Lela

Files
file	DivSigma.cl
	Solid particles interaction with the boundary.
file	DivSigmaShepard.cl
	Renormalization of the deformation gradient.
file	EOS.cl
	Inverse EOS equation application.
file	GradU.cl
	Velocity gradient resulting from the interaction with the boundary.
file	GradUShepard.cl
	Renormalization of the deformation gradient.
file	InitP.cl
	Boundary pressure initialization.
file	Motion.cl
	Fixed boundary elements methods.
file	Stress.cl
	Fixed boundary elements methods.
file	ElasticBounce.cl
	The simplest boundary technique to assert the non-tresspasable boundary condition.
file	Corrector.cl
	Improved Euler time integration scheme corrector stage.
file	deltaSPH.cl
	delta-SPH for the linear elasticity module.
file	DivSigma.cl
	Solid particles interactions computation.
file	DivU.cl
	Velocity divergence computation.
file	GradU.cl
	Solid particles deformation gradient computation.
file	Rates.cl
	Velocity and density variation rates computation.
file	Shepard.cl
	Shepard renormalization factor for the Linear elasticity module.
file	Sigma.cl
	Stress tensor computation.
file	Sort.cl
	Sort all the particle variables by the cell indexes.
file	xSPH.cl
	delta-SPH methods, including the correction terms.

Macros
#define	__DR_FACTOR__   1.5f
	The boundary elements effect is restricted to a quadrangular area of \( R \times R \), where \( R = DR_FACTOR \cdot \Delta r \).
#define	__MIN_BOUND_DIST__   0.3f
	The elastic bounce is not tolerating that a particle becomes closer than this distance (multiplied by \( \Delta r \)).
#define	__ELASTIC_FACTOR__   0.0f
	The amount of kinetic energy conserved in the interaction.
#define	EXCLUDED_PARTICLE(index)   imove[index] != 2
	Restrict the aplication to the solid particles (imove=2)
#define	EXCLUDED_PARTICLE(index)   imove[index] != 2
	Restrict the aplication to the solid particles (imove=2)

