# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 2 FSI Simulation of a Simplified 2D Leaflet Structure  
The academic problem studied in this paper consists of using the data generated by a set of FSI simulations of a moving fluid across a 2D channel with a pair of leaflets fixed at the center and then creating an ARX model with these data. In this section, we explain how the experimental setup found in Ledesma-Alonso [42] was adapted for our simulations to be later used in the SI model. The design presented in Figure 1a shows the leaflets and channel layout. The top and bottom boundary walls of the channel are separated by a distance of 2 h, where h = 15 mm; both walls present a non-slip boundary condition, and the length of the channel is 20 h. The experimental setup considered a 3D environment where the width of the channel was \(w\) = 50 mm; however, this simplified layout neglected this parameter to reduce computational cost, hence a 2D version of the problem has been studied. The location of the fixed tip of the leaflets was at the center of the channel (at \(x\) = 0 mm). This layout also considered the two experimental points located at \(\pm\)5 h horizontal distance from the center, in order to measure the pressures during all the simulation time. Regarding the parameters used for the simulations, they consist of two different groups: the fluid and the solid domain. For the fluid domain \((\Omega_{f})\), the parameters of the input signal or input velocity are the magnitude \((U_{max})\), stroke volume \((V)\), frequency \((f)\), and the fluid parameters are the density \((\rho_{f})\), and dynamic viscosity \((\mu)\). For the solid domain \((\Omega_{s})\), the parameters are the leaflet thickness \((d)\) and length \((l)\), material density \((\rho_{s})\), elastic modulus \((E)\), and Poisson ratio \((\
u)\). One of the materials chosen for the leaflets in the experimental case was silicone rubber; hence, the material properties applied to the solid bodies within the simulation are related to this material, which will be taken as the reference case. Under the circumstances of the proposed system, its dimensions are much larger than the size of red blood cells and, as a consequence, the working fluid can be considered as Newtonian [42]. Figure 1: (**a**) Layout of the system: \(h\) represents the semi-height of the channel. \(P_{0}\) and \(P_{1}\) are the points used to measure pressure. The original setup can be found in [42]. (**b**) The normalized velocity curve \(U_{f}^{*}(t^{*})\) used for the Fluid-Structure Interaction (FSI) simulations. (**b**) represents the fast Fourier transform of the signal with its representative working frequencies. Simulations were conducted using an HP workstation with an Intel Xeon Processor running at 3.40 GHz and 16 GB of RAM. The software selected was LS-DYNA (Livermore Software Technology Corporation, Livermore, CA), which for previous tests has proven its capabilities to handle CFD and FSI simulations by using an ALE approach [43; 44; 45]. The output variables considered for these tests are the leaflets opening area (\(S_{v}\)), defined as the vertical distance between the tips of the leaflets, and the pressure difference (\(\Delta\)P), defined as \(P_{1}-P_{0}\) the pressure difference between points 1 and 0 (see Figure 1a). Then, our first approach of an FSI simulation was conducted using the parameters listed in Table 1.  
The time evolution of the phenomenon, for which \(t\) is the independent variable, takes place along a period \(T\), which is equivalent to \(T=1/f\). For this first scenario, the input velocity curve was a pulsatile flow curve divided into forward time (\(t_{f}\)) and backward time (\(t_{b}\)) stages (see Figure 1b), with \(T=t_{f}+t_{b}\). Using \(T\) as the characteristic time scale to normalize time, thus making \(t^{*}=t/T\), the forward stage occurs during \(t^{*}=[0.05,0.49]\), whereas the backward stage happens during \(t^{*}=(0.49,1]\) and also occurs during a short time before the forward stage begins, i.e., \(t^{*}=[0,0.05)\).