Skip to main content
SpringerLink
Log in

Topology optimization for structure with multi-gradient materials

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

Functionally gradient materials (FGM) widely exist in biological structures, which offer indispensable support and potential protection. In this paper, we expand topology optimization for coated-base materials to multi-gradient structure. In the process of modeling the biological bones, we define three levels of density gradients: a spongy bone as the base in between, wrapped by a layer of compact bone, and multi-layers of periosteum side by side. By defining the properties and thickness of each layer, the base domain is derived by the gradient norm while a modified non-monotone filter function is imposed to separate periosteum and compact bone. Meanwhile, periosteum is separated into multi-gradient layers by Heaviside projection function in order to further mimic the skeletal of bone structure. The method of robust control, using geometric constraints, is applied to the design domain in both solid and void phases to avoid local singularity during the optimal process. Finally, the validity of the algorithm is illustrated by two benchmark problems with a variety of layer combinations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  • Al-Shamary A, Karakuzu R, Özdemir O (2016) Low-velocity impact response of sandwich composites with different foam core configurations. J Sandw Struct Mater 18:754–768

    Article  Google Scholar 

  • Beaupré GS, Orr TE, Carter DR (1990) An approach for time-dependent bone modeling and remodeling–application: a preliminary remodeling simulation. J Orthop 8:662–670

    Google Scholar 

  • Behrou R, Lawry M, Maute K (2017) Level set topology optimization of structural problems with interface cohesion. Internat J Numer Methods Engrg 112:990–1016

    Article  MathSciNet  Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654

    Article  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology optimization. Theory, methods and applications, Springer

  • Chu S, Xiao M, Gao L (2019) Topology optimization of multi-material structures with graded interfaces. Comput Methods Appl Mech Engrg 346:1096–1117

    Article  MathSciNet  Google Scholar 

  • Clausen A, Aage N, Sigmund O (2015) Topology optimization of coated structures and material interface problems. Comput Methods Appl Mech Engrg 290:524–541

    Article  MathSciNet  Google Scholar 

  • Clausen A, Aage N, Sigmund O (2016) Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering 2:250–257

    Article  Google Scholar 

  • Clausen A, Andreassen E (2017) On filter boundary conditions in topology optimization. Struct Multidiscip Optim 56:1147–1155

    Article  MathSciNet  Google Scholar 

  • Dumont E (2010) Bone density and the lightweight skeletons of birds. Proc Royal Soc B Bio Sci 277:2193–2198

    Article  Google Scholar 

  • Fischer KJ, Jacobs CR, Levenston ME (1997) Observation of convergence and uniqueness of node-based bone remodeling simulations. Ann Biomech Eng 25:261–268

    Article  Google Scholar 

  • Groen J, Wu J, Sigmund O (2019) Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill. Comput Methods Appl Mech Engrg 349:722–742

    Article  MathSciNet  Google Scholar 

  • Hägg L, Wadro E (2018) On minimum length scale control in density based topology optimization. Struct Multidiscip Optim 58:1015–1032

    Article  MathSciNet  Google Scholar 

  • Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids 11:127–140

    Article  MathSciNet  Google Scholar 

  • Hoang V, Nguyen N, Nguyen H (2019) Topology optimization of coated structure using moving morphable sandwich bars. Struct Multidiscip Optim 61:419–506

    Google Scholar 

  • Kawamoto A, Matsumori T, Yamasaki S (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim 44:19–24

    Article  Google Scholar 

  • Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86:765–781

    Article  MathSciNet  Google Scholar 

  • Lazarov B, Wang F (2017) Maximum length scale in density based topology optimization. Comput Methods Appl Mech Engrg 318:826–844

    Article  MathSciNet  Google Scholar 

  • Luo Y, Li Q, Liu S (2019) A projection-based method for topology optimization of structures with graded surfaces. Int J Numer Methods Eng 118:654–677

    Article  MathSciNet  Google Scholar 

  • Naebe M, Shirvanimoghaddam K (2016) Functionally graded materials: a review of fabrication and properties. Appl Mater Today 5:223–245

