Acta Mechanica Sinica

, Volume 33, Issue 4, pp 709–724 | Cite as

Numerical stabilities of loosely coupled methods for robust modeling of lightweight and flexible structures in incompressible and viscous flows

Research Paper


The growing interest to examine the hydroelastic dynamics and stabilities of lightweight and flexible materials requires robust and accurate fluid–structure interaction (FSI) models. Classically, partitioned fluid and structure solvers are easier to implement compared to monolithic methods; however, partitioned FSI models are vulnerable to numerical (“virtual added mass”) instabilities for cases when the solid to fluid density ratio is low and if the flow is incompressible. As a partitioned method, the loosely hybrid coupled (LHC) method, which was introduced and validated in Young et al. (Acta Mech. Sin. 28:1030–1041, 2012), has been successfully used to efficiently and stably model lightweight and flexible structures. The LHC method achieves its numerical stability by, in addition to the viscous fluid forces, embedding potential flow approximations of the fluid induced forces to transform the partitioned FSI model into a semi-implicit scheme. The objective of this work is to derive and validate the numerical stability boundary of the LHC. The results show that the stability boundary of the LHC is much wider than traditional loosely coupled methods for a variety of numerical integration schemes. The results also show that inclusion of an estimate of the fluid inertial forces is the most critical to ensure the numerical stability when solving for fluid–structure interaction problems involving cases with a solid to fluid-added mass ratio less than one.


Numerical stability Fluid–structure interaction Loosely hybrid coupling method Incompressible flow Partitioned methods Lightweight structures 


\(A_\mathrm{c}^\mathrm{M} \)

Nondimensional \(C_\mathrm{f}^\mathrm{M} \, (={C_\mathrm{f}^\mathrm{M} }/{\rho _\mathrm{f} Ub^{3}})\)

\(A_\mathrm{k}^\mathrm{M} \)

Nondimensional \(K_\mathrm{f}^\mathrm{M} \, (={K_\mathrm{f}^\mathrm{M} }/{\rho _\mathrm{f} U^{2}b^{2}})\)

\(A_\mathrm{m}^\mathrm{M} \)

Nondimensional \(M_\mathrm{f}^\mathrm{M} \,(={M_\mathrm{f}^\mathrm{M} }/{\rho _\mathrm{f} b^{4}})\)

\(A_\mathrm{c}^\mathrm{P} \)

Nondimensional \(C_\mathrm{f}^\mathrm{P} \,(={C_\mathrm{f}^\mathrm{P} }/{\rho _\mathrm{f} Ub^{3}})\)

\(A_\mathrm{k}^\mathrm{P} \)

Nondimensional \(K_\mathrm{f}^\mathrm{P} \,(={K_\mathrm{f}^\mathrm{P} }/{\rho _\mathrm{f} U^{2}b^{2}})\)

\(A_\mathrm{m}^\mathrm{P} \)

Nondimensional \(M_\mathrm{f}^\mathrm{P} \,(={M_\mathrm{f}^\mathrm{P} }/{\rho _\mathrm{f} b^{4}})\)


Distance to E.A. from the foil’s mid-chord location


Foil’s half chord length


Foil’s nondimensional radius of gyration \((={I_\theta }/{\rho _\mathrm{s} b^{4}})\)


Foil’s center of gravity


Part of the total fluid and solid damping treated explicitly (Eq. (12))


Empirically estimated fluid–induced torsion damping from experiments and viscous simulations


Potential-flow estimation of the fluid–induced torsion damping


Part of the total fluid and solid damping treated implicitly (Eq. (12))

\(C_\theta \)

Foil’s torsional damping value per unit span


Foil’s chord length


Degree of freedom


Foil’s elastic axis




Fluid–structure interaction

\(I_\theta \)

Foil’s mass moment of inertia per unit span


Part of the total fluid and solid damping treated explicitly (Eq. (12))


Empirically estimated fluid–induced torsion stiffness from experiments and viscous simulations


Potential-flow estimation of the fluid–induced torsion stiffness


Part of the total fluid and solid damping treated implicitly (Eq. (12))

\(K_\theta \)

Foil’s torsional stiffness per unit span


Reduced frequency \((={\omega b}/U)\)


Loosely coupled


Loosely hybrid coupled

\(\tilde{M}_\mathrm{e} \)

Part of the total fluid and solid mass treated explicitly (Eq. (12))

\(M_\mathrm{f}^\mathrm{M} \)

Empirically estimated fluid-added moment of inertia from experiments or viscous simulations

\(M_\mathrm{f}^\mathrm{P} \)

Potential-flow estimation of the fluid-added moment of inertia


Fluid–induced moment per unit span

\(M_{\mathrm{fluid}}^\mathrm{E} \)

Estimated fluid–induced moment per unit span


Empirically estimated fluid–induced moment on the foil per unit span from experiments or viscous simulations

\(M_{\mathrm{fluid}}^\mathrm{P} \)

Potential-flow estimation of the fluid–induced moment per unit span

\(\tilde{M}_\mathrm{i} \)

Part of the total fluid and solid mass treated implicitly (Eq. (12))


Discrete time-level \((t=n\Delta t)\)


Reynolds number \((={Uc}/{\upsilon _\mathrm{f} })\)


Foil’s span length




Nondimensional time \((={tU}/b)\)


Inflow speed


Reduced speed \((=U/b\omega _\theta )\)

\(x_\theta \)

Distance to the C.G. from E.A. positive towards the foil trailing edge

\(\alpha \)

Fraction of the fluid-added mass used for \(A_\mathrm{m}^\mathrm{P} \)\(A_\mathrm{m}^\mathrm{M} \)

\(\Delta t\)

Time-step size

\(\Delta \bar{{t}}\)

Nondimensional time-step size

\(\theta \)

Foil’s twist angle at its E.A.

\(\lambda \)

Growth factor (in time) of \(\theta \, (\theta _{n+1} =\lambda \theta _n )\)

\(\mu \)

Solid to fluid density ratio \((={\rho _\mathrm{s} }/{\rho _\mathrm{f} })\)

\(\upsilon _\mathrm{f} \)

Fluid’s kinematic viscosity

\(\xi _\theta \)

Foil’s damping coefficient in air \((={C_\theta }/{2I_\theta \omega _\theta })\)

\(\rho _\mathrm{f} \)

Fluid’s density

\(\rho _\mathrm{s} \)

Foil’s density

\(\phi \)

Theodorsen’s function (Eq. (36))

\(\omega \)

Foil’s primary vibration frequency

\(\omega _\mathrm{n} \)

Foil’s twisting natural frequency in water

\(\omega _\theta \)

Foil’s twisting natural frequency in air



The authors would like to acknowledge the financial support received from the Office of Naval Research (ONR) (Grants N00014-11-1-0833 and N00014-13-1-0383) managed by Ms. Kelly Cooper and Dr. Ki-Han Kim, respectively.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Deniz Tolga Akcabay
    • 1
  • Jian Xiao
    • 1
  • Yin Lu Young
    • 1
  1. 1.Department of Naval Architecture and Marine EngineeringUniversity of MichiganAnn ArborUSA

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