Skip to main content
SpringerLink
Log in

Hammerstein–Wiener based reduced-order model for vortex-induced non-linear fluid–structure interaction

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

Abstract

Fluid–structure interaction (FSI) phenomena are of interest in several engineering fields. It is highly desirable to develop computationally efficient models to predict the dynamics of FSI. The complexity of modeling lies in the highly non-linear response of both the fluid and structure. The current study proposes an overall model containing two blocks corresponding to a force model and a structural model. The force model consists of two submodels: one for the amplitude and one for the frequency, where the latter is composed of an input/output linear model and a non-linear corrector. The amplitude submodel and the non-linear corrector term in the frequency submodel are modeled using an Hammerstein–Wiener modeling technique in which the non-linear input and output functions are determined by training neural networks using a training dataset. The current model is tested on a well-known fluid–structure interaction problem: a suspended rigid cylinder immersed in a flow at a low Reynolds number regime that exhibits a non-linear behavior. First, a training dataset is generated for a given input profile using a high-fidelity numerical simulation and it is used to train the reduced-order model. Subsequently, the trained model is given a different input profile (i.e., a validation profile) to compare its predictive capability against the high-fidelity numerical simulation. The validation profile is significantly different from the one used for training. The predictive performance of the current reduced-order model is further compared with the results obtained from a reduced-order model that uses polynomial fitting. We demonstrate that the current model provides a superior performance for the validation profile, i.e., it results in a better prediction.

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
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

Abbreviations

A, B :

Linear model's state space matrices

\(a_{j}^{k}\) :

Predicted modal coefficient of the jth mode at time t k

\(\tilde{C}\) :

Map between the time evolution of dominant force frequency and \(\tilde{\varvec{V}}\) , \(\tilde{C} \in R^{1 \times 2l}\)

\(c\) :

Structure damping coefficient (kg/s)

\(d(t)\) :

Structural displacement at time t (m)

\(\dot{d}(t)\) :

Structural velocity at time t (m/s)

\(\ddot{d}(t)\) :

Structural acceleration at time t (m/s2)

\(\tilde{\varvec{d}}\) :

Time history of the structural displacement (m)

\(\tilde{d}_{k}\) :

Measured structural displacement at time t k (m)

\(\tilde{\varvec{F}}\) :

Time history of the vertical force (N)

\(F_{k}\) :

Modeled force at time t k (N)

\(\tilde{F}_{k}\) :

Measured force at time t k (N)

\(F_{{M_{K} }}\) :

Modeled force amplitude at time t k (N)

\(\tilde{\varvec{F}}_{u}\) :

Upper envelope of \(\tilde{\varvec{F}}\) (N)

\(f_{M}\) :

Function that relates \(F_{{M_{K} }}\) with \(u_{k}\) and \(t_{k}\)

\(f_{{w_{{a_{j} }} }}\) :

HW model’s input non-linear block for jth modal coefficient

\(f_{{w_{\text{F}} }}\) :

HW model’s input non-linear block for force amplitude

\(f_{{y_{{a_{j} }} }}\) :

HW model’s output non-linear block for jth modal coefficient

\(f_{{y_{\text{F}} }}\) :

HW model’s output non-linear block for force amplitude

\({\hat{\text{f}}\text{r}}_{k}\) :

Compensated (or corrected) dominant force frequency at time t k (Hz)

\(g_{\text{fr}}^{j}\) :

Function that relates \(a_{j}^{k}\) with \(u_{k}\) and time tk

\(H_{{a_{j} }}\) :

HW model’s transfer function for jth modal coefficient

\(H_{\text{F}}\) :

HW model’s transfer function for force amplitude

\(K\) :

Structure stiffness (N/m)

\(m\) :

Structure mass (kg)

\(n_{\text{IN}}\) :

Total number of neurons in the HW model’s input non-linear block

\(n_{{{\text{IN}}_{\text{MAX}} }}\) :

Maximum number of neurons possible in the HW model’s input non-linear block

\(n_{\text{OUT}}\) :

Total number of neurons in the HW model’s output non-linear block

\(n_{\text{p}}\) :

Total number of poles in the HW model’s transfer function

\(n_{\text{PMAX}}\) :

Maximum number of poles possible in the HW model’s transfer function

\(n_{\text{z}}\) :

Total number of zeroes in the HW model’s transfer function

\({\text{NRMSE}}_{\text{F}}\) :

Normalized root mean square error in the modeled force amplitude

\({\text{NRMSE}}_{j}\) :

Normalized root mean square error in the modeled jth modal coefficient

\(o_{j}\) :

jth zero of the HW model’s transfer function

P :

Number of POD modes used in reconstruction

\(p_{j}\) :

jth pole of the HW model’s transfer function

\(t\) :

Time (s)

