An Improved Neural Particle Method for Complex Free Surface Flow Simulation Using Physics-Informed Neural Networks

Abstract

The research on free surface flow is of great interest in fluid mechanics, with the primary task being the tracking and description of the motion of free surfaces. The development of numerical simulation techniques has led to the application of new methods in the study of free surface flow problems. One such method is the Neural Particle Method (NPM), a meshless approach for solving incompressible free surface flow. This method is built on a Physics-Informed Neural Network (PINN), which allows for training and solving based solely on initial and boundary conditions. Although the NPM is effective in dealing with free surface flow problems, it faces challenges in simulating more complex scenarios due to the lack of additional surface recognition algorithms. In this paper, we propose an improved Neural Particle Method (INPM) to better simulate complex free surface flow. Our approach involves incorporating alpha-shape technology to track and recognize the fluid boundary, with boundary conditions updated constantly during operation. We demonstrate the effectiveness of our proposed method through three numerical examples with different boundary conditions. The result shows that: (1) the addition of a surface recognition module allows for the accurate tracking and recognition of the fluid boundary, enabling more precise imposition of boundary conditions in complex situations; (2) INPM can accurately identify the surface and calculate even when particles are unevenly distributed. Compared with traditional meshless methods, INPM offers a better solution for dealing with complex free surface flow problems that involve random particle distribution. Our proposed method can improve the accuracy and stability of numerical simulations for free surface flow problems.

1. Introduction

The term “free surface” refers to a surface without fixed boundary constraints and typically involves one or more nonlinear variables [1]. Fluid flow with a free surface is commonly referred to as free surface flow [2]. This area of study is a challenging field that finds applications in various disciplines such as fundamental physics [3], geophysics [4], and engineering geology [5]. In the context of engineering geology, the free surface typically refers to the interface between a water body and the atmosphere [6]. The identification and tracking of the free surface and the description of its motion are key tasks in this area [7]. In engineering geology, the study of free surface flow is critical in understanding and mitigating fluid-based geological disasters, such as dam breaks [8], landslide surges [9], and tsunamis [10]. Accurate identification and tracking of the free surface are essential for predicting and managing such events. Understanding and modeling complex free surface flow problems can help to improve the design and operation of structures and systems that interact with fluid environments, such as ships, submarines, oil platforms, and dams. In summary, the study of complex free surface flow problems is crucial for advancing our understanding of natural and engineered systems and for developing innovative solutions to complex problems. In the real world, complex boundary conditions have numerous applications in engineering and physics. Examples of such applications include dam breaks and landslide surges, where fluid behavior is subject to complex and constantly changing boundary conditions. Accurately identifying and tracking the free surface is crucial for understanding and mitigating fluid-based geological disasters in engineering geology. Further research in this area is essential for many areas of science and engineering.

Currently, the study of free surface flow primarily includes three approaches: theoretical analysis [11], physical experimentation [12], and numerical simulation [13]. Theoretical analysis involves solving the governing equations of free surface flow and was widely used in the early stages of research on this topic. In 1871, the French scientist de Saint-Venant proposed a set of Partial Differential Equations (PDE) to describe the unsteady flow motion of shallow water with a free surface, known as the shallow water equations [14]. These equations consist of the continuity equation, reflecting the mass conservation law, and the motion equation, reflecting the momentum conservation law. Later, Laitone et al. [15] optimized the shallow water equations. Although theoretical analysis yields highly accurate results, the simplified models used limit their applicability under different conditions, making it challenging to apply them to practical problems under complex conditions. Physical experimentation involves measuring the flow field of free surface flow using related instruments such as the Particle Image Velocimetry (PIV) [16]. This approach provides direct insight into the physical properties of free surface flow and is highly reliable. However, changes in research conditions often require partial or complete rebuilding of the physical model, and the testing period is often long. Additionally, the results are easily limited by laboratory conditions, model size, degree of simplification, and other factors, leading to differences from the actual situation.

