Namespaces
module	fields_mof
	Module containing the fields related to the Moment-of-Fluid method.

module	variables_mof
	Module containing the parameters of the Moment-of-Fluid method.

module	variables_vof_plic
	Module containing the parameters of the Moment-of-Fluid method.

Classes
type	type_mof_phase_extension::t_mof_filament
	Filament structure for MOF extension. More...

type	type_mof_phase_extension::t_mof_phase_extension
	Definition of the MOF extension of the t_phase_geometry type. More...

Functions
subroutine, public	mod_mof2d_analytical_reconstruction::mof2d_symmetric_analytical_reconstruction (polygon, cell_centroid, ref_centroid1, ref_centroid2, ref_volume1, ref_volume2, normal, residual1, residual2, polygon1, polygon2)
	Analytic MOF symmetric reconstruction for two materials in a rectangular cell. More...

subroutine, public	mod_mof2d_analytical_reconstruction::mof2d_analytical_reconstruction (polygon, ref_centroid, ref_volume, normal, residual, polygon_full, polygon_empty)
	Analytic MOF reconstruction for two materials in a rectangular cell. More...

subroutine, public	mod_mof2d_backward_advection::mof2d_backward_advection (mof_phases, boundary_condition, velocity1, velocity2, time_step)
	Advection of the MOF phases using the backward advection algorithm. More...

double precision elemental function	mod_mof2d_coordinate_system_wrapper::cg2w_polygon_volume (polygon)
	Wrapper function for coordinate system abstraction. More...

pure subroutine	mod_mof2d_coordinate_system_wrapper::cg2w_polygon_centroid (polygon, centroid, volume)
	Wrapper function for coordinate system abstraction. More...

pure subroutine	mod_mof2d_coordinate_system_wrapper::cg2w_flood_polygon (polygon, normal, volume, s0, s1, polygon_full, polygon_empty)
	Wrapper function for coordinate system abstraction. More...

subroutine	mod_mof2d_global_reconstruction::mof2d_global_reconstruction (phases)
	Reconstruct the interface on every cells in 2D. More...

pure subroutine, public	mod_mof2d_minimization_reconstruction::mof2d_symmetric_minimization_reconstruction (polygon, ref_centroid1, ref_centroid2, ref_volume1, ref_volume2, angle, normal, residual1, residual2, polygon1, polygon2)
	Find the MOF linear reconstruction in a convex cell using a minimization algorithm. More...

pure subroutine, public	mod_mof2d_minimization_reconstruction::mof2d_minimization_reconstruction (polygon, ref_centroid, ref_volume, angle, normal, residual, polygon_full, polygon_empty)
	Find the MOF linear reconstruction in a convex cell using a minimization algorithm. More...

subroutine, public	mod_mof_mpi_exchange_filaments::mof_mpi_exchange_filaments (mof_phases)
	Exchange the filament structure between neighbor processors. More...

subroutine, public	mod_mof2d_one_cell_reconstruction::mof2d_one_cell_reconstruction (cell, ref_volume, ref_centroid, polygon_list, best_cost)
	Reconstruct the optimal location of the materials in one cell. More...

pure subroutine, public	mod_mof3d_analytic_centroid::mof3d_compute_analytic_gradient (s_angles, ref_centroid1, ref_centroid2, ref_volume, c, objective, gradient)
	Compute the centroid and the gradient of the objective function in rectangular hexahedral cell. More...

pure subroutine, public	mod_mof3d_analytic_centroid::mof3d_transform_angles_to_reference (transformation, orig_angles, ref_angles)
	Transform the spherical angles in the original configuration to the reference configuration. More...

pure subroutine, public	mod_mof3d_analytic_centroid::mof3d_transform_angles_to_original (transformation, ref_angles, orig_angles)
	Transform the spherical angles in the reference configuration to the original configuration. More...

pure subroutine, public	mod_mof3d_analytic_centroid::mof3d_compute_residual_analytic (s_angles, ref_centroid1, ref_centroid2, ref_volume, c, residual, jacobian)
	Compute the residual and its jacobian in a rectangular hexahedral cell. More...

pure subroutine, public	mod_mof3d_bfgs::mof3d_bfgs (cell, cell_volume, cell_centroid, ref_centroid1, ref_centroid2, ref_volume, use_analytic, angles, normal, stat, residual)
	Broyden-Fletcher-Goldfarb-Shanno method to minimize the objective function of MOF. More...