\(\tilde{\varvec{u}}\) :

Time history of the input velocity (m/s)

\(u_{k}\) :

Input value at time t k (m/s)

\(\tilde{u}_{k}\) :

Measured input velocity at time t k (m/s)

\(\tilde{\varvec{V}}\) :

Time history of the two-dimensional velocity vector field (m/s)

\(\tilde{\varvec{v}}_{k}\) :

Measured velocity vector field at time t k (m/s)

\(w_{{{\text{F}}_{{M_{k} }} }}\) :

Output of the HW model’s input non-linear block for force amplitude

\(w_{{a_{j} }}^{k}\) :

Output of the HW model’s input non-linear block for jth modal coefficient

\(\varvec{x}_{k}\) :

State vector of the linear model at time t k

\(y_{{_{{a_{j} }} }}^{k}\) :

Output of the HW model’s transfer function for jth modal coefficient

\(y_{{{\text{F}}_{{M_{k} }} }}\) :

Output of the HW model’s transfer function for force amplitude

\(\varvec{z}_{k}\) :

State vector of the structural model

\(\varPhi\) :

Map between the time evolution of the state vector \(\varvec{x}_{k}\) and \(\tilde{\varvec{V}}\), \(\varPhi \in R^{2l \times r}\)

\(\varvec{\psi}_{j}\) :

jth mode shape (POD)

References

  1. Larsen A (2000) Aerodynamics of the Tacoma narrows bridge—60 years later. Struct Eng Int 10(6):243–248

    Article  Google Scholar 

  2. Kareem A, Kijewski T, Tamura Y (1999) Mitigation of motions of tall buildings with specific examples of recent applications. J Wind Struct 2:201–251

    Article  Google Scholar 

  3. Lin N, Letchford C, Tamura Y, Liang B, Nakamura O (2005) Characteristics of wind forces acting on tall buildings. J Wind Eng Ind Aerodyn 93:217–242

    Article  Google Scholar 

  4. Tamura Y, Kawasana S, Nakamura O, Kanda J, Nakata S (2006) Evaluation perception of wind-induced vibration in buildings. Proc Inst Civ Eng Struct Build 159(5):283–293

    Article  Google Scholar 

  5. Bashor R, Kijewsky-Correa T, Kareem A (2005) On the wind-induced response of tall buildings: the effect of uncertainties in dynamic properties and human comfort thresholds. In: Proceedings of 10th Americas conference on wind engineering, Baton Rouge, Louisiana

  6. Kareem A (1987) Wind effects on structures: a probabilistic viewpoint. Probab Eng Mech 2(4):166–200

    Article  Google Scholar 

  7. Tse KT, Hitchcock PA, Kwok KCS, Thepmongkorn S, Chen CM (2009) Economic perspectives of aerodynamic treatments of square tall building. J Wind Eng Ind Aerodyn 97:455–467

    Article  Google Scholar 

  8. Kwok KCS, Wilhelm PA, Wilkie BG (1988) Effect of edge configuration on wind-induced response of tall buildings. Eng Struct 10(2):135–140

    Article  Google Scholar 

  9. Menicovich D, Vollen J, Amitay M, Letchford C, Dyson A, DeMauro E, Ajith R (2012) A different approach to the aerodynamic performance of tall buildings. In: Council on tall buildings and Urban habitat issue IV

  10. Jonkman JM, Matha D (2011) Dynamics of offshore floating wind turbines analysis of three concepts. J Wind Energy 14:557–569

    Article  Google Scholar 

  11. Ham ND, Garelick MS (1968) Dynamic stall considerations in helicopter rotors. J Am Helicopt 13(2):49–55

    Article  Google Scholar 

  12. Ham ND (1968) Aerodynamic loading on a two-dimensional airfoil during dynamic stall. AIAA J 6(10):1927–1934

    Article  MATH  Google Scholar 

  13. McCroskey WJ, Carr LW, McAlister KW (1976) Dynamic stall experiments on oscillating airfoils. AIAA J 14(1):57–63

    Article  Google Scholar 

  14. Dowell EH, Hall KC (2001) Modeling of fluid–structure interaction. Annu Rev Fluid Mech 33:445–490

    Article  MATH  Google Scholar 

  15. Theodorsen T, Garrick IE (1933) General potential theory of arbitrary wing sections. NACA Report No. 452

  16. Dowell EH, Crawley EF, Curtiss HC Jr, Peters DA, Scanlan RH, Sisto F (1995) A modern course in aeroelasticity, 3rd ed. Kluwer Acad. Pub., Dordrecht

  17. Florea R, Hall KC, Cizmas PG (1998) Reduced-order modeling of unsteady viscous flow in a compressor cascade. AIAA J 36(6):1039–1048

