The Helmholtz decomposition is intended to contain "only mathematical" functions and procedures to compute the Helmholtz decomposition of a generic field. More...

Detailed Description

The Helmholtz decomposition is intended to contain "only mathematical" functions and procedures to compute the Helmholtz decomposition of a generic field.

Given a generic vector field F, it is possibile to define a curl-free component div(phi) and a divergence free component curl(A) such that \( \mathbf{F} = -div(phi) + curl(A) \), where

\[ \phi = \frac{1}{4\pi} \left( \int \frac{\nabla \cdot \mathbf{F}}{|\mathbf{r} - \mathbf{r_1}|} \,dr_1 - \oint \mathbf{n} \cdot \frac{\mathbf{F}}{|\mathbf{r} - \mathbf{r_1}|} \,dS_1 \right) \]

\[\mathbf{A} = \frac{1}{4\pi} \left( \int \frac{\nabla \times \mathbf{F}}{|\mathbf{r} - \mathbf{r_1}|} \,dr_1 - \oint \mathbf{n} \times \frac{\mathbf{F}}{|\mathbf{r} - \mathbf{r_1}|} \,dS_1 \right) \]

We start from a field defined on the system grid, i.e. the grid used for the Maxwell simulation.

VECTOR and SCALAR POTENTIAL

If the user asks to compute the vector potential A or the scalar potential phi, then they are computed on the system grid. All the points up to system_gridnp will have physical meaning

TRANSVERSE and LONGITUDINAL FIELDS

If the user asks to compute the transverse or the longitudinal field, then they are computed on the system grid. However, the points the points contained in a layer of the width of the stencil between the innermost layer of points and last row of points in the grid will be set to zero. This is a visual representation for stencil = 2: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 To understand this, one has to remember that when we get the scalar or vector potential, we know the solution only up to system_gridnp. When we take the final divergence or curl, also the points from system_gridnp+1 to system_gridnp_part matter (due to the way in which the finite differences are computed). Since potentials are set to zero in this non-physical region, there might be spikes at the border. Therefore, we set to zero the points of the fields in the aforementioned mask, so that we do not have spikes

SURFACE CORRECTION

In the equations above, there are two integrals, the volume one (always computed) and the surface one. The surface integral is important when the total field is significantly different from zero at the boundaries of the box

Data Types
type	helmholtz_decomposition_t

Functions/Subroutines
subroutine	helmholtz_decomposition_init (this, namespace, sys_grid, system_mc, space)
	Initialize Helmholtz decomposition object. More...

subroutine	get_indices_from_mask (np, mask, indices)

subroutine	helmholtz_finalize (this)

subroutine	helmholtz_visualize_boxes (this, namespace, space)
	Visualise boxes for use in Helmholtz Decomposition. More...

subroutine	zapply_inner_stencil_mask (this, field)

subroutine	zsubtract_average_from_inner_stencil (this, field)

subroutine	zget_vector_potential (this, namespace, vector_potential, total_field, trans_field, apply_boundary)

subroutine	zget_trans_field (this, namespace, transverse_field, total_field, vector_potential, apply_boundary)

subroutine	zcompute_surface_correction_trans_field (this, namespace, field, surface_correction)

subroutine	zget_scalar_potential (this, namespace, scalar_potential, total_field, apply_boundary)

subroutine	zget_long_field (this, namespace, longitudinal_field, total_field, scalar_potential, apply_boundary)

subroutine	zcompute_surface_correction_long_field (this, namespace, field, surface_correction)

subroutine	dapply_inner_stencil_mask (this, field)

subroutine	dsubtract_average_from_inner_stencil (this, field)

subroutine	dget_vector_potential (this, namespace, vector_potential, total_field, trans_field, apply_boundary)

subroutine	dget_trans_field (this, namespace, transverse_field, total_field, vector_potential, apply_boundary)

subroutine	dcompute_surface_correction_trans_field (this, namespace, field, surface_correction)

subroutine	dget_scalar_potential (this, namespace, scalar_potential, total_field, apply_boundary)

subroutine	dget_long_field (this, namespace, longitudinal_field, total_field, scalar_potential, apply_boundary)

subroutine	dcompute_surface_correction_long_field (this, namespace, field, surface_correction)

Function/Subroutine Documentation

◆ helmholtz_decomposition_init()

subroutine helmholtz_decomposition_m::helmholtz_decomposition_init	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		type(grid_t), intent(in), target	sys_grid,
		type(multicomm_t), intent(in)	system_mc,
		class(space_t), intent(in)	space
	)

private

Initialize Helmholtz decomposition object.

