Skip to main content
SpringerLink
Log in

Inverse internal pressure estimation of functionally graded cylindrical shells under thermal environment

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The time-dependent pressure in long functionally graded cylindrical shells under thermal environment is estimated using the measured strains on their outer surfaces in conjunction with an inverse algorithm. The obtained strains from the solution of the related direct problem are used to simulate the data measurements. The direct problem is formulated based on the three-dimensional thermoelasticity theory under the plane strain conditions. The inverse solution procedure benefits from the discrepancy principle together with the conjugate gradient method as a powerful technique for optimization procedure. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing differential equations subjected to the related boundary and initial conditions in both spatial and temporal domains. The influence of measurement errors on the accuracy of the estimated internal pressure and also the displacement and stress components is investigated. The good accuracy of the results validates the presented approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

T :

Temperature (K)

J :

Functional defined by Eq. (29)

J ′ :

Gradient of functional

K :

Iteration number

k :

Thermal conductivity (Wm−1K−1)

r :

Space coordinate in r direction (m)

t :

Time (s)

h 1 :

Transfer coefficients at the inner surface (1/K)

h 2 :

Transfer coefficients at the outer surface (1/K)

R in :

Inner radius of the cylinder (m)

R out :

Outer radius of the cylinder (m)

u :

Displacement radial component (m)

p :

Pressure at inner surface (MPa)

E :

Elastic modulus (GPa)

T in :

Inner cylinder temperature (K)

T out :

Outer cylinder temperature (K)

\({\sigma _{{\theta}{\theta}}}\) :

Stress tangential component (GPa)

\({\sigma _{\rm rr}}\) :

Stress radial component (GPa)

\({\sigma _{\rm zz}}\) :

Stress axial component (GPa)

\({\varepsilon _{\rm rr}}\) :

Radial strain

\({\varepsilon _{{\theta}{\theta}}}\) :

Tangential strain

\({\varepsilon _{\rm zz}}\) :

Axial strain

\({\varepsilon _T}\) :

Thermal strain

\({\nu}\) :

Poisson’s ratio

\({\omega}\) :

Thermal expansion coefficient (1/K)

\({\Delta}\) :

Small variation quality

\({\lambda}\) :

Variable used in the adjoint problem

\({\delta}\) :

Delta function

\({\beta}\) :

Step size

\({\psi}\) :

Direction of descent

\({\gamma}\) :

Conjugate coefficient

\({\eta}\) :

Stopping criteria

\({\varpi}\) :

Random variable

*:

Dimensionless quantity

\({^\wedge}\) :

Estimated values

_:

Simulated values

References

  1. Shen H.S.: Functionally graded materials: nonlinear analysis of plates and shells. Taylor and Francis Group/CRC Press, Boca Raton (2009)

    Book  Google Scholar 

  2. Turteltaub, S.: Functionally graded materials for prescribed field evolution. Comput. Method Appl. M. 191, 2283–96 (2002)

    Google Scholar 

  3. Marin, L.: Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. Int. J. Solids Struct. 42, 4338–51 (2005)

    Google Scholar 

  4. Sladek, J., Sladek, V., Hon, Y.C.: Inverse heat conduction problems by meshless local Petrov–Galerkin method. Eng. Anal. Bound. Elem. 30, 650–61 (2006)

    Google Scholar 

  5. Golbahar Haghighi, M.R., Eghtesad, M., Malekzadeh, P., Necsulescu, D.S.: Two-dimensional inverse heat transfer analysis of functionally graded materials in estimating time-dependent surface heat flux. Numer. Heat Trans. Part A. Appl. 54, 744–62 (2008)

    Google Scholar 

  6. Golbahar Haghighi, M.R., Eghtesad, M., Malekzadeh, P., Necsulescu, D.S.: Three-dimensional inverse transient heat transfer analysis of thick functionally graded plates. Energ. Convers. Manag. 50, 450–7 (2009)

    Google Scholar 

  7. Marin, L.: Boundary reconstruction in two-dimensional functionally graded materials using a regularized MFS. Comput. Model. Eng. Sci. 46, 221–53 (2009)

    Google Scholar 

  8. Yang, C.Y.: Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems. Appl. Math. Model. 33, 2907–18 (2009)

    Google Scholar 

  9. Chang, W.J., Lee, H.L., Yang, Y.C.: Estimation of heat flux and thermal stresses in functionally graded hollow circular cylinders. J. Therm. Stress. 34, 740–55 (2011)

    Google Scholar 

  10. Lee H.L., Chang W.J., Chen W.L., Yang Y.C.: Inverse heat transfer analysis of a functionally graded fin to estimate time-dependent base heat flux and temperature distributions. Energ. Convers. Manag. 57, 1–7 (2012)

    Article  Google Scholar 

  11. Wu, T.S., Chu, C.L., Char, M.I., Yang, Y.C.: Inverse thermo-mechanical analysis of a functionally graded fin. J. Therm. Stress. 34, 740–55 (2012)

    Google Scholar 

  12. Lee, H.L., Chang, W.J., Sun, S.H., Yang, Y.C.: Estimation of temperature distributions and thermal stresses in a functionally graded hollow cylinder simultaneously subjected to inner-and-outer boundary heat fluxes. Compos. Part B Eng. 43, 786–92 (2012)

