Constructing Embedded Boundaries in MFIX-Exa

MFIX-Exa uses AMReX’s constructive solid geometry framework defined in the namespace amrex::EB2. See the AMReX EB documentation for more details. These are defined in src/eb/mfix_eb.cpp. The function mfix::make_eb_geometry (also defined in src/eb/mfix_eb.cpp) selects one of the following geometries depending on the value of the mfix.geometry setting in the inputs file.

Description	mfix.geometry	Implementation Satus
Planar walls from mfix.dat (on by default)	Don’t specify	Fully implemented
Box (up to six walls)	box	Fully implemented
Cylinder	cylinder	Fully implemented
Hopper	hopper	Fully implemented
Cyclone	cyclone	Fully implemented
General	general cf. note 1 below	Partially implemented
Hourglass	hourglass cf. note 1	Deprecated cf. note 2 below
CLR (chemical looping reactor)	clr cf. note 1	Deprecated cf. note 2
CLR Riser	clr_riser cf. note 1	Deprecated cf. note 2

1. Older (legacy) alternative settings are:

   Value of mfix.geometry	Alternative
   general	mfix.use_poly2 = true or mfix.use_walls = true
   hourglass	mfix.hourglass = true
   clr	mfix.clr = true
   clr_riser	mfix.clr_riser = true

2. These geometries are not ported from AMReX’s old EB system to the new EB2.

Also note that planar boundary conditions can be specified in the mfix.dat file. Even if the user does not specify an mfix.geometry in the inputs, any no-slip or free-slip boundary conditions are expressed as EB walls.

There are also two parameters (specified in the inputs file) that influence the level-set creation:

Parameter	Description
amr.max_level	If greater than 0, MFIX-Exa operates in multi-level mode. The level-set grids follow all other grids. If equal to 1, the level-set has two levels (one additional level with higher refinement).
mfix.levelset__refinement	If amr.max_level > 1 this parameter is ignored. Otherwise it sets the maximum refinement of the level-set

How MFIX-Exa Constructs the EB Geometry

Once a geometry is selected by mfix::make_eb_geometry, the procedure is the same for (almost) all geometries. Also see the AMReX geometry documentation for information on how to construct new geometries:

1. Construct an implicit function representing the geometry (using the language of constructive solid geometry). For example

EB2::CylinderIF my_cyl(radius, height, direction, center, inside);
auto gshop_cyl = EB2::makeShop(my_cyl);

2. (Optional) Construct the implicit function representing the EB seen by the particles. This might deviate from the “standard” EB depending on the specific application. In all standard cases used by mfix, this step is omitted.

3. Call mfix::build_eb_levels(gshop) this function builds the EB levels and fills the implicit function MultiFab (the later being used to construct the level-set function). Note that this also makes the particle EB levels point to the fluid eb levels.

4. (Optional, do this if you did 2.) Call mfix::build_particle_eb_levels(gshop_part) last. This will update the particle EB levels.

MFIX-Exa’s EB Data Structures

The mfix class stores the following EB data:

//! EB levels representing fluid boundary conditions
Vector<const EB2::Level *> eb_levels;
//! EB levels representing particle boundary conditions (same as
//! `mfix::eb_levels` but might include additional walls).
Vector<const EB2::Level *> particle_eb_levels;

//! EB factory that lives on the fluid grids
Vector< std::unique_ptr<amrex::EBFArrayBoxFactory> > ebfactory;
//! EB factory that lives on the particle grids
Vector< std::unique_ptr<amrex::EBFArrayBoxFactory> > particle_ebfactory;

As discussed in the previous sub-section, the difference between mfix::eb_levels and mfix::particle_eb_levels enables the user to specify a modified EB geometry for particles only. Whereas the fluid sees the EB geometry in mfix::eb_levels. If no addition particle EB geometry is specified (point 4 in the previous section), then mfix::particle_eb_levels points to mfix::eb_levels.

In the same spirit, the mfix::ebfactory is constructed over the fluid grid and using the fluid EB levels, whereas mfix::particle_ebfactory is constructed over the particle grid using the particle EB levels.

A note about constructing EB Levels

MFIX-Exa builds EB levels in mfix::build_eb_levels (via LSCore<F>::BuildEBLevel)

EB2::Build(gshop, geom[lev], required_crse_lev, max_crse_level);
const EB2::IndexSpace & ebis = EB2::IndexSpace::top();

When building an EB level, the maximum coarsening level (int max_crse_level) and the required coarsening level (int required_crse_lev) need to be specified. The reason for this is that we need to specify to which level of coarseness the EB is still defined. It might not be immediately obvious, but the Poisson solver (used in the fluid solve) also depends indirectly on these parameters. Thus changing these during EB level creation might restrict how many levels the MLMG solver can use, and therefore give slightly different answers in the fluid solve.

