Control variational method approach to bending and contact problems for Gao beam

Jitka Machalová; Horymír Netuka

Applications of Mathematics (2017)

  • Volume: 62, Issue: 6, page 661-677
  • ISSN: 0862-7940

Abstract

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This paper deals with a nonlinear beam model which was published by D. Y. Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.

How to cite

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Machalová, Jitka, and Netuka, Horymír. "Control variational method approach to bending and contact problems for Gao beam." Applications of Mathematics 62.6 (2017): 661-677. <http://eudml.org/doc/294595>.

@article{Machalová2017,
abstract = {This paper deals with a nonlinear beam model which was published by D. Y. Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.},
author = {Machalová, Jitka, Netuka, Horymír},
journal = {Applications of Mathematics},
keywords = {nonlinear beam; elastic foundation; contact problem; normal compliance condition; control variational method; finite element method},
language = {eng},
number = {6},
pages = {661-677},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Control variational method approach to bending and contact problems for Gao beam},
url = {http://eudml.org/doc/294595},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Machalová, Jitka
AU - Netuka, Horymír
TI - Control variational method approach to bending and contact problems for Gao beam
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 6
SP - 661
EP - 677
AB - This paper deals with a nonlinear beam model which was published by D. Y. Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.
LA - eng
KW - nonlinear beam; elastic foundation; contact problem; normal compliance condition; control variational method; finite element method
UR - http://eudml.org/doc/294595
ER -

