Skip to main content
SpringerLink
Log in

Hierarchical topology optimization addressing material design constraints and application to sandwich-type structures

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

The hierarchical topology optimization model for multiscale design of structures addresses the problem of finding optimal material distributions at different but interconnected structural length scales with the objective of optimally design the structure and its material. In this work some new developments on this model are presented. An algorithm is mounted to address specific features of multiscale design such as multiple material design constrains. Furthermore, previous design parameterizations assume micro design variables associated with each finite element, trying to approximate a pointwise optimal material definition, and leading to very efficient designs but of problematic manufacturability. Here one reduces the total number of problem design variables by assuming a design parameterization where the design is uniform within mechanically consistent larger subdomains—“design subdomains”. This eases applicability, manufacturability, and is a very effective approach for practical design problems involving for example sandwich-type structures, where larger subdomains identify structural constituents such as a soft core between two solid face-sheets. The parameterization to include “design subdomains” and the introduction of local material design constraints requires an appropriate derivation of the optimality conditions. The main structural applications presented here are related to sandwich type of structures. The influence of the designer choices for “design subdomains” characterizing the macrostructure, and “material unit cell” representing the microstructure, will also be studied in the solutions obtained. The examples show the effectiveness of the methodology presented to fully benefit from an enlarged design space incorporating structural and material designs, and thus efficiently maximize the mechanical component structural performance.

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

Similar content being viewed by others

References

  • Andreasen CS, Sigmund O (2012) Multiscale modeling and topology optimization of poroelastic actuators. Smart Mater Struct 21(6):065005

    Article  Google Scholar 

  • Barbarosie C, Toader A-M (2012) Optimization of bodies with locally periodic microstructure. Mech Adv Mater Struct 19:290–301

    Article  Google Scholar 

  • Blasques JP, Stolpe M (2012) Multi-material topology optimization of laminated composite beam cross sections. Compos Struct 94:3278–3289

    Article  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202

    Article  Google Scholar 

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

    Google Scholar 

  • Bourgat JF (1977) Numerical experiments of the homogenization method for operators with periodic coefficients. Lecture notes in mathematics, vol 704. Springer, Berlin, pp 330–356

    Google Scholar 

  • Brittain K, Silva M, Tortorelli DA (2012) Minimax topology optimization. Struct Multidiscip Optim 45:657–668

    Article  MathSciNet  Google Scholar 

  • Burton WS, Noor AK (1997) Assessment of continuum models for sandwich panel honeycomb cores. Comput Methods Appl Mech Eng 145:341–360

    Article  Google Scholar 

  • Cadman JE, Zhou S, Chen Y, Li Q (2013) On design of multi-functional microstructural materials. J Mater Sci 48:51–66

    Article  Google Scholar 

  • Cai Y, Xu L, Cheng G (2014) Novel numerical implementation of asymptotic homogenization method for periodic plate structures. Int J Solids Struct 51(1):284–292

    Article  Google Scholar 

  • Chen Y, Zhou S, Li Q (2009) Computational design for multifunctional microstructural composites. Int J Mod Phys B 23:1345–1351

    Article  Google Scholar 

  • Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimization of three-dimensional structures. Struct Multidiscip Optim 35:107–115

    Article  Google Scholar 

  • Coelho PG, Fernandes PR, Rodrigues H (2011a) Multiscale modeling of bone tissue with surface and permeability control. J Biomech 44(2):321–329

    Article  Google Scholar 

  • Coelho PG, Cardoso JB, Fernandes PR, Rodrigues HC (2011b) Parallel computing techniques applied to the simultaneous design of structure and material. Adv Eng Softw 42:219–227

    Article  Google Scholar 

  • Czarnecki S, Kursa M, Lewinski T (2008) Sandwich plates of minimal compliance. Comput Methods Appl Mech Eng 197:4866–4881

    Article  Google Scholar 

  • Dai G, Zhang W (2009) Cell size effects for vibration analysis and design of sandwich beams. Acta Mech Sinica 25:353–365

    Article  MathSciNet  Google Scholar 

  • Deng J, Yan J, Cheng G (2013) Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct Multidiscip Optim 47:583–597

    Article  MathSciNet  Google Scholar 

  • Ferreira RTL, Rodrigues HC, Guedes JM, Hernandez J (2014) Hierarchical optimization of laminated fibre reinforced composites. Compos Struct 107:246–259

    Article  Google Scholar 

  • Ghiasi H, Fayazbakhsh K, Pasini D, Lessard L (2010) Optimum stacking sequence design of composite materials part II: variable stiffness design. Comput Struct 93:1–13

    Article  Google Scholar 

  • Guedes JM, Lubrano E, Rodrigues HC, Turteltaub (2006) Hierarchical optimization of material and structure for thermal transient problems. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Solid Mechanics and Its Applications 137:527–536

  • Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing fpr materials based on the homogenization method with adaptive finite element method. Comput Methods Appl Mech Eng 83:143–198

    Article  MathSciNet  Google Scholar 

  • Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43:7028–7047

    Article  Google Scholar 

  • Hohe J, Becker W (2002) Optimized structural sandwich panels with minimum delamination hazards. Struct Multidiscip Optim 23:448–459

    Article  Google Scholar 

  • Lions JL (1981) Some methods in mathematical analysis of systems and their control. Kexue Chubanshe Science Press, Beijing

