Though moving meshes are also possible in 3D, consider a 2D geometry for simplicity, where the spatial and material frame coordinates are called (x, y) and (X, Y), respectively. Let (X0, Y0) be the spatial coordinates of a mesh node in the initial material configuration. The spatial coordinates (x0, y0) of the same mesh node at some other time, t, are then given by the functions

These functions can be explicit transformations (expressions) or the solution to a mesh smoothing equation. The mesh node’s material coordinates (X0, Y0) can in turn be seen as functions of an underlying system of geometry coordinates (Xg, Yg) and a parameter, p, such that

with similar options for the transformations. The transformations can also be chained such that (x0, y0) are seen as functions of (Xg, Yg), t, and p.

where f, g, and h are vector-valued functions, t is time, p is some set of parameters controlling a Deformed Geometry interface, and i is number of times the geometry has been remeshed. From the physics point of view, the domain is fixed in the geometry frame coordinates Xg, which are therefore seen as constant in the above formulas.

From the finite elements’ point of view, it is instead the mesh frame coordinates Xm that are constant and Xg = h−1(Xm, i). Therefore when assembling the finite-element matrices, the relation actually used is

where f is a unit map if the spatial and material frames coincide, g is a unit map if the material and geometry frames coincide, and the inverse mapping h−1(Xm, i) is initially a unit map and then updated by interpolation after each remeshing operation.

In addition to the different sets of coordinate variables, some other geometric variables that the software defines are available for both the spatial and material frames (see Geometric Variables, Mesh Variables, and Variables Created by Frames).