1. A new approach in the solution of the thermoelasticity problems by the use of primal-mixed finite elements
dr Dubravka Mijuca
Faculty of Mathematics
Department of Mechanics
University of Belgrade
Studentski trg Belgrade - P.O.Box 550
Serbia and Monte Negro

2. Motive
The present investigation is motivated by idea that any numerical procedures of analysis which threats all variable of interest as fundamental ones (in the present case displacement, stress, temperature) are more reliable and more convenient for real engineering application for several reasons.

3. These reasons are: Better overall accuracy in the numerical results.
Independent control of all field variables
Reliability.
Avoiding a post-processing recovery methods and techniques for calculations of dual variables which entails a lost of accuracy.
In addition, modeling a stress as a independent fundamental variable permits to avoid spurious stress modes.

4. References
D. Mijuca (2002) A New Reliable 3D Finite Element in Elasticity, Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), Editors: Mang, H.A.; Rammerstorfer, F.G.; Eberhardsteiner, J., Publisher: Vienna University of Technology, Austria
A.A. Cannarozzi, F. Ubertini (2001) A mixed variational method for linear coupled thermoelastic analysis, International Journal of Solids and Structures 38,
S. Miranda, F. Ubertini (2001) On the consistency of finite element models, Comput. Methods Appl. Mech. Engrg., 190,
D. N. Arnold (1990) Mixed finite element methods for elliptic problems, Computer Methods in Applied Mechanics and Engineering, No. 82, 281‑300.
J. Jaric, (1988) Mehanika kontinuuma, Gradjevinsa knjiga, Beograd

5. The initial-boundary value problem
We recall that First Low of Motion of Cauchy’s and energy balance equation are given respectively by:
Further boundary conditions per traction, displacement, temperature, thermal flux and convective heat transfer at the surface, are respectively given by: , .

6. The initial-boundary value problem
Hence, constitutive equations in linear elasticity and constitutive equation of vector of heat flux, respectively are given by:
Compatibility equations are given by .
Initial temperature condition

7. Primal-mixed Formulation in Elasticity

8. A finite element implementation

9. System of FE Equations

10. The primal-mixed finite element method
In the present contribution, the hexahedral finite elements HC8/27 of Taylor-Hood type in elasticity, as well as its coarser restriction HC8/9, in primal-mixed finite element approach based on Hellinger-Reissner’s principle, is considered:

11. Basic features of FE HC8/27
Reliable finite element in the full three-dimensional elasticity, with no case dependent numerical tune-ups.
The analysis of bodies of arbitrary geometry, where both stresses and displacements are fundamental variables from continuous finite element sub-spaces
Simultaneous calculations of all components of displacement and stresses

12. Thermal field
In addition, for the present purpose we will assume that temperature field:
is prescribed in
Recall that initial strain
due to temperature is given by:

13. Thermal stresses 
So, known temperature are simply introduced as prescribed temperature deformation.

14. The Mathematical Convergence Requirements
The convergence requirements for shape functions of isoparametric finite element can be grouped into three categories: completeness, compatibility and stability.

15. Consistency
The finite elements HC8/9 and HC8/27 are consistent, which means that they satisfy both consistency criteria:
Completeness: the finite element polinomial functions contain all terms up to k=3
Compatibility: displacement finite element approximation functions are continuous (H1) over the interelement boundaries

16. First Stability Condition
Conclusion: For the primal-mixed procedure the
first Brezzi condition is always satisfied.

17. Discrete LBB (Second Brezzi Condition) Inf-sup Test

18. Numerical Inf-sup Test
The inf-sup test is passed if the inf-sup values mh, for chosen sequence of finite element meshes, do not show decrease toward zero.
On diagrams:
Inf-sup test passed:
Inf-sup test failed:

19. Unit Brick Model Problem. Side a=1. Young Modulus 1.

20. Full Cylinder With Clamped Ends. Thickness 1/100. Poisson’s ratio 0.3

21. Solid Cylinder / Taper / Sphere - Temperature
LOADING: Linear temperature gradient in the radial and axial direction T °C=(x2 + y2)1/2 + z
BOUNDARY CONDITIONS Symmetry on xz-plane, yz-plane, Face on xy-plane: zero z-displacement, Face HIH'I': zero z-displacement
MATERIAL PROPERTIES: Isotropic, E=210E3MPa, n=0.3, a=2.3E-4 /C
ELEMENT TYPES: Solid hexahedra
OUTPUT: Direct stress (szz) at point A, TARGET -105 MPa

22. Solid Cylinder / Taper / Sphere – Temperature MAXIMAL STRESS RESULTS
-99.36MPa

23. Solid Cylinder / Taper / Sphere – Temperature UNDEFORMED AND DEFORMED CONFIGURATION

25. LIME KILN UPGRADE - RECUPERATION

26. Lime Kiln Upgrade PHYSICAL PROPERTIES

27. Rotary Lime Kiln – Recouperator COMPONENT OF MAXIMAL STRESSES

28. Clamped Plate Under Temperature Gradient
LOADING: Linear temperature gradient in the z-direction Tz=0=400, Tz=0.005=300, Tz=0.01=200
BOUNDARY CONDITIONS Symmetry on xz-plane, yz-plane
MATERIAL PROPERTIES: Isotropic, E=210E3MPa, n=0.29, a=11.7E-6 /C
ELEMENT TYPES: Solid hexahedra
OUTPUT: maximal deflection in z wmax= -1mm

29. Clamped plate under temperature gradient MAXIMAL DISPLACEMENTS

31. Main advantages
Main advantages in the accordance to the similar approaches in the literature are:
Solving of the thin solid bodies response by the full 3d theory by the aspect of the geometry and relevant physical low
Full respect of the tensorial character of the physical variables of interest
Treatment of primal and dual variables as fundamental ones

32. Conclusions
The new finite element approach for solving of thermoelasticity problems, when temperature field is known, is proposed.
Results of preliminary investigation encourage the future research in semi coupled thermoelastic analysis

33. Abstract
The results of preliminary investigation of a new three-dimensional finite element scheme in the linear semi-coupled static thermoelastic analysis of solid bodies are given.
The main goal of present contribution is the straightforward calculation of displacements and stresses of a body subjected to the external mechanical forces and prescribed temperature distribution. Therefore, coordinate independent primal-mixed finite element formulation in elasticity is used for calculating the solid body response. For difference to other known approaches, all components of displacements vector and all independent components of stress tensor, are treated as fundamental variables and simultaneously calculated, eliminating the need for dual variables recovery, which entails a lost of accuracy.
Furthermore, obtained finite element scheme is reliable, thus free of locking in the limit of incompressibility and insensitive to the high distortion of finite elements in finite element mesh. Consequently, even very thin solid bodies are treated as three-dimensional bodies. In addition, displacements and stresses are approximated by continuous approximation functions.
In such a manner, genuine characteristics of a real model are preserved as much as possible, at least for always considering full model problem geometry and not neglecting any of the stress components. The numerical examples given enlighten the reliability of the approach proposed.