## 2 Methodology
### Numerical Methodology
#### 2.2.2 Quasi-Steady Model of the Transversal Sloshing Force
The RBF interpolation, including a polynomial of any degree, provides the interpolated function values \(\hat{g}(\hat{x})\) at an evaluation point \(\hat{x}_{E}\),  
\[\hat{g}(\hat{x}_{E})=\sum_{j=1}^{N}\gamma_{i}\phi(|\hat{x}_{E}-\hat{x}_{i}|,r )+\sum_{k=1}^{M}\lambda_{k}p_{k}(\hat{x}_{E}) \tag{8}\]  
where \(N\) is the number of experimental data points, \(p\) the \(M\) monomial coefficients based on the degree of the included polynomial and dimensionality of the data and \(\phi\) a compact basis function (a Wendland C4 is used) dependent on a spatial support length \(r\). Following the standard RBF interpolation procedure [25], model coefficients \(\gamma\) and \(\lambda\) are found by requiring exact recovery of the experimental/sample points, requiring a solution to the linear system  
\[\bigg{(} \begin{array}{cc}\boldsymbol{\phi}&\boldsymbol{P}\\ \boldsymbol{P}^{\intercal}&\boldsymbol{0}\end{array} \bigg{)}\bigg{(} \begin{array} []{c}\gamma\\ \underline{\lambda}\end{array} \bigg{)}=\bigg{(} \begin{array}{c}\underline{ \delta}\\ \underline{0}\end{array} \bigg{)} \tag{9}\]  
where \(\boldsymbol{\phi}\) is an \(N\times N\) matrix of basis function terms evaluated between each data point, \(\boldsymbol{P}\) an \(M\times N\) matrix of the monomial terms, \(\underline{\delta}\) a vector of the known non-dimensional sloshing forces. This interpolation ensures the experimental data points are exactly recovered and returns to a least-squares polynomial fit in the absence of RBF influence. However, in the current experimental data set, the polynomial term has little influence, and only a constant polynomial term is considered to avoid over-fitting and false extrapolation of the experimental data. Additionally, to improve matrix conditioning and simplify dependence on support radius \(r\) in N-dimensional space, the experimental domain (\(\hat{\pi}\)) is scaled to the hypercube. The solution to Equation (9) and subsequent application of Equation (8) to find the sloshing force at each tank can be used directly in the coupled numerical system; however, solving the linear system at each numerical timestep is an unnecessary cost. Therefore, before the time simulation begins, the RBF interpolation procedure performed on a fine grid and the solution stored in memory. For example, \(\hat{\boldsymbol{x}}_{E}=\{\underline{\vartheta}_{E},\underline{\varrho}_{ E},\underline{\omega}_{E}\}\), where \(\underline{\vartheta}_{E}\) and \(\underline{\varrho}_{E}\) cover the interval \([-1.5:1.5]\) in steps of a small resolution. During run-time, a secondary interpolation is performed on this fine grid to determine the sloshing force,  
Figure 7: Interpolated experimental response surfaces. Circular traces shown in the horizontal plane corresponding to the 3.3 Hz response surface. and a cubic interpolation is used here to ensure a smooth interpolation. We denote this final interpolation through the operator \(\hat{g}=G(\hat{y},\hat{v},\omega)\), which provides the final non-dimensional sloshing force. This interpolation is visualised as a series of surfaces within Figure 7 for three distinct values of \(\omega\), the experimental data points for one input frequency are shown in the horizontal displacement-velocity plane. Finally, a mechanism to include the initial transient nature of the fluid is required to avoid the ROM applying non-zero sloshing forces in the deformed (pre-release) condition and large nonphysical forces in the initial cycles after release. To achieve this, a compact time-dependent smoothing function is employed, visually shown in Figure 8 and having the form  
\[S(t)=\left(1-\left|\frac{t}{t_{s}}\right|^{3}\right)\left(3\left|\frac{t}{t_{ s}}\right|+1\right)\left(1-H(t-t_{s})\right)+H(t-t_{s}) \tag{10}\]  
where \(H\) is the Heaviside function, and \(t_{s}\) is the chosen smoothing time.