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