With the rapid advancement of computer science, numerical analysis methods have become widely used in free surface flow studies [17]. These methods can be categorized into two main types, mesh-based and meshless methods [18], based on their different calculation approaches. The mesh-based method involves dividing the continuous space into discrete meshes and using numerical calculation methods such as the Finite Difference Method (FDM) [19] or the Finite Element Method (FEM) [20] to obtain numerical solutions at the nodes of networks [21]. Commonly used mesh-based methods include the Marker and Cell Method (MAC) [22], the Volume of Fluid (VOF) [23], and the Particle in Cell (PIC) [24]. Although the mesh-based method has a long history and mature solutions, it faces some limitations when solving free-surface fluid with complex boundaries and large deformations. In such cases, it is often necessary to encrypt the mesh over a large range due to the fast and large variation of the liquid level, leading to a large number of computational meshes and decreased computational efficiency [25]. To overcome these limitations, the meshless method was developed. Unlike mesh-based methods, the meshless method directly operates on discrete particles without requiring a network or topology. Commonly used meshless methods include the Diffuse Element Method (DEM) [26], the Smoothed Particle Hydrodynamics (SPH) [27], and the Moving Particle Semi-implicit (MPS) [28]. The meshless method is highly adaptive and effectively avoids the construction of networks and topology, making it suitable for studying free surface fluids with complex boundaries and large deformations.

In recent years, deep learning algorithms have been increasingly utilized in the field of fluid mechanics [29,30]. Among various deep learning algorithms, the Physics-Informed Neural Network (PINN) [31] shows a broader potential in fluid mechanics due to its low reliance on data and its capability to incorporate physical laws. The Neural Particle Method (NPM) [32], a fluid dynamics particle tracking method based on the PINN, has demonstrated good performance in practical applications such as dam break. However, it lacks additional surface recognition algorithms to simulate fluid interactions with walls. With the development of computational fluid dynamics, surface recognition has received more attention in recent years. Various methods have been proposed to identify free-surface particles in different numerical schemes.

For example, in the Moving Particle Semi-Implicit Method (MPS) proposed by Koshizuka et al. [33], free surface particles are identified by comparing the particle number density with a specified value to simulate incompressible fluid. Zheng et al. [34] proposed a method to identify free surface particles in an improved Smoothed Particle Hydrodynamics (ISPH), taking into account density ratio variation and three auxiliary functions. The alpha-shape algorithm [35], used to extract the boundary of two-dimensional planar point sets of arbitrary shapes, is employed in the Particle Finite Element Method (PFEM) for boundary identification [36]. S.R. Idelsohn et al. [37] proposed PFEM to solve fluid-structure coupling problems, where the alpha-shape algorithm is used to solve free surfaces and contact points.

In this paper, we propose the INPM to better simulate free surface flow in complex situations. To address the challenges faced by the original method in simulating flows with significant changes in free surfaces, the INPM incorporates the alpha-shape algorithm to recognize the fluid boundary and update the boundary conditions continuously during operation. We demonstrate the effectiveness of the proposed method through three numerical examples with different boundary conditions, showing the advantages of the addition of a surface recognition module for more precise imposition of boundary conditions in complex situations. Furthermore, the INPM can accurately identify the surface and calculate even when particles are unevenly distributed. The proposed method offers a better solution for dealing with complex free surface flow problems that involve random particle distribution.

The rest of this paper is organized as follows. Section 2 will briefly introduce the PINN, the free-surface identification algorithm, and the improved NPM (INPM). In Section 3, the simulation of three numerical examples will be presented to illustrate the effectiveness of our proposed method for complex free surface flow problems. A discussion on the applicability and limitations of the improved NPM in studying free surface flow problems is given in Section 4. Finally, several conclusions are proposed in Section 5.

