1. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION Dubravka Mijuca Bojan Medjo Faculty of Mathematics, University of Belgrade

2. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Literatura The Finite Element Method - Volume 1: The Basis; O.C. Zienkiewicz, R.L. Taylor Finite Element Procedures; K. J. Bathe LUSAS Theory Manual 1, Version 13 STRAUS 7 Verification Manual On hexahedral finite element HC8/27 in elasticity, Mijuca D.

3. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Heat Transfer Modes Conduction Conduction Convection Convection Radiation Radiation

4. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Conduction Fourrier’s Law (1822.) k - Thermal Conductivity

5. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Thermal Conductivities Wood0.05 Wood0.05 Water0.7 Water0.7 Glass0.8 Glass0.8 Steel10-20 Steel10-20 Iron80 Iron80 Copper400 Copper400 Silver450 Silver450 k [W/mK] (Room Temperature)

6. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Convection Convection involves the exchange of Heat between a Fluid and a Surface Convection involves the exchange of Heat between a Fluid and a Surface Natural Convection Forced Convection 1701 – Newton’s “Cooling Law” T,T 0 – Temperatures of the surface and the Fluid T,T 0 – Temperatures of the surface and the Fluid h C – Convective (Film) Coefficient h C – Convective (Film) Coefficient

7. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Convective Coefficient depends on: Temperature Difference; Temperature Difference; Fluid; Fluid; Fluid Speed; Fluid Speed; Geometry of the Surface; Geometry of the Surface; Roughness of the Surface. Roughness of the Surface.

8. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Consequence of the Stefan-Boltzmann’s Law: T – Temperature at the Surface of the Body T – Temperature at the Surface of the Body T 0 – Temperature of the Environment or the other Body T 0 – Temperature of the Environment or the other Body F 1-2 – Shape Factor F 1-2 – Shape Factor – Stefan-Boltzmann Constant – Stefan-Boltzmann Constant – Emissivity of the Surface of the Body – Emissivity of the Surface of the Body Radiation

9. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 1 st Law of Thermodynamics Initial condition: Boundary conditions:

10. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Discretization

11. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Discretization Fourrier’s Law:

12. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

13. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Finite differences time discretization

14. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

15. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Explicit and implicit schemes Explicit scheme: Explicit scheme: Fully implicit scheme: Fully implicit scheme: Crank-Nicholson scheme: Crank-Nicholson scheme: Galerkin scheme: Galerkin scheme:

16. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 1

17. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 1

18. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 E

19. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

20. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 2

21. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 2 B A

22. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

23. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

24. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

25. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 2

26. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005 Example 2

27. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005

28. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION 01.03.2005