pure character(len=:) function, allocatable, public	mod_mof3d_bfgs::mof3d_get_stop_criterion (stat)
	Convert the stop criterion code to a human readable string. More...

pure subroutine	mod_mof3d_gauss_newton::mof3d_gauss_newton2 (cell, cell_volume, cell_centroid, ref_centroid1, ref_centroid2, ref_volume, use_analytic, angles, normal, stat, residual)
	Gauss-Νewton method to minimize the objective function of MOF. More...

subroutine	mod_mof3d_global_reconstruction::mof3d_global_reconstruction (mof_phases)
	Reconstruct the interface on every cells in 3D.

pure subroutine, public	mod_mof3d_gradient::mof3d_compute_gradient (polyhedron, cell_volume, cell_centroid, angles, ref_volume, ref_centroid1, ref_centroid2, use_analytic, objective, gradient)
	Compute the gradient of the objective function of MOF. More...

pure subroutine, public	mod_mof3d_gradient::mof3d_compute_residual (polyhedron, cell_volume, cell_centroid, angles, ref_volume, ref_centroid1, ref_centroid2, use_analytic, residual, jacobian)
	Compute the residual of MOF and its jacobian. More...

pure subroutine	mod_mof3d_gradient::compute_residual_geometric (polyhedron, cell_volume, cell_centroid, angles, ref_volume, ref_centroid1, ref_centroid2, residual, jacobian)
	Compute the residual and the jacobian using geometric approaches. More...

pure subroutine	mod_mof3d_initial_angles::mof3d_tetra_compute_initial_angles (p0, p1, p2, p3, ref_centroid1, ref_centroid2, ref_volume, best_angles)
	Compute the initial angles for the minimization algorithm for tetrahedrons. More...

pure subroutine	mod_mof3d_initial_angles::mof3d_hexa_compute_initial_angles (c, ref_centroid1, ref_centroid2, ref_volume, best_angles)
	Compute the initial angles for the minimization for hexahedrons. More...

pure subroutine, public	mod_mof3d_one_cell_reconstruction::mof3d_one_cell_reconstruction (cell, ref_volume, ref_centroid, polyhedron_list, best_cost)
	Reconstruct the optimal location of the materials in one cell. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_analytic_gradient (p0, p1, p2, p3, angles, ref_centroid1, ref_centroid2, ref_volume, objective, gradient)
	Compute the centroid and the gradient of the objective function in a tetrahedral cell. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_residual (p0, p1, p2, p3, angles, ref_centroid1, ref_centroid2, ref_volume, residual, jacobian)
	Compute the centroid and the derivatives in a tetrahedral cell. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_transformation (p0, p1, p2, p3, transformation)
	Compute the matrix to transform the reference tetrahedron to the original (transformed) tetrahedron. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_transform_angles_to_reference (transformation, orig_angles, ref_angles)
	Transform the spherical angles in the original configuration to the reference configuration. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_transform_angles_to_original (transformation, ref_angles, orig_angles)
	Transform the spherical angles in the reference configuration to the original configuration. More...

pure subroutine, public	mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_analytic_gradient_reference (angles, ref_centroid, volume, objective, gradient)
	Compute the centroid and the gradient of the objective function in the reference tetrahedron. More...

subroutine	mod_mof_advect_phases::mof_advect_phases ()
	Interface to advect phases using the Moment-of-Fluid representation.

subroutine	mod_mof_apply_boundary_conditions::mof_apply_boundary_conditions (mof_phases, mof_bc)
	Apply boundary conditions the MOF phases.

subroutine	mod_mof_compute_cfl::mof_compute_cfl (nb_time_step, time_step)
	Compute the time step and the number of time sub-step according the MOF CFL. More...

Detailed Description

Warning: This method is currently implemented in 2D but not in 3D.

Introduction

Moment-of-Fluid (MOF) is a semi-Lagrangian method to advect interfaces in multiphase flow simulation. This method is the apex of the VOF methods using piecewise linear interface reconstruction. Namely, MOF can manage the reconstruction of a large number of interfaces with automatic ordering using only a one-cell stencil (compared to the 9 (27 in 3D) point stencil of the PLIC method).

Description of the representation

MOF is a sharp interface method to represent n immiscible fluids on a mesh; n can be greater than 2. Each cell contains the volume fraction and the centroid of all the fluids.

