This section describes in detail how the Thompson transformation is implemented. You should familiarize yourself with the basics described above before reading further. When defining boundary conditions for the field, Ansys Polydata verifies that the following conditions are satisfied:

The first requirement ensures that the remeshing domain is mapped onto a rectangular geometry (a rectangle in 2D or a parallelepiped in 3D). The second requirement ensures that no possible singularities in the field are introduced by the imposed boundary conditions. Only one component of can be imposed on a segment. For the other components, natural boundary conditions are used, as in the example of Figure 16.4: Coating Example for Thompson Transformation. Similarly, on a given face , a component is imposed, while natural conditions are used for the remaining components:

(16–8)

(16–9)

For these, Ansys Polydata allows two possible conditions for on . The first condition requires that points on this face be fixed (that is, the corresponding displacement vanishes):

(16–10)

For this condition, the face is referred to as a fixed face: all coordinates remain fixed on . This condition is automatically applied to a face on which neither tangential nor normal remeshing applies. The second possible condition requires that

(16–11)

and

(16–12)

with the further restriction that

(16–13)

For this second condition, the face is referred to as a symmetry face. It is important to note that there is an implicit connection between the imposed component and the component that is fixed in a symmetry face. Imposing , , or requires fixing , , or , respectively. The remaining non-imposed components () define the axis or plane of symmetry. For example, if is imposed, becomes fixed and the face has symmetry about the - plane (or the axis in 2D). Furthermore, the requirement of Equation 16–12 stipulates that this face is parallel to the axis or plane of symmetry.

In addition to the fixed and symmetry faces, another type of condition is required to maintain smooth mesh point distributions. Consider, for example, a face that is a free surface. It is desired that the original distribution of points along the free surface itself be maintained. This face cannot be a fixed face, since it must be free to move in order to satisfy the kinematic condition (Equation 15–2 or Equation 15–3). Moreover, a normal boundary condition (a generalization of the symmetry-type condition) can sometimes lead to an undesirable redistribution of points on the free surface.

The solution chosen for such faces is to create a separate tangential remeshing of the Thompson transformation type on the boundary itself. Ansys Polydata automatically chooses this solution and generates the corresponding problem with the associated boundary conditions. Still, for free surfaces, the displacements of nodal points along the directors remains controlled by the kinematic condition.

Faces that are not free surfaces can also be allowed a tangential remeshing. In such cases, the original tangential distribution of points on the face is maintained, while a zero displacement is imposed in the direction normal to the surface. Such a condition is useful for faces that change size (but not curvature) due to the presence of a free surface or interface. To simplify matters for you, Ansys Polydata automatically makes some decisions about which faces are fixed faces, symmetry faces, and tangentially remeshed faces.

For moving boundaries, Ansys Polydata uses the following rules:

For the remaining faces, Ansys Polydata uses the following rules:

In 3D problems, the intersection of two tangentially remeshed faces forms a line that is given tangential remeshing by a separate one-dimensional Thompson transformation. All boundary conditions associated with this one-dimensional remeshing are inherited from the two- and three-dimensional transformations. Definition of such one-dimensional transformations is handled automatically by Ansys Polydata.