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 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
where f,
g, and
h are vector-valued functions,
t is time,
p is some set of parameters controlling Deformed Geometry features, 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.