal Conductivities
• Wood 0.05• Water 0.7• Glass 0.8• Steel10-20• Iron 80• Copper 400• Silver 450
k [W/mK] (Room Temperature)
Convection
• Convection involves the exchange of Heat between a Fluid and a Surface
Natural Convection
Forced Convection
0( )c cq h T T 1701 – Newton’s “Cooling Law” 1701 – Newton’s “Cooling Law”
• T,T0 – Temperatures of the surface and the Fluid
• hC – Convective (Film) Coefficient
Convective Coefficient depends on:
• Temperature Difference;• Fluid;• Fluid Speed;• Geometry of the Surface;
• Roughness of the Surface.
Radiation
• Consequence of the Stefan-Boltzmann’s Law:
1 2rh F 4 40( )rh T T q n
T - Temperature at the Surface of the Body
T0 - Temperature of the Environment or the other Body
F1-2 - Shape Factor
- Stefan-Boltzmann Constant
- Emissivity of the Surface of the Body
0T
Galerkin Approximation Of The Energy Balance Equation
div / dT
f ct
q
The next identity holds: div =div
Divergence theorem: div
We finaly obtain: div
d d
d d d
q q q
q q n
q q n q
( div ) 0T
c f dt
q
Galerkin Approximation Of The Energy Balance Equation
div 0
0
Tc d d f d
t
Tc d d d f d
t
q
q n q
(1) 0T
c d d d f dt
q n q
Galerkin Approximation of the Fourrier’s Law:
1(2) ( ) 0T d
q Q Qk
1
1
1
0
( ) 0
T
T
T
T d
q
q
q
q Q
k
k
k
k
Symmetric Weak Mixed Formulation
1
q c
c
Tc d d T d
t
d f d hd q d
q Q Q
q
k
12Find , ( ) ( ) such that
TT H L T T
q
12For all , ( ) ( ) such that 0
TH L
Q
Finite Element Approximation Function Spaces that Enables Continuity
_1
1
10
1
( ) : | , ( ),
( ) : | 0, ( ),
( ) : | , | ( ), ( ),
( ) : | 0, ( ),
T
T
q c
q c
Lh L i i h
Mh M i i h
Lh c L i i h
Mh M i i h
T T H T T T T P C
H P C
Q H h h T T V C
H V C
q q n q n q q
Q Q n Q Q
Finite difference time discretization
1n n
n
T T Tc c
t t
Finite Element Matrix Equations
( ) ( )
( ) ,
e
e
ce
e
a b pLLpMr L p L ab M r M e
e
M a MLp L p L a e
e
LM L Mc ce
e
LM L M ee
A g V r g V d q
B g V P d
D h P P
cS P P d
t
1( )
e
he
ce
e
M Me
e
M Mhe
e
M Mc a ce
e
M n ML e
e
F P f d
H P hd
K P h T d
cL T P d
t
1 1
A B
TB D S
00 0 0A B
T 0 S F H KB D T L
Tvvv vv
vvv vv vv
Tpvp vp
t tp vp p p pvp vp p pt
q
0q
Numerical Examples
A Ceramic Strip Model Problem
EA Ceramic Strip Model Problem
A Ceramic Strip Model Problem
animacija_straus_vth2.htm
0 2 4 6 8 10 12460
480
500
520
540
560
580
T
empe
ratu
re [K
]
Time history t[s]
Model VTH2
Target value
l1HC8.15 l2HC8.15
A Ceramic Strip Model Problem
0 2 4 6 8 10 12460
480
500
520
540
560
580
Model VTH2
s1 - HC8/9 s1 - HC8/15
Tem
pera
ture
[K]
Time history t[s]
A Ceramic Strip Model Problem
Transient Temperature Distribution in an Orthotropic Metal Bar
1
2
3
4
Transient Temperature Distribution in an Orthotropic Metal Bar
animacija_ansys_vm113.htm
Transient Temperature Distribution in an Orthotropic Metal Bar
0.0 0.5 1.0 1.5 2.0 2.5 3.0
150
200
250
300
350
400
450
500
N1
N2 N3 N4
Tem
pera
ture
[F
]
Time history t[s]
Transient Temperature Distribution in an Orthotropic Metal Bar
0.0 0.5 1.0 1.5 2.0 2.5 3.0
350
400
450
500
550Model Ansys 113 - Point 2
s2 - HC8/15 s1 - HC8/15
Tem
pera
ture
[K]
Time history t[s]
Transient Temperature Distribution in an Orthotropic Metal Bar
Steel Ball Numerical Example
Steel Ball Numerical Example
Steel Ball Numerical Example
First iteration t=250 Last iteration t=5819
0 1000 2000 3000 4000 5000 6000400
450
500
550
600
650
700
750Model VTH4 (picture q1)
q1 - HC20/21 q2 - HC20/21
A
Target value
T
empe
ratu
re [K
]
Time history t[s]
Steel Ball Numerical Example
0 1000 2000 3000 4000 5000 6000400
450
500
550
600
650
700
750
Model VTH4
Target Value
s1 - HC20/21 s2 - HC20/21Tem
pera
ture
[K]
Time history t[s]
Steel Ball Numerical Example
A Cylindrical Concrete Vessel for Storing the Core of a Nuclear Reactor
• The walls of the cylinder have tubular cooling vents, which carry a cooling fluid.
• Heat flow rate through the walls over a period of 5 hours.
32400
25
1
kg
mJ
ckg K
Wk
m K
2
298
20
473
298i
a
c
r r
ini
T K
Wh
m KT K
T K
Nuclear Reactor – Straus7 Non averaged Results, t=62000s
Nuclear Reactor – Straus7 Results
Nuclear Reactor – Present Results
Conclusion
• A new robust and reliable finite element procedure for calculations of heat transient problem of a solid bodies is presented
• Approach is fully 3d thus enabling possible bridging with nano and micro analysis of regions of interest in the solid body
• Reliable semi-coupling with mechanical analysis is enabled also, which is matter of future report
ADENDUM
Time Integration Schemes
PRIMAL FORMULATIONS
CT KT R 0
Explicit and implicit schemes
• Explicit scheme: • Fully implicit scheme:• Crank-Nicholson scheme:• Galerkin scheme:
1 1/ 2
0
2 / 3
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