Functions
__kernel void	entry (const __global int *imove, const __global vec *r, const __global vec *normal, const __global float *rho, const __global float *m, const __global matrix *sigma, __global vec *div_sigma, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells)
	Solid particles interaction with the boundary.
__kernel void	entry (const __global int *imove, const __global float *shepard, __global vec *div_sigma, __global float *div_u, uint N)
	Renormalization of the deformation gradient.
__kernel void	entry (const __global uint *iset, const __global int *imove, const __global float *p, __global float *rho, __constant float *refd, uint N, float cs, float p0)
	Inverse EOS to get the density from the pressure value.
__kernel void	entry (const __global int *imove, const __global vec *r, const __global vec *u, const __global vec *normal, const __global float *m, __global matrix *grad_u, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells)
	Velocity gradient resulting from the interaction with the boundary.
__kernel void	entry (const __global int *imove, const __global float *shepard, __global matrix *grad_u, uint N)
	Renormalization of the deformation gradient.
__kernel void	entry (const __global int *imove, __global float *p, uint N, float p0)
	Pressure initialization.
__kernel void	interpolation (const __global uint *iset, const __global int *imove, const __global vec *r, const __global float *m, const __global float *rho, __global vec *u, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells, uint BImotion_iset)
	Velocity interpolation at the boundary elements.
__kernel void	shepard (const __global uint *iset, const __global int *imove, const __global float *shepard, __global vec *u, uint N, uint BImotion_iset)
	Velocity renormalization.
__kernel void	euler (const __global uint *iset, const __global int *imove, __global vec *r, const __global vec *u, unsigned int N, float dt, uint BImotion_iset)
	Simple Euler time integration of the velocity.
__kernel void	interpolation (const __global uint *iset, const __global int *imove, const __global vec *r, const __global float *m, const __global float *rho, __global float *p, __global matrix *S, const __global uint *icell, const __global uint *ihoc, __constant float *refd, uint N, uivec4 n_cells, vec g, uint BIstress_iset)
	Pressure and stress deviation interpolation at the boundary elements.
__kernel void	shepard (const __global uint *iset, const __global int *imove, const __global float *shepard, __global float *p, __global matrix *S, uint N, uint BIstress_iset)
	Pressure and stress deviation renormalization.
__kernel void	entry (const __global int *imove, const __global vec *r, const __global vec *normal, __global vec *u, __global vec *dudt, __global uint *icell, __global uint *ihoc, uint N, uivec4 n_cells, float dr, float dt)
	Performs the boundary effect on the fluid particles.
__kernel void	entry (__global int *imove, __global matrix *S, __global matrix *dSdt, __global matrix *S_in, __global matrix *dSdt_in, unsigned int N, float dt)
	Improved Euler time integration scheme corrector stage.
__kernel void	entry (const __global int *imove, const __global vec *r, const __global float *rho, const __global float *m, const __global matrix *sigma, __global vec *div_sigma, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells)
	Solid particles interactions computation.
__kernel void	entry (const __global int *imove, const __global float *rho, const __global matrix *grad_u, __global float *div_u, unsigned int N)
	Velocity divergence computation.
__kernel void	entry (const __global int *imove, const __global vec *r, const __global vec *u, const __global float *rho, const __global float *m, __global matrix *grad_u, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells)
	Solid particles deformation gradient computation.
__kernel void	entry (const __global unsigned int *iset, const __global int *imove, const __global vec *div_sigma, const __global float *div_u, const __global matrix *grad_u, const __global matrix *S, __global vec *dudt, __global float *drhodt, __global matrix *dSdt, __constant float *shear_mod, unsigned int N, vec g)
	Velocity and density variation rates computation.
__kernel void	entry (const __global float *p, const __global matrix *S, __global matrix *sigma, unsigned int N)
	Stress tensor computation.
__kernel void	entry (const __global float *p_in, __global float *p, const __global matrix *S_in, __global matrix *S, const __global matrix *dSdt_in, __global matrix *dSdt, const __global unit *id_sorted, unsigned int N)
	Sort all the particle variables by the cell indexes.
__kernel void	interpolation (const __global int *imove, const __global vec *r, const __global vec *u, const __global float *rho, const __global float *m, __global vec *xSPH_u, const __global uint *icell, const __global uint *ihoc, uint N, uivec4 n_cells)
	Velocity interpolation.
__kernel void	xSPH (const __global int *imove, const __global vec *xSPH_u, const __global float *shepard, __global vec *u, unsigned int N, float xSPH_factor)
	xSPH velocity replacement.

Detailed Description

Macro Definition Documentation

__DR_FACTOR__

#define __DR_FACTOR__   1.5f

The boundary elements effect is restricted to a quadrangular area of \( R \times R \), where \( R = DR_FACTOR \cdot \Delta r \).

__ELASTIC_FACTOR__

#define __ELASTIC_FACTOR__   0.0f

The amount of kinetic energy conserved in the interaction.

A factor of 1 imply that the velocity of the particle will be preserved (except for the direction), while a factor of 0 imply that the particle will loss all its normal to the boundary velocity.

The tangential velocity is not affected.

__MIN_BOUND_DIST__

#define __MIN_BOUND_DIST__   0.3f

The elastic bounce is not tolerating that a particle becomes closer than this distance (multiplied by \( \Delta r \)).

EXCLUDED_PARTICLE [1/2]

#define EXCLUDED_PARTICLE

(

index

)

imove[index] != 2

Restrict the aplication to the solid particles (imove=2)

EXCLUDED_PARTICLE [2/2]

#define EXCLUDED_PARTICLE

(

index

)

imove[index] != 2

Restrict the aplication to the solid particles (imove=2)

Function Documentation

entry() [1/14]

__kernel void entry	(	__global int *	imove,
		__global matrix *	S,
		__global matrix *	dSdt,
		__global matrix *	S_in,
		__global matrix *	dSdt_in,
		unsigned int	N,
		float	dt
	)

Improved Euler time integration scheme corrector stage.

