Add a Matrix Transform node (
) to define variables representing the components of an input matrix transformed to another coordinate system in a 3D vector space. You add it by right-clicking the
Definitions node and choosing
Variable Utilities>Matrix Transform. If
Group by Type has been selected for the
Definitions branch, then you can also add it by right-clicking the
Variable Utilities node and choosing
Matrix Transform.
Use the Name field to select a namespace for the input and output vector components, as well as for components of the transformation matrix.
In the Domain Selection section, select the domains in which this transformation is valid.
In addition, the Settings window for a
Matrix Transform node contains the following sections:
The choice of Matrix format,
Full or
Symmetric, controls the available coordinate system settings. For a
Full matrix (the default), you must specify both the
Row Coordinate System and the
Column Coordinate System in which this tensor is represented. For a
Symmetric matrix, the row and column system must be the same and only a single
Coordinate system setting is needed. When the matrix is entered manually, the default coordinate system settings are always the
Global spatial system; when using
Replace Expression, the default is inferred from the properties of the selected matrix.
If a selected coordinate system is a relative system such that its reference frame cannot be deduced from its definition, select an appropriate
Reference frame. This selection is only shown when a relative input system is selected.
If the relation between an input coordinate system and a corresponding output coordinate system is not a pure rotation, you must specify a component type for each affected index. In the corresponding Treat components as,
Treat row indices as or
Treat column indices as drop down, select whether the corresponding input matrix index is
Covariant or
Contravariant (the default). When using
Replace Expression, the default is inferred from the properties of the selected matrix.
Select the Row Coordinate System and the
Column Coordinate System in which the transformed matrix will be represented. Note that row and column system are set separately also when the input matrix is symmetric. To retain symmetry in the output components, select the same system for both row and column indices. If any of the selected systems is a relative system, also select a corresponding reference frame.
The components of the transformed matrix in those systems become available as variables <name>.V<xi><xj>, where
<name> is the name of the node set in the
Name field, and
<xi> and
<xj> are the coordinates with indices
i and
j. The components of the input matrix also become available as
<name>.U<i><j>. For example, if the
Matrix Transform node has the name
mattr1, the first component of an input matrix being transformed to the spatial frame in both indices becomes
mattr1.U11, while the first component of the output matrix becomes
mattr1.Vxx. Both matrices also become available as matrix objects which can be evaluated using a matrix evaluation node under
Derived Values or selected as input for another Matrix Transform.
The row transformation matrix also becomes available as <name>.Tr, while the column transformation matrix becomes available as
<name>.Tc. These matrices are the composition of all transformation matrices used for the respective index. For example, if the column indices are changed between covariant and contravariant at the same time as the coordinate system for the columns is changed, the matrix
<name>.Tc is the product of the metric and a coordinate transformation matrix.
Choose Transform as, which determines how the input matrix transforms between the chosen coordinate systems. If the option
Generalized tensor density is chosen, the choices of
Input volume reference and
Output volume reference for the change in volume between coordinate systems become available.
Since you can use this option to choose the reference systems for the determinant of the Jacobian, it can be used to transform matrices as either tensor densities or tensor capacities. It can even be used to take into account the volume change between other coordinate systems. The Tensor option transforms matrices as normal tensors, not taking into account the volume change. For convenience, there is one more option, called
Stress Tensor. What it means is that the tensor transforms as a tensor density where the volume reference change is taken from the choice of input and output
column coordinate systems.
As a default, the output matrix will have the same type of basis elements as the input matrix. You can force the representation of the each of the indices of the output matrix to be covariant or contravariant, independent of the form of the input matrix. In order to do this, change Row Component Change or
Column Component Change from the default value
None. Depending on how the indices were set in the
Input section, the other options can be
Covariant->Contravariant and
Contravariant->Covariant. The two indices on the matrix components will be raised or lowered accordingly.
When choosing anything other than None, the choices of which metric to use for the row transformation, as well as which metric to use for the column transformation, become available. Here, choose these metrics as either
Spatial or
Material.