Skip to main content
SpringerLink
Log in

A linear technique for designing optimal rotated shapes

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this article, a simple linear method is established for designing optimal revolving objects with desired physical properties. First, the problem is converted to an optimal control problem by defining an artificial control related to an unknown generator curve. Next, considering a variational presentation and applying an embedding procedure, the problem is transferred into one whose unknown is an optimal Radon measure. Using two stages of approximation determines the optimal control, and the optimal generator rotating curve is obtained from the results of a finite linear programming problem. The accuracy and applications of the method are shown by presenting some classical numerical simulations and comparing the method with the level set method in those examples.

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

Similar content being viewed by others

References

  • Belegun D, Rajan S (1988) Shape optimization approach based on natural design variables and shape functions. Comput Method Appl Mech Eng 66(23):87–106

    Article  Google Scholar 

  • Delfour M, Zolesio J (2001) Shapes and geometries-analysis, differential calculus and optimization. SIAM, Philadelphia

    MATH  Google Scholar 

  • Fakharzadeh AJ (2003) Determining the best domain for a nonlinear wave system. J Appl Math Comput 13:183–194

    Article  MathSciNet  MATH  Google Scholar 

  • Fakharzadeh AJ, Alimorad H (2019) A review of theoretical measure approaches in optimal shape problems. Int J Numer Anal Model 16(4):543–574

    MathSciNet  MATH  Google Scholar 

  • Fakharzadeh AJ, Rubio JE (1999) Shapes and measures. IMA J Math Control Inf 16:207–220

    Article  MathSciNet  MATH  Google Scholar 

  • Fakharzadeh AJ, Rubio JE (2009) Best domain for an elliptic problem in cartesian coordinates by means of shape-measure. Asian J Control 11:536–547

    Article  MathSciNet  Google Scholar 

  • Farahi MH, Mehne HH, Borzabadi AH (2006) Wing drag minimization by using measure theory. Optim Methods Softw 21:1–9

    Article  MathSciNet  MATH  Google Scholar 

  • Farhadinia B, Farahi MH (2005) Optimal shape design of an almost straight nozzle. Int J Appl Math 13:319–334

    MathSciNet  MATH  Google Scholar 

  • Farhadinia B, Farahi MH (2007) On the existence of solution of an optimal shape design problem governed by full Navier–Stoke equations. Int J Contemp Math Sci 2:701–711

    Article  MathSciNet  MATH  Google Scholar 

  • Glashoff K, Gustafson SA (1983) Linear optimization and approximation. Springer, Berlin

    Book  MATH  Google Scholar 

  • Haslinger J, Neittaanamaki P (1988) Finite element approximation for optimal shape design: theory and applications. Wiley, New York

    Google Scholar 

  • Hicks RM, Hanne PA (1977) Wing design by numerical optimization. In: AIAA aircraft systems and technology conference, Seattle, Washington, pp 22–24

  • Leopold F (1976) Advanced calculus. The Williams and Wilkins Company, New York

    MATH  Google Scholar 

  • Mehneh HH, Farahi MH, Esfahani JA (2005) Slot nozzle design with specified pressure in a given subregion by embedding method. J Appl Math Comput 168:1–9

    MathSciNet  Google Scholar 

  • Mohammadi B, Pironneaun O (2002) Applied optimal shape design. Anal Numer

  • Nazemi AR, Farahi MH (2009) Control of the fiber orientation distribution at the outlet of contraction. Acta Appl Math 106:279–292

    Article  MathSciNet  MATH  Google Scholar 

  • Nazemi AR, Farahi MH, Zamirian M (2008) Filtration problem in homogeneous dam by using embedding method. J Appl Math Comput 28:313–332

    Article  MathSciNet  MATH  Google Scholar 

  • Nazemi AR, Farahi MH, Mehneh HH (2009) Optimal shape design of iron pole section of electromagnet. Phys Lett A 372:3440–3451

    Article  MathSciNet  MATH  Google Scholar 

  • Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithm based on Hamilton–Jacobi formulations. J Comput Phys 79:12–49

    Article  MathSciNet  MATH  Google Scholar 

  • Park JJ, Rebelo N, Kobayashi S (1983) A new approach to perform design in metal forming with finite element method. Int J Mach Tool Des Res 23:71–99

    Article  Google Scholar 

  • Pironneau O (1983) Optimal shape design for elliptic system. Springer, Berlin

    Google Scholar 

  • Rosenbloom PC (1956) Qudques classes de problems exteremaux. Bulletin de Societe Mathematique de France 80:183–216

    Google Scholar 

  • Rubio JE (1986) Control and optimization: the linear treatment of nonlinear problems. Manchester University Press, Manchester

    MATH  Google Scholar 

  • Rudin W (1987) Real and complex analysis, 3rd edn. McGraw-Hill Series in Higher Mathematics, New York

    MATH  Google Scholar 

  • Smith DR (1974) Variational methods in optimization. Prentic-Hall Inc, Englewood Cliffs

    MATH  Google Scholar 

Download references

Acknowledgements

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Shiraz University of Technology, Shiraz, Iran

    Alireza Fakharzadeh Jahromi

  2. Department of Mathematics, Jahrom University, P. O. Box: 74135-111, Jahrom, Iran

    Hajar Alimorad

  3. Islamic Azad University Ahvaz Branch, Ahvaz, Iran

    Sedigheh Tuobaei

Authors
  1. Alireza Fakharzadeh Jahromi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hajar Alimorad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sedigheh Tuobaei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hajar Alimorad.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest in preparing this article.

Additional information

Communicated by Frederic Valentin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jahromi, A.F., Alimorad, H. & Tuobaei, S. A linear technique for designing optimal rotated shapes. Comp. Appl. Math. 41, 421 (2022). https://doi.org/10.1007/s40314-022-02134-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-02134-4

Keywords

Mathematics Subject Classification

Search

Navigation