Time integration is based in the following quasi-second order Predictor-Corrector integration scheme:

\( \S_{n+1} = \S_{n} + \Delta t \left. \frac{\mathrm{d}S}{\mathrm{d}t} \right\vert_{n+1/2} + \frac{\Delta t}{2} \left( \left. \frac{\mathrm{d}S}{\mathrm{d}t} \right\vert_{n + 1/2} - \left. \frac{\mathrm{d}S}{\mathrm{d}t} \right\vert_{n - 1/2} \right) \)

Parameters

imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
iset	Set of particles index.
S	Deviatory stress \( S_{n+1} \).
dSdt	Deviatory stress rate of change \( \left. \frac{d S}{d t} \right\vert_{n+1} \).
S_in	Deviatory stress \( S_{n+1/2} \).
dSdt_in	Deviatory stress rate of change \( \left. \frac{d S}{d t} \right\vert_{n+1/2} \).
N	Number of particles.
dt	Time step \( \Delta t \).

See also

lela/Predictor.cl

entry() [2/14]

__kernel void entry	(	const __global float *	p,
		const __global matrix *	S,
		__global matrix *	sigma,
		unsigned int	N
	)

Stress tensor computation.

\[ \sigma = p \mathcal{I} + S \]

See also

https://en.wikipedia.org/wiki/Linear_elasticity

https://en.wikipedia.org/wiki/Hooke's_law

Parameters

p	Pressure \( p \).
S	Deviatory stress \( S \).
sigma	Stress tensor \( \sigma \).
N	Number of particles.

entry() [3/14]

__kernel void entry	(	const __global float *	p_in,
		__global float *	p,
		const __global matrix *	S_in,
		__global matrix *	S,
		const __global matrix *	dSdt_in,
		__global matrix *	dSdt,
		const __global unit *	id_sorted,
		unsigned int	N
	)

Sort all the particle variables by the cell indexes.

Parameters

p_in	Unsorted pressure \( p \).
p	Sorted pressure \( p \).
S_in	Unsorted Deviatory stress \( S \).
S	Sorted Deviatory stress \( S \).
dSdt_in	Unsorted Deviatory stress rate of change \( \frac{d S}{d t} \).
dSdt	Sorted Deviatory stress rate of change \( \frac{d S}{d t} \). \( \left. \frac{d S}{d t} \right\vert_{n+1} \).
id_sorted	Permutations list from the unsorted space to the sorted one.
N	Number of particles.

entry() [4/14]

__kernel void entry	(	const __global int *	imove,
		__global float *	p,
		uint	N,
		float	p0
	)

Pressure initialization.

Since the pressure field is resulting from the density field (applying the EOS), letting free the pressure at the boundary may be dangerous due to the value must be unassigned. Therefore the pressure field will be initialized as the background pressure, letting the user to don't sdpecifically assign a pressure value without crashing the simulation.

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
p	Pressure \( p \).
p0	Background pressure \( p_0 \).
N	Total number of particles and boundary elements.
BImotion_iset	Set of particles affected

entry() [5/14]

__kernel void entry	(	const __global int *	imove,
		const __global float *	rho,
		const __global matrix *	grad_u,
		__global float *	div_u,
		unsigned int	N
	)

Velocity divergence computation.

Since we already computed the gradient of the velocity, we can get the divergence as the sum of the diagonal components:

\[ \nabla \cdot \mathbf{u} = \sum_i^d \left(\nabla \mathbf{u}\right)_ii \]

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors (ignored by this preset). imove < 0 for boundary elements/particles.
rho	Density \( \rho \).
grad_u	Gradient of the deformation \( \nabla \mathbf{r}^{*} \).
div_u	Velocity divergence \( \rho \nabla \cdot \mathbf{u} \).
N	Number of particles.

entry() [6/14]

__kernel void entry	(	const __global int *	imove,
		const __global float *	shepard,
		__global matrix *	grad_u,
		uint	N
	)

Renormalization of the deformation gradient.

The main drawback of the boundary integrals formulation is the requirement of the renormalization of the computed differentiqal operators, which is destroying several conservation properties.

