### 1. Introduction

### 2. LS-DYNA Analysis

### 2.1 LS-DYNA

*X*,

_{i}*x*and

_{i}*i*denote the Lagrangian, Eulerian, and reference coordinate systems, respectively. In addition,

*v*and

_{i}*u*represent the material and space velocities, respectively.

_{i}*ρ*,

*v*,

_{i}*σ*,

_{ij}*∊*,

*e*,

*u*, and

_{i}*t*denote the material’s density, velocity, stress tensor, strain tensor, internal energy, mesh velocity, time, respectively.

### 2.2 Validation of LS-DYNA

^{3}, 32.4 MPa, and 0.475, respectively. In addition, the fluid (water) density was 1000 kg/m

^{3}.

### 2.3 Computation Conditions

### 2.4 Analysis Results

#### 2.4.1 Formation of landslide tsunami

*t*

^{*}is based on the time point at which the maximum water level occurs at

*x*= 0.4 m.

*t*

^{*}= −0.4 s is a feature of the landslide tsunami generation process that was also observed in the experiments conducted by Monaghan and Kos (2000). The wave and flow fields of the landslide tsunami exhibited characteristics similar to a solitary wave at (e)

*t*

^{*}= −0.1 s.

#### 2.4.2 Propagation of landslide tsunami

*A*,

*h*, and

*C*denote the solitary wave’s amplitude, depth, and wave speed (

*λ*,

*∊*, and

*ξ*represent the ratio of the surface displacement to the incident amplitude (=

*η*/

*A*), ratio of incident amplitude to the water depth (=

*A*/

*h*), and ratio vertical position to the water depth (=

*z*/

*h*), respectively.

#### 2.4.3 Flow velocity of landslide tsunami

*u*) computed in LS-DYNA was distributed more narrowly than the solitary wave’s theoretical horizontal flow velocity, while the vertical flow velocity (

*w*) distribution of the computed flow velocity was considerably larger than that of the theoretical flow velocity. This is considered to be because of the difference between the stable theoretical solitary wave and the water surface waveform generated as the falling object’s potential energy is transferred, as illustrated in Fig. 6. Consequently, +

*u*and +

*w*are larger than the solitary wave approximation at WG1, while −

*u*and −

*w*increase because the water level drops below the still water level. As the horizontal and vertical flow velocities propagate throughout the water tank, the difference between the theoretical flow velocities decreases, and a similar distribution is observed, just as the water surface waveform is similar to the solitary wave. However, the horizontal flow velocity is discontinuous owing to the effect of bubbles flowing in when the object falls, and differs from the theoretical flow velocity distribution.

#### 2.4.4 Deformation of landslide tsunami

*V*denotes the volume of the landslide tsunami waveform computed in LS-DYNA.

*V*

_{0}represents the volume of the theoretical solitary waveform. In addition,

*H*denotes the computed wave height, while

*H*

_{0}represents the computed wave height at point WG1 (

*x*= 0.4 m).

*x*= 9.3 m, and the horizontal and vertical flow velocities were also similar to the theoretical solitary wave, as illustrated in Fig. 6(c), 7(c), and 8(c). At

*x*= 9.3 m,

*H*/

*H*

_{0}was 0.58, while

*V*/

*V*

_{0}was 0.91, and it was determined that the landslide tsunami had entered an almost stable/steady state.

### 3. NWT Analysis

### 3.1 Governing Equation

*v*,

_{i}*q*

^{*},

*t*,

*ρ*,

*p*denote the flow velocity component in the

*x-*and

*z-*directions, flow density of the wave source, time, fluid density, and pressure, respectively. In addition,

*ν*represents the sum of the kinematic viscosity coefficient (

_{T}*ν*) and the kinematic eddy viscosity coefficient (

*ν*) calculated from the kinematic eddy viscosity model (Germano et al., 1991; Lilly, 1992) based on the LES technique (Smagorinsky, 1963). Furthermore,

_{t}*D*,

_{ij}*S*,

_{i}*Q*, and

_{i}*g*represent the strain rate velocity tensor, surface tension term calculated in the continuum surface force model by Brackbill et al. (1992), wave generation term, and gravitational acceleration term, respectively.

_{i}### 3.2 Solitary Wave Generator

*η*+

*h*)/(

*η*+

_{s}*h*)) proposed by Ohyama and Nadaoka (1991) was adopted to make solitary waves in a stable manner. where

*V*

_{0}denotes the horizontal flow velocity component based on the wave approximation theory.

*η*

_{0}and

*η*represent the generation point’s approximate and actual water surface displacements, respectively. In addition, this study adopted a constant of “2” because the waves generated by the NWT’s source line propagate ± in both directions.

_{s}*q*

^{*}was adopted for the purpose of eliminating the effect of the intensity (

*q*) according to the grid size (

*Δx*) at the wave source, as expressed in the equation below.

_{s}### 3.3 Computation Conditions

*x*= 9.3 to

*x*= 15 m in the numerical water tank in Fig. 6, for which it was determined that the wave generated by the falling object is in a steady state. As illustrated in Fig. 10, the length of the NWT was 5.7 m and the height was 0.2 m; in addition, an energy absorption layer and open boundary conditions were applied to the offshore side of the wave source. Regarding the boundaries and computation grid of the NWT’s floor and ceiling, impermeable conditions, horizontal and vertical grids of 0.01 m were adopted, similar to the LS-DYNA analysis.

### 3.4 Analysis Results

#### 3.4.1 Water surface waveform

*x*= 10 m) near the wave source, while (b) is the time-domain waveform at WG5 (

*x*= 13 m), which represents the propagation process. Here, the red circles represent the NWT computation results, while the black solid lines depict the LS-DYNA computational results.

*x*= 13 m), the NWT simulation’s wave height was slightly higher than that of LS-DYNA.

#### 3.4.2 Flow velocity

*x*= 9.3 m). Figs. 12 and 13 present the horizontal and vertical flow velocity distributions at WG4 (

*x*= 10 m) and WG5 (

*x*= 13 m), respectively, comparing them with the NWT simulation results that adopt the solitary wave approximation theory.

#### 3.4.3 Maximum wave pressure

*x*= 15 m and the water surface waveform in front of it are presented in Fig. 14, to compare LS-DYNA’s landslide tsunami and NWT computational results. Fig. 14(a) is the water surface waveform in front of the vertical wall, while Fig. 14(b) presents the maximum wave pressure acting on the vertical wall.

### 4. Conclusions

A review and comparison with existing experimental results on landslide tsunamis verified the effectiveness and validity of LS-DYNA.

Via the LS-DYNA analysis, it was possible to understand the generation mechanism of landslide tsunamis caused by falling objects, and even the air inflow phenomena found in hydraulic model experiments (Monaghan and Kos, 2000) were optimally reproduced.

At the beginning of the landslide tsunami generation in LS-DYNA, the tsunami was narrower than a solitary waveform, while the horizontal and vertical flow velocities were larger; however, as the tsunami propagated, the wave height and flow velocities decreased, and the tsunami became similar to a solitary wave.

In a comparison of the analysis results from LS-DYNA and NWT, which considers the region that is 9.3 m away from the falling object where the landslide tsunami enters an almost stable/steady-state, the NWT results were slightly overestimated because a high-wave reduction did not occur during the propagation process; however, the water surface waveform, flow velocity, and maximum wave pressure distribution were similar.