Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Curved hexahedral hybrid-mixed stress elements for structural dynamics

  • Eliseu Lucena Neto (Instituto Tecnologico de Aeronautica, Sao Jose dos Campos) ;
  • Flavio Luiz de Silva Bussamra (Instituto Tecnologico de Aeronautica, Sao Jose dos Campos) ;
  • Giorgio Paciarotti (Capgemini Engineering) ;
  • Felipe Rodrigo Cardoso (Empresa Brasileira de Aeronautica, Sao Jose dos Campos)
  • Received : 2023.12.18
  • Accepted : 2024.10.14
  • Published : 2024.11.10
⟨ Previous Next ⟩

Abstract

Curved hexahedral finite elements based on the hybrid-mixed stress formulation are proposed for structural dynamic analysis of three-dimensional solids. The stress and displacement in the domain of an element and the displacement on its boundary are simultaneously and independently approximated using sets of complete and linearly independent non-nodal Legendre polynomials. The element geometry is given in terms of its corner and mid-edge points using the same interpolation functions of the traditional isoparametric 20-node brick element. Symmetric, highly sparse and well conditioned solving systems are obtained. Numerical tests are carried out using h- and p-refinements to assess the behavior of these new hexahedrons.

Keywords

References

  1. Almeida, J.P.M. and Freitas, J.A.T. (1991), "Alternative approach to the formulation of hybrid equilibrium finite elements", Comput. Struct., 40(4), 1043-1047. https://doi.org/10.1016/0045-7949(91)90336-K.
  2. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A. and Sorensen, D. (1999), LAPACK User's Guide, 3rd Edition, SIAM, Philadelphia.
  3. Arruda, M.R.T. and Castro, L.M.S. (2012), "Structural dynamic analysis using hybrid and mixed finite element models", Finite Elem. Anal. Des., 57, 43-54. https://doi.org/10.1016/j.finel.2012.03.010.
  4. Arruda, M.R.T. and Castro, L.M.S. (2021), "Non-linear dynamic analysis of reinforced concrete structures with hybrid mixed stress finite elements", Adv. Eng. Softw., 153, 102965. https://doi.org/10.1016/j.advengsoft.2020.102965.
  5. Arruda, P.F.T., Arruda, M.R.T. and Castro, L.M.S. (2015), "Computation of critical loads and buckling modes using hybrid-mixed stress finite element models", Comput. Struct., 154, 72-90. https://doi.org/10.1016/j.compstruc.2015.02.012.
  6. Bathe, K.J. (1996), Finite Element Procedures, Prentice-Hall, Englewood Cliffs.
  7. Belytschko, T., Liu, W.K., Moran, B. and Elkhodary, K.I. (2014), Nonlinear Finite Elements for Continua and Structures, 2nd Edition, John Wiley, Chichester.
  8. Beslin, O. and Nicolas, J. (1997), "A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions", J. Sound Vib., 202(5), 633-655. https://doi.org/10.1006/jsvi.1996.0797.
  9. Businaro, F.A. and Bussamra, F.L.S. (2020), "On the sparsity of linear systems of equations for a new stress basis applied to three-dimensional Hybrid-Trefftz stress finite elements", Lat. Am. J. Solid. Struct., 17(7), e307. https://doi.org/10.1590/1679-78256124.
  10. Bussamra, F.L.S., Lucena Neto, E. and Cardoso, F.R. (2018), "Three-dimensional hybrid-mixed stress elements for free vibration analysis", Finite Elem. Anal. Des., 140, 50-58. https://doi.org/10.1016/j.finel.2017.11.004.
  11. Bussamra, F.L.S., Lucena Neto, E. and Raimundo Jr., D.S. (2012), "Hybrid quasi-Trefftz 3-D finite elements for laminated composite plates", Comput. Struct., 92-93, 185-192. https://doi.org/10.1016/j.compstruc.2011.11.005.
  12. Bussamra, F.L.S., Lucena Neto, E. and Rodrigues, M.A.C. (2016), "Simulation of stress concentration problems in laminated plates by quasi-Trefftz finite element models", Lat. Am. J. Solid. Struct., 13(9), 1677-1694. https://doi.org/10.1590/1679-78252698.
