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A stress-based topology optimization method for heterogeneous structures

  • Cian Conlan-SmithEmail author
  • Kai A. James
Research Paper
  • 88 Downloads

Abstract

In this work, we introduce a method to incorporate stress considerations in the topology optimization of heterogeneous structures. More specifically, we focus on using functionally graded materials (FGMs) to produce compliant mechanism designs that are not susceptible to failure. Local material properties are achieved through interpolating between material properties of two or more base materials. Taking advantage of this method, we develop relationships between local Young’s modulus and local yield stress, and apply stress criterion within the optimization problem. A solid isotropic material with penalization (SIMP)–based method is applied where topology and local element material properties are optimized simultaneously. Sensitivities are calculated using an adjoint method and derived in detail. Stress formulations implement the von Mises stress criterion, are relaxed in void regions, and are aggregated into a global form using a p-norm function to represent the maximum stress in the structure. For stress-constrained problems, we maintain local stress control by imposing m p-norm constraints on m regions rather than a global constraint. Our method is first verified by solving the stress minimization of an L-bracket problem, and then multiple stress-constrained compliant mechanism problems are presented. Results suggest that good designs can be produced with the proposed method and that heterogeneous designs can outperform their homogeneous counterparts with respect to both mechanical advantage and reduced stress concentrations.

Keywords

Topology optimization Functionally graded materials Stress-based design Compliant mechanism Heterogeneous structures 

