Tensor type

Type parameters

All tensors in Tensorial.jl are represented by the type Tensor{S, T, N, L}, where each type parameter represents the following:

• S: The size of Tensors which is specified by using Tuple (e.g., 3x2 tensor becomes Tensor{Tuple{3,2}}).
• T: The type of element which must be T <: Real.
• N: The number of dimensions (the order of tensor).
• L: The number of independent components.

Basically, the type parameters N and L do not need to be specified for constructing tensors because it can be inferred from the size of tensor S.

Symmetry

Specifying the symmetry of a tensor can improve performance, as Tensorial.jl provides optimized computations for symmetric tensors. Symmetries can be applied using Symmetry in the type parameter S (e.g., Symmetry{Tuple{3,3}}). The @Symmetry macro simplifies this process by allowing you to omit Tuple, as in @Symmetry{2,2}. Below are some examples of how to specify symmetries:

• $A_{(ij)}$ with 3x3: Tensor{Tuple{@Symmetry{3,3}}}
• $A_{(ij)k}$ with 3x3x2: Tensor{Tuple{@Symmetry{3,3}, 2}}
• $A_{(ijk)}$ with 3x3x3: Tensor{Tuple{@Symmetry{3,3,3}}}
• $A_{(ij)(kl)}$ with 3x3x3x3: Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}

where the bracket $()$ in indices denotes the symmetry.