Boundary Treatment with Additive Boundary Particles for Incompressible Smoothed Particle Hydrodynamics Method

Abstract

A new boundary treatment considering uniform particle distribution using additive boundary particles for the incompressible smoothed particle hydrodynamics (ISPH) method is proposed. With this new boundary treatment, the solid boundary can be easily modeled just following the geometry configuration and consequently the simulations of fluid–structure interaction problems can be simplified. The efficiency of this new boundary treatment is analyzed, and the implementations are compared with other exiting boundary treatments such as repulsive force and ghost particles. Simulations of the 2D dam-breaking case, static tank, and 2D u-tube test are carried out as examples to demonstrate the performance of this new boundary treatment. It is shown that better predictions can be gained with the new boundary treatment for the pressure distribution in these cases than those obtained from other boundary treatments.

1. Introduction

Smoothed particle hydrodynamics (SPH) is a fully Lagrangian mesh-free method initially developed for astrophysics in 1970s [1,2]. It has been widely used in large deformation problems such as fluid motions where the continuum hydrodynamics equations are solved with a set of interacting fluid particles. For incompressible flows, there are generally two ways to impose incompressibility. One is to run the simulations in the quasi-incompressible limit by assuming a small Mach number to ensure the density fluctuation within 1% [3,4,5], which is known as weakly compressible smoothed particle hydrodynamics (WCSPH). This method is conceptually easy as pressure values are computed straightforwardly and are independent of other quantities. However, very small time steps are required, and the pressure values obtained are usually noisy since it is sensitive with density variations [6,7]. The other one is called truly incompressible SPH (ISPH) in which incompressibility is enforced by setting the velocity divergence to be zero, and the pressure is calculated by solving a Poisson’s equation at every time step [8,9,10]. Comparison of these two methods can be found in many works [11] and the ISPH method is selected for the current research.

Since SPH methods are based on integral approximation and further approximated by averaging the neighboring particles within the kernel domain, which will be presented in Section 3, the computations are all based on summation of neighboring particles inside the kernel domain. For the particles without enough neighboring particles, i.e., when the kernel domain is truncated by the boundary, the simulation would not be successful. Hence, it is one of the most challenging problems to find an ideal boundary treatment in the SPH method. According to the existing work, the boundary treatment can be mainly classified into three categories: repulsive force, ghost particles, and semi-analytical boundaries.

In repulsive force boundary treatment, a line of denser particles is introduced along the wall boundaries where forces were generated to push away the approaching fluid particles. The repulsive force treatment is easy to implement and is effective in the prevention of inner fluid particles from penetrating. However, it often leads to spurious effects and other quantities calculated may not be accurate since the kernel domain is still truncated for fluid particles near the near wall [3,12].

In the ghost particle boundary treatment, several lines of virtual particles are placed outside the solid boundary to make a complete smoothing domain for particles near the wall. One type of ghost particles, called mirror particles, possess the same physical properties as those of real fluid particles. They are generated at every time step along the normal direction of the inner fluid particles treating the boundary as a mirror [13,14]. With these mirror particles the spurious effect is eliminated, and the physical values of interior particles could be calculated more accurately. However, special consideration is required to calculate the exact positions of the mirror particles, especially for the angled points, which can be time consuming. Furthermore, fluid particles can move through the wall boundary due to the relatively weak fence effect [15]. To solve this problem, another type of ghost particles called dynamic boundary particles are suggested to be distributed outside the boundary with fixed positions [16]. However, they are not suitable for deforming boundaries. Furthermore, it is not always easy to ensure enough dynamic particles in the kernel domain, and inner particle penetration could thus be found in some cases [17]. Moreover, ghost particles may take up too much space, and sometimes they even overlap with the real fluid particles. In conclusion, ghost particles are useful to keep a complete kernel domain for the near-wall inner particles, so that the physical properties such as density could be calculated correctly. However, some researches show that they are not always effective in the prevention of inner fluid particle penetration [6,18]. This phenomenon is also found in the following 2Dtube flow case study, as shown in Section 6.4.

The semi-analytical boundary is introduced by Kulasegaram et al. [19] and further improved by Ferrand et al. [20]. A correction term is applied to the theoretical formulations to eliminate the effect of lacking neighboring particles [21,22]. The drawback is that extra computation is needed during the simulation, as normally the correction term is not just a simple number but a matrix.

