Wendland kernel definition (2D version).
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| const float | kernelW (const float q) |
| | The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
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| const float | kernelF (const float q) |
| | The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
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| const float | kernelH (const float q) |
| | The kernel partial derivative with respect to the characteristic height \( \frac{\partial W}{\partial h} \).
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| const float | kernelS_P (const float q) |
| | An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.
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| const float | kernelS_D (const float d, const float t, const float b, const float s) |
| | An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.
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Wendland kernel definition (2D version).
◆ _KERNEL_H_INCLUDED_
| #define _KERNEL_H_INCLUDED_ |
◆ iM_PI
| #define iM_PI 0.318309886f |
\( \frac{1}{\pi} \) value.
◆ M_PI
| #define M_PI 3.14159265359f |
◆ kernelF()
| const float kernelF |
( |
const float |
q | ) |
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inline |
The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
The factor \( F \) is defined such that \( \nabla W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) =
\frac{\mathbf{r_j} - \mathbf{r_i}}{h} \cdot
F \left(\mathbf{r_j} - \mathbf{r_i}; h\right)
\).
- Parameters
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| q | Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \). |
- Returns
- Kernel gradient factor value
◆ kernelH()
| const float kernelH |
( |
const float |
q | ) |
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inline |
The kernel partial derivative with respect to the characteristic height \( \frac{\partial W}{\partial h} \).
The result returned by this function should be multiplied by \( \frac{1}{h^{d + 1}} \), where d is 2,3 for 2D and 3D respectively.
- Parameters
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| q | Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \). |
- Returns
- Kernel partial derivative factor
- Note
- This function is useful for variable resolution (non-constant kernel height)
- See also
- Iason Zisis, Bas van der Linden, Christina Giannopapa and Barry Koren, On the derivation of SPH schemes for shocks through inhomogeneous media. Int. Jnl. of Multiphysics (2015).
◆ kernelS_D()
| const float kernelS_D |
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const float |
d, |
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const float |
t, |
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const float |
b, |
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const float |
s |
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) |
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inline |
An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.
For practical purposes, the kernel computation is split in 2 parts: The polynomial part, where trucation errors are acceptable, and the divergent part, which requires an analytical solution. This function computes the divergent part.
The kernel is defined as follows: \( \hat{W} \left(\rho; h\right) =
\frac{1}{\rho^d} \int \rho^{d - 1} W \left(\rho; h\right) d\rho \)
- Parameters
-
| d | Normal distance to the wall, \( (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{n_j} \). |
| t | Tangential distance to the boundary element, \( (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{t_j} \). |
| b | 0 in 2D simulations, distance along the normal direction in 3D simulations, \( (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{b_j} \). |
| s | Area of the boundary element, \( 2 * \Delta r \). |
- Returns
- Equivalent kernel divergent part
- See also
- kernelS_P
- Warning
- Due to the analytical nature of the solution, this kernel should not be multiplied by the element area, nor divided by \( h^2 \)
◆ kernelS_P()
| const float kernelS_P |
( |
const float |
q | ) |
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inline |
An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.
For practical purposes, the kernel computation is split in 2 parts: The polynomial part, where trucation errors are acceptable, and the divergent part, which requires an analytical solution. This function computes the polynomial part.
The kernel is defined as follows: \( \hat{W} \left(\rho; h\right) =
\frac{1}{\rho^d} \int \rho^{d - 1} W \left(\rho; h\right) d\rho \)
- Parameters
-
| q | Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \). |
- Returns
- Equivalent kernel polynomial part
- See also
- kernelS_D
◆ kernelW()
| const float kernelW |
( |
const float |
q | ) |
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inline |
The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
- Parameters
-
| q | Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \). |
- Returns
- Kernel value.
