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## 1.2 Mesh: finite element mesh generation

A finite element mesh is a tessellation of a given subset of the three-dimensional space by elementary geometrical elements of various shapes (in Gmsh’s case: lines, triangles, quadrangles, tetrahedra, prisms, hexahedra and pyramids), arranged in such a way that if two of them intersect, they do so along a face, an edge or a node, and never otherwise. All the finite element meshes produced by Gmsh are considered as “unstructured”, even if they were generated in a “structured” way (e.g., by extrusion). This implies that the elementary geometrical elements are defined only by an ordered list of their nodes but that no predefined order relation is assumed between any two elements.

The mesh generation is performed in the same bottom-up flow as the
geometry creation: lines are discretized first; the mesh of the lines is
then used to mesh the surfaces; then the mesh of the surfaces is used to
mesh the volumes. In this process, the mesh of an entity is only
constrained by the mesh of its boundary. For example, in three
dimensions, the triangles discretizing a surface will be forced to be
faces of tetrahedra in the final 3D mesh only if the surface is part of
the boundary of a volume; the line elements discretizing a curve will be
forced to be edges of tetrahedra in the final 3D mesh only if the curve
is part of the boundary of a surface, itself part of the boundary of a
volume; a single node discretizing a point in the middle of a volume
will be forced to be a vertex of one of the tetrahedra in the final 3D
mesh only if this point is connected to a curve, itself part of the
boundary of a surface, itself part of the boundary of a volume. This
automatically assures the conformity of the mesh when, for example, two
surfaces share a common line. But this also implies that the
discretization of an “isolated” (`n`-1)-th dimensional entity
inside an `n`-th dimensional entity does *not* constrain the
`n`-th dimensional mesh—unless it is explicitly told to do so
(see section Miscellaneous). Every meshing step is
constrained by a “size field” (sometimes called “characteristic
length field”), which prescribes the desired size of the elements in
the mesh. This size field can be uniform, specified by values associated
with points in the geometry, or defined by general “fields” (for
example related to the distance to some boundary, to a arbitrary scalar
field defined on another mesh, etc.).

For each meshing step, all structured mesh directives are executed first, and serve as additional constraints for the unstructured parts (1).

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