Skip to main content
SpringerLink
Log in

Numerical shape optimization based on meshless method and stochastic optimization technique

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.

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
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197:2976–2988

    MathSciNet  MATH  Google Scholar 

  2. Imam MH (1982) Three-dimensional shape optimization. Int J Numer Methods Eng 18:661–673

    MATH  Google Scholar 

  3. Zienkiewicz OC, Campbell JS (1973) Shape optimization and sequential linear programming. In: Gallagher RH, Zienkiewicz OC (eds) Optimum structural design. Wiley, New York, pp 109–126

    Google Scholar 

  4. Bhavikatti SS, Ramakrishnan CV (1977) Optimum design of fillets in flat and round tension bars. Proceedings of the ASME, Design Engineering Technical Conference, Chicago, Illinois, 77-DET-45

  5. Bhavikatti SS, Ramakrishnan CV (1980) Optimum shape design of rotating disks. Comput Struct 11:397–401

    MATH  Google Scholar 

  6. Braibant V, Fleury C (1984) Shape optimal design using B-splines. Comput Methods Appl Mech Eng 44:247–267

    MATH  Google Scholar 

  7. Annicchiarico W, Cerrolaza M (1999) Finite elements, genetic algorithms and b-splines: a combined technique for shape optimization. Finite Elements Anal Des 33:125–141

    MATH  Google Scholar 

  8. Botkin ME (1982) Shape optimization of plate and shell structures. AIAA J 20(2):268–273

    Google Scholar 

  9. Wang SY, Sun Y, Gallagher RH (1985) Sensitivity analysis in shape optimization of continuum structures. Comp Struct 20(5):855–867

    MATH  Google Scholar 

  10. Belegundu AD, Rajan SD (1988) A shape optimization approach based on natural design variables and shape functions. Comput Methods Appl Mech Eng 66:87–106

    MATH  Google Scholar 

  11. Tortorelli DA (1993) A geometric representation scheme suitable for shape optimization. Mech Struct Mach 21(1):95–121

    MathSciNet  Google Scholar 

  12. Firl M, Wuchner R, Bletzinger KU (2013) Regularization of shape optimization problems using FE-based parametrization. Struct Multidisc Optim 47:507–521

    MathSciNet  MATH  Google Scholar 

  13. Stavropoulou E, Hojjat M, Bletzinger KU (2014) In-plane mesh regularization for node-based shape optimization problems. Comput Methods Appl Mech Engrg 275:39–54

    MathSciNet  MATH  Google Scholar 

  14. Haftka RT, Grandhi RV (1986) Structural shape optimization: a survey. Comput Methods Appl Mech Eng 57(1):91–106

    MathSciNet  MATH  Google Scholar 

  15. Ding Y (1986) Shape optimization of structures: a literature survey. Comput Struct 24(6):985–1004

    MathSciNet  MATH  Google Scholar 

  16. Mackerle J (2003) Topology and shape optimization of structures using FEM and BEM-a bibliography (1999–2001). Finite Elem Anal Des 39:243–253

    MathSciNet  MATH  Google Scholar 

  17. Bennett JA, Botkin ME (1985) Structural shape optimization with geometric description and adaptive mesh refinement. AIAA J 23(3):458–464

    Google Scholar 

  18. Kim NH, Choi KK (2005) Structural sensitivity analysis and optimization 1: linear systems. Springer, New York

    Google Scholar 

  19. Lacroix D, Bouillard P (2003) Improved sensitivity analysis by a coupled FE-EFG method. Comput Struct 81:2431–2439

    Google Scholar 

  20. Grindeanu I, Choi KK, Chen JS, Chnag KH (1999) Shape design optimization of Hyperelastic structures using a Meshless method. AIAA J 37(8):990–997

    Google Scholar 

  21. Diaz AR, Kikuchi N, Taylor JE (1983) A method of grid optimization for finite element methods. Comp Methods Appl Mech Eng 41:29–45

    MathSciNet  MATH  Google Scholar 

  22. Ingber MS, Mitra AK (1986) Grid optimization for the boundary element method. Int J Numer Methods Eng 23:2121–2136

    MATH  Google Scholar 

  23. Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  24. Najafi AR, Safdaric M, Tortorelli DA, Geubelle PH (2015) A gradient based shape optimization scheme using an interface-enriched generalized FEM. Comput Methods Appl Mech Eng 296:1–17

    MathSciNet  MATH  Google Scholar 

  25. Kim NH, Chang Y (2005) Eulerian shape design sensitivity analysis and optimization with a fixed grid. Comput Methods Appl Mech Eng 194:3291–3314