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
shepard	Shepard term \( \gamma(\mathbf{x}) = \int_{\Omega} W(\mathbf{y} - \mathbf{x}) \mathrm{d}\mathbf{x} \).
grad_u	Gradient of the velocity \( \nabla \mathbf{u} \).
N	Total number of particles and boundary elements.

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entry() [7/14]

__kernel void entry	(	const __global int *	imove,
		const __global float *	shepard,
		__global vec *	div_sigma,
		__global float *	div_u,
		uint	N
	)

Renormalization of the deformation gradient.

The main drawback of the boundary integrals formulation is the requirement of the renormalization of the computed differentiqal operators, which is destroying several conservation properties.

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
shepard	Shepard term \( \gamma(\mathbf{x}) = \int_{\Omega} W(\mathbf{y} - \mathbf{x}) \mathrm{d}\mathbf{x} \).
div_sigma	Divergence of the stress tensor \( \frac{\nabla \cdot \sigma}{rho} \).
div_u	Velocity divergence \( \rho \nabla \cdot \mathbf{u} \).
N	Total number of particles and boundary elements.

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entry() [8/14]

__kernel void entry	(	const __global int *	imove,
		const __global vec *	r,
		const __global float *	rho,
		const __global float *	m,
		const __global matrix *	sigma,
		__global vec *	div_sigma,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells
	)

Solid particles interactions computation.

Compute the differential operators involved in the numerical scheme, taking into account just the solid-solid interactions.

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors (ignored by this module). imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
rho	Density \( \rho \).
m	Mass \( m \).
sigma	Stress tensor \( \sigma \).
div_sigma	Divergence of the stress tensor \( \frac{\nabla \cdot \sigma}{rho} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction.

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entry() [9/14]

__kernel void entry	(	const __global int *	imove,
		const __global vec *	r,
		const __global vec *	normal,
		__global vec *	u,
		__global vec *	dudt,
		__global uint *	icell,
		__global uint *	ihoc,
		uint	N,
		uivec4	n_cells,
		float	dr,
		float	dt
	)

Performs the boundary effect on the fluid particles.

Parameters

imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
normal	Normal \( \mathbf{n} \).
u	Velocity \( \mathbf{u} \).
dudt	Velocity rate of change \( \left. \frac{d \mathbf{u}}{d t} \right\vert_{n+1} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction
dr	Distance between particle \( \Delta r \).
dt	Time step \( \Delta t \).

entry() [10/14]

__kernel void entry	(	const __global int *	imove,
		const __global vec *	r,
		const __global vec *	normal,
		const __global float *	rho,
		const __global float *	m,
		const __global matrix *	sigma,
		__global vec *	div_sigma,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells
	)

Solid particles interaction with the boundary.

Compute the differential operators involved in the numerical scheme, taking into account just the solid-solid interactions.

Parameters

imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
normal	Normal \( \mathbf{n} \).
rho	Density \( \rho \).
m	Area of the boundary element \( s \).
sigma	Stress tensor \( \sigma \).
div_sigma	Divergence of the stress tensor \( \frac{\nabla \cdot \sigma}{rho} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction.

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entry() [11/14]

__kernel void entry	(	const __global int *	imove,
		const __global vec *	r,
		const __global vec *	u,
		const __global float *	rho,
		const __global float *	m,
		__global matrix *	grad_u,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells
	)

Solid particles deformation gradient computation.

Compute the gradient of the deformation vector:

\[ \nabla \mathbf{r}^{*} = \mathbf{r}^{*} \otimes \nabla \]

See also

https://en.wikipedia.org/wiki/Matrix_calculus

https://en.wikipedia.org/wiki/Outer_product

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors (ignored by this preset). imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
u	Velocity \( \mathbf{u} \).
rho	Density \( \rho \).
m	Mass \( m \).
grad_u	Gradient of the velocity \( \nabla \mathbf{u} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction

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entry() [12/14]

__kernel void entry	(	const __global int *	imove,
		const __global vec *	r,
		const __global vec *	u,
		const __global vec *	normal,
		const __global float *	m,
		__global matrix *	grad_u,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells
	)

Velocity gradient resulting from the interaction with the boundary.