Research has been carried out to find out a more efficient boundary treatment. In this paper, the ISPH method with constant fluid density is applied as it normally provides better pressure evaluations than WCSPH [21]. For the ISPH method, the application of one layer of denser wall particles is supposed to be effective in the prevention of inner particle penetration [22]. However, it is found in some situations, such as violent impact problems, that the distribution of particles could be distorted and the local particle spacing could be shortened largely. This will lead to unrealistic fluid particle exploration [23]. Hence, uniform particle distribution is preferred for a stable and accurate simulation [24,25], where wall particles involved in calculation should be spaced the same as fluid particles.

With all these considerations, a new boundary treatment, using additive particles and dynamic dummy particles is proposed in this work. Dummy particles do not really exist and are only used as neighboring particles. The solid boundary is represented by two types of boundary particles. One type is spaced just the same as fluid particles; the other is used to fill the gap of the first type with smaller intervals, called additive particles. These additive particles are not treated directly as neighboring particles, and thus the uniform particle distribution is retained. They are only used as a medium to create dummy particles. Both types of boundary particles create dummy particles, so that the kernel domain of the inner fluid particle will be complete.

Five different boundary treatments shown in Section 6 are considered including the new boundary treatment, ghost particles, ghost particles plus repulsive force, uniform distributed boundary particles with dynamic dummy particles, as well as uniform distributed particles with repulsive force. Four case studies are carried out, respectively, with the five above different boundary treatment methods, including water motions in dam breaking, 2D and 3D static water tank and u-tube flow. The results are displayed and compared to demonstrate the performance of the proposed boundary treatment. The exact pressure values in dam breaking, 2D, and 3D static water tanks are analyzed. The time history of the impact pressure on a specific point during dam breaking is compared with experimental values [26]. Pressure distributions of u-tube flow are displayed in different colors to provide an intuitive assessment on the performance of different boundary treatments. The results show that the new boundary treatment using additive boundary particles combined with dummy particles is effective to prevent the inner particles from penetrating and could provide accurate pressure evaluations.

2. Governing Equations for Incompressible Fluid

Fluid flows are governed by the classical Navier–Stokes equations including the continuity equation and momentum conservative equation:

$$\frac{1}{\rho }\frac{d\rho }{dt}+\nabla \cdot v=0$$

(1)

$$\frac{dv}{dt}=-\frac{1}{\rho }\nabla P+g+\frac{1}{\rho }\nabla \cdot \tau$$

(2)

where $\rho$ is the density, $v$ is the velocity, $P$ is the pressure, $g$ is the body force (gravitational force in this case), and $\tau$ is the viscous stress tensor. For incompressible fluids, the mass density takes a constant.

The viscosity stress tensor $\tau$ in the momentum conservative equation is related to the velocity by the following equation:

$${\tau }_{ij}={\tau }_{ji}={\mu }_{}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$$

(3)

where $\mu$ is the dynamic viscosity coefficient. The two RHS gradients can be approximated in particle form [8]:

$${\left(\frac{\partial v_i}{\partial x_j}\right)}_a=\left(\frac{v_a^i-v_b^i}{r_{ab}}\right)\left(\frac{x_a^j-x_b^j}{r_{ab}}\right)$$

(4)

The suffix i, j indicates the tensor index of value 1, 2, or 3; a, b indicates different particles; and b must be a neighboring particle of a.

For incompressible fluids, the mass density takes a constant, so that Equations (1) and (2) reduce to

$$\nabla \cdot v=0$$

(5)

$$\frac{dv}{dt}=-\frac{1}{\rho }\nabla P+g+\frac{{\mu }_{eff}}{\rho }{\nabla }^{2}v$$

(6)

where µ_eff is the effective viscosity of the fluid.

Free surface

The pressure on the free surface is assigned to be zero as reference atmospheric pressure.

Boundary conditions

On the velocity boundaries, the velocities are assigned to be a prescribed velocity.

$$v=\widehat{v}$$

(7)

If this prescribed velocity is zero, this equation denotes a fixed fluid boundary condition or an infinite fluid boundary condition, where the fluid is not disturbed in the far field away from the interested finite domain.

