TensorProducts

Mantis.TensorProducts Module

module TensorProducts

Utilities for constructing and working with tensor-products of sets.

A TensorProduct represents the tensor-product of multiple, lower-dimensional factor sets, providing efficient conversion between linear indices and Cartesian indices of the product space. This module provides the shared infrastructure for this pattern: a generic interface for creating and indexing tensor-products of arbitrary sets.

source

The module is intentionally minimal. Its only task is to take any tuple of factor sets, pre-compute the bijection between a global linear index and the tuple of per-set indices, and provide utilities to traverse and map over the factors. Modules such as Geometry and FunctionSpaces implement a small interface and delegate all index bookkeeping to TensorProducts, keeping the tensor-product logic in one place rather than scattered across the codebase.

The TensorProduct Type

The central type in this module is TensorProduct.

Mantis.TensorProducts.TensorProduct Type

TensorProduct{I,CI,LI}

Represents the tensor (Cartesian) product of a tuple of factor sets.

A TensorProduct stores the factor sets together with the corresponding CartesianIndices and LinearIndices, allowing efficient conversion between linear and multi-dimensional indexing.

Fields

• factors::I: Tuple of factor sets.

• cart_ids::CI: Cartesian indices of the tensor product.

• lin_ids::LI: Linear index mapping for the Cartesian indices.

source

For convenience, the module exports the symbolic operator ⊗:

Mantis.TensorProducts.:⊗ Type

Alias for TensorProduct.

source

Two or more sets can also be freely composed. These compositions are automatically flattened: (A ⊗ B) ⊗ C and A ⊗ (B ⊗ C) are both equivalent to A ⊗ B ⊗ C, producing a single TensorProduct with three factors rather than a nested pair.

The get_num_objects requirement

For a set to be used in a TensorProduct, it must implement one function:

Mantis.TensorProducts.get_num_objects Function

get_num_objects(set::Any)

Return the number of objects contained in set.

This function must be implemented for any type that is used as an factor to a TensorProduct.

Defaults are defined in later modules for Geometry.AbstractGeometry and FunctionSpaces.AbstractFESpace, namely Geometry.get_num_elements and FunctionSpaces.get_num_basis.

Examples

julia> using Mantis

julia> TensorProducts.get_num_objects(["a", "b", "c"]) == 3
true

julia> TensorProducts.get_num_objects(Geometry.create_cartesian_box((0.0, 0.0), (1.0, 1.0), (2, 2))) == 4
true

julia> TensorProducts.get_num_objects(word::String) = length(word); nothing

julia> TensorProducts.get_num_objects("Pequod") == 6
true

source

This is the only requirement imposed by the module. Standard Julia containers (Tuple and AbstractArray) satisfy it out of the box using length. Custom types — such as Mantis.Geometry.AbstractGeometry or Mantis.FunctionSpaces.AbstractFESpace — implement this function to expose their element or basis count, respectively. This keeps the TensorProducts module free of any dependency on the rest of Mantis.

Index Bookkeeping

The central service provided by a TensorProduct is the efficient bidirectional mapping between a single linear index and a tuple of per-factor indices. This mapping is pre-computed once at construction time and stored internally. Mantis has a few helpers for this purpose:

Mantis.TensorProducts.get_cart_ids Function

get_cart_ids(tp::TensorProduct)

Return the CartesianIndices associated with the tensor product.

source

Mantis.TensorProducts.get_lin_ids Function

get_lin_ids(tp::TensorProduct)

Return the LinearIndices associated with the tensor product.

source

Mantis.TensorProducts.get_factor_ids Function

get_factor_ids(tp::TensorProduct, id)

Return the tuple of indices corresponding to the linear index id. The tuple can be used to index each factor of tp.

source

Mantis.TensorProducts.get_factor_num_objects Function

get_factor_num_objects(tp::TensorProduct)

Return a tuple containing the number of objects in each factor set.

source

Mantis.TensorProducts.get_num_factors Function

get_num_factors(tp::TensorProduct)

Return the number of factor sets in the tensor product.

source

Mantis.TensorProducts.get_num_objects Method

get_num_objects(set::Any)

Return the number of objects contained in set.

This function must be implemented for any type that is used as an factor to a TensorProduct.

Defaults are defined in later modules for Geometry.AbstractGeometry and FunctionSpaces.AbstractFESpace, namely Geometry.get_num_elements and FunctionSpaces.get_num_basis.

Examples

julia> using Mantis

julia> TensorProducts.get_num_objects(["a", "b", "c"]) == 3
true

julia> TensorProducts.get_num_objects(Geometry.create_cartesian_box((0.0, 0.0), (1.0, 1.0), (2, 2))) == 4
true

julia> TensorProducts.get_num_objects(word::String) = length(word); nothing

julia> TensorProducts.get_num_objects("Pequod") == 6
true

source

An example should make it clear.

julia> using Mantis

julia> tp = TensorProducts.TensorProduct(("a", "b"), ("x", "y", "z"))  # 2 × 3 product
Mantis.TensorProducts.TensorProduct{Tuple{Tuple{String, String}, Tuple{String, String, String}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}((("a", "b"), ("x", "y", "z")), CartesianIndices((2, 3)), LinearIndices((Base.OneTo(2), Base.OneTo(3))))

julia> TensorProducts.get_num_objects(tp)
6

julia> TensorProducts.get_factor_ids(tp, 4)  # (2, 2): second of "a","b", second of "x","y","z"
(2, 2)

julia> TensorProducts.get_factor_num_objects(tp)
(2, 3)

Accessing and Mapping factors

The following functions provide access to the factor sets and allow applying operations to them in a dimension-independent way.