  13. Bussamra, F.L.S., Pimenta, P.M. and Freitas, J.A.T. (2001), "Hybrid-Trefftz stress elements for three-dimensional elastoplasticity", Comput. Assist. Mech. Eng. Sci., 8(2-3), 235-246.
  14. Castro, L.M.S. (2010), "Polynomial wavelets in hybrid-mixed stress finite element models", Int. J. Numer. Meth. Biomed. Eng., 26(10), 1293-1312. https://doi.org/10.1002/cnm.1215.
  15. Castro, L.M.S. and Barbosa, A.R. (2006), "Implementation of a hybrid-mixed stress model based on the use of wavelets", Comput. Struct., 84(10-11), 718-731. https://doi.org/10.1016/j.compstruc.2005.11.012.
  16. Castro, L.M.S. and Freitas, J.A.T. (2001), "Wavelets in hybrid-mixed stress elements", Comput. Meth. Appl. Mech. Eng., 190(31), 3977-3998. https://doi.org/10.1016/S0045-7825(00)00313-3.
  17. Chen, T., Ma, H. and Gao, W. (2013), "A new approach to stability analysis of frame structures using Trefftz-type elements", J. Constr. Steel Res., 82, 153-163. https://doi.org/10.1016/j.jcsr.2012.12.021.
  18. Chopra, A.K. (2019), Dynamics of Structures: Theory and Applications to Earthquake Engineering, 5th Edition, Pearson, London.
  19. Darilmaz, K. (2007) "An assumed-stress hybrid element for static and free vibration analysis of folded plates", Struct. Eng. Mech., 25(4), 405-421. http://doi.org/10.12989/sem.2007.25.4.405.
  20. Darilmaz, K. (2017) "Dynamic behaviour of orthotropic elliptic paraboloid shells with openings", Struct. Eng. Mech., 63(2), 225-235. https://doi.org/10.12989/sem.2017.63.2.225.
  21. Freitas, J.A.T. (1997), "Hybrid-Trefftz displacement and stress elements for elastodynamic analysis in the frequency domain", Comput. Assist. Mech. Eng. Sci., 4, 345-368.
  22. Freitas, J.A.T. (1999), "Hybrid finite element formulations for elastodynamic analysis in the frequency domain", Int. J. Solid. Struct., 36(13), 1883-1923. https://doi.org/10.1016/S0020-7683(98)00064-X.
  23. Freitas, J.A.T. (2008), "Mixed finite element solution of time-dependent problems", Comput. Meth. Appl. Mech. Eng., 197(45-48), 3657-3678. https://doi.org/10.1016/j.cma.2008.02.014.
  24. Freitas, J.A.T. and Bussamra, F.L.S. (2000), "Three-dimensional hybrid-Trefftz stress elements", Int. J. Numer. Meth. Eng., 47(5), 927-950. https://doi.org/10.1002/(SICI)1097-0207(20000220)47:5<927::AID-NME805>3.0.CO;2-B.
  25. Freitas, J.A.T. and Tiago, C. (2020), "Hybrid-Trefftz stress elements for plate bending", Int. J. Numer. Meth. Eng., 121(9), 1946-1976. https://doi.org/10.1002/nme.6294.
  26. Freitas, J.A.T. and Wang, Z.M. (2001), "Elastodynamic analysis with hybrid stress finite elements", Comput. Struct., 79(19), 1753-1767. https://doi.org/10.1016/S0045-7949(01)00106-7.
  27. Freitas, J.A.T., Almeida, J.P.M. and Pereira, E.M.B.R. (1996), "Non-conventional formulations for the finite element method", Struct. Eng. Mech., 4(6), 655-678. https://doi.org/10.12989/sem.1996.4.6.655.
  28. Lavrencic, M. and Brank, B. (2019), "Hybrid-mixed shell finite elements and implicit dynamic schemes for shell postbuckling", Eds. Altenbach, H., Chroscielewski, J., Eremeyev, V. and Wisniewski, K., Recent Developments in the Theory of Shells. Advanced Structured Materials, Vol. 110, 383-412. https://doi.org/10.1007/978-3-030-17747-8_21.
  29. Leissa, A.W. and Jacob, K.I. (1986), "Three-dimensional vibrations of twisted cantilevered parallelepipeds", J. Appl. Mech., 53(3), 614-618. https://doi.org/10.1115/1.3171820.
  30. Leissa, A.W. and So, J. (1995), "Accurate vibration frequencies of circular cylinders from three-dimensional analysis", J. Acoust. Soc. Am., 98(4), 2136-2141. https://doi.org/10.1121/1.414403.
  31. Li, T., Qi, Z., Ma, X. and Chen, W. (2015), "Higher-order assumed stress quadrilateral element for the Mindlin plate bending problem", Struct. Eng. Mech., 54(3), 393-417. http://doi.org/10.12989/sem.2015.54.3.393.
