Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

An isogeometrical level set topology optimization for plate structures

  • Halaku, A. (Civil Engineering Department, Shahrood University of Technology) ;
  • Tavakkoli, S.M. (Civil Engineering Department, Shahrood University of Technology)
  • Received : 2021.07.17
  • Accepted : 2021.09.02
  • Published : 2021.10.10
⟨ Previous Next ⟩

Abstract

This study presents topology optimization of plate structures by employing isogeometrical level set method. For structural analysis of plates, the IsoGeometric Analysis (IGA) approach is applied and Non-Uniform Rational B-Splines (NURBS) basis functions are used for approximation of the design domain geometry as well as the unknown deformation field. In this paper, the level set function is parametrized with Radial Basis Functions (RBFs), which is more efficient than the conventional level set method. This approach along with an approximate re-initialization scheme can maintain a smooth level set function during the optimization process and has less dependency on initial designs because of its ability to nucleate new holes inside the design domain. Due to capability of IGA method in modeling complex design domains while maintaining high accuracy in analysis, combination of IGA with RBFs level set method provides a very useful and effective technique for topology optimization problems. Several numerical examples are prepared to demonstrate the efficiency and accuracy of the method and obtained optimum topologies are compared with the results of other methods in literature.

Keywords

References

  1. Abdelmoety, A.K., Naga, T.H.A. and Rashed, Y.F. (2020), "Isogeometric boundary integral formulation for Reissner's plate problems", Eng. Comput. (Swansea, Wales), 37(1), 21-53. https://doi.org/10.1108/EC-11-2018-0507.
  2. Allaire, G., Jouve, F. and Toader, A.M. (2004), "Structural optimization using sensitivity analysis and a level-set method", J. Comput. Phys., 194(1), 363-393. https://doi.org/10.1016/j.jcp.2003.09.032.
  3. Beirao Da Veiga, L., Hughes, T.J.R., Kiendl, J., Lovadina, C., Niiranen, J., Reali, A. and Speleers, H. (2015), "A locking-free model for Reissner-Mindlin plates: Analysis and isogeometric implementation via NURBS and triangular NURPS", Math. Model. Meth. Appl. Sci., 25(8), 1519-1551. https://doi.org/10.1142/S0218202515500402.
  4. Belblidia, F., Lee, J.E.B., Rechak, 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.
  5. Bendsoe, M.P. (1982), "Some smear-out models for integrally stiffened plates with applications to optimal design", Proc. Int. Symp. on Optimum Structural Design, Tucson, Arizona.
  6. 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.
  7. Bourgeat, A. and Tapiero, R. (1983), "Homogenization of a transversely perforated plate in the frame of mindlin, hencky theory, in the thermoelastic case with non uniformly oscillating coefficients", Comptes Rendus De L Academie Des Sciences Serie I-Mathematique., 297(3), 213-216.
  8. Cheng, K.T. and Olhoff, N. (1981), "An investigation concerning optimal design of solid elastic plates", Int. J. Solid. Struct., 17(3), 305-323. https://doi.org/10.1016/0020-7683(81)90065-2.
  9. Cottrell, J.A., Hughes, T.J. and Bazilevs, Y. (2009), Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons.
  10. Da Veiga, L.B., Buffa, A., Lovadina, C., Martinelli, M. and Sangalli, G. (2012), "An isogeometric method for the Reissner-Mindlin plate bending problem", Comput. Meth. Appl. Mech. Eng., 209-212, 45-53. https://doi.org/10.1016/j.cma.2011.10.009.
  11. Dunning, P.D., Ovtchinnikov, E., Scott, J. and Kim, H.A. (2016), "Level-set topology optimization with many linear buckling constraints using an efficient and robust eigensolver", Int. J. Numer. Meth. Eng., 107(12), 1029-1053. https://doi.org/10.1002/nme.5203.
  12. Dunning, P.D. and Kim, H.A. (2015), "Introducing the sequential linear programming level-set method for topology optimization", Struct. Multidisc. Optim., 51(3), 631-643. https://doi.org/10.1007/s00158-014-1174-z.
  13. Hinton, E. and Owen, D.R. (1981), "Finite elements in plasticity: Theory and practice", Appl. Ocean Res., 3(3), 149. https://doi.org/10.1016/0141-1187(81)90117-6.
  14. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput. Meth. Appl. Mech. Eng., 194(39-41), 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008.
  15. Jahangiry, H.A. and Tavakkoli, S.M. (2017), "An isogeometrical approach to structural level set topology optimization", Comput. Meth. Appl. Mech. Eng., 319, 240-257. https://doi.org/10.1016/j.cma.2017.02.005.
  16. Kansa, E.J., Power, H., Fasshauer, G.E. and Ling, L. (2004), "A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation", Eng. Anal. Bound. Elem., 28, 1191-1206. https://doi.org/10.1016/j.enganabound.2004.01.004.
  17. Lee, S.J. and Kim, H.. (2013), "Vibration and buckling of thick plates using isogeometric approach", Arch. Res., 15(1), 35-42. https://doi.org/10.5659/aikar.2013.15.1.35.