References

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  1. Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  2. Andrews, K. T., Dumont, Y., M'Bengue, M. F., Purcell, J., Shillor, M., 10.1007/s00033-012-0233-9, Z. Angew. Math. Phys. 63 (2012), 1005-1019. (2012) Zbl1261.35093MR3000712DOI10.1007/s00033-012-0233-9
  3. Andrews, K. T., Kuttler, K. L., Shillor, M., 10.1007/978-3-319-14490-0_9, W. Han et al. Advances in Variational and Hemivariational Inequalities Advances in Mechanics and Mathematics 33, Springer, Cham (2015), 225-248. (2015) Zbl1317.74049MR3380538DOI10.1007/978-3-319-14490-0_9
  4. Arnautu, V., Langmach, H., Sprekels, J., Tiba, D., 10.1080/01630560008816960, Numer. Funct. Anal. Optimization 21 (2000), 337-354. (2000) Zbl0976.49025MR1769880DOI10.1080/01630560008816960
  5. Barboteu, M., Sofonea, M., Tiba, D., 10.1002/zamm.201000161, ZAMM, Z. Angew. Math. Mech. 92 (2012), 25-40. (2012) Zbl1304.74033MR2871899DOI10.1002/zamm.201000161
  6. Cai, K., Gao, D. Y., Qin, Q. H., 10.1177/1081286513482483, Math. Mech. Solids 19 (2014), 659-671. (2014) Zbl1298.74085MR3228236DOI10.1177/1081286513482483
  7. Eisley, J. G., Waas, A. M., 10.1002/9781119993278, John Wiley & Sons, Hoboken (2011). (2011) Zbl1232.74001DOI10.1002/9781119993278
  8. Fučík, S., Kufner, A., Nonlinear Differential Equations. Studies in Applied Mechanics 2, Elsevier Scientific Publishing Company, Amsterdam (1980). (1980) Zbl0426.35001MR0558764
  9. Gao, D. Y., 10.1016/0093-6413(95)00071-2, Mech. Res. Commun. 23 (1996), 11-17. (1996) Zbl0843.73042MR1371779DOI10.1016/0093-6413(95)00071-2
  10. Gao, D. Y., 10.1007/978-1-4757-3176-7, Nonconvex Optimization and Its Applications 39, Kluwer Academic Publihers, Dordrecht (2000). (2000) Zbl0940.49001MR1773838DOI10.1007/978-1-4757-3176-7
  11. Gao, D. Y., Machalová, J., Netuka, H., 10.1016/j.nonrwa.2014.09.012, Nonlinear Anal., Real World Appl. 22 (2015), 537-550. (2015) Zbl1326.74075MR3280850DOI10.1016/j.nonrwa.2014.09.012
  12. Gao, D. Y., Ogden, R. W., 10.1007/s00033-007-7047-1, Z. Angew. Math. Phys. 59 (2008), 498-517. (2008) Zbl1143.74018MR2399352DOI10.1007/s00033-007-7047-1
  13. Glowinski, R., Lions, J. L., Trémolières, R., 10.1016/s0168-2024(08)70201-7, Studies in Mathematics and Its Applications 8, North-Holland Publishing, Amsterdam (1981). (1981) Zbl0463.65046MR0635927DOI10.1016/s0168-2024(08)70201-7
  14. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J., 10.1007/978-1-4612-1048-1, Applied Mathematical Sciences 66, Springer, New York (1988). (1988) Zbl0654.73019MR0952855DOI10.1007/978-1-4612-1048-1
  15. Horák, J. V., Netuka, H., Mathematical model of pseudointeractive set: 1D body on non-linear subsoil. I. Theoretical aspects, Engineering Mechanics 14 (2007), 311-325. (2007) 
  16. Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften 170, Springer, Berlin (1971). (1971) Zbl0203.09001MR0271512
  17. Machalová, J., Netuka, H., 10.1063/1.3636963, Numerical Analysis and Applied Mathematics ICNAAM 2011 T. E. Simos et al. AIP Conference Proceedings 1389, AIP-Press, Springer (2011), 1820-1824. (2011) DOI10.1063/1.3636963
  18. Machalová, J., Netuka, H., 10.1155/2015/420649, J. Appl. Math. 2015 (2015), Article ID 420649, 12 pages. (2015) MR3399550DOI10.1155/2015/420649
  19. Machalová, J., Netuka, H., 10.1177/1081286517732382, (to appear) in Math. Mech. Solids. Special issue on Inequality Problems in Contact Mechanics (2017). MR3399550DOI10.1177/1081286517732382
  20. Neittaanmäki, P., Sprekels, J., Tiba, D., 10.1007/b138797, Springer Monographs in Mathematics, Springer, New York (2006). (2006) Zbl1106.49002MR2183776DOI10.1007/b138797
  21. Reddy, J. N., An Introduction to the Finite Element Method, McGraw-Hill Book Company, New York (2006). (2006) Zbl0561.65079MR0033470
  22. Shillor, M., Sofonea, M., Telega, J. J., 10.1007/b99799, Lecture Notes in Physics 655, Springer, Berlin (2004). (2004) Zbl1069.74001DOI10.1007/b99799
  23. Šimeček, R., 10.1007/s10492-013-0016-4, Appl. Math., Praha 58 (2013), 329-346. (2013) Zbl1289.49002MR3066824DOI10.1007/s10492-013-0016-4
  24. Sofonea, M., Matei, A., 10.1017/CBO9781139104166, London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge (2012). (2012) Zbl1255.49002DOI10.1017/CBO9781139104166
  25. Sofonea, M., Tiba, D., The control variational method for contact of Euler-Bernoulli beams, Bull. Trans. Univ. Braşov, Ser. III, Math. Inform. Phys. 2 (2009), 127-136. (2009) Zbl1224.74088MR2642501
  26. Sysala, S., 10.1007/s10492-008-0030-0, Appl. Math., Praha 53 (2008), 347-379. (2008) Zbl1199.49051MR2433726DOI10.1007/s10492-008-0030-0
  27. Sysala, S., 10.1007/s10492-010-0006-8, Appl. Math., Praha 55 (2010), 151-187. (2010) Zbl1224.74011MR2600940DOI10.1007/s10492-010-0006-8
  28. Tröltzsch, F., 10.1090/gsm/112, Graduate Studies in Mathematics 112, American Mathematical Society, Providence (2010). (2010) Zbl1195.49001MR2583281DOI10.1090/gsm/112

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