Compute the gradient of the velocity:

\[ \nabla \mathbf{u} = \mathbf{u} \otimes \nabla \]

See also

https://en.wikipedia.org/wiki/Matrix_calculus

https://en.wikipedia.org/wiki/Outer_product

Parameters

imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors (ignored by this preset). imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
u	Velocity \( \mathbf{u} \).
normal	Normal \( \mathbf{n} \).
m	Area of the boundary element \( s \).
grad_u	Gradient of the velocity \( \nabla \mathbf{u} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction

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entry() [13/14]

__kernel void entry	(	const __global uint *	iset,
		const __global int *	imove,
		const __global float *	p,
		__global float *	rho,
		__constant float *	refd,
		uint	N,
		float	cs,
		float	p0
	)

Inverse EOS to get the density from the pressure value.

Parameters

iset	Set of particles index.
imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
p	Pressure \( p \).
rho	Density \( \rho \).
refd	Density of reference of the fluid \( \rho_0 \).
N	Total number of particles and boundary elements.
cs	Speed of sound \( c_s \).
p0	Background pressure \( p_0 \).

entry() [14/14]

__kernel void entry	(	const __global unsigned int *	iset,
		const __global int *	imove,
		const __global vec *	div_sigma,
		const __global float *	div_u,
		const __global matrix *	grad_u,
		const __global matrix *	S,
		__global vec *	dudt,
		__global float *	drhodt,
		__global matrix *	dSdt,
		__constant float *	shear_mod,
		unsigned int	N,
		vec	g
	)

Velocity and density variation rates computation.

The mass conservation and momentum equations are applied from the already computed differential operators:

• \( \frac{\mathrm{d} \mathbf{u}}{\mathrm{d} t} = - \frac{\nabla \cdot \sigma}{rho} + \mathbf{g}\)
• \( \frac{\mathrm{d} \rho}{\mathrm{d} t} = - \rho \nabla \cdot \mathbf{u}\)

Parameters

iset	Set of particles index.
imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors (ignored by this preset). imove < 0 for boundary elements/particles.
div_sigma	Divergence of the stress tensor \( \frac{\nabla \cdot \sigma}{rho} \).
div_u	Velocity divergence \( \rho \nabla \cdot \mathbf{u} \).
grad_u	Velocity gradient \( \nabla \mathbf{u} \).
S	Deviatory stress \( S \).
dudt	Velocity rate of change \( \frac{d \mathbf{u}}{d t} \).
drhodt	Density rate of change \( \frac{d \rho}{d t} \).
dSdt	Deviatory stress rate of change \( \frac{d S}{d t} \).
shear_mod	Shear modulus \( \mu \).
N	Number of particles.
g	Gravity acceleration \( \mathbf{g} \).

euler()

__kernel void euler	(	const __global uint *	iset,
		const __global int *	imove,
		__global vec *	r,
		const __global vec *	u,
		unsigned int	N,
		float	dt,
		uint	BImotion_iset
	)

Simple Euler time integration of the velocity.

Since the velocity is resulting from the interpolation of the solid particle, it does not make any sense to integrate it with an improved Euler scheme.

\( \mathbf{u}_{n+1} = \mathbf{u}_{n} + \Delta t \left. \frac{\mathrm{d} \mathbf{u}}{\mathrm{d}t} \right\vert_{n+1/2} \right) \)

Parameters

iset	Set of particles index.
imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
u	Velocity \( \mathbf{u} \).
N	Number of particles.
dt	Time step \( \Delta t \).
BImotion_iset	Set of particles affected

interpolation() [1/3]

__kernel void interpolation	(	const __global int *	imove,
		const __global vec *	r,
		const __global vec *	u,
		const __global float *	rho,
		const __global float *	m,
		__global vec *	xSPH_u,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells
	)

Velocity interpolation.