3. SPH Formulations

3.1. Basic SPH Formulas

The SPH method is based on the theory of integral interplant that uses kernel function to approximate Dirac delta function:

$$f(x)\approx 〈f(x)〉={\displaystyle \int f(x)W(x-x^{\prime },h)d{x}^{\prime }}$$

(8)

where $W(x-x^{\prime })$ is the kernel function and h is the smoothing length which defines the influence domain of the “kernel estimation”. A physical property is calculated by the interpolation among a set of particles inside a smoothing area. Those particles carry all the physical properties of the fluid such as density and velocity. The basic idea of this method is to approximate a general function f(x) using a particle form:

$${〈f(x)〉}_a={\displaystyle \sum _{\mathrm{b}=1}^N\frac{m_b}{{\rho }_b}}f(x_b)W(x_a-x_b,h)$$

(9)

where m is the mass of a particle, ρ is the density, a, b indicate different particles,${ W(\mathbf{x}}_a-{\mathbf{x}}_b)$ can be written as $W_{ab}$ for short, and N is the total number of neighboring particles of particles a. The gradient of the function could be approximated as

$${〈\nabla \cdot f(x)〉}_a={\displaystyle \sum _{b=1}^N\frac{m_b}{{\rho }_b}}f(x_b)\cdot \nabla W(x_a-x_b,h)$$

(10)

where

$$\nabla W(x_a-x_b,h)=\frac{x_a-x_b}{r_{ab}}\frac{\partial W_{ab}}{\partial r_{ab}}$$

(11)

and $r_{ab}=r_a-r_b$.

SPH approximation for the motion of a continuum can be found in the literature [18,19,20,21]. One of the most widely used density approximations is derived directly by substituting density into Equation (9):

$${\rho }_a={\displaystyle \sum _{b=1}^N{m}_b}{W}_{ab}$$

(12)

Alternatively, applying the particle approximation to the continuity equation, the gradient of density in terms of time can be computed by the SPH formula as

$$\frac{D{\rho }_a}{Dt}={\displaystyle \sum _{b=1}^N{m}_b}{v}_{ab}^{\beta }\cdot \frac{\partial W_{ab}}{\partial x_a^{\beta }}$$

(13)

where t is time and β is the coordinate direction. To approximate the momentum, the gradient of pressure can be written in a symmetric SPH form as

$$\frac{1}{\rho }\nabla P={\displaystyle \sum _b^N{m}_b}\left(\frac{P_a}{{\rho }_a^2}+\frac{P_b}{{\rho }_b^2}\right)\nabla W_{ab}$$

(14)

Fluid flows are governed by the classical Navier–Stokes equations including the continuity equation and momentum conservative equation.

3.2. SPH Implementation Algorithm for Incompressible Fluid

There are two ways to ensure the incompressibility of fluid in SPH simulations. One is to divide the momentum equation into two parts. In the first part, the momentum equation is solved in the presence of body forces and viscosity forces. The second-part equation is solved in the presence of body force and viscosity forces. The second part considers the effect of the pressure and ensures incompressibility for the mediums. The other method is employed in this paper, including two steps as well [10]. In the former ISPH method, an intermediate density needs to be calculated according to the intermediate velocity values. However, in the present incompressible SPH method, constant density is used in the whole process hence the computation time is saved and no consideration about density oscillation is required. The whole process of the algorithm is shown in Section 5.

$$v^{n+1/2}=v^n+g\Delta t+\left(\frac{1}{\rho }\nabla \cdot {\tau }^n\right)\Delta t$$

(15)

For Newtonian fluids such as water, the viscosity coefficient ${\mu }_{eff}$ (effective viscosity) has a constant value of $\mu$. Hence, the SPH formulation of viscosity term can be written as [8]:

$${\left(\frac{1}{\rho }\nabla \cdot {\tau }^{}\right)}_a=\left({\displaystyle \sum _b\frac{4m_b\mu }{{\left({\rho }_a+{\rho }_b\right)}^2}\frac{r_{ab}\cdot \left(v_a-v_b\right)\nabla W}{{\left|r_{ab}\right|}^2+{\eta }^2}}\right)$$

(16)

where $r_{ab}$ means $r_a-r_b$.

$$\frac{v^{n+1}-v^{n+1/2}}{\Delta t}=-\frac{1}{\rho }\nabla P^{n+1}$$

(17)

Taking the divergence of Equation (17) and substituting it into Equation (5),

$$\Delta P^{n+1}=\frac{\rho }{\Delta t}\nabla \cdot v^{n+1/2}$$

(18)

