TensND.jl

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TensND.jlTensors on any coordinate system

Bases and variance, structured tensor types, symmetry projection, and differential operators in curvilinear coordinates — symbolic or numeric, one implementation.

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Theory
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TensND
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Theory

Bases and variance, the tensor products, Kelvin-Mandel storage, rotations, and the symmetry classes with their Walpole algebra.

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Manual

Every type and operator with the call that produces it — bases, tensors, structured types, projection, coordinate systems, submanifolds.

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Tutorials

Worked sequences, from tensor algebra to differential calculus and applications in mechanics, each runnable end to end.

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Structured types

TensISO, TensTI, TensCubic and TensOrtho store 2, 5, 3 and 9 scalars and compute products and inverses in closed form, orders of magnitude faster than the dense route.

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API reference

Every exported type and function, grouped by topic.

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Developer guide

How the package is put together, and how to add a coordinate system without touching the rest.

What it does ​

TensND handles tensors of any order in any coordinate system. A basis carries its metric, so covariant and contravariant components are related rather than assumed; the differential operators follow from the chart, symbolically or by automatic differentiation.

Three structured types — TensISO, TensTI and TensCubic, TensOrtho — store the 2, 5, 3 and 9 scalars a symmetry class really has, and compute products and inverses in closed form. The same code runs on Float64, ForwardDiff.Dual, SymPy.Sym and Symbolics.Num.

The design is inspired by the Maple library Tens3d of Jean Garrigues.

julia
using TensND, SymPy

Spherical = coorsys_spherical()
θ, ϕ, r = getcoords(Spherical)
𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ = unitvec(Spherical)
@set_coorsys Spherical

σʳʳ = SymFunction("σʳʳ", real = true)(r)
σᶿᶿ = SymFunction("σᶿᶿ", real = true)(r)
𝛔 = σʳʳ * 𝐞ʳ ⊗ 𝐞ʳ + σᶿᶿ * (𝐞ᶿ ⊗ 𝐞ᶿ + 𝐞ᵠ ⊗ 𝐞ᵠ)

pprint(DIV(𝛔))
(Derivative(σʳʳ(r), r) + 2*σʳʳ(r)/r - 2*σᶿᶿ(r)/r)𝐞ʳ

The equilibrium equation of a spherically symmetric stress state, derived rather than transcribed.