  32. Martins, P.H.C., Bussamra, F.L.S. and Lucena Neto, E. (2018), "Three-dimensional hybrid-Trefftz stress finite elements for plates and shells", Int. J. Numer. Meth. Eng., 113(11), 1676-1696. https://doi.org/10.1002/nme.5715.
  33. Moldovan, I.D. and Cismasiu, I. (2018), "FreeHyTE: A hybrid-Trefftz finite element platform", Adv. Eng. Softw., 121, 98-119. https://doi.org/10.1016/j.advengsoft.2018.03.014.
  34. Moldovan, I.D. and Freitas, J.A.T. (2008), "Hybrid-Trefftz stress and displacement elements for dynamic analysis of bounded and unbounded saturated porous media", Comput. Assist. Mech. Eng. Sci., 15(3), 289-303.
  35. Moldovan, I.D., Cao, T.D. and Freitas, J.A.T. (2014), "Elastic wave propagation in unsaturated porous media using hybrid-Trefftz stress elements", Int. J. Numer. Meth. Eng., 97(1), 32-67. https://doi.org/10.1002/nme.4566.
  36. Moldovan, I.D., Climent, N., Bendea, E.D., Cismasiu, I. and Correia, A.G. (2021), "A hybrid-Trefftz finite element platform for solid and porous elastodynamics", Eng. Anal. Bound. Elem., 124, 155-173. https://doi.org/10.1016/j.enganabound.2020.12.014.
  37. NX Nastran (2014), Element Library Reference, Siemens PLM Software, Plano.
  38. Pereira, E.M.B.R. and Freitas, J.A.T. (1996a), "A mixed-hybrid finite element model based on orthogonal functions", Int. J. Numer. Meth. Eng., 39(8), 1295-1312. https://doi.org/10.1002/(SICI)1097-0207(19960430)39:8<1295::AID-NME903>3.0.CO;2-H.
  39. Pereira, E.M.B.R. and Freitas, J.A.T. (1996b), "A hybrid-mixed element model based on Legendre polynomials for Reissner-Mindlin plates", Comput. Meth. Appl. Mech. Eng., 136(1-2), 111-126. https://doi.org/10.1016/0045-7825(96)01061-4.
  40. Pereira, E.M.B.R. and Freitas, J.A.T. (2000), "Numerical implementation of a hybrid-mixed finite element model for Reissner-Mindlin plates", Comput. Struct., 74(3), 323-334. https://doi.org/10.1016/S0045-7949(98)00291-0.
  41. Petrolito, J. (2004), "Vibration and stability of plates using hybrid-Trefftz elements", Int. J. Struct. Stab. Dyn., 4(4), 559-578. https://doi.org/10.1142/S0219455404001380.
  42. Petrolito, J. (2014), "Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements", Appl. Math. Model., 38(24), 5858-5869. https://doi.org/10.1016/j.apm.2014.04.026.
  43. Pian, T.H.H. (1964), "Derivation of element stiffness matrices by assumed stress distributions", AIAA J., 2(7), 1333-1336. https://doi.org/10.2514/3.2546.
  44. Rathore, S.S., Kar, V.R. and Sanjay. (2023), "Thermoelastic eigenfrequency of pre-twisted FG-sandwich straight/curved blades with rotational effect", Struct. Eng. Mech., 86(4), 519-533. https://doi.org/10.12989/sem.2023.86.4.519.
  45. Ray, M.C. (2019), "A novel hybrid-Trefftz finite element for symmetric laminated composite plates", Int. J. Mech. Mater. Des., 15(3), 629-646. https://doi.org/10.1007/s10999-018-9422-9.
  46. Silva, M.V., Castro, L.M.S. and Pereira, E.M.B.R. (2015), "On the parallel implementation of a hybrid-mixed stress formulation", Comput. Struct., 158, 71-81. https://doi.org/10.1016/j.compstruc.2015.05.022.
  47. Toth, B. (2021), "Natural frequency analysis of shells of revolution based on hybrid dual-mixed hp-finite element formulation", Appl. Math. Model., 98, 722-746. https://doi.org/10.1016/j.apm.2021.06.001.
  48. Wang, C.M., Liew, K.M., Xiang, Y. and Kitipornchai, S. (1992), "Application of Trefftz theory in thin-plate buckling with in-plane pre-buckling deformations", Int. J. Mech. Sci., 34(9), 681-688. https://doi.org/10.1016/0020-7403(92)90001-W.
  49. Young, W.C., Budynas, R.G. and Sadegh, A.M. (2012), Roark's Formulas for Stress and Strain, 8th Edition, McGraw-Hill, New York.
  50. Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. (2013), The Finite Element Method: Its Basis and Fundamentals, 7th Edition, Elsevier, Amsterdam.