On a new computational algorithm for impacts of elastic bodies

Hynek Štekbauer; Ivan Němec; Rostislav Lang; Daniel Burkart; Jiří Vala

Applications of Mathematics (2022)

  • Volume: 67, Issue: 6, page 775-804
  • ISSN: 0862-7940

Abstract

top
Computational modelling of contact problems is still one of the most difficult aspects of non-linear analysis in engineering mechanics. The article introduces an original efficient explicit algorithm for evaluation of impacts of bodies, satisfying the conservation of both momentum and energy exactly. The algorithm is described in its linearized 2-dimensional formulation in details, as open to numerous generalizations including 3-dimensional ones, and supplied by numerical examples obtained from its software implementation.

How to cite

top

Štekbauer, Hynek, et al. "On a new computational algorithm for impacts of elastic bodies." Applications of Mathematics 67.6 (2022): 775-804. <http://eudml.org/doc/298528>.

@article{Štekbauer2022,
abstract = {Computational modelling of contact problems is still one of the most difficult aspects of non-linear analysis in engineering mechanics. The article introduces an original efficient explicit algorithm for evaluation of impacts of bodies, satisfying the conservation of both momentum and energy exactly. The algorithm is described in its linearized 2-dimensional formulation in details, as open to numerous generalizations including 3-dimensional ones, and supplied by numerical examples obtained from its software implementation.},
author = {Štekbauer, Hynek, Němec, Ivan, Lang, Rostislav, Burkart, Daniel, Vala, Jiří},
journal = {Applications of Mathematics},
keywords = {computational mechanics; contact problem; finite element method; explicit time integration algorithm},
language = {eng},
number = {6},
pages = {775-804},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a new computational algorithm for impacts of elastic bodies},
url = {http://eudml.org/doc/298528},
volume = {67},
year = {2022},
}

TY - JOUR
AU - Štekbauer, Hynek
AU - Němec, Ivan
AU - Lang, Rostislav
AU - Burkart, Daniel
AU - Vala, Jiří
TI - On a new computational algorithm for impacts of elastic bodies
JO - Applications of Mathematics
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 6
SP - 775
EP - 804
AB - Computational modelling of contact problems is still one of the most difficult aspects of non-linear analysis in engineering mechanics. The article introduces an original efficient explicit algorithm for evaluation of impacts of bodies, satisfying the conservation of both momentum and energy exactly. The algorithm is described in its linearized 2-dimensional formulation in details, as open to numerous generalizations including 3-dimensional ones, and supplied by numerical examples obtained from its software implementation.
LA - eng
KW - computational mechanics; contact problem; finite element method; explicit time integration algorithm
UR - http://eudml.org/doc/298528
ER -

