Steel and Composite Structures

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The level set-based topology optimization for three-dimensional functionally graded plate using thin-plate spline

  • Banh, Thanh T. (Department of Architectural Engineering, Sejong University) ;
  • Luu, Nam G. (Department of Architectural Engineering, Sejong University) ;
  • Lee, Dongkyu (Department of Architectural Engineering, Sejong University)
  • Received : 2021.08.02
  • Accepted : 2022.08.30
  • Published : 2022.09.10
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Abstract

This paper is first implemented with the bending behavior of three-dimensional functionally graded (3DFG) plates in the framework of level set-based topology optimization (LS-based TO). Besides, due to the suitable properties of the current design domain, the thin-plate spline (TPS) is recognized as a RBF to construct the LS function. The overall mechanical properties of the 3DFG plate are assessed using a power-law distribution scheme via Mori-Tanaka micromechanical material model. The bending response is obtained using the first-order shear deformation theory (FSDT). The mixed interpolation of four elements of tensorial components (MITC4) is also implemented to overcome a well-known shear locking problem when the thickness becomes thinner. The Hamilton-Jacobi method is utilized in each iteration to enforce the necessary boundary conditions. The mathematical formulas are expressed in great detail for the LS-based TO using 3DFG materials. Several numerical examples are exhibited to verify the efficiency and reliability of the current methodology with the previously reported literature. Finally, the influences of FG materials in the optimized design are explained in detail to illustrate the behaviors of optimized structures.

Keywords

Acknowledgement

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2C1003776).

References

  1. Bendsoe, M.P. and Kikuchi, N. (1988), "Generating optimal topologies in structural design using a homogenization method", Comput. Meth. Appl. Mech. Eng., 71(2), 197-224. https://doi.org/10.1016/0045-7825(88)90086-2.
  2. Bendsoe, M.P. (1989), "Optimal shape design as a material distribution problem", Struct. Optim., 1(4), 193-202. https://doi.org/10.1007/BF01650949.
  3. Xie, Y.M. and Steven, G.P. (1993), "A simple evolutionary procedure for structural optimization", Comput. Struct., 49(5), 885-896. https://doi.org/10.1016/0045-7949(93)90035-C.
  4. Querin, O.M., Steven, G.P. and Xie, Y.M. (1998), "Evolutionary structural optimisation (ESO) using a bidirectional algorithm", Eng. Comput., 15(8), 1031-1048. https://doi.org/10.1108/02644409810244129.
  5. Chau K.N, Chau K.N., Ngo T., Hackl K. and Nguyen X.H. (2018), "A polytree-based adaptive polygonal finite element method for multi-material topology optimization", Comput. Meth. Appl. Mech. Eng., 332, 712-739. https://doi.org/10.1016/j.cma.2017.07.035.
  6. Osher, S. and Sethian, J.A. (1988), "Fronts propagating with curvature-dependent speed: algorithms based on HamiltonJacobi formulations", J. Comput. Phys., 79(1), 12-49. https://doi.org/10.1016/0021-9991(88)90002-2.
  7. Sethian, J.A. and Wiegmann, A. (2000), "Structural boundary design via level set and immersed interface methods", J. Comput. Phys., 163(2), 489-528. https://doi.org/10.1006/jcph.2000.6581.
  8. Osher, S. and Santosa, F. (2001), "Level set methods for optimization problems involving geometry and constraints: I. frequencies of a two-density inhomogeneous drum", J. Comput. Phys., 171(1), 272-288. https://doi.org/10.1006/jcph.2001.6789.
  9. Wang, M.Y, Wang, X.M. and Guo, D.M. (2003), "A level set method for structural topology optimization", Comput. Meth. Appl. Mech. Eng., 192, 227-324. https://doi.org/10.1016/S0045-7825(02)00559-5.
  10. Wang, M.Y, Wang, X.M. and Guo, D.M. (2004a), "Structural shape and topology optimization in a level-set-based framework of region representation", Struct. Multidiscip. Optimiz., 27, 1-9. https://doi.org/10.1007/s00158-003-0363-y.
