Ice melting simulation with water flow handling

Abstract

In this paper, we propose a new approach based on a particle-based model for ice melting simulation. Each particle has an attribute called virtual water. The amount of the virtual water of an ice particle indicates the amount of water surrounding the ice particle. The transfer of the virtual water is performed between the exterior ice particles so as to simulate the thin layer of water flow on the surface. Our approach also handles the transition between the virtual water and the water particles. We compute the isosurface of a density field defined by the ice particles and the virtual water. A simple ray tracing method is adopted for rendering the objects. We report the experimental results of several ice melting simulations with water flow and water drops.

Appendix A: Density function

The density function Φ(x) is defined in the form

$$\begin{aligned} \varPhi(\mathbf{x})=\sqrt{\sum_{a} \biggl(f(X_a) \biggl(1-\frac{r}{h} \biggr) \biggr)^{2}}, \end{aligned}$$

(5)

where r=|x−x _a |, h is the particle interactive radius, X _a is the total volume of ice particle a (i.e. X _a =V _a +Z _a ); and f is a function of volume and f(V _init)=1. Consider a single ice particle and its total volume is X. The threshold is set to θ. Then the isosurface is a sphere and its radius R is computed as $R = (1-\frac{\theta}{f(X)})h$. Based on the volume ratio of two spheres with and without virtual water, we should have

$$\begin{aligned} \frac{\frac{4}{3} (1-\frac{\theta}{f(X)} )^3h^3\pi}{\frac {4}{3} (1-\frac{\theta}{f(V_{\mathrm{init}})} )^3h^3\pi} = \frac{V}{V_{\mathrm{init}}}. \end{aligned}$$

(6)

We rearrange the terms to obtain f(X):

$$\begin{aligned} f(X)= \biggl(1-\sqrt[3]{\frac{X}{V_{\mathrm{init}}}}(1-\theta) \biggr)^{-1} \theta. \end{aligned}$$

(7)

For X close to V _init and θ=0.5, $f(X) \approx\sqrt[3]{\frac{X}{V_{\mathrm{init}}}}$. This can be shown by using Taylor expansion.