*3.1. Computer-Aided Engineering (CAE)*

Computer-aided engineering (CAE) is an alternative technology for large-scale experiments, conducted in a room or in situ, using prototypes that have been prepared in a study as part of the development process of "manufacturing". In other words, CAE is a general term for technology that simulates and analyzes prototypes on a computer created by a computer-aided design (CAD) and so on, considering the site conditions [16–19]. At the same time, CAE may refer to computer-aided engineering work or its tools for the prior examination, design, manufacturing, and process design of construction methods and products. In the field of geotechnical engineering, CAE can be used not only to visualize the inside of the ground and the stress loading on the inside of the ground but also to estimate the results of experiments that would entail very high costs and/or phenomena that would be difficult to reproduce. In addition, by performing appropriate post-processing, it is possible to communicate with other people in a visually easy-to-understand manner.

In this study, a simulation using a CAE analysis, based on the moving particle semiimplicit (MPS) method as one of the typical particle-based methods (PBMs), that is an "MPS-CAE analysis", for the RS-DMM, is performed as a kind of ground-improvement method, and the visible and measurable performance in the targeted soft ground is evaluated.

The possibility and validity of applying the MPS-CAE analysis to the ground and ground-improvement methods has been clarified by Inazumi et al. [13,18,20–22] and Nakao et al. [23].

#### *3.2. Particle-Based Methods (PBMs) and Moving Particle Semi-Implicit (MPS) Method*

A major feature of particle-based methods (PBMs) is that, unlike the finite element method (FEM) and the finite difference method (FDM), they do not use a lattice, but instead discretize the continuum as particles that move each calculation point with a physical quantity. However, this causes a large difference in the governing equation. When describing the behavior of a continuum, the Euler method (with lattice: FEM, FDM, and so on) and the Lagrange method (non-grid: PBM) are available. In the Lagrange method, the calculation point moves as the object moves and deforms, so the convection term

disappears from the governing equation. Equations (1) and (2) show the Navier–Stokes equations by the Euler method and the Lagrange method, respectively [13,21,24–26]:

$$\frac{\partial \mu(\mathbf{x},t)}{\partial t} + (\mu(\mathbf{x},t) \cdot \nabla)\mu(\mathbf{x},t) = -\frac{1}{\rho} \nabla P(\mathbf{x},t) + \nu \nabla^2 \mu(\mathbf{x},t) + \mathbf{g}(\mathbf{x},t) \tag{1}$$

$$\frac{D\mu(X,t)}{Dt} = \frac{1}{\rho}\nabla P(\mathbf{x},t) + \nu \nabla^2 \mu(X,t) + \mathbf{g}(X,t) \tag{2}$$

where *u* is the velocity, *P* is the pressure, *g* is the external force, *ρ* is the density, and *υ* is the kinematic viscosity coefficient.

The moving particle semi-implicit (MPS) method, one of the typical PBMs, is an incompressible flow analysis method that discretizes a continuum with particles, and the basic governing equations are the continuity equation shown in Equation (3) and the Navier–Stokes equation shown in Equation (4):

$$\frac{D\rho}{Dt} = 0\tag{3}$$

$$\frac{D\stackrel{\rightarrow}{\boldsymbol{\mu}}}{Dt} = -\frac{\nabla P}{\rho} + \mathbf{v}\nabla^2 \stackrel{\rightarrow}{\boldsymbol{\mu}} + \stackrel{\rightarrow}{\boldsymbol{\mathcal{g}}} \tag{4}$$

where *D*/*Dt* represents the Lagrange derivative, *ρ* is the density, <sup>→</sup> *u* is the velocity, *P* is the pressure, ν is the coefficient of kinematic viscosity, and <sup>→</sup> *g* is the gravity. In PBMs, the Navier–Stokes equation is divided into two stages, namely, the pressure term is calculated explicitly from Equation (5) and the pressure gradient term is implicitly calculated from Equations (6) and (7):

$$\frac{\stackrel{\rightarrow}{u^\*} - \stackrel{\rightarrow}{u^k}}{\Delta t} = \nu \stackrel{\rightarrow}{\nabla^2 u^k} + \stackrel{\rightarrow}{\mathcal{g}} \tag{5}$$

$$
\nabla^2 P^{k+1} = \frac{\rho}{\Delta t^2} \frac{n^\*-n^0}{n^0} \tag{6}
$$

$$\frac{\stackrel{\rightarrow}{u^{k+1}} - \stackrel{\rightarrow}{u^\*}}{\Delta t} = -\frac{\nabla P^{k+1}}{\rho} \tag{7}$$

where *n* is the particle number density (dimensionless quantity representing the density of the particle arrangement), *n*<sup>0</sup> is the standard particle number density, and *k* is the time step. The \* indicates the physical quantity at the stage when the explicit calculation is completed. Figure 3 shows the calculation algorithm of the MPS.