    Google Scholar 

  13. Golbahar Haghighi, M.R., Malekzadeh, P., Rahideh, H., Vaghefi, M.: Inverse transient heat conduction problems of a multilayered functionally graded cylinder. Numer. Heat Trans. Part A Appl. 61, 717–33 (2012)

    Google Scholar 

  14. Yang, Y.C., Chen, W.L., Chou, H.M., Salazar, J.L.L.: Inverse hyperbolic thermoelastic analysis of a functionally graded hollow circular cylinder in estimating surface heat flux and thermal stresses. Int. J. Heat Mass Trans. 60, 125–33 (2013)

    Google Scholar 

  15. Okada, H., Fukui, Y., Kumuzawa, N., Nakagawa, Y.: An inverse analysis approach to identify the elastic-plastic material properties of Al-based functionally graded materials. Key Eng. Mater. 149, 913–8 (1998)

    Google Scholar 

  16. Nakamura, T., Wang, T., Sampath, S.: Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Mater. 48, 4293–306 (2000)

    Google Scholar 

  17. Han, X., Liu, G.R.: Computational inverse technique for material characterization of functionally graded materials. AIAA J. 41, 288–95 (2003)

    Google Scholar 

  18. Zhou, S., Li, Q.: Design of graded two-phase microstructures for tailored elasticity gradients. J. Mater. Sci. 43, 5157–67 (2008)

    Google Scholar 

  19. Sladek J., Sladek V., Wen P.H., Hon Y.C.: Inverse problem for determination of heat transfer coefficients by local Petrov–Galerkin method. Comput. Model. Eng. Sci. 48, 191–218 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Nematollahi, M.A., Hematiyan, M.R., Farid, M.A.: Two-stage inverse method for the evaluation of volume fraction distributions in 2D and 3D functionally graded materials. Proceedings of the Institution of Mechanical Engineers. J. Mech. Eng. Sci. Part C 225, 1550–64 (2011)

    Google Scholar 

  21. Chen, W.L., Chou, H.M., Yang, Y.C.: An inverse problem in estimating the space-dependent thermal conductivity of a functionally graded hollow cylinder. Compos. Part B Eng. 50, 112–9 (2013)

    Google Scholar 

  22. Kazemzadeh-Parsi, M.J., Daneshmand, F.: Inverse geometry heat conduction analysis of functionally graded materials using smoothed fixed grid finite elements. Inv. Prob. Sci. Eng. 21, 235–50 (2013)

    Google Scholar 

  23. Heydarpour Y., Malekzadeh P., Golbahar Haghighi M.R., Vaghefi M.: Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method. Acta Mech. 223, 81–93 (2012)

    Article  MATH  Google Scholar 

  24. Golbahar Haghighi M.R., Eghtesad M., Malekzadeh P.: Coupled DQ–FE methods for two dimensional transient heat transfer analysis of functionally graded material. Energ. Convers. Manag. 49, 995–1001 (2008)

    Article  Google Scholar 

  25. Beck J.V., Blackwell B., Clair C.R.S.: Inverse Heat Conduction. Wiley, New York (1985)

    MATH  Google Scholar 

  26. Alifanov, O.M.: Solution of an inverse problem of heat conduction by iteration methods. J. Eng. Phys. 26, 471–6 (1974)

    Google Scholar 

  27. Hsu, P.T., Chen, C.K., Yang, Y.T.: A 2-D inverse method for simultaneous estimation of the inlet temperature and wall heat flux in a laminar circular duct flow. Numer. Heat Trans. Part A Appl. 34, 731–45 (1998)

    Google Scholar 

  28. Jarny, Y., Özisik, M.N., Bardon, J.P.: A general optimization method using adjoint equation for solving multidimensional inverse heat conduction. Int. J. Heat Mass Trans. 34, 2911–9 (1999)

    Google Scholar 

  29. Huang, C.H., Wang, S.P.: A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method. Int. J. Heat Mass Trans. 42, 3387–403 (1999)

    Google Scholar 

  30. Niliot, L.C., Lefevre, F.: Multiple transient point heat source identification in heat diffusion: application to numerical two- and three-dimensional problem. Numer. Heat Trans. B-Fund. 39, 277–301 (2001)

    Google Scholar 

  31. Chen, H.T., Wang, H.C.: Estimation of heat-transfer characteristics on a fin under wet conditions. Int. J. Heat Mass Trans. 51, 2123–38 (2008)

    Google Scholar 

  32. Shao, Z.S., Wang, T.J., Ang, K.K.: Transient thermo-mechanical analysis of functionally graded hollow circular cylinders. J. Therm. Stress. 30, 81–104 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Persian Gulf University, 7516913798, Bushehr, Iran

    M. R. Golbahar Haghighi, P. Malekzadeh & M. Afshari

Authors
  1. M. R. Golbahar Haghighi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Malekzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Afshari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Malekzadeh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haghighi, M.R.G., Malekzadeh, P. & Afshari, M. Inverse internal pressure estimation of functionally graded cylindrical shells under thermal environment. Acta Mech 225, 3377–3393 (2014). https://doi.org/10.1007/s00707-014-1138-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-014-1138-9

Keywords

Search

Navigation