  11. Wang, M.Y, and Wang, X.M. (2004b), "PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization", Comput. Model. Eng. Sci., 6(4), 373-396. https://doi.org/10.1115/DETC2004-57038.
  12. Wang, M.Y. and Zhou, S.W. (2004c), "Synthesis of shape and topology of multi-material structures with a phase-field method", J. Comput.-Aided Mater. Des., 11, 117-138. https://doi.org/10.1007/s10820-005-2323-x.
  13. Wang, M.Y. and Wang, X.M. (2004d), "Color level sets: A multiphase method for structural topology optimization with multiple materials", Comput. Meth. Appl. Mech. Eng., 193, 469-496. https://doi.org/10.1016/j.cma.2003.10.008.
  14. Luo, J.Z., Luo, Z., Chen, S.K. and Tong, L.Y. (2008a), "A new level set method for systematic design of hinge-free compliant mechanisms", Comput. Meth. Appl. Mech. Eng., 198, 318-331. https://doi.org/10.1016/j.cma.2008.08.003.
  15. Luo, Z., Tong, L.Y., Wang, M.Y. and Wang, S.Y. (2008b), "A level set-based parameterization method for structural shape and topology optimization", Int. J. Numer. Meth. Eng., 76, 1-26. https://doi.org/10.1002/nme.2092.
  16. Wei, P. and Wang, M.Y. (2009), "Piecewise constant level set method for structural topology Optimization", Int. J. Numer. Meth. Eng., 78(4), 379-402. https://doi.org/10.1002/nme.2478.
  17. Wang, S.Y. and Wang, M.Y. (2006), "Radial basis functions and level set method for structural topology optimization", Int. J. Numer. Meth. Eng., 65, 2060-2090. https://doi.org/10.1002/nme.1536.
  18. Luo, Z., Tong, L.Y., Wang, M.Y. and Wang, S.Y. (2007), "Shape and topology optimization of compliant mechanisms using a parameterization level set method", J. Comput. Phys., 227, 680-705. https://doi.org/10.1016/j.jcp.2007.08.011.
  19. Liu, T., Li, B., Wang, S.Y. and Gao, L. (2014), "Eigenvalue topology optimization of structures using a parameterized level set method", Struct. Multidiscip. Optimiz., 50, 573-591. https://doi.org/10.1007/s00158-014-1069-z.
  20. Cui, M.T., Luo, C.C., Li, G. and Pan, M. (2021), "The parameterized level set method for structural topology optimization with shape sensitivity constraint factor", Eng. Comput., 37, 855-872. https://doi.org/10.1007/s00366-019-00860-8.
  21. Islam, S.U., Khan, W., Ullah, B. and Ullah, Z. (2020), "The localized radial basis functions for parameterized level set based structural optimization", Eng. Anal. Bound. Elements, 113, 296-305. https://doi.org/10.1016/j.enganabound.2020.01.008.
  22. Wei, P., Li, Z., Li, Z.P. and Wang, M.Y. (2018), "An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions", Struct. Multidiscipl. Optimiz., 58, 831-849. https://doi.org/10.1007/s00158-018-1904-8.
  23. Liu, H., Zhong, H.M., Tian, Y. and Ma, Q.P. (2019), "A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design", Structural and Multidisciplinary Optimization, 60, 2221-2247. https://doi.org/10.1007/s00158-019-02318-3
  24. Jiang, Y.T., Zhao, M., (2020), "Topology optimization under design-dependent loads with the parameterized level-set method based on radial-basis functions", Comput. Meth. Appl. Mech. Eng., 369, 113235. https://doi.org/10.1016/j.cma.2020.113235.
  25. Wei, P. and Paulino, G.H. (2020), "A parameterized level set method combined with polygonal finite elements in topology optimization", Struct. Multidiscip. Optimiz., 61, 1913-1928. https://doi.org/10.1007/s00158-019-02444-y.
  26. Sha, W., Xiao, M., Gao, L. and Zhang, Y. (2021), "A new level set based multi-material topology optimization method using alternating active-phase algorithm", Comput. Meth. Appl. Mech. Eng., 377, 113674. https://doi.org/10.1016/j.cma.2021.113674.