Equation (18) can be expressed into SPH form:

$${\displaystyle \sum _{b=1}^N\frac{m_b{P}_{ab}^{n+1}{\mathbf{r}}_{ab}^n}{{\mathbf{r}}_{ab}^2+{\eta }^2}\nabla W_{ab}^n}=-\frac{\rho }{2\Delta t}{\displaystyle \sum _{b=1}^N{m}_b{v}_{ab}^{n+1/2}}\nabla W_{ab}^n$$

(19)

where n indicates a time step; η is a random small number to avoid zero denominator. The pressure P of particle “a” can be updated by Equation (19) which can be solved efficiently by the BI-CGSTAB method. Solid particles which are the neighboring particles of fluid particles are involved in the Poisson’s equation and pressure values are calculated simultaneously with fluid particles.

Finally, the velocity and position of each fluid particle can be renewed for the next time step:

$$\begin{array}{ll}{v}^{n+1}& =v^{n+1/2}-\frac{1}{\rho }\nabla P^{n+1}\Delta t\\ & =v^{n+1/2}-{\displaystyle \sum _{b=1}^N{m}_b\left(\frac{P^{n+1}{}_a}{{\rho }_a^2}+\frac{P^{n+1}{}_b}{{\rho }_b^2}+{\Pi }_{ab}\right)}\cdot \nabla W(x_a-x_b,h)\end{array}$$

(20)

where ${\Pi }_{ab}$ is artificial viscosity. It is added into the momentum equation to avoid unphysical particle oscillation [5]:

$${\prod }_{ij}=\left\{\begin{array}{l}\frac{-\alpha {\overline{c}}_{ab}{\mu }_{ab}+\beta {\mu }_{ab}^2}{{\overline{\rho }}_{ab}};\dots v_{ab}\cdot r_{ab}<0\\ 0;\dots \dots \dots \dots \dots \dots v_{ab}\cdot r_{ab}>0\end{array}\right.$$

(21)

where ${\mu }_{ab}=\frac{h{v}_{ab}\cdot r_{ab}}{r_{ab}^2+{\eta }^2}$. Any value ${\overline{f}}_{ab}$ denotes $\frac{f_a+f_b}{2}$.

For fluid particles near the boundary, the velocity calculation is affected by the pressure values of the solid particles, so that the fluid particles could not penetrate the solid boundary:

$$x^{n+1}=x^n+v^{n+1}\Delta t$$

(22)

4. Boundary Treatments

Normally, the solid walls are simulated by particles where the inner particles are prevented from penetrating the wall. Repulsive force could be exerted on boundary particles to keep the inner fluid particle away from the boundary. Alternatively, several layers of ghost particles can be arranged or mirror particles can be created dynamically outside the boundary to make a complete kernel domain for the near-wall inner particles. The pressure force from the wall particles together with the ghost particles could prevent the inner particles from penetrating.

4.1. Repulsive Force

Repulsive force boundary treatment is widely used in WCSPH because of its simplicity of implementation. Repulsive force is exerted on the fixed boundary particles to keep the adjacent inner fluid particles away and prevent them from penetrating the boundaries, as shown in Figure 1. The repulsive force is calculated by

$$f(r)=D\left({\left(\frac{r_0}{r}\right)}^{P_1}-{\left(\frac{r_0}{r}\right)}^{P_2}\right)\frac{r}{r^2}, r<r_0$$

(23)

f(r) is set to be zero when $r>r_0$ so that the force is purely repulsive. Mostly, $P_1$ = 4 and $P_2$ = 2, D = 5 gH, and H is the total water height [3]. The length scale $r_0$ is taken to be the initial spacing among the particles.

The penetration of fluid particles could be avoided by applying a repulsive force. Yet the kernel domain is still truncated, and other modification of particle approximations is necessary to obtain a correct computation. Furthermore, this repulsive force is largely dependent on the distance between two particles. Thus, in the situation of violent fluid impact, this force can be extremely large when particles approach each other instantly. Consequently, the inner fluid particle might be forced to move fiercely away from the boundary with a large velocity. All the surrounding fluid particles will then be influenced, and unphysical particle explosion might happen.

4.2. Ghost Particles

In order to make a complete kernel domain for the near-wall fluid particles, several layers of ghost particles are needed outside the boundary, as shown in Figure 2.

