## 2 Methodology
### Numerical Methodology
#### 2.2.2 Quasi-Steady Model of the Transversal Sloshing Force
The applicability of this analysis for fluid-structure coupled simulation is immediately noticeable; as a structure undergoes transverse motion, the vertical kinematics of the tank can be tracked, and the relevant nonlinear sloshing force returned to the structure. Further, the range of amplitudes in the experimental data ensures multiple fluid regimes are included, from little free-surface motion (R3), low amplitude Faraday waves (R2) and violent fluid impacting (R1). However, a series of caveats to this particular method must be addressed, namely  
* Scaling between the experimental and numerical (MiniWoT) tank dimensions. * Producing a continuous and smooth sloshing force for numerical analysis, which can not be directly extracted from the experimental data. A result of the finite sampling frequency producing discrete data points in each cycle, plus discrete amplitudes and frequencies of the harmonic cycles considered. * Identified forces assume the fluid has reached a quasi-steady state under the excitation and, therefore, can not account for sharp transients (i.e., free-release of the structure). A mechanism must be introduced to account for the initial energy input into the fluid to excite the identified motions within the data-set. Here, scaling laws are based purely on non-dimensional kinematic and force terms, namely displacement \(\dot{y}=y/h\), velocity \(\dot{v}=\dot{y}/(\omega_{c}h)\) and force \(\dot{F}_{s}=F_{s}/(m_{l}h\omega^{2})\), where the characteristic length is tank height \(h\), \(m_{l}\) the liquid mass and \(\omega\) the characteristic frequency of excitation for that specific excitation condition. Within the previous experimental work [7], normalisation of dissipated energy via \(\dot{E}=E/(m_{l}\omega^{2}A^{2})\), where A is the amplitude of harmonic excitation, was shown to provide a consistent scaling across excitation conditions. However, generally in the transient setting, a characteristic amplitude is not known a priori; hence, the characteristic length of tank height is used. The significant assumption made is that the fluid forcing (and thus, resulting dissipation characteristics) scales with these non-dimensional terms. No scaling for the tank geometry (aside from height) is considered, which is generally less problematic for the violent impacting regime (R1) but becomes more important during the R2 regime, where the formation of standing waves is sensitive to the base geometry of the tank. This is further discussed later and has a direct influence on the chosen fluid parameters used within this particular analysis. Further, the dissipation resulting from the hydrodynamic response is likely to increase with tank scales (increasing \(m_{l}\)) due to increased free-surface fragmentation and volumetric dissipation forming from turbulent eddies in the liquid. Whilst not included here, a correctly introduced Froude scaling could approximate these effects [24]. Regarding the second point, the discrete experimental data must be applied to the numerical model in a continuous and smooth fashion to ensure successful time integration. Therefore, radial basis function interpolation is used, an established method for providing accurate and robust scattered-data interpolation of n-dimensional data [25]. Initially, consider the total experimental data set across multiple amplitudes and frequencies of excitation, the function mapping excitation kinematics to sloshing force can take the form \(\hat{\underline{F}}_{s}=\hat{g}(\hat{x})\), \(\hat{\pi}=\{\underline{g},\underline{\varrho},\underline{\omega}\}\).