# Potential of Computer-Aided Engineering in the Design of Ground-Improvement Technologies
## 2 CAE_MPS Method
### Particle Methods and Moving Particle Semi-Implicit (MPS) Method  
The space is divided into grids and each grid is assigned a physical quantity as a variable for calculation in the grid method, e.g., finite element method (FEM) and the  
Figure 1: Mechanical agitation and mixing jet-grouting construction.  
difference method (DM) [28]. Meanwhile, the PM discretizes the continuum as particles and it will move with each physical quantity in a Lagrangian manner at each calculation point. Literature review [29] shows the application of the distinct element method (DEM) in the calculation of movement by each particle, in which the interaction forces between particles are expressed by using friction coefficient, spring constant, damping constant, and so on.  
In this study, the authors applied the MPS method [21; 22; 23] to analyze fluid particle behavior. For the incompressible flow used in the analysis, the governing equations are the mass conservation law of Equation (1) and Navier's stroke law of Equation (2) with surface tension considered.  
\[\frac{D\rho}{Dt}=0 \tag{1}\]  
\[\frac{Du}{Dt}=-\frac{1}{\rho}\
abla P+\vartheta\
abla^{2}u+g+\frac{1}{\rho} \sigma k\delta n \tag{2}\]  
where \(u\) is the velocity vector, \(\rho\) is the density of the fluid, \(\frac{Du}{Dt}\) is the rate of change in velocity vector, \(\frac{D\rho}{Dt}\) is the rate of change in density, \(P\) is the pressure, \(\vartheta\) is the kinematic viscosity coefficient, \(g\) is the gravity vector, \(\sigma\) is the surface tension coefficient, \(k\) is the curvature, \(\delta\) is the delta function for the surface tension acting on the surface, and \(n\) is the unit vector in the direction perpendicular to the surface.  
In the MPS method, each differential operator (slope, divergence, and Laplacian) of the governing equation, as shown in Equation (2), is discretized by a weighting function [21; 22; 23]. The weighting function for the governing operator depends on the interparticle distance (\(r\)) and the influence radius (\(R\)), which is equivalent to 2.1- to 4.1-times the interparticle distance in each particle interaction model.  
The weighting function (\(w\)) is expressed by Equation (3); it is a function of distance r between particles and the effective radius (\(R\)) of the support domain.  
\[w(r)=\left\{\begin{array}{cc}\frac{R}{r}-1&(0\leq r\leq R)\\ 0&(R\leq r)\end{array}\right\} \tag{3}\]  
The sum of the weight functions in the support domain is commonly known as the particle number density (\(n_{i}\)); it is defined as Equation (4).  
\[n_{i}=\sum_{j\
eq i}w_{ij} \tag{4}\]  
where \(w_{ij}=w\big{(}r_{ij},\ R\big{)}\) is the weight function between particles (\(i\)) and (\(j\)). The particle number density, when the particles are located on a regular grid whose grid size is the same as the diameter of the particles, is called the criterion of the particle number density (\(n^{*}\)).  
In the derivative models for the traditional MPS method, the differential operators for the gradient, Laplacian, and divergence of a particle (\(i\)) are formulated as Equations (5)-(7).  
\[<\
abla_{\varnothing}>_{i}=\frac{d}{n^{*}}\sum_{j\
eq i}\frac{\varnothing_{ij} }{r_{ij}^{2}}x_{ij}w_{ij} \tag{5}\]  
\[<\
abla_{\varnothing}^{2}>_{i}=\frac{2d}{n^{*}\lambda_{i}}\sum_{j\
eq i} \varnothing_{ij}w_{ij} \tag{6}\]  
\[<\
abla.\Phi>_{i}=\frac{d}{n^{*}}\sum_{j\
eq i}\frac{\varnothing_{ij}.x_{ij}}{ r_{ij}^{2}}w_{ij} \tag{7}\]where \(\Phi\) is an arbitrary vector, \(\varnothing\) is an arbitrary scalar, and \(\lambda\) is a coefficient in the Laplacian model, which is defined as Equation (8).  
\[\overline{\lambda_{i}}=\frac{\sum_{j\
eq i}w_{ij}r_{ij}^{2}}{\sum_{j\
eq i}w_{ij}} \tag{8}\]  
Figure 2 shows the basic calculation algorithm chart for the MPS method.