The physical properties such as density will not decrease too much for those near-wall particles. Wall particles are involved in the Poisson’s equations, so that the pressure values of these particles can be calculated directly. To avoid excessive calculation of the exact positions of mirror particles at every time step, ghost particles can be placed outside the boundary with initially fixed positions. However, additional geometry space will be taken up by these ghost particles, which may cause problems for model configuration. Moreover, it is hard to ensure enough neighboring particles in all cases so particle penetration still might happen.

With the above considerations, a modified ghost particle arrangement using additive particles is proposed in the current work and the details are illustrated below.

4.3. Additive Particles Combined with Dummy Particles

There are two types of particles applied on the boundary, including normal boundary particles and additive boundary particles. The solid boundary is modeled with normal particles spaced the same as inner fluid particles to ensure an initial uniform particle distribution. Then, the gap between two boundary particles is filled by additive particles, as shown in Figure 3. As a result, the boundary is actually modeled with denser particles geometrically, while the additive particles are not involved in the computation of neighboring summations, and the uniform particle distribution is thus not affected. Additive particles are only used to help create enough dummy particles for the near-wall boundary particles, as shown in Figure 4. On the contrary, dummy particles do not really exist, so they cannot be visually seen and no actual space is taken up. Only dummy particles within the kernel domain will be identified and taken into account in the computation. For fluid particle i, j₁ is a normal boundary particle and j₂ is an additive particle. These two particles are inside the kernel domain of particle i with distances as r₁ and r_2, respectively.

The distance between two particles is calculated as the following in 2D cases:

$$\mathrm{r}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2},\mathrm{d}x=x_i-x_j,\mathrm{dy}=y_i-y_j$$

For normal boundary particle j₁, in addition to its contribution to the kernel summation of particle i, a dummy particle j₁′ with twice the distance 2r ($\mathrm{d}\mathrm{x}\prime =2\mathrm{d}\mathrm{x},\mathrm{d}\mathrm{y}\prime =2\mathrm{d}\mathrm{y}$) away from i, as shown in Figure 4, will be checked if it is inside of the kernel domain. If this dummy particle is inside the kernel domain, it will be treated as neighboring particles contributing to the calculations of all the physical properties of particle i. Then, the next dummy particle with three times the distance 3r ($\mathrm{d}\mathrm{x}\prime =3\mathrm{d}\mathrm{x},\mathrm{d}\mathrm{y}\prime =3\mathrm{d}\mathrm{y}$) away from particle i will be checked and the same process will be carried on for other particles on the boundary within the smoothing domain of particle i.

Additive boundary particles such as particle j₂ are in the list of neighboring particles but not involved in the computation directly. Similar to the normal boundary particles, a dummy particle j₂′ with twice the distance away from particle i is checked first. It will contribute to the kernel summation on the condition of being within the kernel domain of i. Then, dummy particles with three times or even four times distance away from particle i will be checked. Similar processes will be carried out to identify all the dummy particles of the additive boundary particles within the kernel domain and to include their contribution in the computation.

The number of the additive particles can be decided practically. Denser additive particles will create more dummy particles. The pressure of a dummy particle is obtained by smoothing over the neighboring particles as

$${\mathrm{p}}_{\mathrm{j}}=\frac{{\displaystyle \sum _i{p}_i{W}_{ij}}}{{\displaystyle \sum _i{W}_{ij}}}$$

(24)

Dummy particles have the same density and mass as fluid particles. Velocities are assigned to be the same as those of the normal boundary particles, which should be ensured specifically to satisfy the fixed no-slip or slip boundary conditions, or the moving flexible boundary condition, according to the case in consideration.

5. Numerical Stability

Since the pressure is calculated implicitly while other properties are calculated explicitly, the size of the time step must be controlled in order to produce stable and accurate results. The following Courant condition must be satisfied [8]:

$$\Delta t\le 0.1\frac{\mathrm{s}}{v_{\max }}$$

(25)

where s is the initial particle spacing and $v_{\max }$ is the maximum particle velocity in the computation. The factor 0.1 ensures that the particle moves only a fraction (in this case 0.1) of the particle spacing per time step. Another constraint is based on the viscous terms

$$\Delta t\le 0.125\frac{{\mathrm{s}}^2}{{\mu }_{eff}/\rho }$$

(26)

where ${\mu }_{eff}$ is the effective viscosity. The allowable time step should satisfy both of the above criteria.

The whole algorithm process is shown in Figure 5.

6. Applications

Five boundary treatments are considered here, and the performances are compared. All the normal boundary particles and ghost particles are included in the Poisson’s equation, which means they are all assigned with certain pressure values. The additive boundary particles are not involved in any of the computations. The five different schemes are illustrated and explained in the following.

