Skip to main content

Multi-resolution methods for the topology optimization of nonlinear electro-active polymers at large strains

Abstract

This work presents a numerical study on the use of multi-resolution strategies for the computationally challenging problem of optimal design of high-resolution nonlinear electro-active shell-type devices using Topology Optimization methods. This paper puts forward the following novelties. First, it presents a tailor-made in-plane multi-resolution technique where a higher level of resolution for the density field is permitted within the in-plane surface of the electro-active material. Second, a numerical study is carried out to elucidate the lower bound for the critical ratio between the density resolution level (or the number of density voxels within every element of the coarse analysis mesh) and the filtering length that can be used to avoid the presence of QR-patterns in the unexplored scenario of nonlinear electromechanics, without compromising the computational efficiency of the multi-resolution scheme. The numerical experiments illustrate the benefits of the proposed methodology to obtain high-resolution designs with a reasonable computational cost and reveal the possibility for more flexible bounds for the critical ratio as those reported in linear elasticity.

This is a preview of subscription content, access via your institution.

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

References

  1. 1.

    Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550:84–86

    Google Scholar 

  2. 2.

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

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bathe KJ (1996) Finite element procedures. Prentice Hall, New York

    MATH  Google Scholar 

  5. 5.

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

    MATH  Google Scholar 

  6. 6.

    Bonet J, Gil AJ, Wood RD (2016) Nonlinear continuum mechanics for finite element analysis: statics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. 7.

    Bortot E, Amir O, Shmuel G (2018) Topology optimization of dielectric elastomers for wide tunable band gaps. Int J Solids Struct 143:262–273

    Google Scholar 

  8. 8.

    Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459

    MATH  Google Scholar 

  9. 9.

    Burger M, Stainko R (2003) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 192:147–1466

    MATH  Google Scholar 

  10. 10.

    de Souza Neto EA, Períc D, Owen DRJ (2008) Computational methods for plasticity. Theory and applications. Wiley, New York

    Google Scholar 

  11. 11.

    Donoso A, Bellido J (2009) Systematic design of distributed piezoelectric modal sensors/actuators for rectangular plates by optimizing the polarization profile. Struct Multidiscip Optim 38:347–356

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Donoso A, Bellido J, Chacón J (2009) Numerical and analytical method for the design of piezoelectric modal sensors/actuators for shell-type structures. Int J Numer Methods Eng 81:1700–1712

    MATH  Google Scholar 

  13. 13.

    Gil AJ, Ortigosa R (2016) A new framework for large strain electromechanics based on convex multi-variable strain energies: variational formulation and material characterisation. Comput Methods Appl Mech Eng 302:293–328

    MathSciNet  MATH  Google Scholar 

  14. 14.

    González O, Stuart AM (2008) A first course in continuum mechanics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  15. 15.

    Groen JP, Langelaar M, Sigmund O, Ruess M (2017) Higher-order multi-resolution topology optimization using the finite cell method. Int J Numer Methods Eng 110(10):903–920

    MathSciNet  Google Scholar 

  16. 16.

    Gupta DK, van der Veen GJ, Aragón AM, Langelaar M, van Keulen F (2017) Bounds for decoupled design and analysis discretizations in topology optimization. Int J Numer Methods Eng 111(1):88–100

    MathSciNet  Google Scholar 

  17. 17.

    Gupta DK, Langelaar M, van Keulen F (2018) Qr-patterns: artefacts in multiresolution topology optimization. Struct Multidiscip Optim 58:1335–1350

    Google Scholar 

  18. 18.

    Gupta DK, van Keulen F, Langelaar M (2020) Design and analysis adaptivity in multiresolution topology optimization. Int J Numer Methods Eng 121(3):450–476

    MathSciNet  Google Scholar 

  19. 19.

    Kang Z, Wang X (2010) Topology optimization of bending actuators with multilayer piezoelectric material. Smart Mater Struct 19(7):075018

    Google Scholar 

  20. 20.

