Abstract
This paper presents new structural sensitivity reanalysis formulations based on the polynomial-type extrapolation methods. In these formulations, the displacement vector of the modified structure is expressed in the form of the vector sequences based on the fixed-point iteration method. By using these vector sequences, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) methods calculate the approximate displacement vector of the modified structure by solving reduced linear least-square problems. Based on the definitions of the MPE and RRE methods, two sensitivity reanalysis formulations are derived, in which the first- and second-order sensitivities of the modified structure are obtained by solving a set of the over-determined least-square problems with much smaller size than the complete set of equations of the exact sensitivity analyses. The performance of the proposed sensitivity reanalysis formulations is evaluated by using four structural sensitivity reanalysis problems under multiple modifications in their initial designs. The results obtained from the numerical test problems indicate that the proposed sensitivity reanalysis formulations approximate the first- and second-order sensitivities of the modified structure with a high level of accuracy and they are able to converge to the exact solutions.
Similar content being viewed by others
References
Adelman HM, Haftka RT (1986) Sensitivity analysis of discrete structural systems. AIAA J 24(5):823–832
Barthelemy J-F, Haftka RT (1993) Approximation concepts for optimum structural design—a review. Struct Multidiscip Optim 5(3):129–144
Fox R, Miura H (1971) An approximate analysis technique for design calculations. AIAA J 9(1):177–179
Haftka RT et al (1987) Two-point constraint approximation in structural optimization. Comput Methods Appl Mech Eng 60(3):289–301
Noor AK (1994) Recent advances and applications of reduction methods. Appl Mech Rev 47(5):125–146
Zuo W et al (2012) A hybrid Fox and Kirsch’s reduced basis method for structural static reanalysis. Struct Multidiscip Optim 46(2):261–272
Wu B, Li Z, Li S (2003) The implementation of a vector-valued rational approximate method in structural reanalysis problems. Comput Methods Appl Mech Eng 192(13–14):1773–1784
Kirsch U (2003) A unified reanalysis approach for structural analysis, design, and optimization. Struct Multidiscip Optim 25(2):67–85
Amir O, Kirsch U, Sheinman I (2008) Efficient non-linear reanalysis of skeletal structures using combined approximations. Int J Numer Methods Eng 73(9):1328–1346
Kirsch U, Bogomolni M (2004) Procedures for approximate eigenproblem reanalysis of structures. Int J Numer Methods Eng 60(12):1969–1986
Leu L.-J. and Huang C.-W. (2000) Reanalysis-based optimal design of trusses. International Journal for, (Numerical): p. 1007–1028
Kirsch U, Bogomolni M, Sheinman I (2006) Nonlinear dynamic reanalysis of structures by combined approximations. Comput Methods Appl Mech Eng 195(33–36):4420–4432
Kirsch U (2010) Reanalysis and sensitivity reanalysis by combined approximations. Struct Multidiscip Optim 40(1–6):1
Kirsch U, Bogomolni M, Sheinman I (2007) Efficient structural optimization using reanalysis and sensitivity reanalysis. Eng Comput 23(3):229–239
Zuo W et al (2017) Sensitivity reanalysis of vibration problem using combined approximations method. Struct Multidiscip Optim 55(4):1399–1405
Sun R et al (2011) New adaptive technique of kirsch method for structural reanalysis. AIAA J 52(3):486–495
Zuo W et al (2011) Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods. Struct Multidiscip Optim 43(6):799–810
Xu T et al (2010) An adaptive reanalysis method for genetic algorithm with application to fast truss optimization. Acta Mech Sinica 26(2):225–234
Zuo W, Fang J, Feng Z (2019) Reanalysis method for second derivatives of static displacement. Int J Comput Methods
Zuo W, Bai J, Yu J (2016) Sensitivity reanalysis of static displacement using Taylor series expansion and combined approximate method. Struct Multidiscip Optim 53(5):953–959
Hosseinzadeh Y, Taghizadieh N, Jalili S (2018) A new structural reanalysis approach based on the polynomial-type extrapolation methods. Struct Multidiscip Optim 58(3):1033–1049
Kirsch U (2000) Combined approximations–a general reanalysis approach for structural optimization. Struct Multidiscip Optim 20(2):97–106
Sidi A (2003) Practical extrapolation methods: Theory and applications. Vol. 10.: Cambridge University Press
Cabay S, Jackson L (1976) A polynomial extrapolation method for finding limits and antilimits of vector sequences. SIAM J Numer Anal 13(5):734–752
Kaniel S, Stein J (1974) Least-square acceleration of iterative methods for linear equations. J Optim Theory Appl 14(4):431–437
Eddy R (1979) Extrapolating to the limit of a vector sequence, in Information linkage between applied mathematics and industry. Elsevier. p. 387–396
Mešina M (1977) Convergence acceleration for the iterative solution of the equations X= AX+ f. Comput Methods Appl Mech Eng 10(2):165–173
Bertelle R, Russo MR, Venturin M (2011) On the application of the minimum polynomial extrapolation method to incompressible flows with heat transfer. Calcolo 48(1):33–45
Duminil S, Sadok H, Silvester D (2014) Fast solvers for discretized Navier-stokes problems using vector extrapolation. Numerical Algorithms 66(1):89–104
Duminil S, Sadok H, Szyld DB (2015) Nonlinear Schwarz iterations with reduced rank extrapolation. Appl Numer Math 94:209–221
Loisel S, Takane Y (2011) Generalized GIPSCAL re-revisited: a fast convergent algorithm with acceleration by the minimal polynomial extrapolation. ADAC 5(1):57–75
Sidi A (1986) Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms. SIAM J Numer Anal 23(1):197–209
Sidi A (1994) Convergence of intermediate rows of minimal polynomial and reduced rank extrapolation tables. Numerical Algorithms 6(2):229–244
Sidi A (2012) Review of two vector extrapolation methods of polynomial type with applications to large-scale problems. J Comput Sci 3(3):92–101
Sidi A, Ford WF, Smith DA (1986) Acceleration of convergence of vector sequences. SIAM J Numer Anal 23(1):178–196
Smith DA, Ford WF, Sidi A (1987) Extrapolation methods for vector sequences. SIAM Rev 29(2):199–233
Süli E and Mayers DF (2003) An introduction to numerical analysis. Cambridge university press
Funding
This research is supported by a research grant of the University of Tabriz (grant No. 3500).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Responsible Editor: Erdem Acar
Publisher’s note
Springer Nature remains neutral withregard to jurisdictional claims in published mapsand institutional affiliations.
Rights and permissions
About this article
Cite this article
Hosseinzadeh, Y., Jalili, S. Structural sensitivity reanalysis formulations based on the polynomial-type extrapolation methods. Struct Multidisc Optim 61, 1027–1050 (2020). https://doi.org/10.1007/s00158-019-02401-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-019-02401-9