Consider a cell \(\Omega\) and a partition \(\{\omega_1,\cdots,\omega_n\}\) of this cell. The volume of a given domain \(\omega\) is denoted \(|\omega|=M_0(\omega)\). It is given by

\[ M_0(\omega) = \int_\omega dx \]

The first momentum of a given domain \(\omega\) is denoted \(M_1(\omega)\). It is given by

\[ M_1(\omega) = \int_\omega x dx \]

The centroid of a given domain \(\omega\) is denoted \(x_c(\omega)\). It is given by

\[ x_c(\omega) = M_1(\omega)/M_0(\omega) \]

Reconstruction

From the volume fractions and the centroids we are able to reconstruct a piecewise linear interface between the n fluids.

Consider a cell \(\Omega\) containing two sub-domains \(\omega_1\) and \(\omega_2\). The reconstruction consists in finding two sub-domains \(\omega_1^\ell\) and \(\omega_2^\ell\) such as:

the intersection of \(\omega_1^\ell\) and \(\omega_2^\ell\) is a line segment
\(|\omega_1^\ell|\) = \(|\omega_1|\) and \(|\omega_2^\ell|\) = \(|\omega_2|\)
the distance between \(x_c(\omega_1^\ell)\) and \(x_c(\omega_1)\) is as small as possible The original algorithm implementation of MOF (Dyadechko & Shashkov 2005) uses a minimization algorithm to find \(\omega_1^\ell\) and \(\omega_2^\ell\), but it is possible to get rid of this minimization on rectangular cells. We have found an analytical solution of the minimization problem. Both minimization and analytic algorithm have been implemented.

This reconstruction is unique (see Dyadechko & Shashkov 2005) and can be generalized to n fluids (see Dyadechko & Shashkov 2008).

Advection

This implementation of MOF uses a backward advection scheme to advect the fluids as proposed by Dyadechko & Shashkov 2005. The description of the algorithm is available in the subroutine mof2d_backward_advection.

References

Dyadechko, V., & Shashkov, M. (2005). Moment-of-fluid interface reconstruction. Los Alamos National Laboratory Report LA-UR-05-7571.
Dyadechko, V., & Shashkov, M. (2008). Reconstruction of multi-material interfaces from moment data. Journal of Computational Physics, 227(11), 5361-5384.

Function Documentation

◆ cg2w_flood_polygon()

pure subroutine mod_mof2d_coordinate_system_wrapper::cg2w_flood_polygon	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(in)	normal,
		double precision, intent(in)	volume,
		double precision, dimension(2), intent(out)	s0,
		double precision, dimension(2), intent(out)	s1,
		type(t_polygon), intent(out)	polygon_full,
		type(t_polygon), intent(out)	polygon_empty
	)

Parameters

[in]	polygon	Any convex polygon whose points are defined in counterclockwise order
[in]	normal	The direction of flood given by a unit vector
[in]	volume	The required volume fraction of the flooded area
[out]	s0,s1	The extreme points of the interface line segment
[out]	polygon_full	Part of the `polygon` full of fluid
[out]	polygon_empty	Part of the `polygon` without fluid

◆ cg2w_polygon_centroid()

pure subroutine mod_mof2d_coordinate_system_wrapper::cg2w_polygon_centroid	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(out)	centroid,
		double precision, intent(out), optional	volume
	)

Parameters

[in]	polygon	Any polygon
[out]	centroid	Centroid of the polygon
[out]	volume	Volume of the polygon

◆ cg2w_polygon_volume()

double precision elemental function mod_mof2d_coordinate_system_wrapper::cg2w_polygon_volume ( type(t_polygon), intent(in) polygon )

Parameters

[in] polygon Any polygon

◆ compute_residual_geometric()

pure subroutine mod_mof3d_gradient::compute_residual_geometric	(	type(t_polyhedron), intent(in)	polyhedron,
		double precision, intent(in)	cell_volume,
		double precision, dimension(3), intent(in)	cell_centroid,
		double precision, dimension(2), intent(in)	angles,
		double precision, intent(in)	ref_volume,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, dimension(:), intent(out)	residual,
		double precision, dimension(:,:), intent(out)	jacobian
	)

Parameters

[in]	polyhedron	Convex polyhedron.
[in]	cell_volume	Volume of the cell.
[in]	cell_centroid	Coordinates of the centroid of the cell.
[in]	angles	Spherical angles.
[in]	ref_volume	Reference volume of the material 1.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[out]	residual	Residual function.