  27. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B: Eng., 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.
  28. Zenkour, A.M. (2006), "Generalized shear deformation theory for bending analysis of functionally graded plates", Appl. Mathem. Modelling, 30, 67-84. https://doi.org/10.1016/j.apm.2005.03.009.
  29. Fares, M.E., Elmarghany, M.K. and Atta, D. (2015), "The influence of the normal strain effect on the control and design optimization of functionally graded plates", Compos. Part B: Eng., 77, 440-453. https://doi.org/10.1016/j.compositesb.2015.03.003.
  30. Mantari, J.L., Ramos, I.A., Carrera, E. and Petrolo, M., (2016), "Static analysis of functionally graded plates using new nonpolynomial displacement fields via Carrera Unified Formulation", Compos. Part B: Eng., 89, 127-142. https://doi.org/10.1016/j.compositesb.2015.11.025.
  31. Nguyen T.N., Ngo T.D. and Nguyen X.H. (2017), "A novel threevariable shear deformation plate formulation: Theory and Isogeometric implementation", Comput. Meth. Appl. Mech. Eng., 326, 376-401. https://doi.org/10.1016/j.cma.2017.07.024.
  32. Nguyen T.N., Thai C.H., Luu A.T, Nguyen X.H. and Lee J. (2019), "NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells", Comput. Meth. Appl. Mech. Eng., 347, 983-1003. https://doi.org/10.1016/j.cma.2019.01.011.
  33. Nguyen L.B., Thai H.C., Nguyen N.D. and Nguyen X.H. (2021), "A size-dependent isogeometric approach for vibration analysis of FG piezoelectric porous microplates using modified strain gradient theory", Eng. Comput. https://link.springer.com/article/10.1007/s00366-021-01468-7.
  34. Lalepalli A.K., Nitin S. and Umut T. (2013), "Isogeometric analysis of functionally graded plates using higher-order shear deformation theory", Comput. Concrete, 25, 303-310. https://doi.org/10.12989/cac.2020.25.4.303.
  35. Das A., Hirwani C.K., Panda S.K., Topal U. and Dede T. (2018), "Prediction and analysis of optimal frequency of layered composite structure using higher-order FEM and soft computing techniques", Steel Compos. Struct., 29, 749-758. https://doi.org/10.12989/scs.2018.29.6.749.
  36. Hirwani C.K. and Panda S.K. (2018), "Numerical nonlinear frequency analysis of pre-damaged curved layered composite structure using higher-order finite element method", Int. J. NonLinear Mech., 102, 14-24. https://doi.org/10.1016/j.ijnonlinmec.2018.03.005.
  37. Hirwani C.K., Panda S.K., Mahapatra T.R. and Mahapatra S.S. (2017), "Nonlinear transient finite-element analysis of delaminated composite shallow shell panels", AIAAJ, 55. https://doi.org/10.2514/1.J055624.
  38. Hirwani C.K. and Panda S.K. (2019), "Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure", Appl. Mathem. Modelling, 65, 303-317. https://doi.org/10.1016/j.apm.2018.08.014.
  39. Hirwani C.K., Panda S.K. and Mahapatra T.R. (2018a), "Thermomechanical deflection and stress responses of delaminated shallow shell structure using higher-order theories", Composi. Struct., 15, 135-145. https://doi.org/10.1016/j.compstruct.2017.09.071.
  40. Hirwani, C.K., Panda, S.K. and Patle, B.K. (2018), "Theoretical and experimental validation of nonlinear deflection and stress responses of an internally debonded layer structure using different higher-order theories", Acta Mechanica, 229(8), 3453-3473. https://doi.org/10.1007/s00707-018-2173-8.
  41. Lieu, X.Q., Lee, D.K., Kang, J.W. and Lee, J.H. (2017), "NURBSbased modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates", Mech. Adv. Mater. Struct., 26, 1-17. https://doi.org/10.1080/15376494.2018.1430273.