Notes

References

  1. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comp Phys 194(1):363–393MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alonso C, Querin OM, Ansola R (2013) A sequential element rejection and admission (SERA) method for compliant mechanisms design. Struct Multidiscip Optim 47(6):795–807CrossRefGoogle Scholar
  3. Alonso C, Ansola R, Querin OM (2014) Topology synthesis of multi-material compliant mechanisms with sequential element rejection and admission. Finite Elem Anal Des 85:11–19CrossRefGoogle Scholar
  4. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Math 69:635–654zbMATHGoogle Scholar
  5. Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods, and applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Bruggi M, Venini P (2008) A mixed FEM approach to stress constrained topology optimization. Int J Numer Methods Eng 73(11):1693–1714MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bruns TE, Totorelli DA (2001) Topology optimization of non-linear structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26-27):3443–3459CrossRefzbMATHGoogle Scholar
  8. Carbonari RC, Silva ECN, Paulino GH (2009) Multi-actuated functionally graded piezoelectric micro-tools design: a multiphysics topology optimization approach. Int J Numer Meth Eng 77(3):301–336CrossRefzbMATHGoogle Scholar
  9. Carbonari RC, Paulino GH, Silva ECN (2010) Integral piezoactuator with optimum placement of functionally graded material - a topology optimization paradigm. J Intel Material Syst Struct 21(16):1653–1668CrossRefGoogle Scholar
  10. Cheng GD, Guo X (1997) Epsilon-relaxed approach in structural topology optimization. Struct Multidisc Optim 13(4):258– 266CrossRefGoogle Scholar
  11. Chu S, Gao L, Xiao M, Luo Z, Li H (2017) Stress-based multi-material topology optimization of compliant mechanisms. Int J Numer Meth Eng 113(7):1021–1044MathSciNetCrossRefGoogle Scholar
  12. Conlan-Smith C, Bhattacharyya A, James KA (2018) Optimal design of compliant mechanisms using functionally graded materials. Struct Multidiscip Optim 57(1):197–212MathSciNetCrossRefGoogle Scholar
  13. De Leon DM, Alexandersen J, Jun JS, Sigmund O (2015) Stress-constrained topology optimization for compliant mechanism design. Struct Multidisc Optim 52(5):929–943MathSciNetCrossRefGoogle Scholar
  14. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Meth Eng 43(8):1453–1478MathSciNetCrossRefzbMATHGoogle Scholar
  15. Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution. In: Proceedings of the 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimizationGoogle Scholar
  16. Gaynor A, Meisel NA, Williams CB, Guest JK (2014) Multiple material topology optimization of compliant mechanisms created via polyjet three-dimensional printing. J Manuf Sci Eng 136(6)Google Scholar
  17. Howell LL (2001) Compliant mechanisms. Wiley, BerlinGoogle Scholar
  18. Jeong SH, Park SH, Choi DH, Yoon GH (2012) Topology optimization considering static failure theories for ductile and brittle materials. Comput Struct 111:116–132CrossRefGoogle Scholar
  19. Kirsch U (1990) On singular topologies in optimum structural design. Struct Multidisc Optim 2(3):133–142CrossRefGoogle Scholar
  20. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620CrossRefGoogle Scholar
  21. Lee E, James KA, Martins JRAA (2012) Stress-constrained topology optimization with design dependent loading. Struct Multidiscip Optim 46(5):647–661MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lipton R (2002) Design of functionally graded structures in the presence of stress constraints. Int J Solids Struct 39(9):2575– 2586MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lipton R, Stuebner Michael (2006) Optimization of composite structures subject to local stress constraints. Comput Methods Appl Mech Eng 196(1-3):66–75MathSciNetCrossRefzbMATHGoogle Scholar
  24. Luo Z, Tong L (2008) A level set method for shape and topology optimization of large-displacement compliant mechanisms. Int J Numer Meth Eng 76(6):862–892MathSciNetCrossRefzbMATHGoogle Scholar
  25. Luo Z, Tong L, Wang MY, Shengin W (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comp Phys 227(1):680–705MathSciNetCrossRefzbMATHGoogle Scholar
  26. Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Int J Numer Meth Eng 59(14):1925–1944MathSciNetCrossRefzbMATHGoogle Scholar
  27. MatWeb (2018) Material Property Data, [Online]. Available: https://www.matweb.com/index.aspx
  28. Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally graded materials: design, processing and applications. Springer, BerlinCrossRefGoogle Scholar
  29. Parìs J, Navarrina F, Colominas I, Casteleiro M (2007) Block aggregation of stress constraints in topology optimization of structures. In: Hernndez S, Brebbia CA (eds) Computer aided optimum design of structures X. Myrtle Beach (SC), USAGoogle Scholar
  30. Parìs J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidisc Optim 39(4):419–437MathSciNetCrossRefzbMATHGoogle Scholar
  31. Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidisc Optim 26(1–2):50–66MathSciNetCrossRefzbMATHGoogle Scholar
  32. Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidisc Optim 21(2):164–172MathSciNetCrossRefGoogle Scholar
  33. Rozvany GIN, Sobieszczanski-Sobieski J (1992) New optimality criteria methods: forcing uniqueness of the adjoint strains by corner-rounding at constraint intersections. Struct Multidisc Optim 4(3–4):244–246CrossRefGoogle Scholar
  34. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach Int J 25(4):493–542CrossRefGoogle Scholar
  35. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067MathSciNetCrossRefGoogle Scholar
  36. Sigmund O (2001) Design of multiphysics actuators using topology optimization ? Part II: two-material structures. Comput Meth Appl Mech 190(49-50):6605?-6627CrossRefzbMATHGoogle Scholar
  37. Stegmann J, Lund E (2005) Discrete material optimization of general shell structures. Int J Numer Methods Eng 62(14):2009–2027CrossRefzbMATHGoogle Scholar
  38. Stump FV, Silva ECN, Paulino GH (2007) Optimization of material distribution in functionally graded structures with stress constraints. Commun Numer Methods Eng 23(6):535551MathSciNetzbMATHGoogle Scholar
  39. Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Meth Eng 24(2):359–373MathSciNetCrossRefzbMATHGoogle Scholar
  40. Svanberg K, Werme M (2007) Sequential integer programming methods for stress constrained topology optimization. Struct Multidisc Optim 34(4):277–299MathSciNetCrossRefzbMATHGoogle Scholar
  41. Taylor GI, Quinney H (1931) The plastic distortion of metals. Philos Trans R Soc 230:323–362CrossRefzbMATHGoogle Scholar
  42. Tresca H (1864) Sur l’Ecoulement des Corps Solides Soumis a de Fortes Pressions. Comptes Rendus de l’Acadè,mie des Sciences 59:754Google Scholar
  43. von Mises R (1913) Mechanik der Festen Körper im Plastisch Deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Gö,ttingen 582–592Google Scholar
  44. Wang MY, hen S, Wang X, Mei Y (2005) Design of multimaterial compliant mechanisms using level-set methods. J Mech Des 127(5):941–956CrossRefGoogle Scholar
  45. Wolf D, Yip S (1993) Material interfaces: atomic-level structure and properties. Springer, NetherlandsGoogle Scholar
  46. Yin L, Ananthasuresh GK (2003) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidiscip Optim 23(1):49–62CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Aerospace EngineeringUniversity of Illinois Urbana-ChampaignUrbanaUSA

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