TensorField topology over FrameBundle ∇ with Grassmann.jl elements
Provides TensorField{B,F,N} <: GlobalFiber{LocalTensor{B,F},N} implementation for both a local ProductSpace and general ImmersedTopology specifications on any FrameBundle expressed with Grassmann.jl algebra.
Many of these modular methods can work on input meshes or product topologies of any dimension, although there are some methods which are specialized.
Building on this, Cartan provides an algebra for FiberBundle sections and associated bundles on a manifold, such as general Connection, LieDerivative, and CovariantDerivative operators in terms of Grassmann elements.
Calculus of Variation fields can also be generated with the combined topology of a FiberProductBundle.
Furthermore, the FiberProduct structure enables construction of HomotopyBundle types.
Utility package for differential geometry and tensor calculus intended for Adapode.jl.
The Cartan package is intended to standardize the composition of various methods and functors applied to specialized categories transformed with a unified representation over a product topology, especially having fibers of the Grassmann algebra.
Initial topologies include ProductSpace types and in general the ImmersedTopology.
Positions{P, G} where {P<:Chain, G} (alias for AbstractArray{<:Coordinate{P, G}, 1} where {P<:Chain, G})
Interval{P, G} where {P<:AbstractReal, G} (alias for AbstractArray{<:Coordinate{P, G}, 1} where {P<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, G})
IntervalRange{P, G, PA, GA} where {P<:Real, G, PA<:AbstractRange, GA} (alias for GridBundle{1, Coordinate{P, G}, <:PointArray{P, G, 1, PA, GA}} where {P<:Real, G, PA<:AbstractRange, GA})
Rectangle (alias for ProductSpace{V, T, 2, 2} where {V, T})
Hyperrectangle (alias for ProductSpace{V, T, 3, 3} where {V, T})
RealRegion{V, T} where {V, T<:Real} (alias for ProductSpace{V, T, N, N, S} where {V, T<:Real, N, S<:AbstractArray{T, 1}})
RealSpace{N} where N (alias for AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G})
AlignedRegion{N} where N (alias for GridBundle{N, Coordinate{P, G}, PointArray{P, G, N, PA, GA}} where {N, P<:Chain, G<:InducedMetric, PA<:(ProductSpace{V, <:Real, N, N, <:AbstractRange} where V), GA<:Global})
AlignedSpace{N} where N (alias for GridBundle{N, Coordinate{P, G}, PointArray{P, G, N, PA, GA}} where {N, P<:Chain, G<:InducedMetric, PA<:(ProductSpace{V, <:Real, N, N, <:AbstractRange} where V), GA})
FrameBundle{Coordinate{B,F},N} where {B,F,N}
GridBundle{N,C,PA<:FiberBundle{C,N},TA<:ImmersedTopology} <: FrameBundle{C,N}
SimplexBundle{N,C,PA<:FiberBundle{C,1},TA<:ImmersedTopology} <: FrameBundle{C,1}
FaceBundle{N,C,PA<:FiberBundle{C,1},TA<:ImmersedTopology} <: FrameBundle{C,1}
FiberProductBundle{P,N,SA<:AbstractArray,PA<:AbstractArray} <: FrameBundle{Coordinate{P,InducedMetric},N}
HomotopyBundle{P,N,PA<:AbstractArray{F,N} where F,FA<:AbstractArray,TA<:ImmersedTopology} <: FrameBundle{Coordinate{P,InducedMetric},N}
Visualizing TensorField reperesentations can be standardized in combination with Makie.jl or UnicodePlots.jl.