◆ mof2d_analytical_reconstruction()

subroutine, public mod_mof2d_analytical_reconstruction::mof2d_analytical_reconstruction	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(in)	ref_centroid,
		double precision, intent(in)	ref_volume,
		double precision, dimension(2), intent(out)	normal,
		double precision, intent(out)	residual,
		type(t_polygon), intent(out)	polygon_full,
		type(t_polygon), intent(out)	polygon_empty
	)

The volume fraction is expected to be in range [0,1/2]

The polygon must be defined in this order:

    p4          p3
     ┌──────────┐
     │          │
     │          │c2   ▲ y
     │    c1    │     ┃
     └──────────┘     ┗━━▶ x
    p1          p2    O

◆ mof2d_backward_advection()

subroutine, public mod_mof2d_backward_advection::mof2d_backward_advection	(	type(t_phase_geometry), dimension(:), intent(inout)	mof_phases,
		type(t_boundary_condition), dimension(:), intent(in)	boundary_condition,
		type(t_face_field), intent(in)	velocity1,
		type(t_face_field), intent(in)	velocity2,
		double precision	time_step
	)

This is a second order advection scheme. It requires the velocity at the beginning of the time step and at the end of the time step.

Algorithm description:

  ┌──────────┐   ┌──────────┐   ┌──────────┐   ┌──────────┐
  │          │   │          │   │          │   │          │
  │  Phase 1 │   │  Phase 2 │   │  Phase 3 │ … │  Phase n │
  │          │   │          │   │          │   │          │
  └──────────┘   └──────────┘   └──────────┘   └──────────┘
       ╰───────────────┴──────┬──────┴───────────────╯
                              │
                   ╓─────────────────────╖
          ╭────────╢ Begin loop on cells ║
          │        ╙─────────────────────╜
          │                   │
          │   ┌────────────────────────────────┐
          │   │ Backward advection of the cell │
          │   └────────────────────────────────┘
          │                   │
          │ ┌────────────────────────────────────┐
          │ │ Intersect the surrounding polygons │
          │ └────────────────────────────────────┘
          │                   │
          │ ┌────────────────────────────────────┐
          │ │ Forward advection of the centroids │
          │ └────────────────────────────────────┘
          │                   │
          │         ┌───────────────────┐
          │         │ Volume correction │
          │         └───────────────────┘
          │                   │
          │         ╓───────────────────╖
          ╰─────────╢ End loop on cells ║
                    ╙───────────────────╜
                              │
                     ┌────────────────┐
                     │ Reconstruction │
                     └────────────────┘

Parameters

[in,out]	mof_phases	list of MOF phases.
[in]	boundary_condition	boundary conditions.
[in]	velocity1	velocity at the beginning of the time step.
[in]	velocity2	velocity at the end of the time step.
[in]	time_step	time step.

◆ mof2d_global_reconstruction()

subroutine mod_mof2d_global_reconstruction::mof2d_global_reconstruction ( type(t_phase_geometry), dimension(:), intent(inout) phases )

Note: the 'is_mixed' flag must be set before reconstruction.

◆ mof2d_minimization_reconstruction()

pure subroutine, public mod_mof2d_minimization_reconstruction::mof2d_minimization_reconstruction	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(in)	ref_centroid,
		double precision, intent(in)	ref_volume,
		double precision, intent(inout)	angle,
		double precision, dimension(2), intent(out)	normal,
		double precision, intent(out)	residual,
		type(t_polygon), intent(out)	polygon_full,
		type(t_polygon), intent(out)	polygon_empty
	)

Parameters

[in]	polygon	convex polygon to split into full and empty part
[in]	ref_centroid	reference centroid
[in]	ref_volume	reference volume
[in,out]	angle	angle first guess. Returns the final angle
[out]	normal	return the interface normal
[out]	residual	return the value of the objective function
[out]	polygon_full	contains the full part of the initial polygon
[out]	polygon_empty	contains the empty part of the initial polygon

◆ mof2d_one_cell_reconstruction()

subroutine, public mod_mof2d_one_cell_reconstruction::mof2d_one_cell_reconstruction	(	type(t_polygon), intent(in)	cell,
		double precision, dimension(:), intent(in)	ref_volume,
		double precision, dimension(:,:), intent(in)	ref_centroid,
		type(t_polygon), dimension(:), intent(out)	polygon_list,
		double precision, intent(out)	best_cost
	)

Warning

This routine depends on mod_mof2d_analytical_reconstruction::mof2d_analytical_reconstruction, cell must be defined the same way:

   4┌──────┐3
    │      │
   1└──────┘2

Parameters

[in]	cell	Convex polyhedral cell.
[in]	ref_volume	Array of reference volumes.
[in]	ref_centroid	Array of reference centroids.
[out]	polygon_list	Array that contains the reconstructed polygons.
[out]	best_cost	Value of the optimal cost.