Boundary treatment 1: additive boundary particles with dynamic dummy particles, as illustrated in Figure 6 (dummy particles are not really created so they cannot be visually seen but they are actually checked and involved in neighboring calculations, as described in Section 4.3).

Boundary treatment 2: ghost particles only, as shown in Figure 7. In this treatment, the pressure of ghost particles is obtained by solving the Poisson’s equation and these pressure forces could affect the velocities of the inner fluid particles so no penetration would be expected.

Boundary treatment 3: ghost particles and repulsive force. The model is the same, as shown in Figure 7. Meanwhile, the repulsive force from the first layer of the normal boundary particles is added to the inner fluid particles in addition to the pressure forces.

Boundary treatment 4: uniform particle distribution with normal boundary particles is illustrated in Figure 8. Only one layer of normal boundary particles is arranged and the pressure values obtained by solving Poisson’s equation are added to the neighboring inner fluid particles, and these boundary particles are checked to determine whether they can contribute to the calculations of the neighboring inner fluid particles).

Boundary treatment 5: uniform particle distribution with boundary pressure and repulsive force together. The model is the same as that shown in Figure 8. Both the pressure values and repulsive forces of the normal boundary particles are calculated and added to the neighboring inner fluid particles.

To verify the efficiency of the proposed boundary treatment, several case studies are carried out.

6.1. Dam-Breaking Testing Case

A 2D dam-breaking problem is simulated for assessment of the efficiency of the proposed boundary treatments. The spacing of fluid particles is set to be 0.01 m, the overall height of the water column is 0.6 m, and its width is 1.2 m. The size of the solid container is 3.22 m long. The initial state of the water tank is shown in Figure 9. The pressure on a point marked red which has a 0.16 m distance above the bottom of the tank on the right-hand-side wall is recorded.

The results obtained from the simulation are compared with experimental values obtained in [25]. Fluid motions at 0.1 s and 0.2 s obtained from five different boundary treatments are provided and compared in Figure 10.

It is evident in Figure 10 that boundary treatment 4 (uniformly distributed boundary particles only) fails to prevent the penetration of inner particles. Furthermore, a gap between the water front and the solid boundary is observed in the results obtained with boundary treatments 2 and 3 using ghost particles.

Figure 11 shows the subsequent results of water motions at 0.3 s. It can be seen that the gap between the fluid front and the solid is still obvious at 0.3 s with all the boundary treatments using ghost particles.

The subsequent processes of water motions at 0.18 s and 1.4 s are provided in Figure 12 and Figure 13. It can be seen that fluid particle penetration occurs in boundary treatment 2, which indicates that the application of ghost particles only is not sufficient in this case. The results obtained from all the boundary treatments are very similar. Time histories of the pressure values at the test point, as marked in Figure 9, are compared in Figure 14 for different boundary treatments as well as the experimental results in [25].

As shown in Figure 14, the value and the happening time of the first pressure peak recorded in the experiment can be well reflected in numerical simulations. The values obtained from numerical methods are lower than the experimental data. This could be contributed to the three-dimensional effect in the experiment. However, most of the numerical methods cannot predict the second peak correctly. This might be caused by the ignorance of air bubbles entrapped inside the fluid after its impacting with the wall [15].

6.2. 2D Static Water Simulation

The water in the tank is 0.6 m high. The initial pressure distribution is shown in Figure 15. Pressure distributions obtained with different boundary treatments 1, 2, 3, 4, and 5 at 3 s are shown in Figure 16 to compare with the initial values.

The pressure distribution at 3 s with boundary treatment 1, 2, 3, 4, and 5 are shown in Figure 16.

From Figure 16, it is clear that boundary treatments 2 and 4 fail to prevent inner particles from penetration. The color layer is continuous at the same level, which indicates that the pressure is the same at the same water height. For other boundary treatments, the pressure values obtained in the corner of the tank are not accurate.

The pressure values of the left corner and the middle point of the tank bottom are traced during the whole simulation and the relative error is exhibited in Figure 17.

From Figure 17, we can see that additive boundary particles with dynamic dummy particles and ghost particles plus a repulsive force produce similar pressure evaluations for the middle tank point. However, for the corner point, only the additive boundary particle treatment provides correct pressure approximation. So far, it is obvious that additive boundary particles with dynamic dummy particles provide the best simulation of the static water tank.

