# MR-WC-MPS: A Multi-Resolution WC-MPS Method for Simulation of Free-Surface Flows

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## Abstract

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## 1. Introduction

## 2. Fundamentals of the MPS Method

#### 2.1. Governing Equations

#### 2.2. MPS Interpolations

#### 2.3. Solution Method

#### 2.4. Weakly Compressible Model

#### 2.5. Boundary Treatment

## 3. Multi-Resolution MPS Method

#### 3.1. Calculation of Kernel Function

#### 3.2. Calculation of Particle Number Density

#### 3.3. Solution Method

## 4. Results and Discussion

#### 4.1. Dam-Break-Induced Water Waves

^{®}Core™ i7-2600 3.40 GHz CPU and a system memory of 16.0 GB. Although the calculation of the interactions between large and small particles adds up to the total computational time compared to WC-MPS computational time, the total computational time is reduced.

#### 4.2. Landslide-Induced Water Waves

#### 4.3. Landslide-Induced Water Waves over Extended Tank

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The particle interaction mechanism in each of the four scenarios. Small and large reference particles are shown in red and blue colors, respectively. Neighboring particles are shown in green.

**Figure 7.**Configuration of the coarse and fine regions in computational domain of the dam-break problem.

**Figure 9.**Comparison for the dam-break wave front position between the experimental, MPS, WC-MPS, and MR-WC-MPS results.

**Figure 13.**A comparison between the WC-MPS, MR-WC-MPS, and experimental results for the water surface profile in submarine landslide simulation.

**Figure 14.**A Comparison between the WC-MPS and MR-WC-MPS computational time for simulation of landslide-induced water waves over extended tank.

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**MDPI and ACS Style**

Nabian, M.A.; Farhadi, L.
MR-WC-MPS: A Multi-Resolution WC-MPS Method for Simulation of Free-Surface Flows. *Water* **2019**, *11*, 1349.
https://doi.org/10.3390/w11071349

**AMA Style**

Nabian MA, Farhadi L.
MR-WC-MPS: A Multi-Resolution WC-MPS Method for Simulation of Free-Surface Flows. *Water*. 2019; 11(7):1349.
https://doi.org/10.3390/w11071349

**Chicago/Turabian Style**

Nabian, Mohammad Amin, and Leila Farhadi.
2019. "MR-WC-MPS: A Multi-Resolution WC-MPS Method for Simulation of Free-Surface Flows" *Water* 11, no. 7: 1349.
https://doi.org/10.3390/w11071349