SPH Discretization

In SPH, the liquid is discretized as a collection of particles. The physical properties of the particles are computed by taking into account the contributions from neighboring particles. The influence of particles on each other within a specified radius is defined by the use of a kernel function.

Points Equations

The solution domain $Ω$ , that is, the SPH liquid phase is subdivided into a set of points, forming a partition $P (Ω)$ . Each point $i$ in the partition has an associated volume $V_{i}, i \in P (Ω)$ . The sum of these volumes is equal to the volume of the solution domain.

The position of each point is calculated as:

图 1. EQUATION_DISPLAY

\frac{{d r}_{i}}{d t} = v_{i}

(1102)

The volume of each point is given by:

图 2. EQUATION_DISPLAY

\frac{{d V}_{i}}{d t} = V_{i} \nabla \cdot v_{i} = 0

(1103)

Continuity Equation: The conservation of mass for point $i$ is is given by:
图 3. EQUATION_DISPLAY

$\frac{{d V}_{i} ρ_{i}}{d t} = \frac{{d m}_{i}}{d t} = 0$

(1104)
Since each point maintains a constant mass over time, these points can be identified and referred to as particles.

Momentum Equation: The conservation of momentum equation for an SPH particle of mass $m_{i}$ is given by:

图 4. EQUATION_DISPLAY

$m_{i} \frac{{d v}_{i}}{d t} + V_{i} \nabla \cdot (p_{i} I) - V_{i} \nabla \cdot T_{i} = V_{i} f_{b, i}$

(1105)

SPH Continuous Approximation

A continuous arbitrary scalar field

ϕ

can be evaluated by:

图 5. EQUATION_DISPLAY

ϕ (r) = \int_{Ω} ϕ (r^{'}) δ (r - r^{'}) d r^{'}

(1106)

where:

$ϕ (r^{'})$ are values of the scalar field at neighboring points $r^{'}$ .
$δ$ is the Dirac distribution.
$Ω$ represents the solution domain.

The continuous SPH approximation $Π_{c}^{R}$ of the scalar field $ϕ$ is given by:

图 6. EQUATION_DISPLAY

Π_{c}^{R} (ϕ) (r) = \int_{Ω} ϕ (r^{'}) W (r - r^{'}, R) d r^{'}

(1107)

where $W$ is the isotropic kernel function with a smoothing radius $R$ . The integral of $W$ over the solution domain is equal to 1. The order of this approximation is $O (R^{2})$ .

The derivative for the SPH continuous approximation is calculated as:

图 7. EQUATION_DISPLAY

\nabla Π_{c}^{R} (ϕ) (r) = Π_{c}^{R} (\nabla ϕ) (r) = \int_{Ω} ϕ (r^{'}) \nabla_{r} W (r - r^{'}, R) d r^{'}

(1108)

SPH Discrete Approximation

The SPH discrete approximation

Π_{c}^{R}

of the scalar field

ϕ

is computed from a Riemann summation over the particles:

图 8. EQUATION_DISPLAY

Π_{d}^{R} (ϕ) (r) = \sum_{j \in P (Ω)} V_{j} ϕ (r_{j}) W (r - r_{j}, R)

(1109)

A normalized (Shepard) approximation is used for post-processing. The interpolation of the field

ϕ

at position

r

in the neighborhood

Ω

is computed as:

图 9. EQUATION_DISPLAY

Π_{d}^{R *} (ϕ) (r_{j}) = \frac{\sum_{j \in P (Ω)} V_{j} ϕ_{j} W (r - r_{j}, R_{j})}{\sum_{j \in P (Ω)} V_{j} W (r - r_{j}, R_{j})}

(1110)

The derivative for the SPH discrete approximation is calculated as:

图 10. EQUATION_DISPLAY

\nabla Π_{d}^{R} (ϕ) (r) = \sum_{j \in P (Ω)} V_{j} ϕ (r_{j}) \nabla W (r - r_{j}, R)

(1111)

To aid with convergence and mass conservation, modified derivation operators are introduced. For example, the gradient

G^{+}

of the scalar field

ϕ

and the divergence

D^{-}

of the vector field

Φ

are given as:

图 11. EQUATION_DISPLAY

G_{i}^{+} (ϕ) (r_{i}) = \sum_{j \in P (Ω)} V_{j} (ϕ (r_{j}) + ϕ (r_{i})) \nabla W (r_{i} - r_{j}, R_{i})

(1112)

图 12. EQUATION_DISPLAY

D_{i}^{-} (Φ) (r_{i}) = \sum_{j \in P (Ω)} V_{j} (Φ (r_{j}) - Φ (r_{i})) \cdot \nabla W (r_{i} - r_{j}, R_{i})

(1113)

The summation is carried out over all particles, $j$ that surround particle $i$ . In general SPH notations, these are simplified as:

图 13. EQUATION_DISPLAY

G_{i}^{+} (ϕ) = \sum_{j} V_{j} (ϕ_{j} + ϕ_{i}) \nabla W_{i j}

(1114)

图 14. EQUATION_DISPLAY

D_{i}^{-} (Φ) = \sum_{j} V_{j} (Φ_{j} - Φ_{i}) \cdot \nabla W_{i j}

(1115)

where

W_{i j} = W (r_{i} - r_{j}, R_{i})

The discrete Laplacian

L_{i}^{M o r r i s}

of the scalar field

ϕ

, [270], is given as:

图 15. EQUATION_DISPLAY

L_{i}^{M o r r i s} (ϕ) = \sum_{j} V_{j} 2 (ϕ_{i} - ϕ_{j}) \frac{r_{i} - r_{j}}{{‖ r_{i} - r_{j} ‖}^{2}} \cdot \nabla W_{i j}

(1116)