    Article  Google Scholar 

  • Pack PE (2001) Anatomy and physiology, Hungry Minds INC, USA

  • Petrudi A, Rahmani M (2020) Numerical and analytical study of fatigue and degradation in multilayer composite plates. Rev des Compos et des Mater Av 30:61–68

    Google Scholar 

  • Rice JC, Cowin SC, Bowman JA (1988) On the dependence of the elasticity and strength of compact bone on apparent density. Biomechanics 155–168:21

    Google Scholar 

  • Sarathchandra D (2016) Functionally graded materials and processing techniques ICSEM. https://doi.org/10.1016/j.matpr.2018.06.536

  • Schevenels M, Sigmund O (2016) On the implementation and effectiveness of morphological close-open and open-close filters for topology optimization. Struct Multidiscip Optim 54:15–21

    Article  MathSciNet  Google Scholar 

  • Sullivan T, Wang B, Espinosa H (2017) Extreme lightweight structures: avian feathers and bones. Mater Today 20:377–391

    Article  Google Scholar 

  • Svanberg K (1986) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    Article  MathSciNet  Google Scholar 

  • Vermaak N, Michailidis G, Parry G (2014) Material interface effects on the topology optimization of multi-phase structures using a level set method. Struct Multidiscip Optim 50:623–644

    Article  MathSciNet  Google Scholar 

  • Vermaak N, Michailidis G, Parry G, Estevez R, Allaire G, Bréchet Y (2014) Material interface effects on the topology optimization of multi-phase structures using a level set method. Struct Multidiscip Optim 50:623–644

    Article  MathSciNet  Google Scholar 

  • Wadbro E, Niu B (2019) Multiscale design for additive manufactured structures with solid coating and periodic infill pattern. Comput Methods Appl Mech Engrg 357:112605

    Article  MathSciNet  Google Scholar 

  • Wang Y, Chen F, Wang M (2017) Concurrent design with connectable graded microstructures. Comput Methods Appl Mech Engrg 317:84–101

    Article  MathSciNet  Google Scholar 

  • Wang Y, Kang Z (2018) A level set method for shape and topology optimization of coated structures, vol 329

  • Wang F, Lazarov O, Sigmund O (2010) On projection methods, convergence and robust formulations in topology optimization, vol 43

  • Wang B, Meyers M (2017) Seagull feather shaft: correlation between structure and mechanical response. Acta Biomater 48:270–288

    Article  Google Scholar 

  • Wu J, Clausen A, Sigmund O (2017) Minimum compliance topology optimization of shell–infill composites for additive manufacturing. Comput Methods Appl Mech Engrg 326:358–375

    Article  MathSciNet  Google Scholar 

  • Xu F, Zhang X, Zhang H (2018) A review on functionally graded structures and materials for energy absorption. Eng Struct 171:309–325

    Article  Google Scholar 

  • Yoon G, Yi B (2019) A new coating filter of coated structure for topology optimization. Struct Multidiscip Optim 60:1527–1544

    Article  Google Scholar 

  • Zhang J, Qin Q, Chen S, et al. (2020) Low-velocity impact of multilayer sandwich beams with metal foam cores: analytical, experimental, and numerical investigations. J Sandw Struct Mater 22:626–657

    Article  Google Scholar 

  • Zhou M, Lazarov B, Wang F (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Engrg 293:266–282

    Article  MathSciNet  Google Scholar 

  • Zhu J, Zhang W, Xia L (2015) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23:595–622

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China under Grant U1613225.

Author information

Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Yihao Dong, Xinfu Liu, Tao Song & Shaoming He

Authors
  1. Yihao Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinfu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shaoming He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Song.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Responsible Editor: Ole Sigmund

Replication of results

This paper proposed a bone-inspired multi-gradient material mode based on SIMP and several known filter methods, which is graphically proposed in Fig. 3. The result code is available as supplementary material for optimal structure in Fig. 12. Other results follow the optimal process and sensitivity analysis in Section 4.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, Y., Liu, X., Song, T. et al. Topology optimization for structure with multi-gradient materials. Struct Multidisc Optim 63, 1151–1167 (2021). https://doi.org/10.1007/s00158-020-02749-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-020-02749-3

Keywords

Search

Navigation