# A Mesh-Free Particle Method for Simulation of Mobile-Bed Behavior Induced by Dam Break
## Moving Particle Semi-Implicit Method
### Particle Interaction Model  
As stated above, all of the differential operators in the governing equation should be replaced by the particle interaction model. The gradient model calculating the averaged local weight of the gradient vectors is expressed below:  
\[\left\langle\
abla\phi\right\rangle_{i}=\frac{d}{n_{0}}\!\!\sum_{j\
eq i}\! \left[\frac{(\phi_{ij})(r_{ij})}{\big{(}|r_{ij}|\big{)}^{2}}\big{(}r_{ij} \big{)}w\big{(}|r_{ij}|\big{)}\right] \tag{4}\]  
where \(\left\langle\right\rangle\) denotes the particle interaction model, \(\phi\) is an arbitrary physical quantity, \(d\) is a number of dimension, and \(n_{0}\) is a particle number density at the initial stage, which will be discussed later. The divergence model represents the divergence of the physical quantity between the center and the neighboring particles. \[\left\langle\
abla\cdot\phi\right\rangle_{i}=\frac{d}{n_{0}}\!\!\sum_{j\
eq i }\!\left[\frac{(\phi_{ij})\cdot(r_{ij})}{\big{(}|r_{ij}|\big{)}^{2}}w\big{(}| r_{ij}|\big{)}\right] \tag{5}\]  
The Laplacian model represents the diffusion of a fluid below. \[\left\langle\
abla^{2}\phi\right\rangle_{i}=\frac{2d}{\lambda n^{0}}\underset{j\
eq i }{\sum}\big{[}(\phi_{ij})w\big{(}|r_{ij}|\big{)}\big{]} \tag{6}\]  
where:  
\[\lambda=\frac{\underset{j\
eq i}{\sum}\big{[}(|r_{ij}|\big{)}^{2}w\big{(}|r_{ ij}|\big{)}\big{]}}{\underset{j\
eq i}{\sum}\big{[}w\big{(}|r_{ij}|\big{)} \big{]}} \tag{7}\]  
\(\lambda\) is an adjustment parameter to equal the Laplacian value to an analytical value. The particle number density corresponds to the density of fluid in the MPS method. It can be calculated by the summation of the kernel function. \[n_{i}=\underset{j\
eq i}{\sum}w\big{(}r_{ij}\big{)} \tag{8}\]  
MPS follows the Lagrangian approach. Therefore, the particle without physical volume is used for simulation, while the conventional grid method uses grid as a volume of fluid. Without a physical volume or a very small volume on the particle, the fluid density may increase to an infinite value. In order to avoid this issue, the particle number density has been used instead of the density of the fluid. Moreover, the particle number density can be calculated by adding the neighboring particle with respect to kernel function. Therefore, the MPS method can avert the violation of continuum and manage the effects of neighboring particles. In order to simulate incompressible flows, the incompressibility model is employed, which has a similar algorithm to the simplified marker-and-cell (SMAC) method in the mesh-based CFD system. The incompressibility model consists of two stages, which include an explicit and an implicit stage. In the explicit stage, the intermediate position and velocity of each particle can be calculated from the viscosity and gravity directly. According to the continuity equation, which is one of governing equations, the particle number density at the intermediate stage, \(n^{*}\), may differ to it at the initial arrangement, \(n^{0}\). It might also lead to the dissatisfaction of the continuity equation. Therefore, the particle number density at an intermediate stage should be adjusted implicitly. \[n^{*}+n^{\prime}=n^{0} \tag{9}\]  
where \(n^{\prime}\) is the corrective particle number density, and it is related to corrective velocity, \(u^{\prime}\), which can be obtained through the following mass conservation equation:  
\[\frac{1}{\Delta t}\frac{n^{\prime}}{n^{0}}=-\
abla\cdot u^{\prime} \tag{10}\]  
The corrective velocity can be derived by the following implicit pressure gradient term:  
\[u^{\prime}=-\frac{\Delta t}{\rho}\
abla P^{n+1} \tag{11}\]  
By recalling Equations (9)-(11), a Poisson equation for pressure calculation can be obtained:  
\[\left\langle\
abla^{2}P^{n+1}\right\rangle_{i}=-\frac{\rho}{\Delta t^{2}} \frac{\left\langle n^{*}\right\rangle_{i}-n^{0}}{n^{0}} \tag{12}\]  
The left-hand side of Equation (12) can be discretized by the Laplacian model, which is given in Equation (7); its right side is expressed by the deviation of the particle number density with the constant value.