    Kofod G, Sommer-Larsen P, Kornbluh R, Pelrine R (2003) Actuation response of polyacrylate dielectric elastomers. J Intell Mater Syst Struct 14(12):787–793

    Google Scholar 

  21. 21.

    Kögl M, Silva E (2005) Topology optimization of smart structures: design of piezoelectric plate and shell actuators. Smart Mater Struct 14:387–399

    Google Scholar 

  22. 22.

    Liu K, Tovar A (2014) An efficient 3d topology optimization code written in matlab. Struct Multidiscip Optim 50:1175–1196

    MathSciNet  Google Scholar 

  23. 23.

    Lundgaard C, Sigmund O (2018) A density-based topology optimization methodology for thermoelectric energy conversion problems. Struct Multidiscip Optim 57:1427–1442

    MathSciNet  Google Scholar 

  24. 24.

    Lundgaard C, Sigmund O (2019) Design of segmented off-diagonal thermoelectric generators using topology optimization. Appl Energy 236:950–960

    Google Scholar 

  25. 25.

    Lundgaard C, Sigmund O (2019) Design of segmented thermoelectric peltier coolers by topology optimization. Appl Energy 239:1003–1013

    Google Scholar 

  26. 26.

    Martínez-Frutos J, Herrero-Pérez D (2018) Evolutionary topology optimization of continuum structures under uncertainty using sensitivity analysis and smooth boundary representation. Comput Struct 205:15–27

    Google Scholar 

  27. 27.

    Miehe C, Vallicotti D, Zäh D (2015) Computational structural and material stability analysis in finite electro-elasto-statics of electro-active materials. Int J Numer Methods Eng 102(10):1605–1637

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Munk DJ, Steven GP (2015) Topology and shape optimization methods using evolutionary algorithms: a review. Struct Multidiscip Optim 52:613–631

    MathSciNet  Google Scholar 

  29. 29.

    Nakasone PH, Silva ECN (2010) Dynamic design of piezoelectric laminated sensors and actuators using topology optimization. J Intell Mater Syst Struct 21(16):1627–1652

    Google Scholar 

  30. 30.

    Nguyen Tam H, Paulino Glaucio H, Le Song Junho, Chau H (2010) A computational paradigm for multiresolution topology optimization (mtop). Struct Multidiscip Optim 41:525–539

    MathSciNet  MATH  Google Scholar 

  31. 31.

    O’Halloran A, O’Malley F, McHugh P (2008) A review on dielectric elastomer actuators, technology, applications, and challenges. J Appl Phys 104(7):071101

    Google Scholar 

  32. 32.

    Ortigosa R, Gil AJ, Lee CH (2016) A computational framework for large strain nearly and truly incompressible electromecahnics based on convex multi-variable strain energies. Comput Methods Appl Mech Eng 310:297–334

    MATH  Google Scholar 

  33. 33.

    Ortigosa R, Martínez-Frutos J, Gil AJ, Herrero-Pérez D (2019) A new stabilisation approach for level-set based topology optimisation of hyperelastic materials. Struct Multidiscip Optim 60:2343–2371

    MathSciNet  Google Scholar 

  34. 34.

    Ortigosa R, Martínez-Frutos J, Ruiz D, Donoso A, Bellido JC Density-based topology optimisation considering nonlinear electromechanics. Struct Multidiscip Optim (n print)

  35. 35.

    Part I, Silva ECN, Fonseca JS, de Espinosa FM (1999) Design of piezocomposite materials and piezoelectric transducers using topology optimization. Archiv Comput Methods Eng 6:117–182

    MathSciNet  Google Scholar 

  36. 36.

    Padoin E, Santos IF, Perondi EA (2019) Topology optimization of piezoelectric macro-fiber composite patches on laminated plates for vibration suppression. Struct Multidiscip Optim 59:941–957

    MathSciNet  Google Scholar 

  37. 37.

    Pelrine R, Kornbluh R, Joseph J (1998) Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation. Sens Actuat A 64(1):77–85

    Google Scholar 

  38. 38.

    Pelrine R, Kornbluh R, Pei Q, Joseph J (2000) High-speed electrically actuated elastomers with strain greater than 100%. Science 287(5454):836–839

    Google Scholar 

  39. 39.