Parameters

[in,out]	this	Helmholtz decomposition type to initialize
[in]	sys_grid	Grid of the system calling Helmholtz
[in]	system_mc	MultiCommunicator of the system calling Helmholtz
[in]	space	Space of the system calling Helmholtz

Definition at line 225 of file helmholtz_decomposition.F90.

◆ get_indices_from_mask()

subroutine helmholtz_decomposition_m::get_indices_from_mask	(	integer, intent(in)	np,
		logical, dimension(:), intent(in)	mask,
		integer, dimension(:), intent(out)	indices
	)

private

Definition at line 335 of file helmholtz_decomposition.F90.

◆ helmholtz_finalize()

subroutine helmholtz_decomposition_m::helmholtz_finalize ( type(helmholtz_decomposition_t), intent(inout) this )

private

Definition at line 354 of file helmholtz_decomposition.F90.

◆ helmholtz_visualize_boxes()

subroutine helmholtz_decomposition_m::helmholtz_visualize_boxes	(	type(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		class(space_t), intent(in)	space
	)

private

Visualise boxes for use in Helmholtz Decomposition.

Definition at line 367 of file helmholtz_decomposition.F90.

◆ zapply_inner_stencil_mask()

subroutine helmholtz_decomposition_m::zapply_inner_stencil_mask	(	class(helmholtz_decomposition_t), intent(inout)	this,
		complex(real64), dimension(:,:), intent(out)	field
	)

private

Definition at line 480 of file helmholtz_decomposition.F90.

◆ zsubtract_average_from_inner_stencil()

subroutine helmholtz_decomposition_m::zsubtract_average_from_inner_stencil	(	class(helmholtz_decomposition_t), intent(inout)	this,
		complex(real64), dimension(:,:), intent(inout)	field
	)

private

Definition at line 497 of file helmholtz_decomposition.F90.

◆ zget_vector_potential()

subroutine helmholtz_decomposition_m::zget_vector_potential	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:,:), intent(out), contiguous	vector_potential,
		complex(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		complex(real64), dimension(:,:), intent(inout), optional, contiguous	trans_field,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	vector_potential	Vector potential obtained from the total_field. The field will be defined on the grid of the Helmholtz type (thisgrid), UP UNTIL THISGRIDNP vector_potential(1:thisgridnp_part, 1:thisgridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	trans_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in]	apply_boundary	Should the curl apply boundary conditions?

Definition at line 526 of file helmholtz_decomposition.F90.

◆ zget_trans_field()

subroutine helmholtz_decomposition_m::zget_trans_field	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:,:), intent(out), contiguous	transverse_field,
		complex(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		complex(real64), dimension(:,:), intent(inout), optional, contiguous	vector_potential,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	transverse_field	Transverse field obtained from the total_field. It will be defined on the inner_grid of the Helmholtz type (thisinner_grid), UP UNTIL INNER_GRNP transverse_field(1:thisinner_gridnp_part, 1:thisinner_gridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	vector_potential	Vector potential from which trans field whould be computed
[in]	apply_boundary	Should the curl apply boundary conditions?

Definition at line 690 of file helmholtz_decomposition.F90.

◆ zcompute_surface_correction_trans_field()

subroutine helmholtz_decomposition_m::zcompute_surface_correction_trans_field	(	class(helmholtz_decomposition_t), intent(in)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:,:), intent(in)	field,
		complex(real64), dimension(:,:), intent(out)	surface_correction
	)

private

Parameters

[in] this Helmholtz object

Definition at line 735 of file helmholtz_decomposition.F90.