2. Methods

2.1. Background and Theory of PINN

2.1.1. Background of PINN

In recent years, breakthroughs in deep learning have provided new ideas for solving fluid mechanics problems. Classical deep learning algorithms mainly include CNN [38], RNN [39], and so on. These algorithms have shown good applicability in fluid dynamics problems, but the accuracy of the solution often depends on a large number of data; missing data or an uneven distribution will significantly reduce the accuracy of the calculation. In addition, fluid mechanics problems usually require some prior knowledge of physical laws, but these knowledge cannot be reflected in classical deep learning algorithms, resulting in a waste of information resources. Raissi et al. [31] presented the PINN. PINNs encode PDEs and other forms of governing equations as a part of the neural network (NN) and achieve accurate solutions by constraining the loss function. Nowadays, the application of PINNs has made great achievements in many fields [40]. Cai et al. [41] applied PINNs to several heat transfer problems that cannot be well handled by traditional deep learning methods, and the results gave evidence of the great application prospects of PINNs in the field of heat transfer. Arzani et al. [42] used PINNs to improve the near-wall blood flow and wall shear stress qualification in diseased arterial flows with their low dependence on large data. Kissas et al. [43] put forth a framework based on PINNs to optimize the prediction of cardiovascular flows.

2.1.2. Theory of PINN

As a new neural network-based method to solve complex problems such as PDEs, the core task of PINN is to approximate the solutions of PDEs by training the NN to minimize the loss function [44]. For the better understanding and application of PINNs, scholars often compared PINNs with the conventional NNs. Similar to conventional NNs, PINNs need to specify the initial conditions and boundary conditions, as well as the residual of partial differential equation at selected points in the domain of the network [45]. Different from the conventional NNs, PINNs combine PDEs and other forms of physics-informed constraints into the loss function to improve the training efficiency and reduce the dependency on data, which is the main improvement [46].

In general, we expect the solution of PDEs obtained by PINN to be a continuous function u(x, t). A parameterized system of PDEs in the general form is defined as

$$\begin{array}{c}{\mathit{u}}_t+{\mathcal{N}}_{\mathit{x}}\left[\mathit{u}\right]=0,\mathit{x}\in \Omega ,t\in [0,T],\hfill \end{array}$$

(1)

$$\begin{array}{c}\mathit{u}\left(\mathit{x},t_0\right)=h_0(\mathit{x},0),\mathit{x}\in \Omega ,\hfill \end{array}$$

(2)

$$\begin{array}{c}\mathit{u}(\mathit{x},t)=h(\mathit{x},t),\mathit{x}\in \partial \Omega ,t\in [0,T],\hfill \end{array}$$

(3)

where $\mathit{x}\in {\mathbb{R}}^D$ represents space coordinates and t represents time coordinates respectively; ${\mathcal{N}}_{\mathit{x}}$ represents a general linear or nonlinear differential operator, and the subscript represents the partial differentiation; $\Omega$ represents the computing domain as the subset of ${\mathbb{R}}^D$; $\partial \Omega$ represents the boundary of $\Omega$; $h_0(\mathit{x},0)$ represents the initial condition; $h(\mathit{x},t)$ represents the boundary condition; and $\mathit{u}(\mathit{x},t)$ represents the solution of the partial differential equation with the initial condition $h_0(\mathit{x},0)$ and the boundary condition $h(\mathit{x},t)$ [47].

According to the argument of Raissi et al. [31], ordinary PINN is a fully connected feed forward neural network composed of multiple hidden layers. The network takes space and time coordinates $(\mathit{x},t)$ as inputs and is trained to predict the solution $\mathit{u}(\mathit{x},t)$ of PDE. For a network with n hidden layers, the transmission form of the input and output of the hidden layer in the network can be generally summarized as

$$\begin{array}{c}\hfill {\mathit{y}}^0=(\mathit{x},t),\end{array}$$

(4)

$$\begin{array}{c}\hfill {\mathit{y}}^i=\sigma \left({\mathit{w}}^i{\mathit{y}}^{i-1}+{\mathit{b}}^i\right),1\le i\le l-1,\end{array}$$

(5)

$$\begin{array}{c}\hfill {\mathit{y}}^i={\mathit{w}}^i{\mathit{y}}^{i-1}+{\mathit{b}}^i,i=l,\end{array}$$

(6)

where ${\mathit{y}}^0$ represents the inputs $(\mathit{x},t)$ of hidden layers, and ${\mathit{y}}^i$ represents the outputs of the next hiding layer after the training of the ith hidden layer; When $i=l$, ${\mathit{y}}^i$ represents the outputs after the training of l hidden layers, which are used to approximate the true solution $\mathit{u}(\mathit{x},t)$; $\sigma (·)$ represents a nonlinear activation function used to enhance the representation and learning ability of the network. In PINNs, the infinitely differentiable $tanh$ function is usually used as the activation function. In the above equation, ${\mathit{w}}^i$ and ${\mathit{b}}^i$ represent the trainable weight matrices and bias vectors at the ith hidden layer [48].