6.3. 3D Static Water Simulation

The water inside the simulation tank is 0.65 m high. The initial pressure distribution color is shown in Figure 18 below:

Pressure distributions at 0.5 s and 0.95 s with five different boundary treatments are shown and compared in Figure 19.

The relative error of the pressure at the middle point of the tank bottom is analyzed and shown in Figure 20 for two different boundary treatments.

It is clear in Figure 19 that boundary treatments 2, 4, and 5 fail to retain the inner fluid particles. From Figure 20, it can be seen that the proposed new boundary treatment obtains a much more accurate pressure evaluation with relative error within 5% compared with other treatments.

6.4. 2D U-Tube Flow Simulation

U-tube is a common piece of equipment in fluid experiments. The initial state of the equipment is shown in Figure 21.

In the initial model, the water column is 0.4 m high and 0.6 m wide, and the tube is 0.4 m long with a radius of 0.1 m. The accuracy of different boundary treatments is discussed in the following, mainly based on the result of the steady state of the U-tube flow, which is obvious in this case.

Water motions at 0.1 s, 0.3 s, 0.4 s, 0.8 s, 1.4 s, 2.0 s, 4.0 s, 10 s, and 20 s with five different boundary treatments are shown in Figure 22.

From Figure 22, it is evident that treatments 2 and 4 fail to retain the inner fluid particles, so water motions are only shown within 2 s in these two cases. The water flow is supposed to reach a balanced steady state after a sufficiently long time as the water state at 20 s shown in boundary treatment 1 of Figure 22. However, it is still at a quite unbalanced and unsteady state at 20 s for boundary treatment 3, which is obvious against the physical law and common sense. The water pressure is supposed to be equal at the same level when the system reaches a balanced and steady state. Hence, the pressure distributions at 20 s are shown in different colors in Figure 23, and a hydrostatic pressure distribution is provided in Figure 24 to make a comparison.

It can be seen that only the pressure distribution obtained from the additive boundary treatment (i.e., treatment 1) is practical in Figure 23 and agrees well with the hydrostatic pressure distribution of the same volume of water, as shown in Figure 24. As a result, we can conclude that the additive boundary treatment proposed could provide a much more accurate pressure calculation than other boundary treatments discussed in this work.

7. Conclusions

A new boundary treatment using additive boundary particles together with dynamic dummy particles for the ISPH method is proposed in this paper. The solid boundary is represented by normal boundary particles which are spaced just the same as fluid particles and additive particles which are filled in the gap of the normal boundary particles with smaller intervals. These additive particles are not treated as neighboring particles so that the uniform particle distribution is retained in the computations. Both the two types of boundary particles create dummy particles under the boundary line, so that the kernel domain of the inner fluid particle is complete. This new boundary treatment is not only efficient in the modeling set up but is also easily implemented.

Four testing case studies, i.e., dam breaking, 2D and 3D static water tank, and 2D U-tube flow are carried out to investigate the performance of different boundary treatments, in terms of the prevention of inner particles from penetration as well as the accuracy of pressure evaluations. In the case of dam breaking, the boundary treatment with one layer of uniformly distributed boundary particles fails to stop the inner fluid particles penetrating if only pressure is taken into account. The penetration could not be prevented even if more layers of ghost particles are applied, unless the repulsive force is added. Moreover, if the repulsive force is added together with the pressure effect, only one layer of uniformly distributed boundary particles is found enough to have a similar performance as that with multi layers in former treatments. In the case of the 2D static water tank, multi layers of ghost particles are found as slightly helpful to retain the inner fluid particles. More layers of ghost particles could produce pressure distribution of a higher accuracy, if the repulsive force is considered simultaneously. However, the most accurate pressure distribution is obtained by additive boundary particles. It is more obvious in the case of the 3D static water tank, where more layers of ghost particles are found useful to retain the inner fluid particles and to calculated pressure distribution with a higher accuracy. The best results are also obtained with additive boundary particles proposed in the current work. In the case of U-tube flow, it is found that the application of multiple layers of ghost particles could disturb the result of pressure distribution, while the additive boundary treatment still provides good prediction of the flow pattern as well as the pressure distribution. From all the results in the case studies, it is evident that the new boundary treatment proposed in the current work is efficient to retain the inner particles and could also provide better prediction of pressure distribution, compared with those by ghost particles and repulsive force treatment.