  18. Lee, U. and Shin, J. (2002), "A frequency response function-based structural damage identification method", Comput. Struct., 80(2), 117-132. https://doi.org/10.1016/S0045-7949(01)00170-5.
  19. Li, X., Zhang, J. and Zheng, Y. (2013), "NURBS-based isogeometric analysis of beams and plates using high order shear deformation theory", Math. Prob. Eng., 2013, Article ID 159027. https://doi.org/10.1155/2013/159027.
  20. Liu, N., Johnson, E.L., Rajanna, M.R., Lua, J., Phan, N. and Hsu, M.C. (2021), "Blended isogeometric Kirchhoff-Love and continuum shells", Comput. Meth. Appl. Mech. Eng., 385, 114005. https://doi.org/10.1016/j.cma.2021.114005.
  21. Liu, N., Beata, P.A. and Jeffers, A.E. (2019), "A mixed isogeometric analysis and control volume approach for heat transfer analysis of nonuniformly heated plates", Numer. Heat Transf., Part B: Fundament., 75(6), 347-362. https://doi.org/10.1080/10407790.2019.1627801.
  22. Liu, N. and Jeffers, A.E. (2017), "Isogeometric analysis of laminated composite and functionally graded sandwich plates based on a layerwise displacement theory", Compos. Struct., 176, 143-153. https://doi.org/10.1016/j.compstruct.2017.05.037.
  23. Liu, N. and Jeffers, A.E. (2018), "A geometrically exact isogeometric Kirchhoff plate: Feature-preserving automatic meshing and C1 rational triangular Bezier spline discretizations", Int. J. Numer. Meth. Eng., 115(3), 395-409. https://doi.org/10.1002/nme.5809.
  24. Liu, N., Ren, X. and Lua, J. (2020), "An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures", Compos. Struct., 237, 111893. https://doi.org/10.1016/j.compstruct.2020.111893.
  25. Lurie, K.A. and Cherkaev, A. (1976), "On applying {Prager}'s theorem to the problems of optimal design of thin plates", Mech. Solid., 11(6), 157-159.
  26. Morse, B.S., Yoo, T.S., Rheingans, P., Chen, D.T. and Subramanian, K.R. (2005), "Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions", ACM SIGGRAPH 2005 Courses, SIGGRAPH 2005. https://doi.org/10.1145/1198555.1198645.
  27. Nakazawa, Y., Kogiso, N., Yamada, T. and Nishiwaki, S. (2016), "Robust topology optimization of thin plate structure under concentrated load with uncertain load position", J. Adv. Mech. Des. Syst. Manuf., 10(4), 1-12. https://doi.org/10.1299/JAMDSM.2016JAMDSM0057.
  28. Olhoff, N., Lurie, K.A., Cherkaev, A.V. and Fedorov, A.V. (1981), "Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates", Int. J. Solid. Struct., 17(10), 931-948. https://doi.org/10.1016/0020-7683(81)90032-9.
  29. Osher, S. and Sethian, J.A. (1988), "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations", J. Comput. Phys., 79(1), 12-49. https://doi.org/10.1016/0021-9991(88)90002-2.
  30. Otomori, M., Yamada, T., Izui, K. and Nishiwaki, S. (2015), "Matlab code for a level set-based topology optimization method using a reaction diffusion equation", Struct. Multidisc. Optim., 51(5), 1159-1172. https://doi.org/10.1007/s00158-014-1190-z.
  31. Piegl, L. and Tiller, W. (1977), The NURBS Book, New York Tech Science Press, USA.
  32. 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.
  33. Soto, C.A. and Diaz, A.R. (1993), "Optimum layout and shape of plate structures using homogenization", Topol. Des. Struct., 407-420. https://doi.org/10.1007/978-94-011-1804-0_29.
  34. Suzuki, K. and Kikuchi, N. (1991), "Generalized layout optimization of three-dimensional shell structures", Proceedings of the Conference on Design Theory, SIAM, Ed. V, Komkov, V., Philadelphia.
  35. Thai, C.H., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T.H., Nguyen-Thoi, T. and Rabczuk, T. (2012). "Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach", Int. J. Numer. Meth. Eng., 91(6), 571-603. https://doi.org/10.1002/nme.4282.
  36. Wang, M.Y., Wang, X. and Guo, D. (2003), "A level set method for structural topology optimization", Comput. Meth. Appl. Mech. Eng., 192(1-2), 227-246. https://doi.org/10.1016/S0045-7825(02)00559-5.
  37. Wang, S.Y., Lim, K.M., Khoo, B.C. and Wang, M.Y. (2007), "An extended level set method for shape and topology optimization", J. Comput. Phys., 221(1), 395-421. https://doi.org/10.1016/j.jcp.2006.06.029.
  38. Wang, Y. and Benson, D.J. (2016), "Isogeometric analysis for parameterized LSM-based structural topology optimization", Comput. Mech., 57(1), 19-35. https://doi.org/10.1007/s00466-015-1219-1.
  39. Wei, P., Li, Z., Li, X. and Wang, M.Y. (2018), "An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions", Struct. Multidisc. Optim., 58(2), 831-849. https://doi.org/10.1007/s00158-018-1904-8.
  40. Wei, P. and Wang, M.Y. (2006), "The augmented lagrangian method in structural shape and topology optimization with RBF based level set method", Proceedings of the Fourth China-Japan-Korea Joint symposium on Optimization of Structural and Mechanical Systems, Kunming, November.
  41. Yamada, T., Izui, K., Nishiwaki, S. and Takezawa, A. (2010), "A topology optimization method based on the level set method incorporating a fictitious interface energy", Comput. Meth. Appl. Mech. Eng., 199(45-48), 2876-2891. https://doi.org/10.1016/j.cma.2010.05.013.