a new original uncoditionaly stable mixed finite element approach in transient heat analysis without...
TRANSCRIPT
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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 [email protected]
Seminar for Rheology, 15 Mart, 2005
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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
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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
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Heat Transfer Modes
• Conduction
• Convection
• Radiation
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Conduction
T q k
( , ( , ))T t r rk k
Fourrier’s Law (1822.)
k - Thermal Conductivity
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Thermal Conductivities
• Wood 0.05• Water 0.7• Glass 0.8• Steel10-20• Iron 80• Copper 400• Silver 450
k [W/mK] (Room Temperature)
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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
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Convective Coefficient depends on:
• Temperature Difference;• Fluid;• Fluid Speed;• Geometry of the Surface;
• Roughness of the Surface.
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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
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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
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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
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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
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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
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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
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Finite difference time discretization
1n n
n
T T Tc c
t t
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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
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Numerical Examples
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A Ceramic Strip Model Problem
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EA Ceramic Strip Model Problem
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A Ceramic Strip Model Problem
animacija_straus_vth2.htm
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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
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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
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Transient Temperature Distribution in an Orthotropic Metal Bar
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1
2
3
4
Transient Temperature Distribution in an Orthotropic Metal Bar
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animacija_ansys_vm113.htm
Transient Temperature Distribution in an Orthotropic Metal Bar
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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
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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
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Steel Ball Numerical Example
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Steel Ball Numerical Example
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Steel Ball Numerical Example
First iteration t=250 Last iteration t=5819
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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
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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
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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
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Nuclear Reactor – Straus7 Non averaged Results, t=62000s
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Nuclear Reactor – Straus7 Results
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Nuclear Reactor – Present Results
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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
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ADENDUM
Time Integration Schemes
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PRIMAL FORMULATIONS
CT KT R 0
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Explicit and implicit schemes
• Explicit scheme: • Fully implicit scheme:• Crank-Nicholson scheme:• Galerkin scheme:
1 1/ 2
0
2 / 3