◆ mof2d_symmetric_analytical_reconstruction()

subroutine, public mod_mof2d_analytical_reconstruction::mof2d_symmetric_analytical_reconstruction	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(in)	cell_centroid,
		double precision, dimension(2), intent(in)	ref_centroid1,
		double precision, dimension(2), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume1,
		double precision, intent(in)	ref_volume2,
		double precision, dimension(2), intent(out)	normal,
		double precision, intent(out)	residual1,
		double precision, intent(out)	residual2,
		type(t_polygon), intent(out)	polygon1,
		type(t_polygon), intent(out)	polygon2
	)

The volume fraction of the first material is expected to be in range [0,1/2].

The polygon must be defined in this order:

    p4          p3
     ┌──────────┐
     │          │
     │          │c2   ▲ y
     │    c1    │     ┃
     └──────────┘     ┗━━▶ x
    p1          p2    O

◆ mof2d_symmetric_minimization_reconstruction()

pure subroutine, public mod_mof2d_minimization_reconstruction::mof2d_symmetric_minimization_reconstruction	(	type(t_polygon), intent(in)	polygon,
		double precision, dimension(2), intent(in)	ref_centroid1,
		double precision, dimension(2), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume1,
		double precision, intent(in)	ref_volume2,
		double precision, intent(inout)	angle,
		double precision, dimension(2), intent(out)	normal,
		double precision, intent(out)	residual1,
		double precision, intent(out)	residual2,
		type(t_polygon), intent(out)	polygon1,
		type(t_polygon), intent(out)	polygon2
	)

Parameters

[in]	polygon	convex polygon to split into full and empty part
[in]	ref_centroid1	reference centroid of material 1
[in]	ref_centroid2	reference centroid of material 2
[in]	ref_volume1	reference volume of material 1
[in]	ref_volume2	reference volume of material 2
[in,out]	angle	angle first guess. Returns the final angle
[out]	normal	return the interface normal
[out]	residual1	return the squared distance of the reference centroid of material 1 to its reference centroid
[out]	residual2	return the squared distance of the reference centroid of material 2 to its reference centroid
[out]	polygon1	contains the polygon of material 1
[out]	polygon2	contains the polygon of material 2

◆ mof3d_bfgs()

pure subroutine, public mod_mof3d_bfgs::mof3d_bfgs	(	type(t_polyhedron), intent(in)	cell,
		double precision, intent(in)	cell_volume,
		double precision, dimension(3), intent(in)	cell_centroid,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		logical, intent(in)	use_analytic,
		double precision, dimension(2), intent(inout)	angles,
		double precision, dimension(3), intent(out)	normal,
		integer, dimension(3), intent(out)	stat,
		double precision, dimension(2), intent(out)	residual
	)

Reference:

R. Fletcher, Practical methods of optimization, John Wiley & Sons, 1987.

Parameters

[in]	cell	Convex polyhedron.
[in]	cell_volume	Cell volume.
[in]	cell_centroid	Cell centroid.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume of the material 1.
[in]	use_analytic	Use analytic formulas.
[in,out]	angles	Initial guess and final angles.
[out]	normal	Unit normal of the reconstructed interface (material 1 → material 2).
[out]	stat	Gradient calls. stat(1): in BFGS. stat(2): in line search. stat(3): exit status.
[out]	residual	Residuals. residual(1): derivative, residual(2): objective function.

◆ mof3d_compute_analytic_gradient()

pure subroutine, public mod_mof3d_analytic_centroid::mof3d_compute_analytic_gradient	(	double precision, dimension(2), intent(in)	s_angles,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, dimension(3), intent(in)	c,
		double precision, intent(out)	objective,
		double precision, dimension(2), intent(out)	gradient
	)

Parameters

[in]	s_angles	Spherical angles in the unit cube.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume.
[in]	c	Dimensions of the cell.
[out]	objective	Objective function.
[out]	gradient	Gradient of the objective function.

