----------------------------
Mesh Extrusion and Mirroring
----------------------------

In ``nek5000/tools``, there is a code ``n2to3.f`` that can be compiled with your
local fortran compiler (preferably not g77).
By running this code, you can extend two dimensional domains to
three dimensional ones with a user-specified number of levels in the
:math:`z`-direction.  Such a mesh can then be modified using the mesh modification
approach. Assuming you have a valid two-dimensional mesh, ``n2to3`` is straightforward
to run.  Below is a typical session, upon typing ``n2to3`` the user is prompted at the command line

.. code-block:: none

    Input old (source) file name:
   h2e
    Input new (output) file name:
   h3e
    input number of levels: (1, 2, 3,... etc.?):
   16
    input z min:
   0
    input z max:
   16
    input gain (0=custom,1=uniform,other=geometric spacing):
   1
    This is for CEM: yes or no:
   n
    Enter Z (5) boundary condition (P,v,O):
   v
    Enter Z (6) boundary condition (v,O):
   0
    this is cbz: v  O   <---

         320 elements written to h3e.rea
   FORTRAN STOP

In this context CEM stands for computational electromagnetics, a spin-off of the original Nek5000 code.

The domain in which the fluid flow/heat transfer
problem is solved consists of two distinct subdomains. The
first subdomain is that part of the region occupied by
fluid, denoted :math:`\Omega_f`, while the second subdomain is that part
of the region occupied by a solid, denoted :math:`\Omega_s`. These two
subdomains are depicted in :numref:`fig-walls`. The entire domain is denoted as :math:`D=\Omega_f \cup \Omega_s`.
The fluid problem is solved in the domain :math:`\Omega_f`, while the
temperature in the energy equation is solved in the
entire domain; the passive scalars can be solved in either
the fluid or the entire domain.

We denote the entire boundary of :math:`\Omega_f` as :math:`\partial \Omega_f`, that part
of the boundary of :math:`\Omega_f` which is not shared by :math:`\Omega_s` as
:math:`\overline{\partial \Omega_f}`, and
that part of the boundary of :math:`\Omega_f` which is shared by :math:`\Omega_s`.
In addition, :math:`\partial \Omega_{s}, \overline{\partial \Omega_s}` are analogously defined.
These distinct portions of the
domain boundary are illustrated in :numref:`fig-walls`.
The restrictions on the domain for Nek5000 are itemized below.

- The domain :math:`\Omega=\Omega_f \cup \Omega_s` must correspond either to a
  planar (Cartesian) two-dimensional geometry, or to the
  cross-section of an axisymmetric region specified by
  revolution of the cross-section about a specified axis, or
  by a (Cartesian) three-dimensional geometry.
- For two-dimensional and axisymmetric geometries, the
  boundaries of both subdomains, :math:`\partial \Omega_f` and
  :math:`\partial \Omega_s`, must be
  representable as (or at least approximated by) the union of
  straight line segments, splines, or circular arcs.
- *Nek5000* can interpret a two-dimensional image as either
  a planar Cartesian geometry, or
  the cross-section of an axisymmetric body. In the case of
  the latter, it is assumed that the :math:`y`-direction is the radial
  direction, that is, the axis of revolution is at :math:`y=0`.
  Although an axisymmetric geometry is, in fact,
  three-dimensional, Nek5000 can assume that the field variables
  are also axisymmetric ( that is, do not depend on azimuth,
  but only :math:`y`, that is, radius, :math:`x`, and :math:`t` ), thus reducing the
  relevant equations to "two-dimensional" form.

Fully general three-dimensional meshes generated by other softwares
packages can be input to *preNek* as imported meshes.