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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)
    • TensISO{order,dim,T,N}: isotropic tensor stored as N scalar parameters (1 for order 2, 2 for order 4)
    • TensWalpole{T,N}: transversely isotropic 4th-order tensor in the Walpole basis (N=5 for major-symmetric, N=6 for general case)
    • TensTI{order,T,N}: transversely isotropic tensor (order 2 or 4)
    • TensOrtho{T}: orthotropic 4th-order tensor with 9 elastic constants in a material frame
    • Projection functions proj_tens and best_sym_tens allow finding the closest tensor of a given symmetry class (ISO, TI, ORTHO) with or without rotation optimization (the latter requires NLopt)

  • coordinate systems

    • Symbolic (CoorSystemSym): a coordinate system contains all information required to perform differential operations on tensor fields: position vector OM expressed in the canonical basis, coordinate names, natural basis, normalized basis, Christoffel coefficients
    • Numerical (CoorSystemNum): pointwise evaluation of differential operators via automatic differentiation (ForwardDiff), without requiring symbolic setup
    • Predefined systems are available for both: cartesian, polar, cylindrical, spherical and spheroidal. The user can also define custom systems

All types support generic scalar types (Sym, Num, Float64, ForwardDiff.Dual, ...) enabling seamless interoperability between symbolic computation and automatic differentiation.

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

For numerical computations (without symbolic overhead), SymPy is not required:

julia> using TensND

The detailed manual is decomposed into the following chapters

Tutorials

Several tutorials demonstrate advanced usage of TensND in various contexts: