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Optimization of three-dimensional truss-like periodic materials considering isotropy constraints

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Abstract

In recent years, much attention has been directed to the study of ultralight periodic cellular materials, such as the so called Periodic Truss Materials (PTMs), which are made up of truss-like unit cells. Application of these materials has its best potential in structures subjected to multifunctional, and sometimes conflicting, engineering requirements. Hence, optimization techniques can be employed to help finding the shape of the optimal unit cell for a given multifunctional application. Although the result can be geometrically complex, this difficulty can be minored in view of modern additive manufacturing technologies. This work presents a parameter/topology optimization procedure to design the particular unit cell geometry (that is, finding the cross sections of the bars) that results in a macroscopic material with optimum homogenized elastic or/and thermal constitutive properties. Emphasis is devoted to analyze the effect of enforcing independently elastic or thermal isotropy in the macroscopic material behavior. Although isotropic behavior can be imposed through adequate cell symmetries, an equivalent effect can be achieved by satisfying equality constraints relating constitutive coefficients. A sequential quadratic programming algorithm (SQP) is adopted, thus enforcing equality constraints gradually and within a tolerance range. This results in an enlarged search space at intermediate stages, rendering an effective strategy to solve the optimization problem. Different 3D cases with engineering appeal are solved and the results discussed.

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References

  • Ai J, Kiasat-Dolatabadi A, Ebrahimi-Barough S, Ai A, Lotfibakhshaiesh N, Norouzi-Javidan A, Arjmand B, Aghayan HR (2013) Polymeric scaffolds in neural tissue engineering: a review. Arch Neurosci 1(1):15–20

    Article  Google Scholar 

  • Amstutz S, Giusti SM, Novotny AA, de Souza Neto EA (2010) Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int J Numer Meth Eng 84(6):733–756

    Article  MATH  Google Scholar 

  • Böhlke T, Brüggemann C (2001) Graphical representation of the generalized Hooke’s law. Tech Mech 21(2):145–158

    Google Scholar 

  • Büchmann T, Stenger N, Kadic M, Kaschke J, Frölich A, Kennerknecht T, Eberl C, Thiel M, Wegener M (2012) Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv Mater 24(20):2710–2714

    Article  Google Scholar 

  • Challis VJ, Roberts AP, Wilkins AH (2008) Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int J Solids Struct 45(14–15):4130–4146

    Article  MATH  Google Scholar 

  • Challis VJ, Roberts AP, Grotowski JF, Zhang L-C, Sercombe TB (2010) Prototypes for bone implant scaffolds designed via topology optimization and manufactured by solid freeform fabrication. Adv Eng Mater 12(11):1106–1110

    Article  Google Scholar 

  • Chen Y, Zhou S, Li Q (2011) Microstructure design of biodegradable scaffold and its effect on tissue regeneration. Biomaterials 32(22):5003–5014

    Article  Google Scholar 

  • Cheng G-D, Cai Y-W, Xu L (2013) Novel implementation of homogenization method to predict effective properties of periodic materials. Acta Mech Sin 29(4):550–556

    Article  MathSciNet  Google Scholar 

  • de Kruijf N, Zhou S, Li Q, Mai YW (2007) Topological design of structures and composite materials with multiobjectives. Int J Solids Struct 44(22–23):7092–7109

    Article  MATH  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  MATH  Google Scholar 

  • Ghasemi-Mobarakeh L, Prabhakaran MP, Morshed M, Nasr-Esfahani MH, Baharvand H, Kiani S, Al-Deyab S, Ramakrishna S (2011) Application of conductive polymers, scaffolds and electrical stimulation for nerve tissue engineering. J Tissue Eng Regen Med 5(4):e17–e35

    Article  Google Scholar 

  • Gibson LJ, Ashby MF (1997) Cellular solids: structure and properties, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids 43(22–23):7028–7047

  • Guth DC, Luersen MA, Muñoz-Rojas PA (2012) Optimization of periodic truss materials including constitutive symmetry constraints. Mater Werkst 43(5):447–456

    Article  Google Scholar 

  • Hassani B, Hinton E (1999) Homogenization and structural topology optimization. Springer Verlag, London

    Book  MATH  Google Scholar 

  • Hollister SJ, Lin CY (2007) Computational design of tissue engineering scaffolds. Comput Methods Appl Mech Eng 196(31–32):2991–2998

    Article  MATH  Google Scholar 

  • Kang H, Lin CY, Hollister SJ (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 

  • Kooistra GW, Wadley HNG (2007) Lattice truss structures from expanded metal sheet. Mater Des 28(2):507–514

    Article  Google Scholar 

  • Lee B, Kang K (2010) A parametric study on compressive characteristics of wire-woven bulk Kagome truss cores. Compos Struct 92(2):445–453

    Article  Google Scholar 

  • Meille S, Garboczi EJ (2001) Linear elastic properties of 2D and 3D models of porous materials made from elongated objects. Model Simul Mater Sci Eng 9(5):371–390