    MathSciNet  MATH  Google Scholar 

  26. Gracia MJ, Gonzalez CA (2004) Shape optimization of continuum structures via evolution strategies and fixed grid finite element analysis. Struct Multidisc Optim 26:92–98

    Google Scholar 

  27. Garcia-Ruiz MJ, Steven GP (1999) Fixed grid finite element in elasticity optimization. Eng Comput 16(2):145–164

    MATH  Google Scholar 

  28. Hughes TJR, Cottrell JA, BazilevsY (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    MathSciNet  MATH  Google Scholar 

  29. Fubeder D, Simeon B, Vuong AV (2015) Fundamental aspects of shape optimization in the context of isogeometric analysis. Comput Methods Appl Mech Eng 286:313–331

    MathSciNet  MATH  Google Scholar 

  30. Ha SH, Choi KK, Cho S (2010) Numerical method for shape optimization using T-spline based isogeometric method. Struct Multidisc Optim 42:417–428

    MATH  Google Scholar 

  31. Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Eng 199:2059–2071

    MathSciNet  MATH  Google Scholar 

  32. Bobaru F, Rachakonda S (2004) Boundary layer in shape optimization of convective fins using a meshless approach. Int J Numer Methods Eng 60(7):1215–1236

    MATH  Google Scholar 

  33. Belytschko TY, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256

    MathSciNet  MATH  Google Scholar 

  34. Mota Soares CA, Radrigues HC, Choi KK (1984) Shape optimal structural design using boundary elements and minimum compliance techniques. ASME 84-DET-57

    Google Scholar 

  35. Mota Soares CA, Rodrigues HC, Oliveira Faria LM, Haug EJ (1984) Optimization of the geometry of shafts using boundary elements. ASME J Mech Trans Autom Des 106(2):199–203

    Google Scholar 

  36. Zhang J, Gong S, Huang Y, Qiu A, Chen R (2008) Structural dynamic shape optimization and sensitivity analysis based on RKPM. Struct Multidisc Optim 36:307–317

    MathSciNet  MATH  Google Scholar 

  37. Kim NH, Choi KK, Botkin ME (2003) Numerical method for shape optimization using meshless method. Struct Multidisc Optim 24:418–429

    Google Scholar 

  38. Bobaru F, Mukherjee S (2001) Shape sensitivity analysis and shape optimization in planar using the element-free Galerkin method. Comput Methods Appl Mech Eng 190:4319–4937

    MATH  Google Scholar 

  39. Phan AV, Mukherjee S, Mayer JRR (1998) Stresses, stress sensitivities and shape optimization in two-dimensional linear elasticity by the boundary contour method. Int J Numer Methods Eng 42:1391:1407

    MathSciNet  MATH  Google Scholar 

  40. Zhao Z (1991) In shape design sensitivity analysis and optimization using the boundary element method. Springer, New York

    MATH  Google Scholar 

  41. Bobaru F, Mukherjee S (2002) Meshless approach to shape optimization of linear thermoelastic solids. Int J Numer Methods Eng 53:765–796

    Google Scholar 

  42. Hou JW, Sheen JS, Chuang CH (1992) Shape-sensitivity analysis and design optimization of linear, thermoelastic solids. AIAA J 30(2):528–537

    MATH  Google Scholar 

  43. Bobaru F, Rachakonda S (2004) Optimal shape profiles for cooling fins of high and low conductivity. Int J Heat Mass Transf 47(23):4953–4966

    MATH  Google Scholar 

  44. Bobaru F, Rachakonda S (2006) E(FG)2: a new fixed-grid shape optimization method based on the element-free Galerkin mesh-free analysis: taking large steps in shape optimization. Struct Multidiscip Optim 32:215–228

    MathSciNet  MATH  Google Scholar 

  45. Zou W, Zhou JX, Zhang ZQ, Li Q (2007) A truly meshless method based on partition of unity quadrature for shape optimization of continua. Comput Mech 39:357–365

    MathSciNet  MATH  Google Scholar 

  46. Zhang ZQ, Zhou JX, Zhou N, Wang XM, Zhang L (2005) Shape optimization using reproducing kernel particle method and an enriched genetic algorithm. Comput Methods Appl Mech Eng 194:4048–4070

    MathSciNet  MATH  Google Scholar 

  47. Banichuk NV, Serra M, Sinitsyn A (2006) Shape optimization of quasi-brittle axisymmetric shells by genetic algorithm. Comput Struct 84:1925–1933

    Google Scholar 

  48. Muc A, Gurba W (2001) Genetic algorithms and finite element analysis in optimization of composite structures. Compos Struct 54:275–281