References

top
  1. Abe, K., Higashimori, N., Kubo, M., Fujiwara, H., Iso, Y., 10.4208/aamm.2014.5.s6, Adv. Appl. Math. Mech. 6 (2014), 693-698. (2014) MR3244370DOI10.4208/aamm.2014.5.s6
  2. Ahmad, M., Ismail, K. A., Mat, F., Impact models and coefficient of restitution: A review, ARPN J. Eng. Appl. Sci. 11 (2016), 6549-6555. (2016) 
  3. Bank, R. E., Dupont, T., 10.1090/S0025-5718-1981-0595040-2, Math. Comput. 36 (1981), 35-51. (1981) Zbl0466.65059MR595040DOI10.1090/S0025-5718-1981-0595040-2
  4. Bathe, K.-J., Finite Element Procedures, Prentice Hall, New Jersey (2009). (2009) 
  5. Castro, A. Bermúdez de, 10.1007/3-7643-7383-0, Progress in Mathematical Physics 43. Birkhäuser, Basel (2005). (2005) Zbl1070.74001MR2145925DOI10.1007/3-7643-7383-0
  6. Brenner, S. C., Scott, L. R., 10.1007/978-1-4757-3658-8, Texts in Applied Mathematics 15. Springer, New York (2002). (2002) Zbl1012.65115MR1894376DOI10.1007/978-1-4757-3658-8
  7. Courant, R., Friedrichs, K., Lewy, H., 10.1007/BF01448839, Math. Ann. 100 (1928), 32-74 German 9999JFM99999 54.0486.01. (1928) MR1512478DOI10.1007/BF01448839
  8. Duriez, C., Andriot, C., Kheddar, A., 10.1109/IROS.2004.1389915, International Conference on Intelligent Robots and Systems (IROS) IEEE, Piscataway (2004), 3232-3237. (2004) DOI10.1109/IROS.2004.1389915
  9. Duvaut, G., Lions, J. L., 10.1007/978-3-642-66165-5, Grundlehren der mathematischen Wissenschaften 219. Springer, Berlin (1976). (1976) Zbl0331.35002MR0521262DOI10.1007/978-3-642-66165-5
  10. Eck, C., Jarušek, J., Sofonea, M., 10.1017/S0956792510000045, Eur. J. Appl. Math. 21 (2010), 229-251. (2010) Zbl1330.74131MR2646670DOI10.1017/S0956792510000045
  11. Francavilla, A., Zienkiewicz, O. C., 10.1002/nme.1620090410, Int. J. Numer. Methods Eng. 9 (1975), 913-924. (1975) MR3618552DOI10.1002/nme.1620090410
  12. (ed.), J. O. Halquist, LS-DYNA Theoretical Manual, Livermore Software Technology Corporation, Livermore (2006). (2006) 
  13. Halquist, J. O., Goudreau, G., Benson, D. J., 10.1016/0045-7825(85)90030-1, Comput. Methods Appl. Mech. Eng. 51 (1985), 107-137. (1985) Zbl0567.73120MR0822741DOI10.1016/0045-7825(85)90030-1
  14. Han, J., Zeng, H., 10.1016/j.jmaa.2018.12.068, J. Math. Anal. Appl. 473 (2019), 712-748. (2019) Zbl1433.49012MR3912849DOI10.1016/j.jmaa.2018.12.068
  15. Hashiguchi, K., 10.1007/978-3-642-35849-4, Lecture Notes in Applied and Computational Mechanics 69. Springer, Berlin (2014). (2014) Zbl1318.74001MR3235845DOI10.1007/978-3-642-35849-4
  16. Hughes, T. J. R., Taylor, R. L., Kanoknukulchai, W., A finite element method for large displacement contact and impact problems, Formulations and Computational Algorithms in Finite Element Analysis M.I.T. Press, Cambridge (1977), 468-495. (1977) MR0475196
  17. Kloosterman, G., Damme, R. M. J. van, Boogaard, A. H. van der, Huétink, J., 10.1002/nme.209, Int. J. Numer. Methods Eng. 51 (2001), 865-882. (2001) Zbl1039.74046MR1837060DOI10.1002/nme.209
  18. Kocur, G. K., Harmanci, Y. E., Chatzi, E., Steeb, H., Markert, B., 10.1007/s00419-020-01751-x, Arch. Appl. Mech. 91 (2021), 47-60. (2021) DOI10.1007/s00419-020-01751-x
  19. Laursen, T. A., 10.1007/978-3-662-04864-1, Springer, Berlin (2002). (2002) Zbl0996.74003MR1902698DOI10.1007/978-3-662-04864-1
  20. Li, J., Yu, K., Li, X., 10.1007/s11071-019-04936-4, Nonlinear Dyn. 96 (2019), 2475-2507. (2019) DOI10.1007/s11071-019-04936-4
  21. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris (1969), French. (1969) Zbl0189.40603MR0259693
  22. Liu, Y. F., Li, J., Zhang, Z. M., Hu, X. H., Zhang, W. J., 10.5194/ms-6-15-2015, Mech. Sci. 6 (2015), 15-28. (2015) DOI10.5194/ms-6-15-2015
  23. Na, J., Chen, Q., Ren, X., 10.1016/C2016-0-04487-X, Emerging Methodologies and Applications in Modelling, Identification and Control. Academic Press, Amsterdam (2018). (2018) Zbl1415.93006DOI10.1016/C2016-0-04487-X
  24. Němec, I., Štekbauer, H., Vaněčková, A., Vlk, Z., 10.1051/matecconf/201710700066, Dynamics of Civil Engineering and Transport Structures and Wind Engineering -- DYN-WIND'2017 MATEC Web of Conferences 107. EDP Sciences, Paris (2017), Article ID 66, 8 pages. (2017) DOI10.1051/matecconf/201710700066
  25. Puso, M. A., 10.1002/nme.865, Int. J. Numer. Methods Eng. 59 (2004), 315-336. (2004) Zbl1047.74065DOI10.1002/nme.865