◆ zget_scalar_potential()

subroutine helmholtz_decomposition_m::zget_scalar_potential	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:), intent(out), contiguous	scalar_potential,
		complex(real64), dimension(:,:), intent(inout), contiguous	total_field,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	scalar_potential	Vector potential obtained from the total_field. It will be defined on the grid of the Helmholtz type (thisgrid), UP UNTIL THISGRIDNP: scalar_potential(1:thisgridnp_part)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART: total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in]	apply_boundary	Should the divergence apply boundary conditions?

Definition at line 785 of file helmholtz_decomposition.F90.

◆ zget_long_field()

subroutine helmholtz_decomposition_m::zget_long_field	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:,:), intent(out), contiguous	longitudinal_field,
		complex(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		complex(real64), dimension(:), intent(inout), optional, contiguous	scalar_potential,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	longitudinal_field	Transverse field obtained from the total_field. It will be defined on the inner_grid of the Helmholtz type (thisinner_grid), UP UNTIL INNER_GRNP longitudinal_field(1:thisinner_gridnp_part, 1:thisinner_gridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr) UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	scalar_potential	Scalar potential from which long field whould be computed
[in]	apply_boundary	Should the divergence apply boundary conditions?

Definition at line 833 of file helmholtz_decomposition.F90.

◆ zcompute_surface_correction_long_field()

subroutine helmholtz_decomposition_m::zcompute_surface_correction_long_field	(	class(helmholtz_decomposition_t), intent(in)	this,
		type(namespace_t), intent(in)	namespace,
		complex(real64), dimension(:,:), intent(in)	field,
		complex(real64), dimension(:), intent(out)	surface_correction
	)

private

Parameters

[in]	this	Helmholtz object
[in]	field	(1:thissys_gridnp, 1:thissys_gridboxdim)
[out]	surface_correction	(1:thissys_gridnp)

Definition at line 880 of file helmholtz_decomposition.F90.

◆ dapply_inner_stencil_mask()

subroutine helmholtz_decomposition_m::dapply_inner_stencil_mask	(	class(helmholtz_decomposition_t), intent(inout)	this,
		real(real64), dimension(:,:), intent(out)	field
	)

private

Definition at line 1004 of file helmholtz_decomposition.F90.

◆ dsubtract_average_from_inner_stencil()

subroutine helmholtz_decomposition_m::dsubtract_average_from_inner_stencil	(	class(helmholtz_decomposition_t), intent(inout)	this,
		real(real64), dimension(:,:), intent(inout)	field
	)

private

Definition at line 1021 of file helmholtz_decomposition.F90.

◆ dget_vector_potential()

subroutine helmholtz_decomposition_m::dget_vector_potential	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:,:), intent(out), contiguous	vector_potential,
		real(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		real(real64), dimension(:,:), intent(inout), optional, contiguous	trans_field,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	vector_potential	Vector potential obtained from the total_field. The field will be defined on the grid of the Helmholtz type (thisgrid), UP UNTIL THISGRIDNP vector_potential(1:thisgridnp_part, 1:thisgridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	trans_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in]	apply_boundary	Should the curl apply boundary conditions?

Definition at line 1050 of file helmholtz_decomposition.F90.

◆ dget_trans_field()

subroutine helmholtz_decomposition_m::dget_trans_field	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:,:), intent(out), contiguous	transverse_field,
		real(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		real(real64), dimension(:,:), intent(inout), optional, contiguous	vector_potential,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	transverse_field	Transverse field obtained from the total_field. It will be defined on the inner_grid of the Helmholtz type (thisinner_grid), UP UNTIL INNER_GRNP transverse_field(1:thisinner_gridnp_part, 1:thisinner_gridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	vector_potential	Vector potential from which trans field whould be computed
[in]	apply_boundary	Should the curl apply boundary conditions?

