Transformation Matrices and Volume Factors
The ALE machinery defines a number of variables based on the relations in Equation 18-3. The variables and component names are listed in Table 18-1, where indices i and j are 1, 2, or 3. A more detailed explanation of the variables follows below.
Table 18-1: Geometric Variables and Mesh variables.
Variable
Components
Description
spatial.F
spatial.Fij
The Jacobian of spatial coordinates x with respect to material coordinates X.
spatial.invF
spatial.invFij
The Jacobian of material coordinates X with respect to spatial coordinates x; the inverse of spatial.F.
spatial.detF
spatial.detF
The ratio of local element volume computed in the spatial frame to volume computed in the material frame; the determinant of spatial.F in elements of the same dimension as the geometry.
spatial.detInvF
spatial.detInvF
The ratio of local element volume computed in the material frame to volume computed in the spatial frame; the determinant of spatial.invF in elements of the same dimension as the geometry.
material.F
material.Fij
The Jacobian of material coordinates X with respect to geometry frame coordinates Xg
material.invF
material.invFij
The Jacobian of geometry frame coordinates Xg with respect to material coordinates X; the inverse of material.F.
material.detF
material.detF
The ratio of local element volume computed in the material frame to volume computed in the geometry frame; the determinant of material.F in elements of the same dimension as the geometry.
material.detInvF
material.detInvF
The ratio of local element volume computed in the geometry frame to volume computed in the material frame; the determinant of material.invF in element of the same dimension as the geometry.
3D, Plane 2D and 1D geometries
spatial.F
The matrix spatial.F contains components of the Jacobian of spatial coordinates with respect to material coordinates:
Rows of the matrix can be interpreted as the global material coordinate system’s covariant base vectors expressed using contravariant components in the global spatial coordinate system. Conversely, columns can be interpreted as the global spatial coordinate system’s contravariant base vectors expressed using covariant components in the global material coordinate system. The matrix therefore transforms contravariant material components of a vector, multiplying the matrix from the left, into contravariant spatial components. It also transforms covariant spatial components of a vector, by multiplication from the right, into covariant material components.
Note that spatial.F is the transpose of the deformation gradient defined by Structural Mechanics.
spatial.invF
The matrix spatial.invF contains the inverse of spatial.F, which consists of components of the Jacobian of material coordinates with respect to spatial coordinates:
Rows of the matrix can be interpreted as the global spatial coordinate system’s covariant base vectors expressed using contravariant components in the global material coordinate system. Conversely, columns can be interpreted as the global material coordinate system’s contravariant base vectors expressed using covariant components in the global spatial coordinate system. The matrix therefore transforms contravariant spatial components of a vector, multiplying the matrix from the left, into contravariant material components. It also transforms covariant material components of a vector, by multiplication from the right, into covariant spatial components.
spatial.detF and spatial.detInvF
The volume/area/length factor spatial.detF and its inverse spatial.detInvF are ratios of local element volume/area/length computed in the spatial and material frames. That is, these variables are the factors necessary to compute a spatial frame integral of quantity over a set of elements of any dimension, using material frame coordinates, or the other way around:
where D is a set of volume, surface, line, or point elements. This means that spatial.detF is a ratio of volumes in domains of 3D models, a ratio of areas in domains of plane 2D models and on boundaries of 3D models, and a ratio of lengths on boundaries of 2D models and edges in 3D models. On points, spatial.detF is equal to 1.
material.F, material.invF, material.detF, and material.detInvF
These variables are defined in complete analogy with the corresponding spatial variables, but based on the relation between material frame coordinates X and geometry frame coordinates Xg.
Axisymmetric Geometries
Axisymmetric geometries are treated like 3-dimensional solids for the purpose of defining transformation matrices and volume factors variables. In particular, radial displacement of the mesh induces a nonunit transformation in the azimuthal direction, in analogy with the azimuthal strain resulting from a radial displacement in an axisymmetric solid object. The transformation matrices spatial.F and spatial.InvF are:
and
The volume/area/length factors spatial.detF and spatial.detInvF also include the azimuthal strain factor r/R, such that they fulfill
where D is any set of surface, line, or point elements in an axisymmetric geometry. This means that spatial.detF is a ratio of volumes in domains, a ratio of areas on boundaries and a ratio of lengths on point entities.