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Designing the Optimal Shape of a Nozzle by the Method of Fundamental Solutions

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Abstract

In this paper, we propose a numerical technique based on the method of fundamental solutions (MFS) for solving a classical optimal shape design problem. The problem contains a free boundary condition which should be approximated to find the optimal domain for the solution of Laplace equation. For solving the considered optimization problem, we introduce a meshless regularization technique based on the combination of the MFS and application of the Tikhonov’s regularization method and reduce the problem to solve a system of nonlinear equations. A brief sensitivity analysis on model parameters including the position and the size of the subregion D as well the error with boundary conditions is discussed. Numerical simulations while solving several test examples are presented to show the applicability of the proposed method in obtaining satisfactory results.

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References

  • Ashrafzadeh A, Raithby GD, Stubley GD (2002) Direct design of shape. Numer Heat Transf B-Fund 41:501–520

    Article  Google Scholar 

  • Butt R (1993) Optimal shape design for a nozzle problem. J Aust Math Soc Ser B 35:71–86

    Article  MathSciNet  Google Scholar 

  • Chen CS, Cho HA, Golberg MA (2006) Some comments on the ill-conditioning of the method of fundamental solutions. Eng Anal Bound Elem 30:405–410

    Article  Google Scholar 

  • Fakharzadeh Jahromi A (1996) Shapes, measure and elliptic equations, Ph. D. thesis mathematics, Leeds University

  • Fakharzadeh Jahromi A (2013) A shape-measure method for solving free boundary elliptic systems with boundary control function. Iran J Numer Anal Optim, 3 47–65

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Fakharzadeh A, Rubio JE (1999b) The global solution of optimal shape design problem. Z Anal Anwend 18(1):143–155

    Article  MathSciNet  Google Scholar 

  • Fakharzadeh A, Rubio JE (2001) Shape-measure method for solving elliptic optimal shape problems (fixed control case). B Iran Math Soc 27(1):41–63

    MathSciNet  MATH  Google Scholar 

  • Fakharzadeh A, Rubio JE (2009) Best domain for an elliptic problem in cartesian coordinates by means of shape-measure. Asian J Control 11(5):536–547

    Article  MathSciNet  Google Scholar 

  • Fakharzadeh A, Rafiei Z (2012) Best minimizing algorithm for Shape-measure method. J Math Comput Sci. 5:176–184

    Article  Google Scholar 

  • Fakharzadeh Jahromi A, Alimorad H, Rafiei Z (2013) Using linearization and penalty approach to solve optimal shape design problem with an obstacle. J Math Comput Sc 7 43–53

    Article  Google Scholar 

  • Farahi MH, Borzabadi AH, Mehne HH, Kamyad AV (2005) Measure theoretical approach for optimal shape design of a nozzle. J Appl Math Comput 17:315–328

    Article  MathSciNet  Google Scholar 

  • Farahi MH, Mehne HH, Borzabadi AH (2006) Wing drag minimization by using measuretheory. Optim Method Softw 21:169–177

    Article  Google Scholar 

  • Garg VK (1998) Applied computational fluid dynamics, CRC Press, Boca Raton

    Book  Google Scholar 

  • Gosset A, Buchlin JM (2003) Comparison of experimental techniques for the measurement of unstable film flow. In: 5th European coating symposium on advances in liquid film coating technology. Fribourg, pp 17–19

  • Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34:561–580

    Article  MathSciNet  Google Scholar 

  • Johansson BT, Lesnic D, Reeve T (2011) Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions. Invese Probl Sci Eng 19:659–677

    Article  MathSciNet  Google Scholar 

  • Johansson BT, Lesnic D, Reeve T (2014) A meshless method for an inverse two-phase onedimensional nonlinear Stefan problem. Math Comput Simulation 101:61–77

    Article  MathSciNet  Google Scholar 

  • Karageorghis A, Lesnic D, Marin L (2011) A survey of applications of the MFS toinverse problems. Inverse Probl Sci Eng 19:309–336

    Article  MathSciNet  Google Scholar 

  • Keuthen M, Kraft D (2016) Shape optimization of a breakwater. Inverse Probl Sci Eng 25 936-956

    Article  MathSciNet  Google Scholar 

  • Kirsch A (2011) An introduction to the mathematical theory of inverse problems. Springer, New York

    Book  Google Scholar 

  • Kita E, Kamiyia N (1995) Trefftz method: An overview. Adv Eng Softw 24:3–12

    Article  Google Scholar 

  • Laumen M (2000) Newton’s method for a class of optimal shape design problems. SIAM J Optim 10:503–533

    Article  MathSciNet  Google Scholar 

  • Mehne HH, Farahi MH, Esfahani JA (2005) Slot nozzle design with specified pressure in a given subregion by using embedding method. Appl Math Comput 168:1258–1272

    MathSciNet  MATH  Google Scholar 

  • Mohammadi B, Pironneau O (2010) Applied shape optimization for fluids. Oxford University Press, New York

    MATH  Google Scholar 

  • Pironneau O (1984) Optimal shape design for elliptic systems. Springer, New York

    Book  Google Scholar 

  • Rad JA, Rashedi K, Parand K, Adibi H (2017) The meshfree strong form methods for solving one dimensional inverse Cauchy–Stefan problem. Eng Comput 33:547–571

    Article  Google Scholar 

  • Rashedi K, Sarraf A (2018) Heat source identification of some parabolic equations based on the method of fundamental solutions. Eur Phys J Plus 133:403. https://doi.org/10.1140/epjp/i2018-12193-8

    Article  Google Scholar 

  • Rashedi K, Adibi H, Dehghan M (2013) Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation. Comput Math Appl 65:1990–2008

    Article  MathSciNet  Google Scholar 

  • Rashedi K, Adibi H, Amani Rad J, Parand K (2014) Application of the meshfree methods for solving the inverse one-dimensional Stefan problem. Eng Anal Bound Elem 40 1–21

    Article  MathSciNet  Google Scholar 

  • Rashedi K, Adibi H, Dehghan M (2015) Efficient numerical methods for boundary data and righat-hand side reconstructions in elliptic partial differential equations. Numer Methods Partial Differ Equ 40:1995–2026

    Article  Google Scholar 

  • Reeve TH (2013) The method of fundamental solutions for some direct and inverse problems. Ph. D. thesis mathematics, University of Birmingham

  • Sarabadan S, Rashedi K, Adibi H (2018) Boundary determination of the inverse heat conduction problem in one and two dimensions via the collocation method based on the satisfier functions. Iran J Sci Technol A 42:827–840

    Article  MathSciNet  Google Scholar 

  • Stoer J, Bulirsch R (1980) Introduction to numerical analysis. Springer, New York

    Book  Google Scholar 

  • Trefftz E (1926) Ein Gegenstuck zum Ritzschen Verfahren 2nd Int Kongr fur Techn Mech Zurich, pp 131–137

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Acknowledgements

The author would like to thank anonymous reviewer for the careful reading of this manuscript and constructive comments which have helped improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

    Kamal Rashedi

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  1. Kamal Rashedi
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Correspondence to Kamal Rashedi.

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Rashedi, K. Designing the Optimal Shape of a Nozzle by the Method of Fundamental Solutions. Iran J Sci Technol Trans Sci 44, 1863–1873 (2020). https://doi.org/10.1007/s40995-020-00991-4

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  • DOI: https://doi.org/10.1007/s40995-020-00991-4

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