◆ mof3d_compute_gradient()

pure subroutine, public mod_mof3d_gradient::mof3d_compute_gradient	(	type(t_polyhedron), intent(in)	polyhedron,
		double precision, intent(in)	cell_volume,
		double precision, dimension(3), intent(in)	cell_centroid,
		double precision, dimension(2), intent(in)	angles,
		double precision, intent(in)	ref_volume,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		logical, intent(in)	use_analytic,
		double precision, intent(out)	objective,
		double precision, dimension(2), intent(out)	gradient
	)

Parameters

[in]	polyhedron	Convex polyhedron.
[in]	cell_volume	Volume of the cell.
[in]	cell_centroid	Coordinates of the centroid of the cell.
[in]	angles	Spherical angles.
[in]	ref_volume	Reference volume of the material 1.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	use_analytic	Use analytic gradient.
[out]	objective	Objective function.
[out]	gradient	Coordinates of the gradient.

◆ mof3d_compute_residual()

pure subroutine, public mod_mof3d_gradient::mof3d_compute_residual	(	type(t_polyhedron), intent(in)	polyhedron,
		double precision, intent(in)	cell_volume,
		double precision, dimension(3), intent(in)	cell_centroid,
		double precision, dimension(2), intent(in)	angles,
		double precision, intent(in)	ref_volume,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		logical, intent(in)	use_analytic,
		double precision, dimension(:), intent(out)	residual,
		double precision, dimension(:,:), intent(out)	jacobian
	)

Parameters

[in]	polyhedron	Convex polyhedron.
[in]	cell_volume	Volume of the cell.
[in]	cell_centroid	Coordinates of the centroid of the cell.
[in]	angles	Spherical angles.
[in]	ref_volume	Reference volume of the material 1.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	use_analytic	Use analytic gradient.
[out]	residual	Residual function.
[out]	jacobian	Jacobian of the residual.

◆ mof3d_compute_residual_analytic()

pure subroutine, public mod_mof3d_analytic_centroid::mof3d_compute_residual_analytic	(	double precision, dimension(2), intent(in)	s_angles,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, dimension(3), intent(in)	c,
		double precision, dimension(:), intent(out)	residual,
		double precision, dimension(:,:), intent(out)	jacobian
	)

Parameters

[in]	s_angles	Spherical angles in the unit cube.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume of the material 1.
[in]	c	Dimensions of the cell.
[out]	residual	Residual function.
[out]	jacobian	Jacobian of the residual function.

◆ mof3d_gauss_newton2()

pure subroutine mod_mof3d_gauss_newton::mof3d_gauss_newton2	(	type(t_polyhedron), intent(in)	cell,
		double precision, intent(in)	cell_volume,
		double precision, dimension(3), intent(in)	cell_centroid,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		logical, intent(in)	use_analytic,
		double precision, dimension(2), intent(inout)	angles,
		double precision, dimension(3), intent(out)	normal,
		integer, intent(out)	stat,
		double precision, dimension(2), intent(out)	residual
	)

Reference:

R. Fletcher, Practical methods of optimization, John Wiley & Sons, 1987.

Parameters

[in]	cell	Convex polyhedron.
[in]	cell_volume	Cell volume.
[in]	cell_centroid	Cell centroid.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume of the material 1.
[in]	use_analytic	Use analytic formulas.
[in,out]	angles	Initial guess and final angles.
[out]	normal	Unit normal of the reconstructed interface (material 1 → material 2).
[out]	stat	Gradient calls.
[out]	residual	Residuals. residual(1): derivative, residual(2): objective function.

◆ mof3d_get_stop_criterion()

pure character(len=:) function, allocatable, public mod_mof3d_bfgs::mof3d_get_stop_criterion ( integer, dimension(3), intent(in) stat )

Parameters

[in] stat Statistic array from the BFGS.

Returns: criterion: String containing the description of the stop criterion.

◆ mof3d_hexa_compute_initial_angles()

pure subroutine mod_mof3d_initial_angles::mof3d_hexa_compute_initial_angles	(	double precision, dimension(3), intent(in)	c,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, dimension(2), intent(out)	best_angles
	)

Parameters

[in]	c	Dimensions of the hexahedron.
[in]	ref_centroid1	Coordinates of the first reference centroid.
[in]	ref_centroid2	Coordinates of the second reference centroid.
[in]	ref_volume	Reference volume.
[out]	best_angles	Initial angles.

