Mesh#

This subsection provides information of the simulation geometry and its mesh. The simulation geometry shape and size, number of refinements and other required information can be provided here. It should be mentioned that meshes from gmsh can also be defined in this section by setting type = gmsh.

subsection mesh
  # Type of mesh. Choices are <gmsh|dealii|lethe>
  set type = dealii

  # GMSH file name
  set file name = none

  # Grid arguments for dealii and lethe grids.
  set grid type      = hyper_cube
  set grid arguments = -1 : 1 : false

  # Initial refinement of the mesh
  set initial refinement = 0

  # Initial refinement of the mesh near user-specified boundaries
  set initial boundary refinement = 0

  # List of boundaries next to which the mesh should be refined. The list must contain integers separated by commas.
  set boundaries refined = 0, 1

  # Enable initial refinement until target size is reached
  set enable target size = false
  set target size        = 1

  # Indicates that the mesh is a simplex mesh
  set simplex = false

  # DEM-specific mesh parameters

  # Enables checking the input grid for diamond-shaped cells
  set check diamond cells = false

  # Enables adding the boundary neighbor cells of boundary cells to the particle-wall contact search list
  set expand particle-wall contact search = false

  # Mesh modification parameters

  # Translation to apply to the mesh
  set initial translation = 0, 0, 0

  # Rotation to apply to the mesh
  set initial rotation axis  = 1, 0, 0
  set initial rotation angle = 0

  # Mesh scaling factor
  set scale = 1
end

The following choices for the mesh type are available:
- gmsh: if this type is chosen, a .msh file generated from GMSH can be used. In this case, the grid file name must be specified in the file name variable.
- dealii: if this type is chosen, the deal.II grid generator class can be used. For additional documentation on these grids, you can consult the deal.II documentation for the GridGenerator . The type of grid is specified by the grid type parameter and the arguments used for grid generation by the grid arguments parameter.
- lethe: if this type is chosen, the grid will be generated by Lethe built-in mesh. These custom meshes are generally used for benchmark problems and are constructed using deal.II meshing tools. The type of grid is specified by the grid type parameter and the arguments used for grid generation by the grid arguments parameter.
The following lethe built-in mesh types, defined by the grid type parameter, can be generated:

Name	Geometry
Periodic Hills
Fichera Oven
Uniform Channel with Meshed Cylinder
Uniform Channel with Meshed Square Prism
Birmingham Fluidized Bed
Impinging Jet Mixer
Cylinder
Merged Cube

The initial refinement number determines the number of refinements the grid will undergo in the simulation before the simulation is run. This allows one to refine a coarse grid automatically. By default, most deal.II grids will be as coarse as possible and need to be refined. This is a desirable behavior for parallel simulations, since for quad/hex meshes, the coarsest level of the grid is shared amongst all cores. Consequently, using a coarse grid with too many cells will lead to a prohibitive memory consumption.
The initial boundary refinement determines the number of refinements the grid will undergo in the simulation in the vicinities of the boundary specified by the boundaries refined parameter.
The enable target size and target size respectively enable and provide a maximal target size that initial refinement cycles must lead towards, in contrast to the more common way of specifying the number of refinement cycles to apply.
The check diamond cells and expand particle-wall contact search are parameters used in particles simulations. The former is used to verify the quality of the background mesh; detecting diamond cells is important as they should be avoided. The latter serves a purpose in contact detection when the background mesh is concave.
simplex. If simplex is set to true, it indicates that the mesh being read is made of only simplex elements. If the mesh is of type = dealii it will be converted from a quad/hex mesh to a simplex mesh. If the mesh is of type = gsmh, it will be read from a file as long as it is only made of simplices.
The initial translation parameter provides a way to move the mesh in space prior to simulating the problem. It can be useful when space-dependent functions are used, but that generating a translated mesh is inconvenient or impossible.

Attention

Regardless of the dimension of the problem, 3 components (\(x\), \(y\), and \(z\)) must be specified for the initial translation. If the mesh is defined in a 2D space, the third component of initial translation (\(z\)-component) is ignored and the mesh translates in \(x\) and \(y\) only.
The initial rotation axis and initial rotation angle parameters provide another way to move the mesh prior to simulating the problem.

Attention

If the mesh is defined in a 2D space, initial rotation axis is ignored and the mesh rotates counter-clockwise around the origin of the coordinate system.
The scale parameter is used to scale the mesh. This is useful when the mesh is made in a different set of unit than what is desired by the simulation.

Warning

When scale, translation and rotation are used together, the scaling is applied first, then the rotation, then the translation.