The essence of PINN training is to train network parameters to make the predicted values approximate the real values. The network parameters that can be trained generally refer to the weight matrix $\mathit{w}$ and the bias vector $\mathit{b}$, usually denoted by $\theta$. Network parameters can be trained by minimizing a compound loss function. The loss function is a non-negative real-value function used to measure the difference between the predicted values and the real values of the model. The general form is as follows [41]:

$$\begin{array}{c}\hfill L=L_p+L_b+L_0+L_d,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill L_p=\frac{1}{N_p}\sum _{i=1}^{N_p}{\left|{\mathit{u}}_t^i+{\mathcal{N}}_{\mathit{x}}\left[{\mathit{u}}^i\right]\right|}^2,\end{array}$$

(8)

$$\begin{array}{c}\hfill L_b=\frac{1}{N_b}\sum _{i=1}^{N_b}{\left|{\mathit{u}}^i-h^i\right|}^2,\end{array}$$

(9)

$$\begin{array}{c}\hfill L_0=\frac{1}{N_0}\sum _{i=1}^{N_0}{\left|{\mathit{u}}^i-h_0^i\right|}^2,\end{array}$$

(10)

$$\begin{array}{c}\hfill L_d=\frac{1}{N_d}\sum _{i=1}^{N_d}{\left|{\mathit{u}}^i-{\mathit{u}}_d^i\right|}^2.\end{array}$$

(11)

Here, $L_p$, $L_b$, $L_0$, and $L_d$ represent the loss terms of the governing equation, boundary conditions, initial conditions, and actual observed values, respectively; $N_p$, $N_b$, $N_0$, and $N_d$ represent the number of data corresponding to different loss items, respectively. In order to simplify the equation, the expressions above do not emphasize the relationship between the loss term and the space coordinate $\mathit{x}$, the time coordinate t, the weight $\mathit{w}$, and the bias term $\mathit{b}$. It should be noted that when there is no reliable observation data in the domain, the loss term between the predicted values and the actual observation values can be omitted.

In general, PINNs consist of neural networks, physical information, and feedback mechanisms [31]. PINNs use NNs to predict the solution of PDEs, substitute the predicted solution into the loss function containing physical information, evaluate the predicted results based on the values of the loss function, and determine if the loss function is convergent. The neural network outputs the convergent predicted values and trains the non-convergent predicted values through feedback until they converge to minimize the training loss.

2.2. Free Surface Identification Methods

The accurate recognition of free surface boundaries is crucial for correctly applying pressure boundary conditions and obtaining accurate calculation results. Several methods are available for studying free-surface flow problems, including MAC, VOF, and SPH. The MAC method, developed by Harlow and Welch in 1965, combines an Euler mesh with Lagrangian markers to describe the free boundary’s shape while avoiding difficulties caused by rapid interface deformation. The VOF method tracks the free surface’s evolution by measuring the volume of the grid cell occupied by the fluid. Compared with MAC, VOF has the advantages of less memory consumption, easy implementation, and extension to three dimensions. However, grid-based methods have high computational requirements and may become unstable when dealing with violent impact and deformation. SPH, a relatively new meshless method, uses analytic differential expressions in the interpolation kernel function rather than grid discretization, avoiding high computational requirements and mesh entanglements. However, SPH may lack stability when dealing with complex problems.

To recognize and update boundary conditions in NPM, the alpha-shape algorithm is an ideal method to reconstruct arbitrary random points’ geometric shape. The alpha-shape algorithm has various mature implementations, including the Ball-Pivoting Algorithm (BPA) proposed by Bernardini et al. [49]. The BPA mainly includes the following steps. First, set the unique threshold alpha. Take any point p in the random point set S, and denote the point whose distance from p is less than 2 alpha as point set Q. After that, the point p is connected with any point Q in the point set q, and two external circles with radius alpha pass that through p and q at the same time are generated. If no other point in point set S exists in any of the outer circles, then points p and q are connected. By repeating the above steps, the boundary points can be recognized. In this study, we implement the alpha-shape algorithm using the Alpha Shape Toolbox [50]. An example of surface particle identification based on two-dimensional alpha-shape technology is given in Figure 1.