    Article  Google Scholar 

  • Meng S, Rouabhia M, Zhang Z (2011) Electrical stimulation in tissue regeneration, Applied Biomedical Engineering, Dr. Gaetano Gargiulo (Ed.), ISBN: 978-953-307-256-2, InTech, Available from: http://www.intechopen.com/books/applied-biomedical-engineering/electrical-stimulation-in-tissueregeneration

  • Muñoz-Rojas PA, Carniel TA, Silva ECN, Öchsner A (2010) Optimization of a unit periodic cell in lattice block materials aimed at thermo-mechanical applications. In: Öchsner A, Murch G (eds) Heat transfer in multi-phase materials. Springer, Berlin

    Google Scholar 

  • Muñoz-Rojas PA, Luersen MA, Carniel TA, Bertoti E (2011) Design of multifunctional truss-like periodic materials using a global–local optimization method. Defect Diffus Forum 312–315:1073–1078

    Article  Google Scholar 

  • Neves MM, Rodrigues H, Guedes JM (2000) Optimal design of periodic linear elastic microstructures. Comput Struct 76(1–3):421–429

    Article  Google Scholar 

  • Ravari MRK, Kadkhodaei M, Badrossamay M, Rezaei R (2014) Numerical investigation on mechanical properties of cellular lattice. Int J Mech Sci 88:154–161

    Article  Google Scholar 

  • Schittkowski K (2009) NLPQLP: a Fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search - user’s guide, version 3.0. Report, Department of Computer Science, University of Bayreuth

  • Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20(4):351–368

    Article  MathSciNet  Google Scholar 

  • Subramanian A, Krishnan UM, Sethuraman S (2009) Development of biomaterial scaffold for nerve tissue engineering: biomaterial mediated neural regeneration. J Biomed Sci 16:108–118

    Article  Google Scholar 

  • Torquato S (2001) Random heterogeneous materials: microstructure and macroscopic properties. Springer, New York

    Google Scholar 

  • Tromans D (2011) Elastic anisotropy of HCP metal crystals and polycrystals. Int J Res Rev Appl Sci 6(4):462–483

    MathSciNet  Google Scholar 

  • Valdevit L, Jacobsen AJ, Greer JR, Carter WB (2011) Protocols for the optimal design of multi-functional cellular structures: from hypersonics to micro-architected materials. J Am Ceram Soc 94(s1):s15–s34

    Article  Google Scholar 

  • Vasiliev VV, Barynin VA, Razin AF (2012) Anisogrid composite lattice structures – development and aerospace applications. Compos Struct 94(3):1117–1127

    Article  Google Scholar 

  • Wang HV (2005) A unit cell approach for lightweight structure and compliant mechanism. Ph.D. Thesis, Georgia Institute of Technology

  • Wang B, Yan J, Cheng G (2011) Optimal structure design with low thermal directional expansion and high stiffness. Eng Optim 43(6):581–595

    Article  Google Scholar 

  • Whulanza Y (2011) Engineering composite conductive scaffolds for locomotor tissue regeneration. Ph. D. Thesis, University of Pisa

  • Wieding J, Wolf A, Bader R (2014) Numerical optimization of open-porous bone scaffold structures to match the elastic properties of human cortical bone. J Mech Behav Biomed Mater 37:56–68

    Article  Google Scholar 

  • Williams CB, Cochran JK, Rosen DW (2011) Additive manufacturing of mettalic cellular materials via three-dimensional printing. Int J Adv Manuf Technol 53(1–4):231–239

    Article  Google Scholar 

  • Yan J, Cheng G, Liu L (2006) Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure. Int J Mech Sci 48:400–413

    Article  MATH  Google Scholar 

  • Yan J, Cheng G, Liu L (2008) 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

The authors acknowledge the use of the NLPQLP optimization code, kindly provided by prof. K. Schittkowski for research and academic purposes. They also wish to recognize and acknowledge the supportive collaboration given by M. Sc. Jorge Luís Suzuki who helped handling technical details in the elaboration of the article.

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Authors and Affiliations

  1. Laboratório de Mecânica Computacional, Departamento de Engenharia Mecânica, Universidade do Estado de Santa Catarina, Joinville, SC, Brazil

    D. C. Guth & P. A. Muñoz-Rojas

  2. Laboratório de Mecânica Estrutural, Universidade Tecnológica Federal do Paraná, Avenida Sete de Setembro, 3165, Curitiba, PR, Brazil

    M. A. Luersen

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  1. D. C. Guth
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  2. M. A. Luersen
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  3. P. A. Muñoz-Rojas
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Correspondence to P. A. Muñoz-Rojas.

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Guth, D.C., Luersen, M.A. & Muñoz-Rojas, P.A. Optimization of three-dimensional truss-like periodic materials considering isotropy constraints. Struct Multidisc Optim 52, 889–901 (2015). https://doi.org/10.1007/s00158-015-1282-4

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  • DOI: https://doi.org/10.1007/s00158-015-1282-4

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