Definition at line 1214 of file helmholtz_decomposition.F90.

◆ dcompute_surface_correction_trans_field()

subroutine helmholtz_decomposition_m::dcompute_surface_correction_trans_field	(	class(helmholtz_decomposition_t), intent(in)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:,:), intent(in)	field,
		real(real64), dimension(:,:), intent(out)	surface_correction
	)

private

Parameters

[in] this Helmholtz object

Definition at line 1259 of file helmholtz_decomposition.F90.

◆ dget_scalar_potential()

subroutine helmholtz_decomposition_m::dget_scalar_potential	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:), intent(out), contiguous	scalar_potential,
		real(real64), dimension(:,:), intent(inout), contiguous	total_field,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	scalar_potential	Vector potential obtained from the total_field. It will be defined on the grid of the Helmholtz type (thisgrid), UP UNTIL THISGRIDNP: scalar_potential(1:thisgridnp_part)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr), UP UNTIL SYS_GRNP_PART: total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in]	apply_boundary	Should the divergence apply boundary conditions?

Definition at line 1309 of file helmholtz_decomposition.F90.

◆ dget_long_field()

subroutine helmholtz_decomposition_m::dget_long_field	(	class(helmholtz_decomposition_t), intent(inout)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:,:), intent(out), contiguous	longitudinal_field,
		real(real64), dimension(:,:), intent(inout), optional, contiguous	total_field,
		real(real64), dimension(:), intent(inout), optional, contiguous	scalar_potential,
		logical, intent(in), optional	apply_boundary
	)

private

Parameters

[in,out]	this	Helmholtz object
[out]	longitudinal_field	Transverse field obtained from the total_field. It will be defined on the inner_grid of the Helmholtz type (thisinner_grid), UP UNTIL INNER_GRNP longitudinal_field(1:thisinner_gridnp_part, 1:thisinner_gridboxdim)
[in,out]	total_field	Total field to decompose. To ensure the correct behaviour this field must be correctly defined on the grid of the system that calls Helmholtz (sys_gr) UP UNTIL SYS_GRNP_PART total_field(1:sys_grnp_part, 1:sys_grboxdim)
[in,out]	scalar_potential	Scalar potential from which long field whould be computed
[in]	apply_boundary	Should the divergence apply boundary conditions?

Definition at line 1357 of file helmholtz_decomposition.F90.

◆ dcompute_surface_correction_long_field()

subroutine helmholtz_decomposition_m::dcompute_surface_correction_long_field	(	class(helmholtz_decomposition_t), intent(in)	this,
		type(namespace_t), intent(in)	namespace,
		real(real64), dimension(:,:), intent(in)	field,
		real(real64), dimension(:), intent(out)	surface_correction
	)

private

Parameters

[in]	this	Helmholtz object
[in]	field	(1:thissys_gridnp, 1:thissys_gridboxdim)
[out]	surface_correction	(1:thissys_gridnp)

Definition at line 1404 of file helmholtz_decomposition.F90.

Detailed Description

VECTOR and SCALAR POTENTIAL

TRANSVERSE and LONGITUDINAL FIELDS

SURFACE CORRECTION

Data Types

Functions/Subroutines

Function/Subroutine Documentation

◆ helmholtz_decomposition_init()

◆ get_indices_from_mask()

◆ helmholtz_finalize()

◆ helmholtz_visualize_boxes()

◆ zapply_inner_stencil_mask()

◆ zsubtract_average_from_inner_stencil()

◆ zget_vector_potential()

◆ zget_trans_field()

◆ zcompute_surface_correction_trans_field()

◆ zget_scalar_potential()

◆ zget_long_field()

◆ zcompute_surface_correction_long_field()

◆ dapply_inner_stencil_mask()

◆ dsubtract_average_from_inner_stencil()

◆ dget_vector_potential()

◆ dget_trans_field()

◆ dcompute_surface_correction_trans_field()

◆ dget_scalar_potential()

◆ dget_long_field()

◆ dcompute_surface_correction_long_field()