The following section describes the equations governing the different methods of
deformation or motion that can be applied on the geometric entities of a mesh (that is,
volumes, surfaces, lines, and points). All these methods compute new coordinates
(
) by modifying the initial coordinates (
) that are read from the mesh files. The new coordinates will be used
by the flow problem after the mesh deformation has been applied.
Mesh deformation is mainly governed by four equations:
The first equation sets the new coordinates of a node to be equal to the original coordinates, so that the node does not move:
(35–1)
The second equation defines a rigid translation, which is characterized by a direction (
) and an amplitude (
). The new coordinates 
are given by: 
(35–2)
During a rigid translation, all nodes experience the same displacement. An evolution function can be used to define the amplitude, and hence a series of meshes may be generated at different magnitudes of translation. Note that an evolution function cannot be used to define the direction.
The third equation defines an elastic remeshing, governed by the pseudo-elastic equation for the new coordinates
: 
(35–3)
The previous equation represents an elastic material analogy, where
functions in the role of the Young’s modulus and the
Poisson’s ratio is ignored (that is, set equal to 0). The reference
configuration is related to the initial values of the coordinates
(
). Ansys Polyflow allows you to specify how 
is calculated; by default, the values that make up this field
are functions of the quality of the individual mesh elements, in order to
control mesh distortion during the remeshing (see Element Stiffness During Elastic Remeshing for further details).
Note that in terms of defining the remeshing, it is not the absolute value of
for an element that is critical, but rather its value relative
to the values of the other elements. Equation 35–3 requires boundary conditions that prescribe the motion of the domain boundaries. In order to have a well-posed problem, the boundary conditions must be such that the whole domain does not undergo rigid motion (that is, translation and rotation).
The fourth equation defines a constraint on the normal displacement. This equation imposes a vanishing normal displacement:
(35–4)
where
is the vector normal to the surface.
By combining Equation 35–1 – Equation 35–4 for volumes, surfaces, lines, and points, you can obtain a large set of mesh transformations. For the sake of simplicity, the methods are organized by the geometric dimension on which they are applied, and they are combined to result in predefined motions. In order to have greater flexibility, there are no predefined boundary conditions. The boundary conditions are imposed by defining the motion of the geometry of lower dimension. Thus, a volume may have boundary conditions defined on its surfaces, which may have boundary conditions defined on its lines, which may have boundary conditions defined on its points.
For this reason, the methods are described on the basis of the geometric dimension of the domain. Domain Transformations is related to the domain transformation, including 2D and 3D domains. Surface Transformations is related to the surface transformations used to impose the boundary conditions on a 3D domain transformation. Line Transformations concerns the line transformations used to define the boundary conditions for either a 2D domain transformation or a surface transformation. Finally, Point Displacement describes the motion of points that set the boundary conditions for line transformations.
With this approach, a surface, line, or point may have more than one equation applied to it. Hierarchy of Equations describes how multiple equations on a single entity are addressed.