2.3. Improved Neural Particle Method

Wesels et al. [32] proposed the NPM, which offers a new approach for applying PINN in inviscid-free fluid dynamics. In the NPM, the fluid is treated as a collection of several particles, and the Lagrangian method is used to track and update the position of the particles. When using the NPM, the loss function typically needs to include the conservation laws and boundary conditions, among which the conservation laws mainly involve the conservation of momentum, mass, and energy. The loss function can be defined as follows [32,51]:

$$\begin{array}{c}\hfill L=L_{\mathit{v}}+L_{\mathrm{div}\mathit{v}}+L_{\mathit{p}},\end{array}$$

(12)

where $L_{\mathit{v}}$, $L_{\mathrm{div}\mathit{v}}$, and $L_{\mathit{p}}$ are respectively the loss terms of momentum conservation, mass conservation, and boundary conditions. The automatic differential algorithm is used to calculate the differential terms in the loss terms of the momentum conservation and mass conservation. Based on momentum conservation, $L_{\mathit{v}}$ [52] can be described as

$$\begin{array}{c}\hfill L_{\mathit{v}}=\frac{1}{N_{\mathit{v}}}\sum _{n=1}^{N_{\mathit{v}}}\sum _{j=1}^{m+1}{\left|{\mathit{v}}_s^{n,j}-{\mathit{v}}_s^n\right|}^2,\end{array}$$

(13)

where

$$\begin{array}{c}\hfill {\mathit{v}}_s^j={\mathit{v}}_s-\Delta t\sum _{i=1}^m{b}^{ji}{\mathit{a}}^{c^i},\end{array}$$

(14)

$$\begin{array}{c}\hfill {\mathit{v}}_s^{m+1}={\mathit{v}}_{s+1}-\Delta t\sum _{j=1}^m{d}^j{\mathit{a}}^{c^j}.\end{array}$$

(15)

The mth order Runge-Kutta (RK) method [53] is applied to obtain the fluid particle velocity ${\mathit{v}}_s$ at the timestep s. $N_{\mathit{v}}$ is the number of the fluid particles. The indices i and j denote the RK stage, ranging from 1 to m. The coefficients of the RK method are shown as b, c, and d. The predicted acceleration is presented by $\mathit{a}$, which can be expressed by

$$\begin{array}{c}\hfill {\mathit{a}}^{c^j}=-\frac{1}{\rho }\nabla \mathit{p}+\frac{1}{\rho }\nabla \left(\mu \nabla {\mathit{v}}^{c^j}\right)+\mathit{g}.\end{array}$$

(16)

Here, $\rho$ denotes the fluid density, $\mathit{p}$ the fluid pressure, $\mu$ the fluid viscosity, and $\mathit{g}$ the gravitational acceleration.

The conservation of mass is controlled by the mean square of the divergence of the predicted velocity, which is expressed as

$$\begin{array}{c}\hfill L_{\mathrm{div}\mathit{v}}=\frac{1}{N_{\mathit{v}}}\sum _{i=1}^{N_{\mathit{v}}}{|div\mathit{v}|}^2.\end{array}$$

(17)

The loss term of the boundary conditions is the mean square of the error between the predicted boundary pressure values and the ones derived from Dirichlet boundary conditions, which can be described as

$$\begin{array}{c}\hfill L_{\mathit{p}}=\frac{1}{N_{\mathit{p}}}\sum _{i=1}^{N_{\mathit{p}}}{\left|{\mathit{p}}^{i+1}-{\mathit{p}}^i\right|}^2.\end{array}$$

(18)

Here, $N_{\mathit{p}}$ is the fluid particles on the boundaries.

The addition of a surface recognition algorithm based on meshes to the original meshless numerical simulation method represents a meaningful improvement. Accurate surface recognition is crucial in simulating complex free fluid motion. In the original approach, the boundary particles were determined during initialization, and there were no significant changes in the free-surface particles during the outflow process, so the updating of the boundary particles could be neglected in the simulation process. However, this approach lacked additional surface recognition algorithms to simulate fluid interaction with walls.