Due to the versatility of the TensorField type instances, it's possible to disambiguate them into these type alias specifications with associated methods:
ElementMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:ElementBundle, A})
SimplexMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:SimplexBundle, A})
FaceMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:FaceBundle, A})
IntervalMap (alias for TensorField{B, F, 1, P, A} where {B, F, P<:(AbstractArray{<:Coordinate{P, G}, 1} where {P<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, G}), A})
RectangleMap (alias for TensorField{B, F, 2, P, A} where {B, F, P<:(AbstractMatrix{<:Coordinate{P, G}} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
HyperrectangleMap (alias for TensorField{B, F, 3, P, A} where {B, F, P<:(AbstractArray{<:Coordinate{P, G}, 3} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
ParametricMap (alias for TensorField{B, F, N, P, A} where {B, F, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G}), A})
Variation (alias for TensorField{B, F, N, P, A} where {B, F<:TensorField, N, P, A})
RealFunction (alias for TensorField{B, F, 1, P, A} where {B, F<:AbstractReal, PA<:(AbstractVector{<:AbstractReal}), A})
PlaneCurve (alias for TensorField{B, F, N, P, A} where {B, F, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {N, P<:(Chain{V, 1, <:Real} where V), G}), A})
SpaceCurve (alias for TensorField{B, F, 1, P, A} where {B, F<:(Chain{V, G, Q, 3} where {V, G, Q}), P<:(AbstractVector{<:Coordinate{P, G}} where {P<:AbstractReal, G}), A})
AbstractCurve (alias for TensorField{B, F, 1, P, A} where {B, F<:Chain, P<:(AbstractVector{<:Coordinate{P, G}} where {P<:AbstractReal, G}), A})
SurfaceGrid (alias for TensorField{B, F, 2, P, A} where {B, F<:AbstractReal, P<:(AbstractMatrix{<:Coordinate{P, G}} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
VolumeGrid (alias for TensorField{B, F, 3, P, A} where {B, F<:AbstractReal, P<:(AbstractArray{<:Coordinate{P, G}, 3} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
ScalarGrid (alias for TensorField{B, F, N, P, A} where {B, F<:AbstractReal, N, P<:(AbstractArray{<:Coordinate{P, G}, N} where {P<:(Chain{V, 1, <:Real} where V), G}), A})
DiagonalField (alias for TensorField{B, F, N, P, A} where {B, F<:DiagonalOperator, N, P, A})
EndomorphismField (alias for TensorField{B, F, N, P, A} where {B, F<:(TensorOperator{V, V, T} where {V T<:(TensorAlgebra{V, <:TensorAlgebra{V}})}), N, P, A})
OutermorphismField (alias for TensorField{B, F, N, P, A} where {B, F<:Outermorphism, N, P, A})
CliffordField (alias for TensorField{B, F, N, P, A} where {B, F<:Multivector, N, P, A})
QuaternionField (alias for TensorField{B, F, N, P, A} where {B, F<:(Quaternion), N, P, A})
ComplexMap (alias for TensorField{B, F, N, P, A} where {B, F<:(Union{Complex{T}, Single{V, G, B, Complex{T}} where {V, G, B}, Chain{V, G, Complex{T}, 1} where {V, G}, Couple{V, B, T} where {V, B}, Phasor{V, B, T} where {V, B}} where T<:Real), N, P, A})
PhasorField (alias for TensorField{B, F, N, P, A} where {B, F<:Phasor, N, P, A})
SpinorField (alias for TensorField{B, F, N, P, A} where {B, F<:AbstractSpinor, N, P, A})
GradedField{G} where G (alias for TensorField{B, F, N, P, A} where {G, B, F<:(Chain{V, G} where V), N, P, A})
ScalarField (alias for TensorField{B, F, N, P, A} where {B, F<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, N, P, A})
VectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 1} where V), N, P, A})
BivectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 2} where V), N, P, A})
TrivectorField (alias for TensorField{B, F, N, P, A} where {B, F<:(Chain{V, 3} where V),N, P, A})In the Cartan package, a technique is employed where a TensorField is constructed from an interval, product manifold, or topology, to generate an algebra of sections which can be used to compose parametric maps on manifolds.
Constructing a TensorField can be accomplished in various ways,
there are explicit techniques to construct a TensorField as well as implicit methods.
Additional packages such as Adapode build on the TensorField concept by generating them from differential equations.
Many of these methods can automatically generalize to higher dimensional manifolds and are compatible with discrete differential geometry.
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developed by chakravala with Grassmann.jl
