A Lagrangian particle model on GPU for contaminant transport in groundwater

Abstract

To simulate contaminant transport in groundwater, this paper proposes a parallelized Lagrangian particle model using compute unified device architecture (CUDA) on graphics processing unit (GPU) based on smoothed particle hydrodynamics (SPH) method. The solved governing equation is the advection–diffusion equations (ADEs) with retardation factor for given typical flow fields. To solve the inherent particle inconsistency problem of traditional SPH method, the corrective smoothed particle method (CSPM) is applied. The speedup ratio of the parallelized SPH solver for ADEs is analyzed. The consistency and convergence of the proposed model are theoretically analyzed and numerically tested. The reduction of its computational cost and storage requirement is discussed. Numerical examples including one-dimensional (1D) and two-dimensional (2D) cases are simulated, and the results are compared with the analytical solutions and those obtained by the high-resolution monotonic upstream schemes for conservation laws (MUSCL) scheme. To further verify the practicality of the model, two engineering cases of contaminant transport through soil into groundwater are investigated. It is shown that the solutions of the developed model are in good agreement with measured data.

Appendices

Appendix A: Stability analysis

In this appendix, the von Neumann analysis is applied to the SPH method for 1D advection–diffusion problems with diffusion coefficient $k$. Suppose that the particles are uniformly distributed. When smoothing length is $1 < h/ \Delta x \le 1.5$, the discrete format can be written as:

$$ c_{j}^{n + 1} = c_{j}^{n} + \frac{k\Delta t}{{\Delta x^{2} }}\left( {a_{ - 2} c_{j - 2}^{n} + a_{ - 1} c_{j - 1}^{n} + a_{0} c_{j}^{n} + a_{1} c_{j + 1}^{n} + a_{2} c_{j + 2}^{n} } \right). $$

(37)

Due to anti-symmetry of the kernel gradient, we have $a_{ - 2} = a_{2} ,a_{ - 1} = a_{1} ,a_{0} = - 2\left( {a_{2} + a_{1} } \right),a_{2} \ge 0,a_{1} \ge 0$. According to von Neumann assumption, we know $c_{j}^{n} = e^{ij\hbar \xi } ,\;c_{j - 1}^{n} = e^{{i\left( {j - 1} \right)\hbar \xi }} ,\;c_{j}^{n + 1} = Ge^{ij\hbar \xi }$ ($\hbar = \Delta x$, $i$ is the imaginary number unit). Substituting them into Eq. (37) gives

$$\begin{aligned} &Ge^{ij\hbar \xi } = e^{ij\hbar \xi } + \frac{k\Delta t}{{\Delta x^{2} }}\\ & \quad \left( {a_{ - 2} e^{{i\left( {j - 2} \right)\hbar \xi }} + a_{ - 1} e^{{i\left( {j - 1} \right)\hbar \xi }} + a_{0} e^{ij\hbar \xi } + a_{1} e^{{i\left( {j + 1} \right)\hbar \xi }} + a_{2} e^{{i\left( {j + 2} \right)\hbar \xi }} } \right)\end{aligned} $$

(38)

$$ G = 1 + \frac{k\Delta t}{{\Delta x^{2} }}\left( {a_{ - 2} e^{ - 2i\hbar \xi } + a_{ - 1} e^{ - i\hbar \xi } + a_{0} + a_{1} e^{i\hbar \xi } + a_{2} e^{2i\hbar \xi } } \right) $$

(39)

$$ G = 1 + \frac{k\Delta t}{{\Delta x^{2} }}\left( {a_{2} \left( {e^{ - 2i\hbar \xi } + e^{2i\hbar \xi } - 2} \right) + a_{1} \left( {e^{ - i\hbar \xi } + e^{i\hbar \xi } - 2} \right)} \right) $$

(40)

$$ G = 1 + \frac{k\Delta t}{{\Delta x^{2} }}\left( {a_{2} \left[ {\left( {e^{ - i\hbar \xi } + e^{i\hbar \xi } } \right)^{2} - 4} \right] + 2a_{1} \left( {\cos \left( {\hbar \xi } \right) - 1} \right)} \right) $$

(41)

$$ G = 1 + \frac{k\Delta t}{{\Delta x^{2} }}\left( {4a_{2} \left[ {\left( {\cos \left( {\hbar \xi } \right)} \right)^{2} - 1} \right] + 2a_{1} \left( {\cos \left( {\hbar \xi } \right) - 1} \right)} \right) $$

(42)

where $G$ is the amplification factor.

According to the stability criterion of von Neumann, if the discrete form is stable, the amplification factor must satisfy the condition of $|G| \le 1$, and then Eq. (42) can be transformed into:

$$ - 2 \le \frac{k\Delta t}{{\Delta x^{2} }}\left( {4a_{2} \left[ {\left( {\cos \left( {\hbar \xi } \right)} \right)^{2} - 1} \right] + 2a_{1} \left( {\cos \left( {\hbar \xi } \right) - 1} \right)} \right) \le 0. $$

(43)

With the setting $\tau = \frac{k\Delta t}{{\Delta x^{2} }}$, Eq. (43) becomes:

$$ \tau \le \frac{1}{{2a_{2} \left[ {1 - \left( {\cos \left( {\hbar \xi } \right)} \right)^{2} } \right] + a_{1} \left( {1 - \cos \left( {\hbar \xi } \right)} \right)}} $$

