4.4.10. Local Coordinate System

Local cartesian, cylindrical, and spherical coordinates systems are available in Aria. The figure below shows each local coordinate system definition.

../../_images/localCoordSys_Cartesian.png

../../_images/localCoordSys_Cylindrical.png

../../_images/localCoordSys_Spherical.png

Local cartesian, cylindrical, and spherical coordinate systems.

Local coordinate systems are particularly useful for defining tensor material properties in terms of their local principal values. The input deck below shows the incremental change needed to define a tensor thermal conductivity for a generalized heat conduction model using a local cartesian coordinate system.

Begin Sierra myJob

  ...

  Begin Aria Material mat_cartesian
    ...
    # Define principal values of tensor thermal conductivity
    tensor thermal conductivity = constant T11=100.0 T22=45.0 T33=20.0
    heat conduction = generalized
  End

  # Define a local cartesian coordinate system
  Begin local coordinate system my_cartesian
    TYPE   = cartesian
    ORIGIN =  0.0  0.0  0.0
    POINT  =  5.0  0.0  0.0
    VECTOR =  0.0  0.0  1.0
  End

  Begin Finite Element Model cube
    ...
    coordinate system is cartesian

    Begin parameters for block block_2
      material mat_cartesian

      # Define a local cartesian coordinate system only for block_2
      local coordinate system = my_cartesian
    End
  End
  ...
End

Here, the line command local coordinate system in the parameters block refers to a local coordinate system command block. This command block defines the TYPE of coordinate system (i.e., cartesian, cylindrical, or spherical), ORIGIN of the coordinate system, as well as a VECTOR, and POINT. Together, this information is used to define the principal directions ${\bf e}_{1}$ , ${\bf e}_{2}$ and ${\bf e}_{3}$ as is discussed below.

In the material block, the tensor thermal conductivity material model defines the principal values $K_{11}$ , $K_{22}$ and $K_{33}$ corresponding to principal directions ${\bf e}_{1}$ , ${\bf e}_{2}$ and ${\bf e}_{3}$ respectively. With ${\bf K}$ defined locally, the tensor components in global coordinates are defined as

(4.21) ${\bf K}^{'} = {\bf Q} {\bf K} {\bf Q}^T$

where the transformation matrix ${\bf Q}$ is defined as

(4.22) ${\bf Q}_{ij} = {\bf e}_i^{'} \cdot {\bf e}_j.$

Here, ${\bf e}_i^{'}$ is the global cartesian basis, resulting in a transformation matrix ${\bf Q}$ whose columns are the principal directions ${\bf e}_j$ . Physically, ${\bf Q}$ transforms local vector coordinates to global vectors coordinates, whereas ${\bf Q}^T$ transforms global vector coordinates to local vector coordinates.

The line commands ORIGIN, POINT, and VECTOR define vectors ${\bf o}$ , ${\bf p}$ , and ${\bf v}$ respectively. For the cartesian coordinate system, the principal directions do not vary with position and are defined as

(4.23) ${\bf e}_{1} = \dfrac{{\bf d}}{||{\bf d}||_2} \quad \quad {\bf e}_{2} = {\bf e}_{3} \times {\bf e}_{1} \quad \quad {\bf e}_{3} = \dfrac{\bf v}{||\bf v||}_2 ,$

where ${\bf d} = \left({\bf I} - {\bf e}_3 {\bf e}_3^T\right) \cdot \left({\bf p} - {\bf o}\right)$ is independent of position. Here, VECTOR defines the local z-axis and POINT determines the local x-axis such that it is orthogonal to the local z-axis.

For the cylindrical and spherical coordinate systems, the principal directions vary from point to point within a material block. At a given point ${\bf r}$ , the cylindrical principal directions are defined as:

(4.24) ${\bf e}_{1} = \dfrac{{\bf d}\left({\bf r}\right)}{||{\bf d}\left({\bf r}\right)||_2} \quad \quad {\bf e}_{2} = {\bf e}_{3} \times {\bf e}_{1} \quad \quad {\bf e}_{3} = \dfrac{\bf v}{||\bf v||}_2 ,$

where ${\bf d}\left({\bf r}\right) = \left({\bf I} - {\bf e}_3 {\bf e}_3^T\right) \cdot \left({\bf r} - {\bf o}\right)$ depends on the position now. VECTOR defines the local z-axis of the cylinder and so these principal directions correspond to $r$ , $\theta$ , and $z$ directions of a cylinder, respectively.

Similarly, at a given point ${\bf r}$ , the spherical principal directions are defined as:

(4.25) ${\bf e}_{1} = \dfrac{{\bf r} - {\bf o}}{||{\bf r} - {\bf o}||}_2 \quad \quad {\bf e}_2 = {\bf v} \times {\bf e}_1 \quad \quad {\bf e}_{3} = {\bf e}_1 \times {\bf e}_2.$

Here, the principal directions correspond to $r$ , $\theta$ , and $\phi$ directions of the sphere. The command reference for the local coordinate system can be found here.