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Application of the Variational Method of Homogeneous Solutions for the Optimal Control of the Axisymmetric Thermoelastic State of a Cylinder

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We propose a variational approach to the solution of the problem of optimization of a stationary axisymmetric thermal stressed state of a finite solid cylinder by controlling the distribution of volumetric heat sources. The proposed approach is based on the variational method of homogeneous solutions developed earlier for the solution of axisymmetric problems of the theory of elasticity for a cylinder. We study the influence of the height-to-radius ratio of the cylinder on the optimal values of the objective functional and the stressed state.

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References

  1. V. M. Vigak, Control of Temperature Stresses and Displacements [in Russian], Naukova Dumka, Kiev (1988).

    Google Scholar 

  2. V. S. Deineka and A. A. Aralova, “Optimal control of the thermal stressed state of a hollow long cylinder,” Dop. Nats. Akad. Nauk Ukr., No. 5, 38–42 (2012).

    MathSciNet  MATH  Google Scholar 

  3. M. M. Drobinin and I. N. Boyarshinova, “On one procedure of optimal control over the process of cooling of products made of vitrifying polymers with an aim of lowering residual stresses,” Vest. Perm. Nats. Issled. Politekh. Univ., Prikl. Mat. Mekh., No. 10, 52–62 (2012).

  4. A. D. Kovalenko, Fundamentals of Thermoelasticity [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  5. R. M. Kushnir, V. S. Popovych, and A. V. Yasins’kyi, “Optimization and identification in the thermomechanics of inhomogeneous bodies,” in: Ya. Yo. Burak and R. M. Kushnir (editors), Modeling and Optimization in the Thermomechanics of Conducting Inhomogeneous Bodies [in Ukrainian], Vol. 5, Spolom, Lviv (2011).

  6. V. V. Meleshko, Yu. V. Tokovyy, and J. R. Barber, “Axially symmetric temperature stresses in an elastic isotropic cylinder of finite length,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 1, 120–137 (2010); English translation:J. Math. Sci., 176, No. 5, 646–669 (2011).

  7. G. Ya. Popov, “Axisymmetric boundary-value problems of the theory of elasticity for cylinders and cones of finite length,” Dokl. Ros. Akad. Nauk, 439, No. 2, 192–197 (2011).

    MathSciNet  Google Scholar 

  8. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970).

    MATH  Google Scholar 

  9. V. Chekurin and L. Postolaki, “Variational method of homogeneous solutions in the axisymmetric problem of the theory of elasticity for a piecewise-homogeneous cylinder,” Fiz.-Mat. Model. Inform. Tekhnol., Issue 24, 118–129 (2016).

  10. V. K. Agarwal, “Axisymmetric solution of the end-problem for a semiinfinite elastic circular cylinder and its application to joined dissimilar cylinders under uniform tension,” Int. J. Eng. Sci., 16, No. 12, 985–998 (1978).

    Article  Google Scholar 

  11. V. F. Chekurin and L. I. Postolaki, “A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder,” Math. Model. Comput., 2, No. 2, 128–139 (2015).

    Article  Google Scholar 

  12. V. Chekurin and L. Postolaki, “Application of the least-squares method in axisymmetric biharmonic problems,” Math. Probl. Eng., 2016, Article ID 3457649, 9 pages (2016). https://doi.org/10.1155/2016/3457649

    Article  MathSciNet  Google Scholar 

  13. D. Clever and J. Lang, “Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient,” Optim. Contr. Appl. Meth., 33, No. 2, 157–175 (2012).

    Article  MathSciNet  Google Scholar 

  14. H. Huang and S. Liang, “Thermal-stress reduction for a Czochralski grown single crystal,” J. Eng. Math., 59, No. 1, 1–23 (2007).

    Article  MathSciNet  Google Scholar 

  15. H. K. Kuiken, “The cooling of low-heat-resistance cylinders by radiation,” J. Eng. Math., 13, No. 2, 97–106 (1979).

    Article  MathSciNet  Google Scholar 

  16. R. Sburlati, “Three-dimensional analytical solution for an axisymmetric biharmonic problem,” J. Elasticity, 95, No. 1–2, 79–97 (2009).

    Article  MathSciNet  Google Scholar 

  17. V. M. Vihak, A. V. Yasinskyy, Yu. V. Tokovyy, and A. V. Rychahivskyy, “Exact solution of the axisymmetric thermoelasticity problem for a long cylinder subjected to varying with-respect-to-length loads,” J. Mech. Behav. Mater., 18, No. 2, 141–148 (2007).

    Article  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    V. F. Chekurin & L. I. Postolaki

Authors
  1. V. F. Chekurin
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  2. L. I. Postolaki
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 60, No. 2, pp. 105–116, April–June, 2017.

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Chekurin, V.F., Postolaki, L.I. Application of the Variational Method of Homogeneous Solutions for the Optimal Control of the Axisymmetric Thermoelastic State of a Cylinder. J Math Sci 243, 128–144 (2019). https://doi.org/10.1007/s10958-019-04531-3

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  • DOI: https://doi.org/10.1007/s10958-019-04531-3

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