Scalars, Vectors, and Tensors

Most physical quantities in Simcenter STAR-CCM+ are either scalars, vectors, or 2nd-order tensors.

Scalar: A scalar is a quantity that is completely defined by its magnitude.; For example, the temperature at a particular point in time and space is completely defined by a dimensional number, such as $T = - 273 ° C$ .; In general, the value of a scalar quantity can depend on other variables, such as space and time. In this case, the scalar field of values at different locations and times can be described using a scalar function.
Vector: A vector is a quantity that is completely defined by its magnitude and direction. For example, the electric field and velocity are vector quantities.; A vector can be represented geometrically as an arrow, which defines the vector direction, with the length of the arrow representing the vector magnitude.; Like scalars, the magnitude and direction of a vector can depend on other variables, such as space and time. In this case, the vector quantity can be described using a vector function.
Tensor: A tensor is a quantity that is completely defined by its magnitude and direction on different planes of action. For example, stress and strain are tensor quantities. Like scalars and vectors, tensors can depend on other variables, such as space and time. In this case, the tensor quantity can be described using a tensor function.

Vector and Tensor Representations

As vectors and tensors are directional quantities, it is convenient to define them with respect to a coordinate system (see Coordinate Systems).

Considering the 3D Euclidean space, a vector $a$ can be written as:

Figure 1. EQUATION_DISPLAY

\begin{matrix} a = \end{matrix} a_{i} e_{i}

(5184)

where

\begin{matrix} e_{i} & i = 1, 2, 3 \end{matrix}

are the basis vectors in some coordinate system (for example, the unit vectors

i, j, k

of a Cartesian coordinate system) and

\begin{matrix} a_{i} \end{matrix}

are the components of

a

with respect to the basis vectors.

This formulation uses the Einstein summation convention, which implies summation over a set of repeated indices:

Figure 2. EQUATION_DISPLAY

a_{i} e_{i} \equiv \sum_{i = 1}^{3} a_{i} e_{i} = a_{1} e_{1} + a_{2} e_{2} + a_{3} e_{3}

(5185)

Similarly, a 2nd-order tensor $A$ can be written as:

Figure 3. EQUATION_DISPLAY

\begin{matrix} A = \end{matrix} A_{i j} e_{i} \otimes e_{j}

(5186)

where $e_{i} \otimes e_{j}$ is the dyadic product, or tensor product, between the basis vectors (see Eqn. (5196)).

Eqn. (5184) and Eqn. (5186) are in index notation, which is also called tensor notation. The number of indices that are required to identify the tensor components define the order of the tensor. In fact, scalars and vectors can be considered as 0th-order and 1st-order tensors, respectively.

For a tensor component

A_{i j}

, the first index identifies the plane the component acts on, whereas the second index identifies the direction. For example, consider the components of the stress tensor

σ_{i j}

in a Cartesian coordinate system

X, Y, Z

The component $σ_{x x}$ identifies the stress component in the $X$ direction acting on the plane normal to the $X$ axis (that is, the plane $Y Z$ ). The component $σ_{z y}$ identifies the stress component in the $Y$ direction acting on the plane normal to the $Z$ axis (that is, the plane $X Y$ ).

A convenient notation consists of representing the components of vectors and tensors, with respect to a vector basis, as matrices. In matrix representation, the vector basis is implied.

In the 3D Euclidean space, the components of a vector $a$ can be represented as a one-dimensional array of length 3:

Figure 4. EQUATION_DISPLAY

\begin{matrix} a = (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) \end{matrix}

(5187)

which is called the column vector representation of $a$ . The vector $a$ can also be defined using the row vector representation:

Figure 5. EQUATION_DISPLAY

\begin{matrix} a = (\begin{matrix} a_{1} & a_{2} & a_{3} \end{matrix}) \end{matrix}

(5188)

The choice of row or column representation is purely conventional. Defining row and column representations allows you to write the scalar product and the dyadic product between two vectors as matrix products (see Common Operations).

In the 3D Euclidean space, the components of a 2nd-order tensor $A$ can be represented as a 3x3 matrix:

Figure 6. EQUATION_DISPLAY

A = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix})

(5189)

Tensors can be defined as covariant or contravariant based on how their components transform under a change of coordinate system. This formal description is beyond the purpose of this overview.