Wendland3D.hcl File Reference

Wendland kernel definition (3D version). More...

Macros
#define	_KERNEL_H_INCLUDED_
#define	M_PI   3.14159265359f
#define	iM_PI   0.318309886f

Functions
const float	kernelW (const float q)
	The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
const float	kernelF (const float q)
	The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
const float	kernelH (const float q)
	The kernel partial derivative with respect to the characteristic height \( \frac{\partial W}{\partial h} \).
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.
const float	_Omega (const float a, const float b)
	Helper function to compute the solid angle of a rectangular patch, with a corner placed in the projection of the origin into the patch plane.
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.

Detailed Description

Wendland kernel definition (3D version).

Macro Definition Documentation

_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

\( \pi \) value.

Function Documentation

_Omega()

const float _Omega ( const float a, const float b )

inline

Helper function to compute the solid angle of a rectangular patch, with a corner placed in the projection of the origin into the patch plane.

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Parameters

d	Normal distance of the patch to the origin.
a	Width of the rectangular patch.
b	Height of the rectangular patch.

Returns

Equivalent kernel divergent part

See also

kernelS_D

kernelF()

const float kernelF ( const float q )

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

q	Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \).

Returns

Kernel amount

kernelH()

const float kernelH ( const float q )

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

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 ( const float d, const float t, const float b, const float s )

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	Unsigned normal distance to the wall, \( \vert (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{n_j} \vert \).
t	Unsigned Tangential distance to the boundary element, \( \vert (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{t_j} \vert \).
b	0 in 2D simulations, unsigned distance along the binormal direction in 3D simulations, \( \vert (\mathbf{r_j} - \mathbf{r_i}) \cdot \mathbf{b_j} \vert \).
s	Area of the boundary element, \( \Delta r^2 \).

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 \( h^2 \)

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kernelS_P()

const float kernelS_P ( const float q )

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 )

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.