Lethe Grid Generators#

Periodic Hills#

subsection mesh
  set type            = lethe
  set grid type       = periodic_hills
  # Grid arguments for 2D case
  set grid arguments  = spacing_y : alpha : repetitions_x : repetitions_y
  # Grid arguments for 3D case
  set grid arguments  = spacing_y : alpha : repetitions_x : repetitions_y : repetitions_z

If this grid type is chosen, a mesh for the periodic hills CFD benchmark is created. For more details on this type of mesh and its grid arguments, refer to Flow over Periodic Hills.

Fichera Oven#

subsection mesh
  set type            = lethe
  set grid type       = fichera_oven
  set grid arguments  = bottom_left : top_right : colorize

If this grid type is chosen, a mesh for the Fichera oven time-harmonic Maxwell benchmark is created. For more details on this type of mesh, refer to Fichera Oven.

The grid arguments accepts up to three colon-separated values as listed above.

The first two fields are required, where bottom_left and top_right are x,y,z coordinates defining the bounding box of the mesh. They follow deal.II’s convention for a hyper rectangle, where the bottom left corner is the point with the smallest coordinates in each direction, and the top right corner is the point with the largest coordinates in each direction.
The last field is optional, where colorize is a boolean that applies the following boundary ID convention: 0 for all the oven walls, 1 for the waveguide inlet which is the furthest face from the bottom_left corner with the normal pointing in the positive z direction.

Uniform Channel with Meshed Cylinder#

subsection mesh
  set type            = lethe
  set grid type       = uniform_channel_with_meshed_cylinder
  set grid arguments  = bottom_left : top_right : center : inner_radius : outer_radius : pad_bottom : pad_top : pad_left : pad_right : height : n_slices : use_transfinite_region : mesh_obstacle : colorize

If this grid type is chosen, a mesh for a channel (similar to the deal.II uniform_channel_with_cylinder) is created. The difference is the cylinder in the channel is meshed enabling the calculation of fields inside the cylinder which is useful for multiphysics simulations.

The grid arguments accepts up to fourteen colon-separated values as listed above.

The first five fields are required, where bottom_left, top_right, and center are x,y coordinates, and inner_radius and outer_radius are real numbers defining the cylinder and transition radii.
The next four optional integer values (defaulting to 0) are pad_bottom, pad_top, pad_left, and pad_right, which control the number of cells in each padding direction.
In 3D, height (default 1.0) sets the extrusion height and n_slices (default and minimum 2) sets the number of layers in the \(z\) direction.
The last 3 parameters are boolean: use_transfinite_region, mesh_obstacle and colorize which are by default all set to false.
- use_transfinite_region is use to enable the use of transfinite interpolation for the transition region between the inner and outer radii.
- mesh_obstacle is used to indicate whether the obstacle is meshed and given a material ID for the solid region or if the obstacle is not meshed and a new boundary is created.
- colorize applies the following boundary ID convention: 0 for the obstacle (if not meshed), then follows the deal.II subdivided_hyper_rectangle convention for the outer boundaries but shifted by +1 for all of them to accommodate the obstacle boundary ID.

Note

If the obstacle is meshed, the boundary IDs follow the standard deal.II convention for a subdivided hyper rectangle: 0 for left (-x), 1 for right (+x), 2 for bottom (-y), and 3 for top (+y), if 3D, 4 for front (-z) and 5 for back (+z).

Uniform Channel with Meshed Square Prism#

subsection mesh
  set type            = lethe
  set grid type       = uniform_channel_with_meshed_square_prism
  set grid arguments  = bottom_left : top_right : center : inner_half_side : outer_half_side : rotation_deg : pad_bottom : pad_top : pad_left : pad_right : height : n_slices : mesh_obstacle : colorize

This mesh follows the same channel-and-padding concept as uniform_channel_with_meshed_cylinder, but replaces the circular obstacle by a square prism and adds an in-plane obstacle rotation. The grid arguments accepts up fourteen colon-separated values as listed above.

As in the cylinder case, bottom_left, top_right, and center are x,y coordinates, the padding parameters are optional integers with default 0, and the 3D extrusion parameters are height (default 1.0) and n_slices (default and minimum 2).
The square-specific geometric fields are inner_half_side and outer_half_side (instead of radii), and rotation_deg is an optional angle in degrees (default 0) that rotates the inner square region.
The last 2 parameters are boolean: mesh_obstacle and colorize which are by default all set to false.
- mesh_obstacle is used to indicate whether the obstacle is meshed and given a material ID for the solid region or if the obstacle is not meshed and a new boundary is created.
- colorize applies the following boundary ID convention: 0 for the obstacle (if not meshed), then follows the deal.II subdivided_hyper_rectangle convention for the outer boundaries but shifted by +1 for all of them to accommodate the obstacle boundary ID.