Mantis.TensorProducts.get_factors Function

get_factors(tp::TensorProduct)

Return the tuple of factor sets defining the tensor product.

source

Mantis.TensorProducts.mapfactors Method

mapfactors(f, tp::TensorProduct)

Apply f to each factor set and return the resulting tuple.

source

Mantis.TensorProducts.mapfactors Method

mapfactors(f, tp::TensorProduct, id)

Apply f to each factor set and factor id, converted from the global id, and return the resulting tuple.

source

Base.map Method

map(f, tp::TensorProduct, args...)

Apply f to each factor set and construct a new TensorProduct from the resulting sets.

source

Usage in Other Modules

TensorProducts is used by several modules in Mantis. The two most prominent use-cases are Mantis.Geometry.TensorProductGeometry and Mantis.FunctionSpaces.TensorProductSpace.

Tensor-Product Geometry

In Geometry, a Mantis.Geometry.TensorProductGeometry is formed by assembling lower-dimensional geometries along independent coordinate directions. The interface with TensorProducts is valid since Mantis.Geometry.AbstractGeometry implements get_num_objects as an alias of Mantis.Geometry.get_num_elements. As a result, a global element ID in the product geometry can be decomposed into per-factor element IDs, and geometry evaluation, Jacobian computation, and Hessian computation all reduce to per-factor operations with no special-casing.

julia> using Mantis

julia> line_x = Geometry.create_cartesian_box((0.0,), (2.0,), (4,))
Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 2.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),))

julia> line_y = Geometry.create_cartesian_box((0.0,), (1.0,), (3,))
Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),))

julia> geo = Geometry.TensorProductGeometry(line_x, line_y)
Mantis.Geometry.TensorProductGeometry{2, 2, 1, 2, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}((Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 2.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),))), CartesianIndices((4, 3)), LinearIndices((Base.OneTo(4), Base.OneTo(3)))), (12,))

julia> Geometry.get_num_elements(geo)
12

julia> Geometry.get_factor_num_elements(geo)
(4, 3)

Tensor-Product Function Spaces

In FunctionSpaces, a Mantis.FunctionSpaces.TensorProductSpace assembles a multi-dimensional finite element space from lower-dimensional ones. Here the TensorProducts interface uses the number of basis functions, aliasing get_num_objects with Mantis.FunctionSpaces.get_num_basis. As such, a global basis ID decomposes into per-factor basis IDs.

The local basis functions on a product element are assembled via Kronecker products of the per-factor local bases, and supports and extraction operators follow the same structure.

julia> using Mantis

julia> Bx = FunctionSpaces.create_bspline_space((0.0,), (1.0,), (4,), (3,), (2,))
Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]])

julia> By = FunctionSpaces.create_bspline_space((0.0,), (1.0,), (2,), (3,), (2,))
Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), 3, [4, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5 0.25],), ([0.25 0.5 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,))], 2, 5), [[[1], [2, 3, 4], [5]]])

julia> Bz = FunctionSpaces.create_bspline_space((0.0,), (1.0,), (3,), (3,), (2,))
Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),)), 3, [4, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.24999999999999994],), ([0.16666666666666669 0.5833333333333333 0.24999999999999994 0.0; 0.0 0.5 0.49999999999999994 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,))], 3, 6), [[[1], [2, 3, 4, 5], [6]]])

julia> B3D = FunctionSpaces.TensorProductSpace(Bx, By, Bz)
Mantis.FunctionSpaces.TensorProductSpace{3, 1, 3, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Mantis.Geometry.TensorProductGeometry{3, 3, 1, 3, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.TensorProductGeometry{3, 3, 1, 3, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Vector{Vector{Vector{Int64}}}}(Mantis.TensorProducts.TensorProduct{Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}((Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), 3, [4, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5 0.25],), ([0.25 0.5 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,))], 2, 5), [[[1], [2, 3, 4], [5]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),)), 3, [4, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.24999999999999994],), ([0.16666666666666669 0.5833333333333333 0.24999999999999994 0.0; 0.0 0.5 0.49999999999999994 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,))], 3, 6), [[[1], [2, 3, 4, 5], [6]]])), CartesianIndices((7, 5, 6)), LinearIndices((Base.OneTo(7), Base.OneTo(5), Base.OneTo(6)))), Mantis.Geometry.TensorProductGeometry{3, 3, 1, 3, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}((Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),))), CartesianIndices((4, 2, 3)), LinearIndices((Base.OneTo(4), Base.OneTo(2), Base.OneTo(3)))), (24,)), Mantis.Geometry.TensorProductGeometry{3, 3, 1, 3, Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(Mantis.TensorProducts.TensorProduct{Tuple{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}}((Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 3),),), (CartesianIndices((2,)),), (LinearIndices((Base.OneTo(2),)),)), Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 4),),), (CartesianIndices((3,)),), (LinearIndices((Base.OneTo(3),)),))), CartesianIndices((4, 2, 3)), LinearIndices((Base.OneTo(4), Base.OneTo(2), Base.OneTo(3)))), (24,)), [[[1], [2, 3, 4, 5, 6], [7], [8, 15, 22], [9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27], [14, 21, 28], [29], [30, 31, 32, 33, 34], [35], [36, 71, 106, 141]  …  [70, 105, 140, 175], [176], [177, 178, 179, 180, 181], [182], [183, 190, 197], [184, 185, 186, 187, 188, 191, 192, 193, 194, 195, 198, 199, 200, 201, 202], [189, 196, 203], [204], [205, 206, 207, 208, 209], [210]]])

julia> FunctionSpaces.get_num_basis(B3D)
210

julia> FunctionSpaces.get_factor_num_basis(B3D)
(7, 5, 6)