Getting started

Brief description of the package

The package relies on the definition of

• bases which can be of the following types (T denotes the scalar type, subtype of Number)

  • CanonicalBasis{dim,T}: fundamental canonical basis in ℝᵈⁱᵐ in which the metric tensor is the second-order identity
  • RotatedBasis{dim,T}: orthonormal basis in ℝᵈⁱᵐ obtained by rotation of the canonical basis by means of one angle if dim=2 or three Euler angles if dim=3, the metric tensor is again the second-order identity
  • OrthogonalBasis{dim,T}: orthogonal basis in ℝᵈⁱᵐ obtained from a given orthonormal rotated basis by applying a scaling factor along each unit vector, the metric tensor is then diagonal
  • Basis{dim,T}: arbitrary basis not entering the previous cases

• tensors

  • a tensor is determined by a set of data (array or synthetic parameters) corresponding to its order, a basis and a tuple of variances
  • depending on the type of basis, the type of tensor can be TensCanonical{order,dim,T,A}, TensRotated{order,dim,T,A}, TensOrthogonal{order,dim,T,A} or Tens{order,dim,T,A} if the data are stored under the form of an array or a Tensor object (see Tensors.jl), or the type can be TensISO{order,dim,T,N} if the tensor is isotropic and data are stored under the form of a set of parameters (one for order 2 and two for order 4).

  NOTE: More material symmetry types such as transverse isotropy or orthotropy will be added in the future.

• coordinate systems

  • a coordinate system contains all information required to perform differential operations on tensor fields: position vector OM expressed in the canonical basis, coordinate names xⁱ, natural basis aᵢ=∂ᵢOM, normalized basis eᵢ=aᵢ/||aᵢ||, Christoffel coefficients Γᵢⱼᵏ=∂ᵢaⱼ⋅aᵏ where (aⁱ)(1≤i≤dim) form the dual basis associated to the natural one
  • predefined coordinate systems are available: cartesian, polar, cylindrical, spherical and spheroidal but the user can define new systems

  NOTE: Note that for the moment the coordinate systems and differential operators have been implemented only for symbolic calculations (using SymPy.jl). Numerical coordinate systems and differential operators based on automatic differentiation will be implemented in the future.

Detailed manual

Before detailing explanations about the main features of TensND, it is worth recalling that the use of the libraries TensND and SymPy requires starting scripts by

julia> using TensND, SymPy

The detailed manual is decomposed into the following chapters

• Bases
• Tensors
• Coordinate systems and differential operators