  42. Lieu, X.Q., Lee, S.H., Kang, J.W. and Lee, J.H. (2018), "Bending and free vibration analyses of inplane bi-directional functionally graded plates with variable thickness using isogeometric analysis", Compos. Struct., 192, 434-451. https://doi.org/10.1016/j.compstruct.2018.03.021.
  43. Lieu, X.Q. and Lee, J.H. (2019), "A reliability-based optimization approach for material and thickness composition of multidirectional functionally graded plates", Compos. Part B: Eng., 164, 599-611. https://doi.org/10.1016/j.compositesb.2019.01.089.
  44. Do, V.T., Nguyen, D.K., Nguyen, D.D., Doan, H.D. and Bui, T.Q. (2017), "A reliability-based optimization approach for material and thickness composition of multidirectional functionally graded plates", Compos. Part B: Eng., 164, 599-611. https://doi.org/10.1016/j.compositesb.2019.01.089.
  45. Wu, C.P. and Yu, L.T. (2019), "Free vibration analysis of bidirectional functionally graded annular plates using finite annular prism methods", J. Mech. Sci. Technol., 33, 2267-2279. https://doi.org/10.1007/s12206-019-0428-5.
  46. Belabed Z., Selim M.M., Slimani O., Taibi N., Tounsi A. and Hussain M. (2021), "An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells", Steel Compos. Struct., 40, 307-321. http://dx.doi.org/10.12989/scs.2021.40.2.307.
  47. Merazka B., Bouhadra A., Menasria A., Selim M.M., Bousahla A.A., Bousahla F., Tounsi A., Benrahou K.H., Tounsi, A. and Al-Zahrani M.M. (2021), "Hygro-thermo-mechanical bending response of FG plates resting on elastic foundations", Steel Compos. Struct., 39, 631-643. http://dx.doi.org/10.12989/scs.2021.39.5.631.
  48. Belblidiaa, F., Lee, J.E.B, Rechakb. S. and Hinton, E. (2001), "Topology optimization of plate structures using a single- or three-layered artificial material model", Adv. Eng. Softw., 32(2), 159-168. https://doi.org/10.1016/S0045-7949(00)00141-3.
  49. El-Sabbagh, A., Akl, W. and Baz, A. (2008), "Topology optimization of periodic Mindlin plates", Finite Element. Anal. Des., 44(8), 439-449. https://doi.org/10.1016/j.finel.2008.01.016.
  50. Sharma, N., Lalepalli, A.K. and Hirwani, C.K. (2021), "Optimal deflection and stacking sequence prediction of curved composite structure using hybrid (FEM and soft computing) technique", Eng. Comput., 37, 477-487. https://doi.org/10.1007/s00366-019-00836-8.
  51. Banh, T.T. and Lee, D.K. (2019), "Topology optimization of multidirectional variable thickness thin plate with multiple materials", Struct. Multidiscip. Optimiz., 59, 1503-1520. https://doi.org/10.1007/s00158-018-2143-8.
  52. Banh, T.T., Nguyen, X.Q., Herrmann, M., Filippou, F.C. and Lee, D.K. (2020), "Multiphase material topology optimization of Mindlin-Reissner plate with nonlinear variable thickness and Winkler foundation", Steel Compos. Struct., 35(1), 129-145. http://dx.doi.org/10.12989/scs.2020.35.1.129.
  53. Bathe K.J. and Dvorkin E. (1985), "A four node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation", Int. J. Numer. Meth. Eng., 21, 367-383. https://doi.org/10.1002/nme.1620210213.
  54. Nguyen M.N. and Bui T.Q. (2022), "Multi-material gradient-free proportional topology optimization analysis for plates with variable thickness", Struct. Multidiscip. Optimiz., 65.
  55. Duchon, J. (1977), Splines Minimizing Rotation-Invariant SemiNorms in Sobolev Spaces, In: Schempp W., Zeller K. Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, 571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086566.
  56. Franke, R. (1982), "Scattered data interpolation: tests of some methods", Mathem. Comput., 38, 181-200. http://dx.doi.org/10.1090/S0025-5718-1982-0637296-4.