(44)

Set $\eta = 2a_{2} \left[ {1 - \left( {\cos \left( {\hbar \xi } \right)} \right)^{2} } \right] + a_{1} \left( {1 - \cos \left( {\hbar \xi } \right)} \right),\;\left( {\eta \ge 0} \right)$, and then $\tau$ takes the minimum value when $\eta$ takes the maximum value. After simplification, $\eta$ can be obtained as:

$$ \eta = \left( {a_{1} + a_{2} + \frac{{a_{1}^{2} }}{{8a_{2} }}} \right) - 2a_{2} \left[ {\cos \left( {\hbar \xi } \right) + \frac{{a_{1} }}{{4a_{2} }}} \right]^{2} . $$

(45)

If $a_{2} = 0$ or $\frac{{a_{1} }}{{4a_{2} }} > 1$, $\eta$ takes the maximum value when $\cos (\hbar \xi ) = - 1$ and $\tau \le \frac{1}{{2a_{1} }}$. If $\frac{{a_{1} }}{{4a_{2} }} \le 1$, $\eta$ takes the maximum value when $\cos (\hbar \xi ) = - \frac{{a_{1} }}{{4a_{2} }}$ and $\tau \le \frac{1}{{a_{1} + 2a_{2} + \frac{{a_{1}^{2} }}{{8a_{2} }}}}$. For the applied cubic spline kernel function, when $h = \Delta x$, we have $a_{2} = 0$, and when $\Delta x < h \le 1.5\Delta x$, we have $\frac{{a_{1} }}{{4a_{2} }} > 1$ as shown in Fig. 13

Fig. 13

Variation of a₁ and 4a₂ against h/dx in SPH

. Therefore, the stability condition is $\Delta t \le \frac{{\Delta x^{2} }}{{2ka_{1} }}$. Note that when the particles are not evenly distributed, the stability condition will be determined by the minimum particle spacing.

Appendix B: Corrective smoothed particle method (CSPM)

As a correction to the traditional SPH, corrective smoothed particle method (CSPM) is based on Taylor series expansion. Taking 1D case into consideration, firstly, the field function $f(x)$ is expanded by Taylor expansion at particle $i$ as

$$ f\left( x \right) = f\left( {x_{i} } \right) + \left( {x - x_{i} } \right)f^{\prime}\left( {x_{i} } \right) + \frac{{\left( {x - x_{i} } \right)^{2} }}{2!}f^{\prime\prime}\left( {x_{i} } \right) + \cdots $$

(46)

Then, multiplying the kernel function, and integrating in the support domain of particle $i$ yields

$$ \int_{\Omega } {f\left( x \right)W_{i} {\rm d}x} = \int_{\Omega } {f\left( {x_{i} } \right)W_{i} {\rm d}x} + \int_{\Omega } {\left( {x - x_{i} } \right)f^{\prime}\left( {x_{i} } \right)} W_{i} {\rm d}x + \int_{\Omega } {\frac{{\left( {x - x_{i} } \right)^{2} }}{2}f^{\prime\prime}\left( {x_{i} } \right)W_{i} {\rm d}x} + \cdots $$

(47)

If the derivative terms contained in Eq. (47) are ignored, the corrected kernel approximation of the field function $f(x)$ at particle $i$ can be obtained as

$$ f\left( {x_{i} } \right) = \frac{{\int_{\Omega } {f\left( x \right)W_{i} {\rm d}x} }}{{\int_{\Omega } {W_{i} {\rm d}x} }}. $$

(48)

Similarly, if we substitute $\nabla W\left( {x_{i} } \right) = \partial W\left( {x_{i} } \right)/\partial x$ for $W_{i}$ in Eq. (47) and ignore the second-order derivative, then we can obtain the corrected kernel approximation of the first derivative at particle $i$ as

$$ f^{\prime}\left( {x_{i} } \right) = \frac{{\int_{\Omega } {\left( {f\left( x \right) - f\left( {x_{i} } \right)} \right)\nabla W\left( {x_{i} } \right){\rm d}x} }}{{\int_{\Omega } {\left( {x - x_{i} } \right)\nabla W\left( {x_{i} } \right){\rm d}x} }}. $$

(49)

The above ideas can also be extended to higher-order derivative approximations. It can be seen from the above equations that the difference between CSPM and SPH lies in the construction of kernel approximation. In Eqs. (48) and (49), the denominator is the description of regularization properties of smoothing function, while the numerator is the traditional SPH expression. According to the above CSPM results, the corrected particle approximations can be obtained as:

$$ f\left( {x_{i} } \right) = \frac{{\sum\limits_{j = 1}^{N} {\frac{{m_{j} }}{{\rho_{j} }}f_{j} W_{ij} } }}{{\sum\limits_{j = 1}^{N} {\frac{{m_{j} }}{{\rho_{j} }}W_{ij} } }} $$

(50)

$$ f_{x} \left( {x_{i} } \right) = \frac{{\sum\limits_{j = 1}^{N} {\frac{{m_{j} }}{{\rho_{j} }}\left( {f_{j} - f_{i} } \right)\nabla W_{ij} } }}{{\sum\limits_{j = 1}^{N} {\frac{{m_{j} }}{{\rho_{j} }}\left( {x_{j} - x_{i} } \right)\nabla W_{ij} } }}. $$

(51)