#### *3.3. Visualization in Ground by MPS-CAE Analysis*

Conventionally, when performing ground-improvement work, the construction management is carried out by repeating measures based on the empirical rules of the site. This is because the target of the construction is the ground, and it is impossible to directly and visually check the ground and its internal conditions during the construction. Therefore, what is required is the visualization of the ground. By simulating the ground improvement in advance and visualizing the construction status, it is possible to carry out surveys, designs, and construction management based on scientific methods. In addition, ground improvement situations can be simulated by applying various conditions, such as physical property values and the movements of the stirring wing, which can be useful for improving the economic efficiency and quality of the ground-improvement work.

Nakao et al. (2021) used an MPS-CAE analysis to three-dimensionally model the process of penetration and stirring by the RS-DMM on a computer, and examined the behavior inside a targeted soft ground and the effect of the RS-DMM on the surrounding ground. They succeeded in visually capturing the process of penetration and stirring by the RS-DMM and the inside of the targeted soft ground in the MPS-CAE analysis. It has also been confirmed that the simulation results by the MPS-CAE analysis are in good agreement with the results of the mixture model experiment. However, a quantitative evaluation based on the results of an MPS-CAE analysis has not been achieved, and the inside of the targeted soft ground must be measurably reproduced and evaluated. After that, if various external factors inside the ground can be incorporated into the MPS-CAE analysis, it is expected that the co-rotation mechanism can be successfully investigated and that an effective ground-improvement method can be designed and developed.

**Figure 3.** Calculation algorithm for MPS.

In this study, the visualization of the RS-DMM, a kind of ground-improvement method, was attempted using an MPS-CAE analysis, which is one of the numerical analysis methods. By expressing the targeted soft ground and the solidifying material as collections of particles, the purpose is to evaluate how they are stirred and the quality of the improved body in the MPS-CAE analysis. In addition, the performance of the displacement reduction and the quality of the improved body when the stirring wing of the DRT was used were compared with those when the stirring wing of the normal type (NT) was used, and the performances were evaluated.

#### **4. Parameter Setting and Analysis Model**

## *4.1. Bingham Fluid Model*

The MPS-CAE analysis requires the determination of the rheological parameters for each fluid. In this analysis, the rheological properties of the cement slurry and targeted soft ground are assumed as Bingham fluid models, which are non-Newtonian fluids because they are mixtures of various substances.

The Bingham fluid model is a fluid in which the strain ratio remains 0 until the shear stress exceeds the yield value. The authors adopted the bi-viscosity model, which behaves as a viscoelastic fluid when flowing and as a highly viscous fluid when immobile (see Figure 4). The apparent viscosity when flowing is shown in Equation (8), and the apparent viscosity when immobile is shown in Equation (9):

$$
\eta = \eta\_p + \frac{\tau\_y}{\dot{\gamma}} \qquad \dot{\gamma} \ge \dot{\gamma}\_c \tag{8}
$$

$$
\eta = \eta\_p + \frac{\tau\_y}{\dot{\gamma}\_c} \qquad \dot{\gamma} < \dot{\gamma}\_c \tag{9}
$$

where *<sup>τ</sup><sup>y</sup>* is the yield value, *<sup>η</sup><sup>p</sup>* is the plastic viscosity, . *<sup>γ</sup>* is the average shear velocity, and . *γ<sup>c</sup>* is the yield reference value at the boundary between the fluid state and the immobility state. A preliminary analysis was performed in this analysis, and the result was . *γ<sup>c</sup>* = 1. In the preliminary analysis, the value of . *γ<sup>c</sup>* was constantly changed and set in terms of flow time, solution stability, and calculation time.

**Figure 4.** Concept of bi-viscosity model.
