## 4 Partitioned Strong Coupling Formulation  
The structural dynamic system regarding fluid-structure interaction discretized by a finite element method gives coupled linear equations. The system of Equations (17)-(26) includes 12 equations and 12 unknown terms concerning displacement, velocity, acceleration and force. All quantities associated with time instant \(t^{n}\) are known. Eliminating all quantities associated with fractional time instants between \(t^{n}\) and \(t^{n+1}\), together with the Dirichlet-to-Neumann mapping along the interface  
\[\mathbf{d}^{\prime}=\mathbf{d}^{s}=\mathbf{d}^{\prime},\ \hat{\mathbf{d}}^{\prime}=\hat{\mathbf{d}}^ {s},\ \mathbf{\mathcal{R}}=\mathbf{\mathcal{R}}_{s}=-\mathbf{\mathcal{R}}_{f} \tag{27}\]  
yields  
\[\left\{ \begin{array}{c}\mathbf{d}^{n+1}\\ \hat{\mathbf{d}}^{n+1}_{s}\Delta t\\ \hat{\mathbf{d}}^{n+1}_{s}\Delta t^{2}\\ \mathbf{\mathcal{R}}^{n+1}\Delta t^{2}\end{array} \right\}=\left[ \begin{array}{c} \mathbf{\mathcal{A}}_{\text{strong}}\\ \mathbf{\mathcal{A}}_{\text{strong}}\\ \mathbf{\mathcal{A}}_{s}^{n}\Delta t^{2}\\ \mathbf{\mathcal{A}}_{f}^{n}\Delta t^{2}\\ \mathbf{\mathcal{R}}^{n}\Delta t^{2}\end{array} \right\} \tag{28}\]  
where \(\mathbf{\mathcal{A}}_{\text{strong}}\) is a five-dimensional amplification matrix. The coefficients of \(\mathbf{\mathcal{A}}_{\text{strong}}\) are dependent on the mass, damping and stiffness matrices, as well as on the time integration parameters \(\rho_{\infty}^{s}\) and \(\Delta t\). The system (26) is unconditionally stable if the spectral radius \(\rho_{\infty}\leq 1\) for any time step size \(\Delta t\). According to ([PERSON] and [PERSON], 2012), the spectral radius of an amplification matrix of dimension \(\kappa\) is defined as  
\[\rho_{\infty}=\ \text{max}(|\lambda_{1}|,|\lambda_{2}|,\ \ldots\,\ |\lambda_{ \kappa}|\ ) \tag{29}\]  
Here, \(\lambda_{i}\) are the eigenvalues of the amplification matrix. The eigenvalues of \(\mathbf{\mathcal{A}}_{\text{strong}}\) were evaluated by [PERSON] et al. (2013) for various parameters. Their evaluations suggested that unconditional stability could be achieved in all cases if the spectral radius \(\rho_{\infty}\leq 1\). Thus, in this paper, we specify \(\rho_{\infty}^{s}=0.5\) for elastic bodies ([PERSON] and [PERSON], 2006). The partitioned coupling scheme considered in the present work is based on the force predictor for the solid problem ([PERSON] and [PERSON], 2013; 2013). The coupling formulation is Dirichlet-Neumann coupling, in which the interface traction from the fluid solver is considered as a Neumann boundary condition for the solid domain, and the interface velocity is applied as a Dirichlet boundary condition for the fluid problem. The procedure of the scheme is presented in Algorithm 1. ```

1. Perform the traction force predictor for interface traction pressure \(\mathbf{P}_{s}^{n}\) and velocity \(\mathbf{u}_{s}^{n}\). 2. Each FSI iteration between \(t\in[t^{n},\ t^{n+1}]\):

(a) Solve structural Eq. (15) using known interface traction \(\mathbf{P}_{s}^{n}\);

(b) Apply Dirichlet velocity continuity condition

\(\mathbf{u}_{f}^{n+1}=\mathbf{u}_{f}^{n+1}\) on interface \(\Gamma\);

(c) Set \(\hat{\mathbf{d}}_{f}^{n+1}=\mathbf{u}_{f}^{n+1}\) and update the fluid dynamic mesh;

(d) Solve ALE fluid Equation (10) for new interface traction \(\mathbf{P}_{s}^{n+1}\) and apply Neumann condition \(\mathbf{P}_{s}^{n+1}=\mathbf{P}_{f}^{n+1}\) on interface \(\Gamma\). In Algorithm 1, a structural solver is employed to accept external fluid forces. A fluid solver is used to accept the prescribed motion of the interface boundary. The structural and fluid solvers are used to proceed in a predictor-corrector format by constructing an iterative interface force correction and the force transfer at each sub-iteration. The schematic for the partitioned iterative coupling of the ALE fluid solver with the structural solver is depicted in Figure 1.

It is worth mentioning that the partitioned coupling scheme in this paper has second-order accuracy in time.