In the improved NPM, a surface recognition algorithm based on meshes is applied. After initializing the fluid particles, their positions and other information are used to determine the boundary conditions for the fluid particles in the current time step and predict their motion state in the next time step. The position information of the fluid particles is input into a fully connected neural network to predict their velocity and pressure. At the same time, an alpha-shape technology-based surface recognition algorithm is applied to identify the free surface of the fluid particles, and the recognition results are passed to the boundary conditions used for neural network training. This allows the boundary conditions to be updated in real-time and flexibly according to the recognition results of the free surface in the improved NPM. After ensuring the accuracy of the boundary conditions, a loss function is constructed based on the conservation of mass and momentum and the boundary conditions. Finally, the position and velocity information of the fluid particles are updated by continuously adjusting the feedback to minimize the loss and input into the next time step. The flowchart of the PINN for a two-dimensional free surface flow problem is given in Figure 2. And the process of free surface particles identifying and updating boundary conditions is given in Figure 3.

In this study, the effectiveness of the method is verified through a simulated dam break example. The neural network is trained and solved solely based on the governing equations and boundary conditions. After each time step, the position of fluid particles is updated, and the free surface is recognized, allowing for the boundary conditions to be updated accordingly. By adding this surface recognition algorithm to the original approach, the improved NPM can accurately simulate fluid interaction with walls, providing a more comprehensive and precise simulation of complex fluid motion. The improved NPM can effectively simulate the flow of incompressible fluid on a free surface, even in cases of disordered particle distribution.

3. Numerical Examples of the Improved Neural Particle Method

In this section, we present three numerical examples to illustrate the effectiveness of our proposed method for complex free surface flow problems. Firstly, we demonstrate the differences and similarities between the improved NPM and the original method by simulating the outflow of a dam break and its interaction with the wall. Secondly, we simulate large sloshing and climbing processes of the fluid, which further demonstrates the accuracy of the improved NPM in identifying and applying boundary conditions. Moreover, we also analyze the influence of various factors, such as the uniform and random distribution of particles, particle density, and other factors, on the surface recognition results. The aforementioned research validates the effectiveness and stability of our proposed method.

3.1. Dam Break

Dam break [54] is a classic test case of free surface flow.To begin with, we present a dam break example to demonstrate the improved NPM’s performance in handling free surface flow problems. We simulate the outflow of a dam break and its interaction with the wall using the improved NPM and compare the results with those obtained using the original method. The simulation results reveal the significant differences and similarities between the two methods. Our proposed method demonstrates improved accuracy in capturing the fluid’s behavior near the wall, which is due to the modification of the free surface particle distribution near the wall region.

Figure 4 demonstrates the front-end position of the fluid simulated by the improved NPM before impact with the SPH and experimental data [55]. In Figure 4, X and T respectively represent the non-dimensional leading edge position and the non-dimensional time. Here, x refers to the position of the fluid leading edge corresponding to time t, L is the initial dam length, and g is the gravitational acceleration.

Table 1. Comparison of the non-dimensional leading-edge position versus the non-dimensional time from INPM, SPH, and experimental data.

Method	Non-Dimensional Time
	0.65	0.78	1.02	1.20	1.40	1.60
INPM Random	1.29	1.38	1.55	1.67	1.80	1.93
INPM Uniform	1.29	1.37	1.54	1.69	1.86	1.99
SPH	1.27	1.37	1.54	1.71	1.91	1.96
Experimental	1.10	1.20	1.40	1.60	1.80	2.00

shows the experimental data and the results obtained by different numerical simulation methods. In Table 1, a curve-fitting approach will be used to obtain a continuous representation of the discrete numerical simulation data, in order to facilitate comparison with the experimental results. The resulting continuous function will be evaluated at the same set of discrete points as the experimental data, providing a more intuitive evaluation of the effectiveness of the method. The simulation results show that our proposed method is consistent with traditional meshless methods when the particles are distributed uniformly and randomly.

