CubicSpline2D.hcl File Reference

Cubic spline kernel definition (2D version). More...

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

Functions
float	kernelW (float q)
	The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
float	kernelF (float q)
	The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
float	kernelH (float q)
	The kernel partial derivative with respect to the characteristic height \( \frac{\partial W}{\partial h} \).
float	kernelS (float q)
	An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.

Detailed Description

Cubic spline kernel definition (2D 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

kernelF()

float kernelF

(

float

q

)

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()

float kernelH

(

float

q

)

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()

float kernelS

(

float

q

)

An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.

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

kernelW()

float kernelW

(

float

q

)

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.