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Computation of Design Sensitivities in Steady-State Incompressible Laminar Flows Based on New Semi-Analytical Method

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Abstract

In this paper, a new semi-analytic method (SAM) is proposed as a robust and efficient approach based on complex variables to compute the sensitivity of steady-state incompressible laminar flows. This method combines the complex variable method (CVM) via the discrete sensitivity analysis in order to obtain the sensitivity of response accurately and efficiently. The governing Navier–Stokes equations are solved using the finite element method and then new SAM is employed. The proposed procedure retains the computational efficiency of SAM with higher accuracy. In addition, the scheme is not sensitive to the step size, a characteristic that eases its application in practical problems. It is proved that the discrete sensitivity analysis is equivalent to CVM and solves the same equation. Finally, the accuracy of the method is investigated through various numerical cases compared to other methods and reveals that this scheme is reliable and independent to the step size. The proposed approach is applicable to a wide range of engineering problems.

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References

  1. Ta, T.T.M., Le, V.C., Pham, H.T.: Shape optimization for Stokes flows using sensitivity analysis and finite element method. Appl. Numer. Math. 126, 160–179 (2018)

    Article  MathSciNet  Google Scholar 

  2. Ballarin, F., Manzoni, A., Rozza, G., Salsa, S.: Shape optimization by free-form deformation: existence results and numerical solution for Stokes flows. J. Sci. Comput. 60(3), 537–563 (2014)

    Article  MathSciNet  Google Scholar 

  3. Lambert, J., Gosselin, L.: Sensitivity analysis of heat exchanger design to uncertainties of correlations. Appl. Therm. Eng. 136, 531–540 (2018)

    Article  Google Scholar 

  4. Jafari, M., Jafari, M.: Thermal stress analysis of orthotropic plate containing a rectangular hole using complex variable method. Eur. J. Mech. A. Solids 73, 212–223 (2019)

    Article  MathSciNet  Google Scholar 

  5. Voorhees, A., Millwater, H., Bagley, R., Golden, P.: Fatigue sensitivity analysis using complex variable methods. Int. J. Fatigue 40, 61–73 (2012)

    Article  Google Scholar 

  6. Kavvadias, I.S., Papoutsis-Kiachagias, E.M., Giannakoglou, K.C.: On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization. J. Comput. Phys. 301, 1–18 (2015)

    Article  MathSciNet  Google Scholar 

  7. Hu, G., Kozlowski, T.: Application of continuous adjoint method to steady-state two-phase flow simulations. Ann. Nucl. Energy 117, 202–212 (2018)

    Article  Google Scholar 

  8. Liu, G., Geier, M., Liu, Z., Krafczyk, M., Chen, T.: Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method. Comput. Math. Appl. 68(10), 1374–1392 (2014)

    Article  MathSciNet  Google Scholar 

  9. Ding, Z., Li, L., Li, X., Kong, J.: A comparative study of design sensitivity analysis based on adjoint variable method for transient response of non-viscously damped systems. Mech. Syst. Signal Process. 110, 390–411 (2018)

    Article  Google Scholar 

  10. Kiendl, J., Schmidt, R., Wüchner, R., Bletzinger, K.U.: Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Comput. Methods Appl. Mech. Eng. 274, 148–167 (2014)

    Article  MathSciNet  Google Scholar 

  11. Wang, W., Clausen, P.M., Bletzinger, K.-U.: Improved semi-analytical sensitivity analysis using a secant stiffness matrix for geometric nonlinear shape optimization. Comput. Struct. 146, 143–151 (2015)

    Article  Google Scholar 

  12. Cho, M., Kim, H.: Improved semi-analytic sensitivity analysis combined with a higher order approximation scheme in the framework of adjoint variable method. Comput. Struct. 84(29), 1827–1840 (2006)

    Article  Google Scholar 

  13. Lyness, J.N., Moler, C.B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal. 4(2), 202–210 (1967)

    Article  MathSciNet  Google Scholar 

  14. Lyness, J.N.: Numerical algorithms based on the theory of complex variable. In: Proceedings of the 1967 22nd National Conference, pp. 125–133. ACM (1967)

  15. Squire, W., Trapp, G.: Using complex variables to estimate derivatives of real functions. SIAM Rev. 40(1), 110–112 (1998)

    Article  MathSciNet  Google Scholar 

  16. Martins, J.R., Sturdza, P., Alonso, J.J.: The complex-step derivative approximation. ACM Trans. Math. Softw. (TOMS) 29(3), 245–262 (2003)

    Article  MathSciNet  Google Scholar 

  17. Martins, J. Kroo, I., Alonso, J.: An automated method for sensitivity analysis using complex variables. In: 38th Aerospace Sciences Meeting and Exhibit, p. 689 (2000)

  18. Anderson, W.K., Newman, J.C., Whitfield, D.L., Nielsen, E.J.: Sensitivity analysis for Navier–Stokes equations on unstructured meshes using complex variables. AIAA J. 39(1), 56–63 (2001)

    Article  Google Scholar 

  19. Rodriguez, D.: A multidisciplinary optimization method for designing inlets using complex variables. In: 8th Symposium on Multidisciplinary Analysis and Optimization, p. 4875 (2000)

  20. Jin, W., Dennis, B.H., Wang, B.P.: Improved sensitivity analysis using a complex variable semi-analytical method. Struct. Multidiscip. Optim. 41(3), 433–439 (2010)

    Article  Google Scholar 

  21. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics. OUP Oxford, Oxford (2014)

    Book  Google Scholar 

  22. Grove, A.S., Shair, F.H., Petersen, E.E.: An experimental investigation of the steady separated flow past a circular cylinder. J. Fluid Mech. 19(1), 60–80 (1964)

    Article  Google Scholar 

  23. Tritton, D.: Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6(4), 547–567 (1959)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Kordkuy Center, Gorgan Branch, Islamic Azad University, Kordkuy, Iran

    Mahdi Hassanzadeh

  2. School of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran

    Mahmood Mazare

Authors
  1. Mahdi Hassanzadeh
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  2. Mahmood Mazare
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Correspondence to Mahdi Hassanzadeh.

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Hassanzadeh, M., Mazare, M. Computation of Design Sensitivities in Steady-State Incompressible Laminar Flows Based on New Semi-Analytical Method. J Sci Comput 83, 12 (2020). https://doi.org/10.1007/s10915-020-01205-0

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  • DOI: https://doi.org/10.1007/s10915-020-01205-0

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