Parameters

imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
u	Velocity \( \mathbf{u} \).
rho	Density \( \rho \).
m	Mass \( m \).
xSPH_u	Interpolated velocity (Shepard renormalization is not applied yet).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction

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interpolation() [2/3]

__kernel void interpolation	(	const __global uint *	iset,
		const __global int *	imove,
		const __global vec *	r,
		const __global float *	m,
		const __global float *	rho,
		__global float *	p,
		__global matrix *	S,
		const __global uint *	icell,
		const __global uint *	ihoc,
		__constant float *	refd,
		uint	N,
		uivec4	n_cells,
		vec	g,
		uint	BIstress_iset
	)

Pressure and stress deviation interpolation at the boundary elements.

The values are computed using just the fluid information. The resulting interpolated values are not renormalized yet.

Just the elements with the flag imove = -3 are considered boundary elements.

Parameters

iset	Set of particles index.
imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
m	Mass \( m \).
rho	Density \( \rho \).
p	Pressure \( p \).
S	Deviatory stress \( S \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
refd	Density of reference of the fluid \( \rho_0 \).
N	Number of particles.
n_cells	Number of cells in each direction
g	Gravity acceleration \( \mathbf{g} \).
BIstress_iset	Set of particles affected

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interpolation() [3/3]

__kernel void interpolation	(	const __global uint *	iset,
		const __global int *	imove,
		const __global vec *	r,
		const __global float *	m,
		const __global float *	rho,
		__global vec *	u,
		const __global uint *	icell,
		const __global uint *	ihoc,
		uint	N,
		uivec4	n_cells,
		uint	BImotion_iset
	)

Velocity interpolation at the boundary elements.

The values are computed using just the solid information. The resulting interpolated values are not renormalized yet.

Just the elements with the flag imove = -3 are considered boundary elements.

Parameters

iset	Set of particles index.
imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
r	Position \( \mathbf{r} \).
m	Mass \( m \).
rho	Density \( \rho \).
u	Velocity \( \mathbf{u} \).
icell	Cell where each particle is located.
ihoc	Head of chain for each cell (first particle found).
N	Number of particles.
n_cells	Number of cells in each direction
BImotion_iset	Set of particles affected

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shepard() [1/2]

__kernel void shepard	(	const __global uint *	iset,
		const __global int *	imove,
		const __global float *	shepard,
		__global float *	p,
		__global matrix *	S,
		uint	N,
		uint	BIstress_iset
	)

Pressure and stress deviation renormalization.

Parameters

iset	Set of particles index.
imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
shepard	Shepard term \( \gamma(\mathbf{x}) = \int_{\Omega} W(\mathbf{y} - \mathbf{x}) \mathrm{d}\mathbf{x} \).
p	Pressure \( p \).
S	Deviatory stress \( S \).
N	Total number of particles and boundary elements.
BIstress_iset	Set of particles affected

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shepard() [2/2]

__kernel void shepard	(	const __global uint *	iset,
		const __global int *	imove,
		const __global float *	shepard,
		__global vec *	u,
		uint	N,
		uint	BImotion_iset
	)

Velocity renormalization.

Parameters

iset	Set of particles index.
imove	Moving flags. imove = 2 for regular solid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
shepard	Shepard term \( \gamma(\mathbf{x}) = \int_{\Omega} W(\mathbf{y} - \mathbf{x}) \mathrm{d}\mathbf{x} \).
u	Velocity \( \mathbf{u} \).
N	Total number of particles and boundary elements.
BImotion_iset	Set of particles affected

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xSPH()

__kernel void xSPH	(	const __global int *	imove,
		const __global vec *	xSPH_u,
		const __global float *	shepard,
		__global vec *	u,
		unsigned int	N,
		float	xSPH_factor
	)

xSPH velocity replacement.

Parameters

imove	Moving flags. imove > 0 for regular fluid particles. imove = 0 for sensors. imove < 0 for boundary elements/particles.
xSPH_u	Interpolated velocity (Shepard renormalization is not applied yet).
shepard	Shepard term \( \gamma(\mathbf{x}) = \int_{\Omega} W(\mathbf{y} - \mathbf{x}) \mathrm{d}\mathbf{x} \).
u	Velocity \( \mathbf{u} \).
N	Number of particles.
xSPH_factor	Ratio between the instantaneous velocity, and the interpolated one.

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