◆ mof3d_one_cell_reconstruction()

pure subroutine, public mod_mof3d_one_cell_reconstruction::mof3d_one_cell_reconstruction	(	type(t_polyhedron), intent(in)	cell,
		double precision, dimension(:), intent(in)	ref_volume,
		double precision, dimension(:,:), intent(in)	ref_centroid,
		type(t_polyhedron), dimension(:), intent(out)	polyhedron_list,
		double precision, intent(out)	best_cost
	)

Parameters

[in]	cell	Convex polyhedral cell.
[in]	ref_volume	Array of reference volumes.
[in]	ref_centroid	Array of reference centroids.
[out]	polyhedron_list	Array that contains the reconstructed polyedrons.
[out]	best_cost	Value of the optimal cost.

◆ mof3d_tetra_compute_analytic_gradient()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_analytic_gradient	(	double precision, dimension(3), intent(in)	p0,
		double precision, dimension(3), intent(in)	p1,
		double precision, dimension(3), intent(in)	p2,
		double precision, dimension(3), intent(in)	p3,
		double precision, dimension(2), intent(in)	angles,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, intent(out)	objective,
		double precision, dimension(2), intent(out)	gradient
	)

Parameters

[in]	p0,p1,p2,p3	Vertices of the tetrahedron.
[in]	angles	Spherical angles.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume.
[out]	objective	Objective function.
[out]	gradient	Gradient of the objective function.

◆ mof3d_tetra_compute_analytic_gradient_reference()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_analytic_gradient_reference	(	double precision, dimension(2), intent(in)	angles,
		double precision, dimension(3), intent(in)	ref_centroid,
		double precision, intent(in)	volume,
		double precision, intent(out)	objective,
		double precision, dimension(2), intent(out)	gradient
	)

Parameters

[in]	angles	Spherical angles.
[in]	ref_centroid	Coordinates of the reference centroid.
[in]	ref_volume	Reference volume.
[out]	objective	Objective function.
[out]	gradient	Gradient of the objective function.

◆ mof3d_tetra_compute_initial_angles()

pure subroutine mod_mof3d_initial_angles::mof3d_tetra_compute_initial_angles	(	double precision, dimension(3), intent(in)	p0,
		double precision, dimension(3), intent(in)	p1,
		double precision, dimension(3), intent(in)	p2,
		double precision, dimension(3), intent(in)	p3,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, dimension(2), intent(out)	best_angles
	)

Parameters

[in]	p0,p1,p2,p3	Vertices of the tetrahedron.
[in]	ref_centroid1	Coordinates of the first reference centroid.
[in]	ref_centroid2	Coordinates of the second reference centroid.
[in]	ref_volume	Reference volume.
[out]	best_angles	Initial angles.

◆ mof3d_tetra_compute_residual()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_residual	(	double precision, dimension(3), intent(in)	p0,
		double precision, dimension(3), intent(in)	p1,
		double precision, dimension(3), intent(in)	p2,
		double precision, dimension(3), intent(in)	p3,
		double precision, dimension(2), intent(in)	angles,
		double precision, dimension(3), intent(in)	ref_centroid1,
		double precision, dimension(3), intent(in)	ref_centroid2,
		double precision, intent(in)	ref_volume,
		double precision, dimension(:), intent(out)	residual,
		double precision, dimension(:,:), intent(out)	jacobian
	)

Parameters

[in]	p0,p1,p2,p3	vertices of the tetrahedron.
[in]	angles	Spherical angles.
[in]	ref_centroid1	Coordinates of the reference centroid of the material 1.
[in]	ref_centroid2	Coordinates of the reference centroid of the material 2.
[in]	ref_volume	Reference volume.
[out]	residual	Residual function.
[out]	jacobian	Partial derivatives of the residual.

◆ mof3d_tetra_compute_transformation()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_compute_transformation	(	double precision, dimension(3), intent(in)	p0,
		double precision, dimension(3), intent(in)	p1,
		double precision, dimension(3), intent(in)	p2,
		double precision, dimension(3), intent(in)	p3,
		double precision, dimension(3,3), intent(out)	transformation
	)

The transformation is given by X(ξ) = Aξ + b. This routine returns the matrix A providing the vertices of the original (transformed) tetrahedron.