    Article  Google Scholar 

  18. Tijdeman H, Seebass R (1980) Transonic flow past oscillating airfoils. Annu Rev Fluid Mech 12(1):181–222

    Article  MATH  Google Scholar 

  19. Seebass AR, Fung KY, Przybytkowski SM (1986) Advances in the understanding and computation of unsteady transonic flow. Recent advances in aerodynamics. Springer, New York, pp 3–37

    Chapter  Google Scholar 

  20. Nixon D (1989) Unsteady transonic aerodynamics. In: Progress in Astronautics and Aeronautics, vol 120. AIAA, Washington DC

  21. Benaroya H, Gabbai RD (2007) Modeling vortex-induced fluid–structure interaction. Philos Trans R Soc 366:1231–1274

    Article  Google Scholar 

  22. Karhunen Kari (1947) Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann Acad Sci Fennicae Ser A I Math Phys 37:1–79

    MathSciNet  MATH  Google Scholar 

  23. Loève M (1978) Probability theory II. In: Graduate Texts in Mathematics, vol 46. Springer, Berlin

  24. Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25:539–575

    Article  MathSciNet  Google Scholar 

  25. Aradag S, Siegel S, Seidel J, Cohen K, McLaughlin T (2011) Filtered POD-based low dimensional modeling of the 3D turbulent flow behind a circular cylinder. Numer Meth Fluids 66:1–16

    Article  MATH  Google Scholar 

  26. Seidel J, Siegel S, Fagley C, Cohen K, McLaughlin T (2009) Feedback control of a circular cylinder wake. J Aerosp Eng 223:379–392

    Google Scholar 

  27. Siegel SG, Seidel J, Fagley C, Luchtenburg DM, Cohen K, McLaughlin T (2008) Low-dimensional modeling of a transient cylinder wake using double proper orthogonal decomposition. J Fluid Mech 610:1–42

    Article  MathSciNet  MATH  Google Scholar 

  28. Siegel SG, Cohen K, Seidel J, McLaughlin T (2007) State estimation of transient flow fields using double proper orthogonal decomposition (DPOD). In: Active Flow Control, vol 95. Springer, Berlin, pp 105–118

  29. Schmid PJ (2010) Dynamic mode decomposition of numerical and experimental data. J Fluid Mech 656:5–28

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen KK, Tu JH, Rowley CW (2012) Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J Nonlinear Sci 22(6):887–915

    Article  MathSciNet  MATH  Google Scholar 

  31. Siegel S, Cohen K, McLaughlin T (2006) Numerical simulations of a feedback-controlled circular cylinder wake. AIAA J 44:1266–1276

    Article  Google Scholar 

  32. Gallardo D, Bevilacqua R, Sahni O (2014) Data-based hybrid reduced order modeling for vortex-induced nonlinear fluid-structure interaction at low Reynolds numbers. J Fluids Struct 44:115–128

    Article  Google Scholar 

  33. Ge X, Wen JT (2011) Hybrid model reduction for compressible flow controller design. In: Proceedings of the 50th IEEE conference on decision and control and European control conference, Orlando, Florida

  34. Zhu Y (2002) Estimation of an N-L-N Hammerstein–Wiener model. Automatica 38:1607–1614

    Article  MathSciNet  MATH  Google Scholar 

  35. Bai EW (2002) A blind approach to the Hammerstein–Wiener model identification. Automatica 38:967–979

    Article  MathSciNet  MATH  Google Scholar 

  36. Bloemen HHJ, Van Der Boom TJJ, Verbruggen HB (2001) Model-based predictive control for Hammerstein–Wiener systems. Int J Control 74(5):482–495

    Article  MathSciNet  MATH  Google Scholar 

  37. Janczak A (2004) Identification of nonlinear systems using neural networks and polynomial models: a block-oriented approach. In: Lecture Notes in Control and Information Sciences, vol 310. Springer, Berlin

  38. Murphy MJ, Dieterich S (2006) Comparative performance of linear and nonlinear neural networks to predict irregular breathing. Phys Med Biol 51(22):5903–5914

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical, Aerospace and Nuclear Engineering (MANE), Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180, USA

    Daniele Gallardo & Onkar Sahni

  2. Department of Mechanical and Aerospace Engineering, University of Florida, 939 Sweetwater Drive, Gainesville, FL, 32608, USA

    Riccardo Bevilacqua

Authors
  1. Daniele Gallardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Onkar Sahni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Riccardo Bevilacqua
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Onkar Sahni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gallardo, D., Sahni, O. & Bevilacqua, R. Hammerstein–Wiener based reduced-order model for vortex-induced non-linear fluid–structure interaction. Engineering with Computers 33, 219–237 (2017). https://doi.org/10.1007/s00366-016-0467-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-016-0467-9

Keywords

Search

Navigation