  26. Rek, V., Vala, J., On a distributed computing platform for a class of contact-impact problems, Seminar on Numerical Analysis (SNA'21) Institute of Geonics CAS, Ostrava (2021), 64-67. (2021) 
  27. Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, Mathematics and Its Applications (East European Series) 4. D. Reidel, Dordrecht (1982). (1982) Zbl0505.65029MR0689712
  28. Roubíček, T., 10.1007/978-3-0348-0513-1, ISNM. International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005). (2005) Zbl1087.35002MR2176645DOI10.1007/978-3-0348-0513-1
  29. Sanz-Serna, J. M., Spijker, M. N., 10.1007/BF01389633, Numer. Math. 49 (1986), 319-329. (1986) Zbl0574.65106MR0848530DOI10.1007/BF01389633
  30. Schwab, A. L., On the interpretation of the Lagrange multipliers in the constraint formulation of contact problems; or why are some multipliers always zero?, Proc. ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE ASME, New York (2014), 1-5. (2014) 
  31. Sewerin, F., Papadopoulos, P., 10.1016/j.cma.2020.113108, Comput. Methods Appl. Mech. Eng. 368 (2020), Article ID 113108, 24 pages. (2020) Zbl07337970MR4114140DOI10.1016/j.cma.2020.113108
  32. Shi, P., 10.1016/S0895-7177(98)00132-0, Math. Comput. Modelling 28 (1998), 427-435. (1998) Zbl1122.74429MR1648769DOI10.1016/S0895-7177(98)00132-0
  33. Sofonea, M., Danan, D., Zheng, C., 10.1007/s00009-014-0504-0, Mediterr. J. Math. 13 (2016), 857-872. (2016) Zbl1335.74048MR3483867DOI10.1007/s00009-014-0504-0
  34. Štekbauer, H., 10.1515/tvsb-2016-0027, Trans. VŠB-TU Ostrava 16 (2016), 161-164. (2016) DOI10.1515/tvsb-2016-0027
  35. Štekbauer, H., Lang, R., Zeiner, M., Burkart, D., A correct and efficient algorithm for impacts of bodies, Seminar on Numerical Analysis (SNA'21) Institute of Geonics CAS, Ostrava (2021), 55-58. (2021) 
  36. Štekbauer, H., Němec, I., 10.1063/5.0031396, AIP Conf. Proc. 2293 (2020), Article ID 340013, 4 pages. (2020) DOI10.1063/5.0031396
  37. Štekbauer, H., Vlk, Z., 10.1051/matecconf/201710700060, Dynamics of Civil Engineering and Transport Structures and Wind Engineering -- DYN-WIND'2017 MATEC Web of Conferences 107. EDP Sciences, Paris (2017), Article ID 60, 6 pages. (2017) DOI10.1051/matecconf/201710700060
  38. Vala, J., Kozák, V., 10.1016/j.tafmec.2020.102486, Theor. Appl. Fracture Mech. 107 (2020), Article ID 102486, 8 pages. (2020) DOI10.1016/j.tafmec.2020.102486
  39. Vala, J., Kozák, V., 10.21136/AM.2021.0281-20, Appl. Math., Praha 66 (2021), 815-836. (2021) Zbl07442408MR4342610DOI10.21136/AM.2021.0281-20
  40. Weyler, R., Oliver, J., Sain, T., Cante, J. C., 10.1016/j.cma.2011.01.011, Comput. Methods Appl. Mech. Eng. 205-208 (2012), 68-82. (2012) Zbl1239.74075MR2872027DOI10.1016/j.cma.2011.01.011
  41. Wriggers, P., 10.1007/BF02736195, Arch. Comput. Methods Eng. 2 (1995), 1-49. (1995) MR1367147DOI10.1007/BF02736195
  42. Wu, S. R., 10.1016/j.cma.2008.12.013, Comput. Methods Appl. Mech. Eng. 198 (2009), 2009-2015. (2009) Zbl1227.74041DOI10.1016/j.cma.2008.12.013
  43. Wu, S. R., 10.1016/j.cma.2009.10.003, Comput. Methods Appl. Mech. Eng. 199 (2009), 220. (2009) Zbl1231.74332MR1344385DOI10.1016/j.cma.2009.10.003
  44. Xu, D., Hjelmstad, K. D., A new node-to-node approach to contact/impact problems for two dimensional elastic solids subject to finite deformation, Newmark Structural Laboratory Report Series University of Illinois, Urbana-Champaign (2008), Available at http://hdl.handle.net/2142/5318. (2008) 
  45. Yang, B., Laursen, T. A., 10.1016/j.cma.2007.09.004, Comput. Methods Appl. Mech. Eng. 197 (2008), 756-772. (2008) Zbl1169.74513MR2397015DOI10.1016/j.cma.2007.09.004
  46. Yang, B., Laursen, T. A., Meng, X., 10.1002/nme.1222, Int. J. Numer. Methods Eng. 62 (2005), 1183-1225. (2005) Zbl1161.74497MR2120292DOI10.1002/nme.1222
  47. Yastrebov, V. A., 10.1002/9781118647974, J. Wiley & Sons, London (2013). (2013) Zbl1268.74003DOI10.1002/9781118647974
  48. Zavarise, G., Lorenzis, L. de, 10.1002/nme.2559, Int. J. Numer. Methods Eng. 79 (2009), 379-416. (2009) Zbl1171.74455DOI10.1002/nme.2559
  49. Zhong, Z.-H., Finite Element Procedures for Contact-Impact Problems, Oxford University Press, Oxford (1993). (1993) MR1206475

NotesEmbed ?

top

You must be logged in to post comments.