Structural and Multidisciplinary Optimization

, Volume 59, Issue 4, pp 1033–1051 | Cite as

Shape preserving design of geometrically nonlinear structures using topology optimization

  • Yu LiEmail author
  • Jihong ZhuEmail author
  • Fengwen Wang
  • Weihong Zhang
  • Ole Sigmund
Research Paper


Subparts of load carrying structures like airplane windows or doors must be isolated from distortions and hence structural optimization needs to take such shape preserving constraints into account. The paper extends the shape preserving topology optimization approach from simple linear load cases into geometrically nonlinear problems with practical significance. Based on an integrated deformation energy function, an improved warpage formulation is proposed to measure the geometrical distortion during large deformations. Structural complementary elastic work is assigned as the objective function. The average distortion calculated as the integrated deformation energy accumulated in the incremental loading process is accordingly constrained to obtain warpage control. In the numerical implementation, an energy interpolation scheme is utilized to alleviate numerical instability in low stiffness regions. An additional loading case avoids isolation phenomena. Optimization results show that shape preserving design is successfully implemented in geometrically nonlinear structures by effectively suppressing local warping deformations.


Shape preserving design Topology optimization Geometrical nonlinearity Integrated deformation energy 



Yu Li received financial support from CSC (China Scholarship Council). Fengwen Wang and Ole Sigmund received support from the Villum foundation through the VILLUM Investigator project InnoTop. This work is also supported by the National Key Research and Development Program (2017YFB1102800) and the National Natural Science Foundation of China (11722219, 11620101002, 51790171, 5171101743).


  1. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055MathSciNetCrossRefGoogle Scholar
  2. Guo X, Cheng G-D (2010) Recent development in structural design and optimization. Acta Mech Sinica 26(6):807–823MathSciNetCrossRefzbMATHGoogle Scholar
  3. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRefGoogle Scholar
  4. Zhu J-H, Zhang W-H, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Meth Eng 23(4):595–622MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cho S, Jung H-S (2003) Design sensitivity analysis and topology optimization of displacement–loaded non-linear structures. Comput Methods Appl Mech Eng 192(22):2539–2553CrossRefzbMATHGoogle Scholar
  6. Zhu J-H, Zhao Y-B, Zhang W-H, Gu X-J, Gao T, Kong J, Shi G-H, Xu Y-J, Quan D-L (2018) Bio-inspired feature-driven topology optimization for rudder structure design, Engineered Science.
  7. Paris J, Navarrina F, Colominas I, Casteleiro M (2010) Stress constraints sensitivity analysis in structural topology optimization. Comput Methods Appl Mech Eng 199(33-36):2110–2122MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: a b-spline finite cell method combined with level set function. Comput Methods Appl Mech Eng 278:361–387MathSciNetCrossRefzbMATHGoogle Scholar
  9. Zhu J-H, Hou J, Zhang W-H, Li Y (2014) Structural topology optimization with constraints on multi-fastener joint loads. Struct Multidiscip Optim 50(4):561–571CrossRefGoogle Scholar
  10. Gao T, Qiu L, Zhang W (2017) Topology optimization of continuum structures subjected to the variance constraint of reaction forces. Struct Multidiscip Optim 56(4):755–765MathSciNetCrossRefGoogle Scholar
  11. Krog L, Tucker A, Kemp M, Boyd R (2004) Topology optimization of aircraft wing box ribs. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference, pp 1–11Google Scholar
  12. Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84CrossRefGoogle Scholar
  13. Maute K, Allen M (2004) Conceptual design of aeroelastic structures by topology optimization. Struct Multidiscip Optim 27(1):27–42CrossRefGoogle Scholar
  14. Zuo ZH, Xie YM (2014) Evolutionary topology optimization of continuum structures with a global displacement control. Comput Aided Des 56:58–67CrossRefGoogle Scholar
  15. Zhu J-H, Li Y, Zhang W-H, Hou J (2016) Shape preserving design with structural topology optimization. Struct Multidiscip Optim 53(4):893–906MathSciNetCrossRefGoogle Scholar
  16. Li Y, Zhu JH, Zhang WH, Wang L (2017) Structural topology optimization for directional deformation behavior design with the orthotropic artificial weak element method. Struct Multidiscip Optim 57(3):1251–1266MathSciNetCrossRefGoogle Scholar
  17. Castro MS, Silva OM, Lenzi A, Neves MM (2018) Shape preserving design of vibrating structures using topology optimization. Struct Multidiscip Optim 58(3):1109–1119MathSciNetCrossRefGoogle Scholar
  18. Buhl T, Pedersen CB, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optim 19(2):93–104CrossRefGoogle Scholar
  19. Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57(10):1413–1430CrossRefzbMATHGoogle Scholar
  20. Yoon GH, Kim YY (2005) Element connectivity parameterization for topology optimization of geometrically nonlinear structures. Int J Solids Struct 42(7):1983–2009MathSciNetCrossRefzbMATHGoogle Scholar
  21. Klarbring A, Strömberg N (2013) Topology optimization of hyperelastic bodies including non-zero prescribed displacements. Struct Multidiscip Optim 47(1):37–48MathSciNetCrossRefzbMATHGoogle Scholar
  22. Wang F, Lazarov BS, Sigmund O, Jensen JS (2014) Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Comput Methods Appl Mech Eng 276:453–472MathSciNetCrossRefzbMATHGoogle Scholar
  23. Luo Y, Wang MY, Kang Z (2015) Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique. Comput Methods Appl Mech Eng 286:422–441MathSciNetCrossRefzbMATHGoogle Scholar
  24. Pedersen P (1998) Elasticity-anisotropy-laminates. Department of Mechanical Engineering Technical University of DenmarkGoogle Scholar
  25. Bendsøe MP, Sigmund O (2004) Topology optimization - theory, methods and applications. Springer Verlag, BerlinzbMATHGoogle Scholar
  26. Zienkiewicz OC, Taylor RL, Taylor RL, Zhu J (2013) Finite Element Method: Its basis and fundamentals, The: Its basis and fundamentals. Elsevier, AmsterdamzbMATHGoogle Scholar
  27. Kemmler R, Lipka A, Ramm E (2005) Large deformations and stability in topology optimization. Struct Multidiscip Optim 30(6):459–476MathSciNetCrossRefzbMATHGoogle Scholar
  28. Wallin M, Ivarsson N, Tortorelli D (2018) Stiffness optimization of non-linear elastic structures. Comput Methods Appl Mech Eng 330:292–307MathSciNetCrossRefGoogle Scholar
  29. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43(1):1–16CrossRefzbMATHGoogle Scholar
  30. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784CrossRefzbMATHGoogle Scholar
  31. Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefzbMATHGoogle Scholar
  32. Lahuerta RD, Simões ET, Campello EM, Pimenta PM, Silva EC (2013) Towards the stabilization of the low density elements in topology optimization with large deformation. Comput Mech 52(4):779–797MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wang F (2018) Systematic design of 3d auxetic lattice materials with programmable Poisson’s ratio for finite strains. J Mech Phys Solids 114:303–318MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State IJR Center of Aerospace Design and Additive ManufacturingNorthwestern Polytechnical UniversityXianChina
  2. 2.MIIT Laboratory of Metal Additive Manufacturing and Innovative DesignNorthwestern Polytechnical UniversityXianChina
  3. 3.Department of Mechanical Engineering, Section for Solid MechanicsTechnical University of DenmarkLyngbyDenmark

Personalised recommendations