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

Abstract

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.

Appendix

1.1 A1 Time-discretization of Eq. (12) for Scheme B shown in Eq. (14)

Scheme B uses the Crank–Nicholson method to approximate $\dot{\bar{{\theta }}}$ and $\ddot{\bar{{\theta }}}$ as

$$\begin{aligned} \frac{\hbox {d}\theta }{\hbox {d}\bar{{t}}}= & {} \dot{\bar{{\theta }}}\,\rightarrow \,\theta _{n+1} \approx \theta _n +\frac{{\Delta }\bar{{t}}}{2}\left( {\dot{\bar{{\theta }}}_{n+1} +\dot{\bar{{\theta }}}_n } \right) , \end{aligned}$$

(A1)

$$\begin{aligned} \frac{\hbox {d}\left( {\dot{\bar{{\theta }}}} \right) }{\hbox {d}\bar{{t}}}= & {} \ddot{\bar{{\theta }}}\,\rightarrow \,\dot{\bar{{\theta }}}_{n+1} \approx \dot{\bar{{\theta }}}_n +\frac{{\Delta }\bar{{t}}}{2}\left( {\ddot{\bar{{\theta }}}_{n+1} +\ddot{\bar{{\theta }}}_n } \right) . \end{aligned}$$

(A2)

Re-arranging Eqs. (A1) and (A2) yield

$$\begin{aligned} \dot{\bar{{\theta }}}_{n+1}\approx & {} \frac{2}{\Delta \bar{{t}}}\left( {\theta _{n+1} -\theta _n } \right) -\dot{\bar{{\theta }}}_n, \end{aligned}$$

(A3)

$$\begin{aligned} \ddot{\bar{{\theta }}}_{n+1}\approx & {} \frac{2}{\Delta \bar{{t}}}\left( {\dot{\bar{{\theta }}}_{n+1} -\dot{\bar{{\theta }}}_n } \right) -\ddot{\bar{{\theta }}}_n , \end{aligned}$$

(A4)

and, hence,

$$\begin{aligned} \ddot{\bar{{\theta }}}_{n+1} \approx \frac{4}{\Delta \bar{{t}}^{2}}\left( {\theta _{n+1} -\theta _n } \right) -\frac{4\dot{\bar{{\theta }}}_n }{\Delta \bar{{t}}}-\ddot{\bar{{\theta }}}_n. \end{aligned}$$

(A5)

Using Eqs. (A3) and (A5) in Eq. (12), in addition to Eqs. (A3) and (A5) themselves result in the linear system shown in Eq. (14), which could be solved in terms of $\theta _{n+1}$, $\dot{\bar{{\theta }}}_{n+1}$, and $\ddot{\bar{{\theta }}}_{n+1}$.

1.2 A2 Time-discretization of Eq. (12) for Scheme C shown in Eq. (15)

Scheme C is constructed by converting the second-order differential equation (Eq. (12)) into two first-order differential equations for ($\theta$, $\dot{\bar{{\theta }}})$ as

$$\begin{aligned}&\frac{\hbox {d}\theta }{\hbox {d}\bar{{t}}}=\dot{\bar{{\theta }}}, \end{aligned}$$

(A6)

$$\begin{aligned}&\tilde{M}_\mathrm{i} \frac{\hbox {d}\dot{\bar{{\theta }}}}{\hbox {d}\bar{{t}}}+\tilde{C}_\mathrm{i} \dot{\bar{{\theta }}}+\tilde{K}_\mathrm{i} \theta =-\tilde{M}_\mathrm{e} \frac{\hbox {d}\dot{\bar{{\theta }}}}{\hbox {d}\bar{{t}}}-\tilde{C}_\mathrm{e} \dot{\bar{{\theta }}}-\tilde{K}_\mathrm{e} \theta . \end{aligned}$$

(A7)

Scheme C discretizes Eqs. (A6) and (A7) as

$$\begin{aligned}&\theta _{n+1} =\theta _n +\frac{\Delta \bar{{t}}}{2}\left( {\dot{\bar{{\theta }}}_{n+1} +\dot{\bar{{\theta }}}_n } \right) , \end{aligned}$$

(A8)

$$\begin{aligned}&\dot{\bar{{\theta }}}_{n+1} \left( {1+\frac{\Delta \bar{{t}}\tilde{C}_\mathrm{i} }{\tilde{M}_\mathrm{i} }} \right) + \frac{\Delta \bar{{t}}\tilde{K}_\mathrm{i} }{\tilde{M}_\mathrm{i} }\theta _{n+1} \nonumber \\&\quad =\dot{\bar{{\theta }}}_n -\frac{\tilde{M}_\mathrm{e} }{\tilde{M}_\mathrm{i} }\left( {\dot{\bar{{\theta }}}_n -\dot{\bar{{\theta }}}_{n-1} } \right) -\frac{\Delta \bar{{t}}\tilde{C}_\mathrm{e} }{\tilde{M}_\mathrm{i} }\dot{\bar{{\theta }}}_n -\frac{\Delta \bar{{t}}\tilde{K}_\mathrm{e} }{\tilde{M}_\mathrm{i}}\theta _n, \nonumber \\&\dot{\bar{{\theta }}}_{n+1} \left( {1+\frac{\Delta \bar{{t}}\tilde{C}_\mathrm{i} }{\tilde{M}_\mathrm{i} }} \right) +\frac{\Delta \bar{{t}}\tilde{K}_\mathrm{i} }{\tilde{M}_\mathrm{i} }\theta _{n+1} \nonumber \\&\quad =\dot{\bar{{\theta }}}_n -\frac{\tilde{M}_\mathrm{e} }{\tilde{M}_\mathrm{i} }\left( {\dot{\bar{{\theta }}}_n -\dot{\bar{{\theta }}}_{n-1} } \right) -\frac{\Delta \bar{{t}}\tilde{C}_\mathrm{e} }{\tilde{M}_\mathrm{i} }\dot{\bar{{\theta }}}_n -\frac{\Delta \bar{{t}}\tilde{K}_\mathrm{e}}{\tilde{M}_\mathrm{i} }\theta _n. \end{aligned}$$

(A9)

Equation (A8) is the Crank–Nicholson discretization of Eq. (A6). To get Eq. (A9), Eq. (A7) has been discretized by treating its left-hand side of Eq. (A7) implicitly, and right-hand side explicitly. The combined linear system of Eqs. (A8) and (A9) is given as Eq. (15).