    Google Scholar 

  49. Sonmez FO (2007) Shape optimization of 2D structures using simulated annealing. Comput Methods Appl Mech Eng 196:3279–3299

    MATH  Google Scholar 

  50. Shim PY, Manoochehri S (1997) Generating optimal configurations in structural design using simulated annealing. Int J Numer methods Eng 40:1053–1069

    MATH  Google Scholar 

  51. Karaboga D (2005) An Idea Based on Honey Bee Swarm for Numerical Optimization. Technical Report-TR06, Erciyes University

  52. Karaboga D, Akay B (2009) A comparative study of Artificial Bee Colony algorithm. Appl Math Comput 214:108–132

    MathSciNet  MATH  Google Scholar 

  53. Sonmez M (2011) Discrete optimum design of truss structures using artificial bee colony algorithm. Struct Multidisc Optim 43:85–97

    Google Scholar 

  54. Park JY, Han SY (2013) Application of artificial bee colony algorithm to topology optimization for dynamic stiffness problems. Comput Math Appl 66:1879–1891

    MathSciNet  MATH  Google Scholar 

  55. Zhu T, Atlure SN (1998) A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput Mech 21:211–222

    MathSciNet  MATH  Google Scholar 

  56. Lu YY, Belytschko T, Gu L (1994) A new implementation of the element free Galerkin method. Comput Methods Appl Mech Eng 113(3–4):397–414

    MathSciNet  MATH  Google Scholar 

  57. Krongauz Y, Belytschko T (1996) Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput Methods Appl Mech Eng 131(1–2):133–145

    MathSciNet  MATH  Google Scholar 

  58. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139:49–71

    MathSciNet  MATH  Google Scholar 

  59. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466

    MATH  Google Scholar 

  60. Carpinteri A, Ferro G, Ventura G (2002) The partition of unity quadrature in meshless methods. Int J Numer Methods Eng 54:987

    MathSciNet  MATH  Google Scholar 

  61. DuflotM HungND (2002) A truly meshless Galerkin method based on a moving least squares quadrature. Commun Numer Methods Eng 18:441–449

    MathSciNet  Google Scholar 

  62. Zuohiui P (2000) Treatment of point loads in element free Galerkin method (EFGM). Commun Numer Meth Engrg 16:335–341

    Google Scholar 

  63. Karaboga D, Gorkemli B, Ozturk C, Karaboga N (2012) A comprehensive survey: artificial bee colony (ABC) algorithm and applications. Artf Intell Rev 42:21–57

    Google Scholar 

  64. Karaboga D, Basturk B (2007) On the performance of artificial bee colony (ABC) algorithm. Appl Soft Comput 8:687–697

    MATH  Google Scholar 

  65. Tereshko V (2000) Reaction-diffusion model of a honeybee colony’s foraging behaviour. In: Schoenauer M et al (eds) Parallel problem solving from nature VI. Lecture Notes in Computer Science, vol 1917. Springer, Berlin, pp 807–816

    Google Scholar 

  66. Jordehi AR (2014) A review on constraint handling strategies in particle swarm optimization. Neural Comput Appl 26:1265–1275

    Google Scholar 

  67. Akima H (1970) A new method of interpolation and smooth curve fitting based on local procedures. J Assoc Comput Mach 17(4):589–602

    MATH  Google Scholar 

  68. Wang C, Yu T, Shao G, Nguyen TT, Bui TQ (2018) Shape optimization of structures with cutouts by an efficient approach based on XIGA and chaotic particle swarm optimization. Eur J Mech A Solids 74:176–187

    MathSciNet  MATH  Google Scholar 

  69. Lian H, Kerfriden P, Bordas SPA (2016) Implementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity. Int J Numer Meth Eng 106:972–1017

    MathSciNet  MATH  Google Scholar 

  70. Sun SH, Yu TT, Nguyen TT, Atroshchenko E, Bui TQ (2018) Structural shape optimization by IGABEM and particle swarm optimization algorithm. Eng Anal Bound Elem 88:26–40

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Babaria Institute of Technology, Gujarat Technological University, Vadodara, Gujarat, India

    S. D. Daxini

  2. Department of Mechanical Engineering, M. S. University, Vadodara, Gujarat, India

    J. M. Prajapati

Authors
  1. S. D. Daxini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Prajapati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. D. Daxini.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daxini, S.D., Prajapati, J.M. Numerical shape optimization based on meshless method and stochastic optimization technique. Engineering with Computers 36, 565–586 (2020). https://doi.org/10.1007/s00366-019-00714-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-019-00714-3

Keywords

Search

Navigation