Symmetric Tensor Field Functions

STAR-CCM+ provides support for symmetric tensor-valued field functions. These functions are stored internally as six floating point values for a 3 x 3 symmetric matrix.

Symmetric tensor-valued field functions are currently only produced by the finite-element solid stress module, but they can be constructed and used for general calculations. Similar to vector-valued field functions, you can access individual components of symmetric tensors, perform coordinate transformations on them, and compute common algebraic operations on them.

Construction

In the field function expression editor, a symmetric tensor-valued variable must be prefixed with $$$.

Symmetric tensor variables are constructed in the following way:


Operation	Expression Syntax	Result
3 x 3, lower-triangular specification	[1 ; 2,3 ; 4,5,6]	$A = [\begin{matrix} 1 & 2 & 4 \\ 2 & 3 & 5 \\ 4 & 5 & 6 \end{matrix}]$
3 x 3, upper-triangular specification	[1,2,3 ; 4,5 ; 6]	$A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{matrix}]$

Note	Symmetric tensors support all basic arithmetic operations, except for multiplication of two symmetric tensors. Multiplication of two symmetric tensors can result in a non-symmetric tensor and so the current version of STAR-CCM+ does not support this operation.

Coordinate System Transformations

Symmetric tensors are transformed into a local coordinate system through similarity transformation. For example, the user-defined field function expression

$$$A( @CoordinateSystem("Laboratory.Sph1") )

transforms the symmetric tensor $A$ into the local spherical coordinate system Sph1, whose parent is the Laboratory coordinate system. Let $R$ denote the 3 x 3 orthogonal matrix which transforms a vector $v_{s}$ from Sph1 into Laboratory coordinates. Then the mathematical form of this transformation is:

Figure 1. EQUATION_DISPLAY

A_{s} = R^{T} A R

(10)

The columns of $R$ are the (position-dependent) local basis in Sph1.

The components of $A_{s}$ are now with respect to the local basis at that particular location within Sph1. (Recall that the unit basis vectors of a local curvilinear coordinate system depend on the coordinate within that local frame.) The matrix-vector product of $A_{s}$ with $v_{s}$ ,

Figure 2. EQUATION_DISPLAY

A_{s} v_{s} = R^{T} A R v_{s}

(11)

is therefore equivalent to transforming $v_{s}$ into the Laboratory frame, applying $A$ , and then transforming the result back into Sph1. In STAR-CCM+, it is possible to verify that for any symmetric tensor $A$ and any vector $v$ (both in the Laboratory frame), the expression

$$$A(@CoordinateSystem("Laboratory.Sph1")) *
$$v(@CoordinateSystem("Laboratory.Sph1"))

produces values that are identical to the ones that are produced by

($$$A*$$v)(@CoordinateSystem("Laboratory.Sph1"))

since

Figure 3. EQUATION_DISPLAY

(R^{T} A R) (R^{T} v) = R^{T} (A v)

(12)

Coordinate System Components

When selecting a component of a symmetric tensor in each coordinate system, the following labels correspond to the appropriate matrix entry:


Coordinate System Type	Tensor Component
Laboratory	$\begin{matrix} {i i = a}_{00} & i j = a_{01} & i k = a_{02} \\ j j = a_{11} & j k = a_{12} & k k = a_{22} \end{matrix}$
Local Cartesian	$\begin{matrix} {i i = a}_{00} & i j = a_{01} & i k = a_{02} \\ j j = a_{11} & j k = a_{12} & k k = a_{22} \end{matrix}$
Cylindrical	$\begin{matrix} {r r = a}_{00} & r θ = a_{01} & r z = a_{02} \\ θ θ = a_{11} & θ z = a_{12} & z z = a_{22} \end{matrix}$
Spherical	$\begin{matrix} {r r = a}_{00} & r θ = a_{01} & r ϕ = a_{02} \\ θ θ = a_{11} & θ ϕ = a_{12} & ϕ ϕ = a_{22} \end{matrix}$