Local Mesh Refinement at Walls

MFIX-Exa has the capability of locally refining the computational grid near EBs. This is done by tagging (in mfix::ErrorEst) any cells with volume fraction between 0 and 1. To enable local mesh refinement, set amr.max_level to a value greater than 1. Note that the parameter mfix.levelset__refinement is ignored on all cases except when amr.max_level = 1.

MFIX-Exa Initialization Process

Since MFIX-Exa requires the volume fraction when building grids (because this is needed by mfix::ErrorEst), the EB geometries need to be built before calling mfix::Init. The recommended procedure therefore is

// Default constructor (geom[lev] is defined here)
mfix my_mfix;

// Initialize internals from ParamParse database
my_mfix.InitParams(solve_fluid, solve_dem, call_udf);

// Initialize memory for data-array internals
my_mfix.ResizeArrays();

// Construct EB (must be done _before_ mfix::Init)
my_mfix.make_eb_geometry();

// Initialize derived internals. Grids are create here.
my_mfix.Init(dt, time);

// Create EB factories on new grids
my_mfix.make_eb_factories();

if (solve_dem)
{
    // Fill level-sets on each level (must be done _after_ mfix::Init)
    my_mfix.fill_eb_levelsets();
}

// Finish constructing levels
my_mfix.InitLevelData(dt,time);

// Regrid (ensure all MultiFabs are on their correct grids)
my_mfix.Regrid();

Also note that mfix defines boundary conditions in Fortran also (via the mfix.dat). Since these are potentially needed to build EB walls, mfix::make_eb_geometry also calls mfix_set_bc_type.

The grids for each level are build in the mfix::Init by invoking the initialization functions inherited from amrex::AmrCore.

// This tells the AmrMesh class not to iterate when creating the initial
// grid hierarchy
SetIterateToFalse();

// This tells the Cluster routine to use the new chopping routine which
// rejects cuts if they don't improve the efficiency
SetUseNewChop();

// This Builds the new Grids
InitFromScratch(0.);

The Level-Set Function

MFIX-Exa uses a level-set function to resolve particle-wall collisions. See the AMReX Level-Set documentation for more details. The level-set function is stored on the nodal Vector<std::unique_ptr<MultiFab>> mfix::level_sets. The level-set data is always stored on the particle grids. Depending on the input amr.max_level The level-set can be in one of two modes:

1. MFIX-Exa is running in single-level mode (nlev == 1). Then mfix::level_sets[0] will be at the same resolution as the fluid (except that it is stored on the particle grid). Even though nlev == 1, there is a second level, level_sets[1]. This level is the same as level_sets[0] but refined by mfix::levelset__refinement. This way the level-set always has the appropriate resolution to resolve structures in the EB, even if the fluid is defined on a fairly coarse grid.

2. MFIX-Exa is running in multi-level mode (nlev > 1). The the parameter mfix::levelset__refinement is ignored. mfix::level_sets then follows the rest of MFIX-Exa, i.e. it is defined on the particle grids on all levels.

The level-set is used in two places:

1. The function MFIXParticleContainer::EvolveParticles interpolates the level-set onto each particle’s position in order to resolve collisions with the EBs. If nlev == 1, level_sets[1] is used to evolve the particle positions. Otherwise level_sets[lev] is used for each level.

2. The fluid-particle coupling can sometimes rely on neighbor stencils where one or more cell is covered by an EB. In order to avoid values that do not conform with the boundary conditions, the fluid velocity is reconstructed in those cells. The algorithm relies on the level-set, and uses level_sets[lev] on each level.

Special Cases Involving Level-Sets

The level-set function is filled by the mfix::fill_eb_levelsets() function. There are two special cases involving level-sets:

1. Mass-Inflow boundary conditions are not given EB walls. However, we don’t want particles to fall out of a MI either, so at the very end of the mfix::fill_eb_levelsets() function we call mfix::intersect_ls_walls(). This performs an intersection operation with the level-set representing a wall at each MI.

2. Box geometries and regular geometries are comprised entirely out of planar surfaces. Therefore the levelset is not construction out of an EB factory (as would be the case for all other geometries). But out of an intersection with all planar surfaces. This has the advantage of correctly describing corners.