    Pelrine R, Kornbluh R, Pei Q, Stanford S, Oh S, Eckerle J, Full RJ, Rosenthal MA, Meijer K (2002) Dielectric elastomer artificial muscle actuators: toward biomimetic motion. Smart Structures and Materials 2002: electroactive polymer actuators and devices (EAPAD). volume 4695. International Society for Optics and Photonics, SPIE, pp 126–137

  40. 40.

    Ruiz D, Bellido J, Donoso A (2015) Design of piezoelectric modal filters by simultaneously optimizing the structure layout and the electrode profile. Struct Multidiscip OptimStruct Multidiscip OptimStruct Multidiscip Optim 53:715–730

    MathSciNet  Google Scholar 

  41. 41.

    Ruiz D, Bellido J, Donoso A, Sanchez-Rojas JL (2013) Design of in-plane piezoelectric sensors for static response by simultaneously optimizing the host structure and the electrode profile. Struct Multidiscip Optim 48:1023–1026

    Google Scholar 

  42. 42.

    Ruiz D, Alex Díaz-Molina, Sigmund O, Donoso A, Bellido J, Sánchez-Rojas JL (2018) Optimal design of robust piezoelectric unimorph microgrippers. Appl Math Model 55:1–12

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Ruiz D, Sigmund O (2018) Optimal design of robust piezoelectric microgrippers undergoing large displacements. Struct Multidiscip Optim 55:71–82

    MathSciNet  Google Scholar 

  44. 44.

    Sánchez-Rojas JL, Hernando J, Donoso A, Bellido J, Manzaneque T, Ababneh A, Seidel H, Schmid U (2010) Modal optimization and filtering in piezoelectric microplate resonators. J Micromech Microeng 20:055027

    Google Scholar 

  45. 45.

    Skatulla S, Sansour C, Arockiarajan A (2012) A multiplicative approach for nonlinear electro-elasticity. Comput Methods Appl Mech Eng 245–246:243–255

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Skov AL, Pei O, Opris D, Spontak RJ, Gallone G, Shea H, Benslimane MY (2016) Dielectric elastomers (DEs) as EAPs materials. Sringer, Cham, pp 1–28

    Google Scholar 

  47. 47.

    Sokolowski J, Zochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37:1251–1272

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Svanberg C (1987) The method of moving asymptotes: a new method for structural optimization. Appl Energy 24:359–373

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Svanberg K (1987) The method of moving asymptotes: a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784

    MATH  Google Scholar 

  51. 51.

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

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Wang N, Guo H, Chen B, Zhang X (2017) Design of a rotary dielectric elastomer actuator using topology optimization method. In: 2017 international conference on manipulation, automation and robotics at small scales (MARSS)

  53. 53.

    Zhang X, Takezawa A, Kang Z (2018) Topology optimization of piezoelectric smart structures for minimum energy consumption under active control. Struct Multidiscip Optim 58:185–199

    MathSciNet  Google Scholar 

  54. 54.

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

    Google Scholar 

Download references

Acknowledgements

The first author acknowledges the financial support through the contract 21132/SF/19, Fundación Séneca, Región de Murcia (Spain). Both authors acknowledge the support provided through project supported by the Autonomous Community of the Region of Murcia, Spain through the programme for the development of scientific and technical research by competitive groups (20911/PI/18), included in the Regional Program for the Promotion of Scientific and Technical Research of Fundación Séneca - Agencia de Ciencia y Tecnología de la Región de Murcia.

Author information

Affiliations

Authors

Corresponding author

Correspondence to R. Ortigosa.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Ortigosa, R., Martínez-Frutos, J. Multi-resolution methods for the topology optimization of nonlinear electro-active polymers at large strains. Comput Mech 68, 271–293 (2021). https://doi.org/10.1007/s00466-021-02030-4

Download citation

Keywords

  • Topology optimisation
  • Nonlinear electroelasticity
  • Electro-active polymers
  • Multi-resolution