To further illustrate the accuracy of the improved NPM in identifying and imposing boundary conditions, we performed more detailed simulations and analyses of the fluid process from the dam break to interaction with the opposite wall. Figure 5 shows the simulation results of fluid morphology at different times, and Figure 6 shows the pressure before and after the fluid interacts with a vertical wall. In the results presented in Figure 5 and subsequent figures, the labels X and Y refer to the horizontal and vertical positions of fluid particles, respectively.

It can be observed from Figure 5a,b and Figure 6 that the fluid morphology and pressure simulation results are in good agreement before the front end of the fluid reaches the opposite wall. However, after the fluid impinges on the opposite wall, the NPM is unable to recognize and update the free surface, resulting in a deviation in the calculation results of the pressure at the boundary. Compared with the simulation results obtained using SPH, the fluid pressure on the right wall is significantly lower (Figure 6f,h), which also leads to a lower fluid climbing height on the right wall in the subsequent simulation of NPM (Figure 5c,d).

The proposed method can more accurately apply boundary conditions to ensure the accuracy of the calculation results. As shown in Figure 6, when the improved NPM simulated the interaction between the fluid and the wall, the pressure at the point where the fluid impacted the wall increased significantly. The fluid profile is very consistent with the pressure distribution and the results obtained using SPH. The difference is that the simulation results obtained by our method are much smoother, whereas in the SPH method, the particle morphology and pressure distribution have higher discontinuities. These results validate the effectiveness of the proposed method in identifying and tracking free surfaces accurately, making it a valuable tool for practical applications. It is worth noting that the simulation results of the improved NPM and SPH methods are still highly consistent when the particles are not uniformly distributed. We will discuss the stability and accuracy of the improved NPM for a highly uneven distribution of particles in a subsequent section in more detail.

3.2. Violent Sloshing

Unlike the dam break scenario, where the free surface morphology and extent change gradually, the free surface in large sloshing of the fluid changes more dramatically as the fluid undergoes violent oscillations. To demonstrate the accuracy of the improved NPM, we conducted simulations and obtained the results of fluid contour recognition and pressure distribution using the improved NPM with random distribution, as shown in Figure 7. The accuracy of free surface particle identification has a significant impact on pressure distribution calculation results, and we can observe from Figure 7 that the shape of the pressure contour is reasonable. These results further underscore the effectiveness of our improved NPM in identifying free surface particles.

Additionally, we found that although the morphology and particle density of the free surface differ significantly between the late and early phases of fluid movement, the accuracy of free surface recognition remains consistent. Our method can effectively track and describe the change in fluid-free surface, whether particles are tightly or discontinuously distributed. This advantage enables the accurate application of boundary conditions in computation, allowing for the description of more complex fluid motions. By accurately identifying free surface particles and tracking changes in the free surface, our method provides a useful tool for describing complex fluid motions.

3.3. Slope Climbing

When dealing with free surface flow problems, it is necessary to consider complex boundary conditions that go beyond the motion of fluid along horizontal and vertical walls in order to more accurately describe the motion of the fluid. To address this issue, we utilized the improved NPM to simulate fluid climbing an inclined plane, as shown in Figure 8 and Figure 9. The simulation results include external contour recognition and velocity distribution, which demonstrate the accuracy of our method in identifying free surface particles and imposing boundary conditions.

At the same time, to further analyze the accuracy of our method, we compared the simulation results of three randomly distributed particles with different densities. As Figure 8 shows, the calculated fluid profile and velocity distribution are consistent, even when particle density varies. However, it is worth noting that in Figure 9a, errors in identifying free surface particles may occur due to some spilled particles. To address this issue, we plan to further study this phenomenon in future work and improve the surface recognition module of the algorithm to ensure accurate simulation of complex fluid motion.

In the simulation of various free surface flow problems using the improved NPM, we have observed that it demonstrates robustness in handling highly unevenly distributed particles. In order to further illustrate and analyze the stability of our proposed method, we compared the improved NPM with the traditional meshless method for randomly distributed particles. Figure 10 presents the calculation results obtained using the improved NPM and SPH, respectively, when particles have the same random distribution and initial velocity. Through comparison, it is evident that the improved NPM remains stable and accurate even under highly uneven particle distribution conditions. This stability is evident in the identification of free surface particles and the application of boundary conditions, as well as the stability of particle coordinates and velocities. In contrast, SPH tends to generate significant errors when handling highly uneven particle distribution.

