The Euclidean method extends the concepts of the 1D spine technique to two dimensions. The displacement of an internal node is a linear function of the displacements of all boundary nodes of the current slice. The weighting depends on the Euclidean distance between the internal node being relocated and the corresponding boundary node. The weighting factor equals 1 when the node to be relocated coincides with a boundary node, and it equals 0 when the node is very far from a given boundary node. This can be mathematically expressed as follows:

(16–4)

where are weighting factors defined as

(16–5)

NBD is the number of boundary nodes in a given section and and are indices of the internal and boundary nodes, respectively.

Before the internal nodes are relocated according to Equation 16–4, the mesh nodes on the free surface or moving interface are relocated according to the kinematic condition. Nodal displacements propagate into tangential displacement on adjacent faces. This results in a tangential remeshing technique applicable for small deformations. Figure 16.1: Tangential and Domain Remeshing illustrates how sections will be remeshed.

An advantage of the Euclidean remeshing technique is that the number of position unknowns is small compared to the velocity and pressure unknowns. Moreover, this technique has a complete topological generality.

Its drawback is that it is robust only for moderately small displacements of the free surface or moving interface, because of its linearity. It has been found that only elliptic techniques (based on partial differential equations of the elliptic type), such as the Thompson transformation (described in Thompson Transformation) or Optimesh (described in Optimesh) are robust enough to deal with large boundary displacements. See User Inputs for Remeshing for information about using the Euclidean remeshing method.