Constructive Solid Geometry

Theory

As stated, a CSG representation is defined minimally as a series of surfaces, cells, and universes.

Surfaces are defined explicitly through surface equations (such as equations of a plane, sphere, etc.). Each surface inherently separates two half-space regions: positive and negative half-spaces. For example, for a plane with the equation the positive half-space represents the region , while the negative half-space represents the region . Similarly, for a spherical surface defined by the equation , the negative half-space represents the region within the sphere while the positive half-space represents the region outside the sphere. Example half-spaces are shown in Figure 1.

Figure 1: Example depiction of the positive and negative half-spaces defined by a plane (left) and sphere (right).

These half-space regions defined by the surfaces can be combined further as series of boolean operators for unions, intersections, and complements to further define more complex regions. For example, if we wanted to use the surfaces from Figure 1 to define just the left hemisphere, we would define the cell region as the intersection of the negative half-space of the plane and the negative half-space of the sphere (Figure 2).

Figure 2: Example depiction of a closed region defined by an intersection of two half-spaces.

Cells are defined by two main characteristics: a region and a fill. The region is defined as described above and defines the domain of the cell. The fill can typically be set as void (i.e., nothing), a material (placeholder type for now, this will be expanded in future works), a universe, or a lattice (note, lattices are not yet supported for MOOSE implementation).

Universes can then be optionally defined as a collection of cells, which can then be used to either fill other cells, or used repeatedly throughout a geometry (such as in a repeated lattice). By default, every model will have a root universe, which is the singular overarching universe that all other universes can be traced back to through the tree defined by universes containing cells and cells filled with universes.

For a more detailed description of how surfaces, cells, and universes are represented within the MOOSE mesh generator system, please refer to CSGBase.

How to Generate a CSG Model

The CSG model generation can be invoked on the command line using the --csg-only option with any MOOSE mesh input file. The JSON file that is generated will be called the name of the input file with _csg appended by default. An optional output file name can be provided on the command line (--csg-only <output_file_name.json>).

Output

Calling --csg-only will produce a JSON file containing the complete geometric description of the mesh generator input. There are three main sections in the file:

• surfaces

• cells

• universes

Each item within each section is keyed (a JSON is a dictionary) by its unique name identifier, and the value is the corresponding definition for that item. Detailed description of each type of item in the section follows.

  "surfaces": {
    "inf_square_surf_minus_x": {
      "coefficients": {
        "a": -1.0,
        "b": 0.0,
        "c": 0.0,
        "d": 2.0
      },
      "type": "CSG::CSGPlane"
    },
    "inf_square_surf_minus_y": {
      "coefficients": {
        "a": 0.0,
        "b": 1.0,
        "c": 0.0,
        "d": -2.0
      },
      "type": "CSG::CSGPlane"
    },

  "cells": {
    "cube_box_cell": {
      "fill": "square_material",
      "filltype": "CSG_MATERIAL",
      "region":
          "(+inf_square_surf_plus_x & -inf_square_surf_minus_x & -inf_square_surf_plus_y & +inf_square_surf_minus_y & -cube_surf_plus_z & +cube_surf_minus_z)"
    }
  },

  "universes": {
    "ROOT_UNIVERSE": {
      "cells": [
        "cube_box_cell"
      ],
      "root": true
    }
  }
}