4. Discussion

In this work, we present an improved NPM for simulating free surface flows. The proposed method offers several advantages over other methods, as demonstrated in the following aspects. First, it is a deep learning-based approach that builds upon the PINN. Unlike traditional machine learning algorithms, the improved NPM can effectively incorporate known physical laws, resulting in better training results without requiring an extensive amount of data. When the initial and boundary conditions are clear, and the relevant physical laws are known, PINN has a significant advantage. In addition, the improved NPM includes a free surface recognition algorithm, allowing for flexible and precise boundary condition application in complex scenarios. In the simulation of multiple numerical examples of free surface flows, the improved NPM accurately identifies free surfaces and applies boundary conditions. Furthermore, compared with traditional meshless methods, the improved NPM is better suited to handle the uneven distribution of particles. The proposed method offers a more robust solution for complex free surface flow problems that involve random particle distribution. Even in cases where SPH and other methods struggle to deal with the random distribution of particles, the simulation results obtained using the proposed method are still stable and accurate. The proposed method offers a more robust solution for complex free surface flow problems that involve random particle distribution.

Despite the fact that the use of meshes in the recognition algorithm may seem like a step backward towards a fully meshless particle approach, the alpha-shape technology sets itself apart from traditional meshing techniques and has proven to be highly effective in areas such as shape analysis and surface reconstruction. The proposed method combines the advantages of grid-based and grid-free approaches. Specifically, the grid-based method accurately captures the overall geometry of the surface, while the grid-free method handles the complex physics involved in the simulation. Consequently, the mesh provides a reliable framework for the meshless method to operate on, which helps avoid errors and inaccuracies that may arise in purely meshless methods. The use of alpha-shape techniques in meshless methods allows the flexibility of meshless methods to be combined with the accuracy and efficiency of grid-based methods. The proposed method can be valuable for studying free surface flow and other problems involving complex geometry.

Although the proposed method has excellent performance in the simulation of free surface flow problems, it still has some limitations. The accuracy of the calculation results is significantly affected by the network structure since the improved NPM relies on PINN. To make it more practical for solving real-world problems, further research on neural networks is necessary. Additionally, the improved NPM can only be applied to the simulation of two-dimensional cases. However, its principle and the surface recognition algorithm used can be extended to three-dimensional cases. Moreover, due to the relatively complex process of identifying free surfaces and calculating the proposed method, the computational process needs further refinement to achieve more efficient computation. In the future, we plan to conduct more research on the aforementioned issues to enhance the applicability of the improved NPM.

5. Conclusions

In this paper, we present an improved NPM based on PINN to simulate complex free surface flows. The proposed method incorporates a surface recognition module based on alpha-shape technology to track and identify fluid boundaries. In the improved NPM, the neural network is trained and solved solely based on the governing equations and boundary conditions. After each time step, the position of fluid particles is updated, and the free surface is recognized, allowing for the boundary conditions to be updated accordingly. The results of numerical experiments show that (1) the proposed method can accurately impose boundary conditions in more complex cases compared with NPM. (2) The proposed method converges well with the particle density and distribution used for calculation, the calculated fluid profile and velocity distribution remain consistent even when the particle density varies. (3) The proposed method accurately identifies the free surface and provides stable simulation results even when particles have a highly uneven distribution, which is challenging to deal with by SPH and other methods. For incompressible fluid dynamics, the proposed method accurately identifies the free surface and predicts the pressure and velocity fields.

In summary, the improved NPM based on PINN and alpha-shape technology accurately identifies free surfaces and applies boundary conditions in complex free surface flow problems. The numerical verification results demonstrate its effectiveness in dealing with randomly distributed particles and its potential for broader applications. Our research effectively addresses the limitations of NPM in dealing with complex free surface flow problems and provides assistance for the use of meshless methods in dealing with randomly distributed particles. In future work, we plan to improve the proposed method and explore its application in three-dimensional cases to enhance its applicability.