  57. Siraj-ul-Islam, Khan, W., Ullah, B. and Ullah, Z. (2020), "The localized radial basis functions for parameterized level set based structural optimization", Eng. Anal. with Bound. Elements, 113, 296-305. http://dx.doi.org/10.1016/j.enganabound.2020.01.008.
  58. Roberts, S., Hegland, M. and Irfan Altas. (2004), "Approximation of a thin plate spline smoother using continuous piecewise polynomial functions", SIAM J. Numer. Anal., 41(1), 208-234. http://dx.doi.org/10.1137/S0036142901383296.
  59. Mori T. and Tanaka K. (1973), "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metallurgica, 21, 571-5747. http://doi.org/10.1016/0001-6160(73)90064-3.
  60. Thompson L.L. and Thangavelu S.R. (2002), "A stabilized MITC element for accurate wave response in Reissner-Mindlin plates", Comput. Struct., 80, 769-789. https://doi.org/10.1016/S0045-7949(02)00046-9
  61. Do T.T.D, Nguyen X.H. and Lee J. (2020), "Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach", Appl. Mathem. 501-533. https://doi.org/10.1016/j.apm.2020.06.002.
  62. Vel, S.S. and Batra, R.C. (2002), "Exact solution for thermoelastic deformations of functionally graded thick rectangular plates", AIAA J., 40(7), 1421-1433. https://doi.org/10.2514/2.1805.
  63. Qian L.F., Batra R.C. and Chen L.M. (2004), "Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method", Compos. Part B: Eng., 35, 685-697. https://doi.org/10.1016/j.compositesb.2004.02.004.
  64. Reddy, J.N. (1999), Theory and Analysis of Elastic Plates, Taylor and Francis, Philadelphia.
  65. Zhuang, Z., Zhanli L., Binbin C. and Jianhui L., (2014), Extended Finite Element Method, Tsinghua University Press Computational Mechanics Series, Elsevier Inc.
  66. Reddy, J.N. (2002), Energy Principles and Variational Methods in Appl. Mech., John Wiley, New York, NY, USA.
  67. Cheng, A.D., Golberg, M.A., Kansa, E.J. and Zammito, G. (2003), "Exponential convergence and H-c multiquadric collocation method for partial differential equations", Numer. Meth. Partial Equ., 19, 571-594. https://doi.org/10.1002/num.10062.
  68. Morse, B.S., Yoo, T.S., Chen, D.T., Rheingans, P. and Subramanian, K.R. (2001), "Interpolating implicit surfaces from scattered surface data using compactly supported radial basic functions", Int Conf Shape Model Appl, 15, 89-98. 10.1109/SMA.2001.923379.
  69. Gilhooley D.F., Batra R.C., Xiao J.R., McCarthy M.A. and Gillespie J.W. (2007), "Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions", Comput. Struct., 80, 539-552. https://doi.org/10.1016/j.compstruct.2006.07.007.
  70. Tran V.L., Ferreira A.J.M. and Nguyen-Xuan H. (2013), "Isogeometric analysis of functionally graded plates using higher-order shear deformation theory", Compos Part B: Eng., 51, 368-383. https://doi.org/10.1016/j.compositesb.2013.02.045.
  71. Nguyen X.H., Thai H.C. and Nguyen T.T. (2013), "Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory", Compos. Part B: Eng., 55, 558-574. https://doi.org/10.1016/j.compositesb.2013.06.044.
  72. Thai H.C., Ferreira A.J.M., Rabczuk T., Bordas S.P.A. and Nguyen-Xuan H. (2014), "Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory", Eur J Mech - A/Solids, 55, 89-108. https://doi.org/10.1016/j.euromechsol.2013.09.001.
  73. Nguyen X.H., Tran L.V., Thai C.H. Kulasegaram S. and Bordas S.P.A. (2014), "Isogeometric analysis of functionally graded plates using a refined plate theory", Compos. Part B: Eng., 64, 222-234. https://doi.org/10.1016/j.compositesb.2014.04.001.
  74. Arnold, D. and Falk, R. (1989), Analytical and Computational Models for Shells, American Society of Mechanical Engineers, New York, New York, United States.