Note

Birmingham Fluidized Bed#

subsection mesh
  set type            = lethe
  set grid type       = birmingham_fluidized_bed
  set grid arguments  = enable_chimney : inlet_offset

If this grid type is chosen a 3D mesh for the Birmingham fluidized bed geometry is created. The geometry is taken from Fede et al. (2016). The mesh is composed of a bottom cylinder (radius 0.077 m), a truncated cone, and a top cylinder (radius 0.127 m) joined along the x-axis. An optional rectangular chimney pipe can be attached to an off-center cell on the top end cap to serve as the outlet.

The grid arguments accepts two colon-separated values as mentioned above.

enable_chimney is true or false to enable or disable the chimney (defaults to true)
inlet_offset is a positive real number (in meters) that extends the bottom cylinder into negative x, so that the original geometry starts at x = 0 regardless of the offset (defaults to 0)

Boundary IDs are: 0 for walls, 1 for the inlet (x = -inlet_offset), and 2 for the outlet.

Impinging Jet Mixer#

subsection mesh
  set type            = lethe
  set grid type       = impinging_jet_mixer
  set grid arguments  = R_chamber : r_inlet : r_outlet : H_chamber : L_cone : L_outlet : L_inlet : z_inlet

If this grid type is chosen a 3D mesh for an impinging-jet mixer is created. The geometry (mixing-chamber axis aligned with \(z\)) consists of a vertical cylindrical mixing chamber closed at the top by a hemispherical dome, a conical reduction followed by a straight outlet pipe below the chamber, and two opposing horizontal inlet pipes entering the chamber side wall from \(+x\) and \(-x\). The two inlet pipes share a common axis, and the mesh is datumed so that this axis lies on the \(z = 0\) plane (the plane where the opposing jets impinge).

The grid arguments accepts either an empty string (which uses the default dimensions) or exactly eight colon-separated dimensions (all in meters), in the order listed above.

R_chamber is the chamber radius
r_inlet and r_outlet are the inlet- and outlet-pipe radii
H_chamber is the chamber height
L_cone is the axial length of the conical reduction
L_outlet is the length of the straight outlet pipe
L_inlet is the length of each inlet pipe
z_inlet is the height of the inlet axis above the chamber floor (which must satisfy \(0 < z_\text{inlet} < H_\text{chamber}\))

The default dimensions are 0.05 : 0.02 : 0.025 : 0.16 : 0.06 : 0.05 : 0.08 : 0.08. The mesh discretisation is not exposed and is fixed internally. Boundary IDs are: 0 for the \(+x\) inlet, 1 for the \(-x\) inlet, 2 for the outlet, and 3 for all walls.

Cylinder#

subsection mesh
  set type            = lethe
  # Cylinder type. Choices are <cylinder_classic | cylinder_balanced | cylinder_squared | cylinder_regularized>
  set grid type       = cylinder_classic
  set grid arguments  = x_subdivisions : radius : half_length

If this grid type is chosen, one of the following grid type options must be declared : cylinder_classic | cylinder_balanced | cylinder_squared | cylinder_regularized.

The cylinder_classic type is equivalent to a subdivided cylinder from deal.II and grid arguments must follow their related deal.II documentation for a cylinder . The other cylinder types follow the same grid arguments:

x_subdivisions is the number of cells to generate in the x direction
radius is the radius of the circle in the yz-plane
half_length is the half-length of the cylinder in the x direction

Warning

The squared cylinder may eventually leads to ill-posed jacobian transformation. As said in the deal.II documentation : “The four cells that were originally the corners of a square will give you some troubles during computations, as the jacobian of the transformation from the reference cell to those cells will go to zero, affecting the error constants of the finite element estimates”. This type of mesh is available, but its usage is not recommended.

Merged Cube#

subsection mesh
  set type            = lethe
  set grid type       = cube_merged
  set grid arguments  = R_cylinder : L_plate : H : n_subdivisions

This three-dimensional mesh is a combination of an extruded square plate into a cube with a circular hole at the center and a cylinder that fits perfectly in the hole. The following grid arguments can be prescribed:

R_cylinder is the radius of the cylinder mesh and of the hole in the extruded square plate;
L_plate is the half of the edge length of the square;
H is the height in the z direction;
n_subdivisions is the number of subdivisions along the z direction.

The resulting cube will contain only the boundary ID 0 to identify its whole external boundary.

Upcoming update

A generalization to allow the prescription of different boundary conditions at each cube face will be done in the future.