Designing stress for optimizing and toughening truss-like structures


Optimization of materials and structures is a crucial step in the design of man-made mechanical components for a wide field of engineering applications. It also plays a key role in mechanobiology of living systems, being involved by nature across the scales, from single-cell to tissues and organs, as a strategy to minimize metabolic cost and maximize biomechanical performances. The synergy between  the continuously increasing development of high-resolution 3D printing technologies  and the possibility to predict chemical and physical properties through molecular dynamics-based numerical analyses has recently contributed to boost the use of both design and topology optimization procedures. They are employed in ab initio simulations as key strategies for deciding microstructures to improve mechanical performances and, concretely, to achieve prototypes of new material components. With this in mind, we here propose to abandon the classical approach of using a single scalar objective function employed in the classical design and topology optimization strategies, to introduce multiple quantities to be minimized, identified as the differences between material yield stress and the maximum von Mises stress. After mathematically justifying the well-posedness  of this unconventional choice for the case at hand, it is highlighted that the proposed strategy is based on the concept of "equalizing" a proper stress measure at any point of the body and, for this reason, it is baptized as Galilei’s optimization, in honor of the Italian scholar who somehow first wondered about the possibility of changing sizes of beams to have uniform internal forces and, in turn, minimum weight. By exploiting analytical solutions and ad hoc implementing a parametric finite element algorithm to be applied to a wide variety of solids with arbitrary complex structural geometries, including nested or hierarchically organized architectures, it is first demonstrated that the proposed optimization strategy roughly retraces principles invoked by nature to guide growth, remodeling and shaping of biomaterials. More importantly, by means of several benchmark examples, we finally show the proposed procedure might be also helpfully employed to conceive a new class of micro-structured, eventually 3D-printed materials exhibiting surprising post-elastic properties, such as high overall resilience and toughness, in particular obtaining a decrease of stress concentration and a slowing down of crack propagation as direct effects of the optimization, which de facto minimizes stress gradients wherever in the solid domain.

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  1. 1.

    Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Ahrari A, Atai AA, Deb K (2015) Simultaneous topology, shape and size optimization of truss structures by fully stressed design based on evolution strategy. Eng Optim 47(8):1063–1084

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ahrari A, Deb K (2016) An improved fully stressed design evolution strategy for layout optimization of truss structures. Comput Struct 164:127–144

    Article  Google Scholar 

  4. 4.

    Andreasen CS, Sigmund O (2013) Topology optimization of fluid–structure-interaction problems in poroelasticity. Comput Methods in Appl Mech Eng 258:55–62

  5. 5.

    Balduzzi G, Aminbaghai M, Sacco E, Füssl J, Eberhardsteiner J, Auricchio F (2016) Non-prismatic beams: a simple and effective Timoshenko-like model. Int J Solids Struct 90:236–250

    Article  Google Scholar 

  6. 6.

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

    Article  Google Scholar 

  7. 7.

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. 8.

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

    MATH  Article  Google Scholar 

  9. 9.

    Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications, 2nd edn. Springer, Berlin

    Google Scholar 

  10. 10.

    Borrvall T, Petersen J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Bruns TE (2007) Topology optimization of convection-dominated, steady-state heat transfer problem. Int J Heat Mass Transf 50:2859–2873

    MATH  Article  Google Scholar 

  12. 12.

    Byun JK, Hahn SY (2001) Application of topology optimization to electromagnetic system. Int J Appl Electrom 13:25–33

    Google Scholar 

  13. 13.

    Carotenuto AR, Cutolo A, Petrillo A, Fusco R, Arra C, Sansone M, Larobina D, Cardoso L, Fraldi M (2018) Growth and in vivo stresses traced through tumor mechanics enriched with predator-prey cells dynamics. J Mech Behav Biomed Mat 86:55–70

  14. 14.

    Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37(5):447–465

    MATH  Article  Google Scholar 

  15. 15.

    Deng Y, Korvink JG (2018) Self-consistent adjoint analysis for topology optimization of electromagnetic waves. J Comput Phys 361:353–376

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Dühring MB, Jensen JS, Sigmund O (2008) Acoustic design by topology optimization. J Sound Vib 317:557–575

    ADS  Article  Google Scholar 

  17. 17.

    Esposito L, Cutolo A, Barile M, Lecce L, Mensitieri G, Sacco E, Fraldi M (2019) Topology optimization-guided stiffening of composites realized through automated fiber placement. Compos Part B Eng 164:309–323

    Article  Google Scholar 

  18. 18.

    Feury C, Geradin M (1978) Optimality criteria and mathematical programming in structural weight optimization. Comput Struct 8(1):7–17

    MATH  Article  Google Scholar 

  19. 19.