    Google Scholar 

  • Jayachandran KP, Guedes JM, Rodrigues HC (2008) Piezoelectricity enhancement in ferroelectric ceramics due to orientation. Appl Phys Lett 92(23):232901-232901-3

    Article  Google Scholar 

  • Jones RM (1975) Mechanics of composites materials. McGraw-Hill, Washington DC

    Google Scholar 

  • Kang H, Lin C-Y, Hollister S (2010) Topology optimization of three dimensional tissue engineering scaffold architectures for prescribed bulk modulus and diffusivity. Struct Multidiscip Optim 42(4):633–644

    Article  Google Scholar 

  • Khanoki SA, Pasini D (2012) Multiscale design and multiobjective optimization of orthopedic hip implants with functionally graded cellular material. J Biomech Eng – Trans ASME 134(3) 031004

  • Le C, Bruns TE, Tortorelli DA (2012) Material microstructure optimization for linear electrodynamics energy wave management. J Mech Phys Solids 60:351–378

    Article  MathSciNet  Google Scholar 

  • Lebée A, Sab K (2012) Homogenization of cellular sandwich panels. CR Mecanique 340:320–337

    Article  Google Scholar 

  • Lee C-Y, Yu W (2011) Homogenization and dimensional reduction of composite plates with in-plane heterogeneity. Int J Solids Struct 48:1474–1484

    Article  Google Scholar 

  • Liu L, Yan J, Cheng G (2008) Optimum structure with homogeneous optimum truss-like material. Comput Struct 86:1417–1425

    Article  Google Scholar 

  • Luenberger DG (1989) Liner and nonlinear programming, 2nd edn. Addison-Wesley Publishing Company, Reading

    Google Scholar 

  • Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parameterization. Wind Energy 8:109–124

    Article  Google Scholar 

  • Mallick PK (1997) Composites engineering handbook. Marcel Dekker, Inc., New York

    Google Scholar 

  • Moses E, Fuchs MB, Ryvkin M (2003) Topological design of modular structures under arbitrary loading. Struct Multidiscip Optim 24:407–417

    Article  Google Scholar 

  • Nakshatrala PB, Tortorelli DA, Nakshatrala KB (2013) Nonlinear structural design using multiscale topology optimization. Part I: static formulation. Comput Methods Appl Mech Eng 261–262:167–176

    Article  MathSciNet  Google Scholar 

  • Niu B, Yan J, Cheng G (2009) Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscip Optim 39:115–132

    Article  Google Scholar 

  • Noor AK, Burton WS, Bert CW (1996) Computational models for sandwich panels and shells. Appl Mech Rev 49(3):155–199

    Article  Google Scholar 

  • Rodrigues H, Guedes JM, Bendsøe MP (2002) Hierarchical optimization of material and structure. Struct Multidiscip Optim 24:1–10

    Article  Google Scholar 

  • Sigmund O (1999) On the optimality of bone structure. In: Proceedings of IUTAM Symposium on Synthesis in Bio Solid Mechanics, Copenhagen, Denmark, pp 221–233

  • Sørensen SN, Lund E (2013) Topology and thickness optimization of laminated composites including manufacturing constraints. Struct Multidiscip Optim 48:249–265

    Article  MathSciNet  Google Scholar 

  • Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62:2009–2027

    Article  Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    Article  MathSciNet  Google Scholar 

  • Sypeck DJ (2005) Cellular truss core sandwich structures. Appl Compos Mater 12:229–246

    Article  Google Scholar 

  • Swan CC, Arora JS (1997) Topology design of material layout in structured composites of high stiffness and strength. Struct Optim 13:45–59

    Article  Google Scholar 

  • Taylor JE, Bendsøe MP (2001) A mutual energy formulation for optimal structural design. Struct Multidiscip Optim 22:95–101

    Article  Google Scholar 

  • Vinson JR (2001) Sandwich structures. Appl Mech Rev 54(3):201–214

    Article  MathSciNet  Google Scholar 

  • Xu S, Cheng G (2010) Optimum material design of minimum structural compliance under seepage constraint. Struct Multidiscip Optim 41:575–587

    Article  MathSciNet  Google Scholar 

  • Yan J, Cheng G, Liu L, Liu S (2008a) Concurrent material and structural optimization of hollow plate with truss-like material. Struct Multidiscip Optim 35:153–163

    Article  Google Scholar 

  • Yan J, Cheng G, Liu L (2008b) A uniform optimum material based model for concurrent optimization of thermoelastic structures and materials. Int J Simul Multidiscip Des Optim 2:259–266

    Article  Google Scholar 

Download references

Acknowledgments

Funding provided by FCT – Fundação para a Ciência e a Tecnologia, Portugal, through project PEST-OE/EME/LA0022/2013. Authors wish to thank also Professor Krister Svanberg (Royal Institute of Technology, Stockholm, Sweden) for the MMA codes. The simulations presented here were produced using the ISTCluster (Instituto Superior Técnico/Portugal).

Author information

Authors and Affiliations

  1. Mechanical and Industrial Engineering Department, Universidade Nova de Lisboa, Lisbon, Portugal

    P. G. Coelho

  2. LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    P. G. Coelho & H. C. Rodrigues

Authors
  1. P. G. Coelho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. C. Rodrigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. G. Coelho.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coelho, P.G., Rodrigues, H.C. Hierarchical topology optimization addressing material design constraints and application to sandwich-type structures. Struct Multidisc Optim 52, 91–104 (2015). https://doi.org/10.1007/s00158-014-1220-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-014-1220-x

Keywords

Search

Navigation