The following guidelines presents the domain and some relative points associated to the partitioning of the domain and the boundary conditions.

Domain definitions

Physical domain and immersed boundaries

Notus solves fluid flows in a global physical domain that is a Cartesian part of the 2D or 3D real space delimited by global boundaries. Figure 1 illustrates this definition: the global physical domain is outlined in gray and the global boundaries are drawn with black lines.

When physics requires non-Cartesian boundaries, Notus uses immersed boundaries: the global domain is split into two sub-domains: the inner sub-domain contains the physics of interest, while the outer sub-domain adapts the physical problem to the Cartesian nature of Notus. Figure 1 also illustrates immersed boundaries drawn with blue lines; the inner sub-domain is shaded in white while the outer sub-domain is shaded in blue.

Figure 1: Global physical domain and immersed boundaries.

Notus can handle periodical domain. When the domain is periodic in some direction, the domain extent in that direction must match the period.

The module variables_domain defines a handful of variables that describes the global domain.

Local domains and process boundaries

When the code is executed on more than one processor, the global physical domain is partitioned in local physical domains, and each one is solved by a process unit. Notus automatically finds optimal partitioning ensuring a balance between processor loads and a minimisation of the data exchange between processors. Similarly to the global physical domain, each local domain has its own local boundaries. A local boundary may match a part of a physical boundary, but most of them are boundaries between processors where a special treatment must be done to solve the equations in parallel. Figure 2 illustrates a partition of the global physical domain with local boundaries drawn with black lines.

Figure 2: Domain decomposition with local boundaries.

Part of the code that deals with parallelism topics can be found in several directorie directory:

core/mpi directory
core/openmp directory
geometry/grid/partitioning directory

Mesh definitions

Numerical domain and process ghost cells

The global physical domain is represented by a Cartesian grid composed of cells. The local physical domain is exactly a subset of theses cells as shown in Figure 3.

Figure 3: Global mesh.

For different numerical reasons, the local physical domain is actually extended across each local boundary by nghost rows of cells; the extended domain is called the numerical domain. It has nx, ny, nz cells in the x, y or z direction.

Figure 4: Local mesh and numerical domain.

The Fortran arrays are based on the numerical domain. Some index variables help to locate the start or the end of the numerical domain (for instance is, ie and js, je in Figure 4).

The ghost boundary cells that are located outside the physical domain are used to apply the boundary conditions. The ghost cells that are locate inside the physical domain create ovelapping regions between processors which are used to exchange data explicitly with the MPI library. This way, each processor has the necessary information to compute the solution on its local physical domain.

The number of ghost cells is automatically adapted to the schemes used. Whatever the number of processor, the solution is the same up to computer precision (or nearly, depending on the problem solved and the obtained solver residual).

Part of the code that deals with grid generation can be found in geometry/grid directory. Especially, one can find grid coordinates arrays and various grid indices.

Node types

Within a cell, different nodes are defined:

the cell node center,
the x-face nodes at the left and right faces center,
the y-face nodes at the bottom and top faces center,
the z-face nodes at the rear and front faces center,
the vertex nodes

The set of cell nodes, of x-face nodes, etc. forms a staggered arrangement of Cartesian grids named cell grid, x-face grid, etc.

Figure 5 illustrates (in 2D) the node types: cell nodes are represented by circles, x-face nodes are represented by rightward triangles, and y-face nodes are represented by upward triangles.

Figure 5: Node types.

More generally, the adjectives cell and face are used to designate things related to the corresponding grid. The term node used without these adjective designates a cell node or a face node indistinctly. For example, scalar fields will be named a cell field represents a field discretized on cell nodes, a x-face field represents a field discretized on x-face nodes, etc. Additionally, face fields designates vector fields whose components are discretized on x-face, y-face, and z-face nodes.

Staggered grid are used to solve the velocity/pressure coupling. Components u, v and w of the velocity vector are respectively associated to the x-face, y-face and z-face grids. Pressure and other scalar fields are defined at the centrer of the cells.