    Fraldi M, Esposito L, Perrella G, Cutolo A, Cowin SC (2010) Topological optimization in hip prosthesis design. Biomech Model Mechanobiol 9(4):389–402

    Article  Google Scholar 

  20. 20.

    Jang IG, Kim IY (2008) Computational study of Wolff’s law with trabecular architecture in the human proximal femur using topology optimization. J Biomech 41:2353–2361

    Article  Google Scholar 

  21. 21.

    Kato J, Hoshiba H, Takase S, Terada K, Kyoya T (2015) Analytical sensitivity in topology optimization for elastoplastic composites. Struct Multidiscip Optim 52(3):507–526

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kirsch G (1898) Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Zeitschrift des Vereines Deutscher Ingenieure 42:797–807

    Google Scholar 

  23. 23.

    Minutolo V, Ruocco E, Ciaramella S (2009) Isoparametric FEM vs. BEM for elastic functionally graded materials. CMES 41(1):27–48

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Noilublao N, Bureerat S (2011) Simultaneous topology, shape and sizing optimisation of a three-dimensional slender truss tower using multiobjective evolutionary algorithms. Comput Struct 89(23–24):2531–2538

    Article  Google Scholar 

  25. 25.

    Panagant N, Bureerat S (2018) Truss topology, shape and sizing optimization by fully stressed design based on hybrid grey wolf optimization and adaptive differential evolution. Eng Optim 50(10):1645–1661

    MathSciNet  Article  Google Scholar 

  26. 26.

    Papadrakakis M, Lagaros N, Plevris V (2002) Multi-objective optimization of skeletal structures under static and seismic loading conditions. Eng Optim 34(6):645–669

    MATH  Article  Google Scholar 

  27. 27.

    Rahami H, Kaveh A, Gholipour Y (2008) Sizing, geometry and topology optimization of trusses via force method and genetic algorithm. Eng Struct 30(9):2360–2369

    Article  Google Scholar 

  28. 28.

    Saka M (1990) Optimum design of pin-jointed steel structures with practical applications. J Struct Eng ASCE 116(10):2599–2620

    Article  Google Scholar 

  29. 29.

    Seki Y, Kad B, Benson D, Meyers MA (2010) The toucan beak: structure and mechanical response. Mater Sci Eng C 26:1412–1420

    Article  Google Scholar 

  30. 30.

    Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055

    MathSciNet  Article  Google Scholar 

  31. 31.

    Stolpe M (2010) On some fundamental properties of structural topology optimization problems. Struct Multidiscip Optim 41:661–670

    MATH  Article  Google Scholar 

  32. 32.

    Subramanian V, Harion JL (2018) Topology optimization of conductive heat transfer devices—an experimental investigation. Appl Therm Eng 131:390–411

    Article  Google Scholar 

  33. 33.

    Takezawa A, Yonekura K, Koizumi Y, Zhang X, Kitamura M (2018) Isotropic Ti–6Al–4V lattice via topology optimization and electron-beam melting. Addit Manuf 22:634–642

    Google Scholar 

  34. 34.

    Topping BHV (1983) Shape optimization of skeletal structures: a review. J Struct Eng 109:1933–1951

    Article  Google Scholar 

  35. 35.

    Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Wang X, Xu S, Zhou S, Xu W, Leary M, Choong P, Qian M, Brandt M, Xie YM (2016) Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: a review. Biomaterials 83:127–141

    Article  Google Scholar 

  37. 37.

    Wu CY, Tseng KY (2010) Truss structure optimization using adaptive multi-population differential evolution. Struct Multidiscip Optim 42(4):575–590

    Article  Google Scholar 

  38. 38.

    Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896

    Article  Google Scholar 

  39. 39.

    Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336

    ADS  Article  Google Scholar 

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ES acknowledges the support of this work through the Project PRIN2015 “Multi-scale mechanical models for the design and optimization of micro-structured smart materials and metamaterials” (prot. 2015LYYXA8_002, CUP H32F15000090005). MF gratefully acknowledges the support of the Grants by the Italian Ministry of Education, Universities and Research (MIUR) through the Grants PRIN-20177TTP3S and PON-ARS01-01384. MF also acknowledges the support by the University of Napoli “Federico II” through the funded Project E62F17000200001-NAPARIS.

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Minutolo, V., Esposito, L., Sacco, E. et al. Designing stress for optimizing and toughening truss-like structures. Meccanica 55, 1603–1622 (2020).

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  • Optimization
  • Microstructure
  • Stress intensity factor
  • Crack propagation
  • Bio-inspired materials