Parameters

[in]	p0,p1,p2,p3	Vertices of the tetrahedron.
[out]	transformation	Tranformation matrix.

◆ mof3d_tetra_transform_angles_to_original()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_transform_angles_to_original	(	double precision, dimension(3,3), intent(in)	transformation,
		double precision, dimension(2), intent(in)	ref_angles,
		double precision, dimension(2), intent(out)	orig_angles
	)

Parameters

[in]	transformation	Tranformation matrix.
[in]	ref_angles	Angles in the reference configuration.
[out]	orig_angles	Angles in the original configuration.

◆ mof3d_tetra_transform_angles_to_reference()

pure subroutine, public mod_mof3d_tetra_analytic_centroid::mof3d_tetra_transform_angles_to_reference	(	double precision, dimension(3,3), intent(in)	transformation,
		double precision, dimension(2), intent(in)	orig_angles,
		double precision, dimension(2), intent(out)	ref_angles
	)

Parameters

[in]	transformation	Tranformation matrix.
[in]	orig_angles	Angles in the original configuration.
[out]	ref_angles	Angles in the reference configuration.

◆ mof3d_transform_angles_to_original()

pure subroutine, public mod_mof3d_analytic_centroid::mof3d_transform_angles_to_original	(	double precision, dimension(3), intent(in)	transformation,
		double precision, dimension(2), intent(in)	ref_angles,
		double precision, dimension(2), intent(out)	orig_angles
	)

Parameters

[in]	transformation	Tranformation matrix.
[in]	ref_angles	Angles in the reference configuration.
[out]	orig_angles	Angles in the original configuration.

◆ mof3d_transform_angles_to_reference()

pure subroutine, public mod_mof3d_analytic_centroid::mof3d_transform_angles_to_reference	(	double precision, dimension(3), intent(in)	transformation,
		double precision, dimension(2), intent(in)	orig_angles,
		double precision, dimension(2), intent(out)	ref_angles
	)

Parameters

[in]	transformation	Tranformation matrix.
[in]	orig_angles	Angles in the original configuration.
[out]	ref_angles	Angles in the reference configuration.

◆ mof_compute_cfl()

subroutine mod_mof_compute_cfl::mof_compute_cfl	(	integer, intent(out)	nb_time_step,
		double precision, intent(out)	time_step
	)

Parameters

[out]	nb_time_step	Number of time sub-steps
[out]	time_step	Time step

◆ mof_mpi_exchange_filaments()

subroutine, public mod_mof_mpi_exchange_filaments::mof_mpi_exchange_filaments ( type(t_phase_geometry), dimension(:), intent(inout) mof_phases )

Parameters

[in,out] mof_phases Array of phases

Namespaces

Classes

Functions

Detailed Description

Introduction

Description of the representation

Reconstruction

Advection

References

Function Documentation

◆ cg2w_flood_polygon()

◆ cg2w_polygon_centroid()

◆ cg2w_polygon_volume()

◆ compute_residual_geometric()

◆ mof2d_analytical_reconstruction()

◆ mof2d_backward_advection()

◆ mof2d_global_reconstruction()

◆ mof2d_minimization_reconstruction()

◆ mof2d_one_cell_reconstruction()

◆ mof2d_symmetric_analytical_reconstruction()

◆ mof2d_symmetric_minimization_reconstruction()

◆ mof3d_bfgs()

◆ mof3d_compute_analytic_gradient()

◆ mof3d_compute_gradient()

◆ mof3d_compute_residual()

◆ mof3d_compute_residual_analytic()

◆ mof3d_gauss_newton2()

◆ mof3d_get_stop_criterion()

◆ mof3d_hexa_compute_initial_angles()

◆ mof3d_one_cell_reconstruction()

◆ mof3d_tetra_compute_analytic_gradient()

◆ mof3d_tetra_compute_analytic_gradient_reference()

◆ mof3d_tetra_compute_initial_angles()

◆ mof3d_tetra_compute_residual()

◆ mof3d_tetra_compute_transformation()

◆ mof3d_tetra_transform_angles_to_original()

◆ mof3d_tetra_transform_angles_to_reference()

◆ mof3d_transform_angles_to_original()

◆ mof3d_transform_angles_to_reference()

◆ mof_compute_cfl()

◆ mof_mpi_exchange_filaments()