## 2 Model
### Computational Set Up  
In this work, we focus on the non-symmetrical collapse induced by an anisotropic pressure field. The anisotropy is generated by non-symmetric water domain shown in Figure 1. The domain is divided in three concentric regions, delimited by three different radii. In each region, different types of computational particles are used:  
* Cavity (\(r<R_{0}\)): inside the yellow region in Figure 1, particles are removed to generate an empty cavity with \(P_{b}=0\).
* Liquid (\(R_{0}<r<R_{S}\)): inside the blue region in Figure 1, particles are modeled as water following a liquid EOS. The density is set as \(\rho_{L}=1000\) [kg m\({}^{-3}\)] with initial pressure \(P_{\infty}\).
* Shell (\(r>R_{S}\)): inside the red region of Figure 1, are modeled as in the liquid region. Moreover, they have fixed position and density to keep constant pressure as boundary condition. The extension of shell region has been discussed in previous work [17]. The lower part of this region also acts as a wall inducing anisotropy in the pressure field during the collapse.  
The green region in Figure 1 does not refer to a different type of particle. We highlight it because, later, the pressure generated during the collapse will be monitored in the green region. As explained in next section, when a non-symmetric collapse is studied, it is necessary to quantify the anisotropy of the collapse. When the anisotropy is induced by the presence of a nearby solid surface, is common to use the stand-off, \(\gamma\)[6] defined as  
\[\gamma=\frac{d}{R_{0}}, \tag{5}\]  
where \(d\) is the distance between the cavity center and the wall (see Figure 1) and \(R_{0}\) its initial radius. The dynamic of a cylindrical cavity in a Rayleigh collapse, where the driving force is the pressure difference between the pressure in the liquid, \(P_{\infty}=P_{L}\), and the pressure in the cavity, \(p_{b}\), is described by a 2D Rayleigh-Plesset (2 DRP) equation [46]:  
\[\frac{P_{\infty}-P_{b}}{\rho}=\left(\left(\frac{dR}{dt}\right)^{2}+R\frac{d^{2 }R}{dt}\right)ln\left(\frac{R}{r_{\infty}}\right)+\frac{1}{2}\bigg{(}R\frac{ dR}{dt}\bigg{)}^{2}\bigg{(}\frac{1}{R^{2}}-\frac{1}{r_{\infty}^{2}}\bigg{)}. \tag{6}\]  
However, the 2 DRP equation only describe the dynamic of a symmetrical collapse. Form our knowledge an equation to validate the dynamic of a non-symmetrical collapse has still to  
Figure 1: Geometry of the simulation box.  
be developed. For this reason, as commonly done in the literature [15; 17; 46; 47], the model presented as been validated for the case of symmetric collapse against Equation (6) [17].  
In the simulations we use the Lucy Kernel [13] and the Tait EOS [13] with a smoothing length of \(h=1.3\cdot dL\) where \(dL\) is the initial particle spacing. The dimensionless dissipation factor, time step and speed of sound were set as \(\alpha=1\), \(t_{s}=1e-10\) [s] and \(c_{0}=1484\) [m s\({}^{-1}\)]. The sensitivity of the results to parameters such as the kernel function, EOS or \(h\), was investigated in a previous publication for the case of symmetric collapse [17]. In the next section, the particle resolution for the specific case of non-symmetric collapse is discussed.