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SymmetricTensors.jl
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#=
In this module we construct totally symmetric rank 2, 3, and 4 tensors (arrays) with indices which run from 1, 2, 3. In these functions,
the tensors are functions of time, so each element in the 3x3, 3x3x3, or 3x3x3x3 arrays will be a vector, but the type declarations
can easily be amended to change the use of these functions.
=#
module SymmetricTensors
# Note that a note that a totally-symmetric tensor with r indices, each running from 1,...,d has (d-1+r)!/((d-1)! r!) independent components. We therefore need only
# specify 6, 10, and 15 components of totally symmetric tensors with d=3 and r = 2, 3, and 4, respectively. Below we specify the indices of the independent components
# we choose to input to construct for each tensor
const two_index_components::Vector{Tuple{Int64, Int64}} = [(1, 2), (1, 3), (2, 3), (1, 1), (2, 2), (3, 3)];
const three_index_components::Vector{Tuple{Int64, Int64, Int64}} = [(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 1, 3), (1, 3, 3),
(1, 2, 3), (2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)];
const four_index_components::Vector{Tuple{Int64, Int64, Int64, Int64}} = [(1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 2, 2), (1, 2, 2, 2), (1, 1, 1, 3), (1, 1, 3, 3),
(1, 3, 3, 3), (1, 1, 2, 3), (1, 2, 2, 3), (1, 2, 3, 3), (2, 2, 2, 2), (2, 2, 2, 3),
(2, 2, 3, 3), (2, 3, 3, 3), (3, 3, 3, 3)];
const traj_indices::Vector{Tuple} = [(1, 2), (1, 3), (2, 3), (1, 1), (2, 2), (3, 3),
(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 1, 3), (1, 3, 3),
(1, 2, 3), (2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)]
const waveform_indices::Vector{Tuple} = [(1, 2), (1, 3), (2, 3), (1, 1), (2, 2), (3, 3),
(1, 1, 1), (1, 1, 2), (1, 2, 2), (1, 1, 3), (1, 3, 3),
(1, 2, 3), (2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3),
(1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 2, 2), (1, 2, 2, 2), (1, 1, 1, 3), (1, 1, 3, 3),
(1, 3, 3, 3), (1, 1, 2, 3), (1, 2, 2, 3), (1, 2, 3, 3), (2, 2, 2, 2), (2, 2, 2, 3),
(2, 2, 3, 3), (2, 3, 3, 3), (3, 3, 3, 3)]
# constructs totally symmetric two-index tensor from the specified six independent components
@views function ConstructTwoIndexTensor(A11::AbstractVector{Float64}, A12::AbstractVector{Float64}, A13::AbstractVector{Float64}, A22::AbstractVector{Float64},
A23::AbstractVector{Float64}, A33::AbstractVector{Float64})
A = [Float64[] for i=1:3, j=1:3]
@inbounds for (i, j) in two_index_components
if i == 1 && j == 1
A[i, j] = A11
elseif i == 1 && j == 2
A[i, j] = A12
A[j, i] = A12
elseif i == 1 && j == 3
A[i, j] = A13
A[j, i] = A13
elseif i == 2 && j == 2
A[i, j] = A22
elseif i == 2 && j == 3
A[i, j] = A23
A[j, i] = A23
elseif i == 3 && j == 3
A[i, j] = A33
end
end
return A
end
# symmetrizes a two-index tensor with its six independent components already specified
@views function SymmetrizeTwoIndexTensor!(A::AbstractArray)
@inbounds for (i, j) in two_index_components
if i == 1 && j == 2
A[2, 1] = A[1, 2]
elseif i == 1 && j == 3
A[3, 1] = A[1, 3]
elseif i == 2 && j == 3
A[3, 2] = A[2, 3]
end
end
end
# constructs totally symmetric three-index tensor from the specified six independent components
@views function ConstructThreeIndexTensor(A111::AbstractVector{Float64}, A112::AbstractVector{Float64}, A122::AbstractVector{Float64}, A113::AbstractVector{Float64}, A133::AbstractVector{Float64},
A123::AbstractVector{Float64}, A222::AbstractVector{Float64}, A223::AbstractVector{Float64}, A233::AbstractVector{Float64}, A333::AbstractVector{Float64})
A = [Float64[] for i=1:3, j=1:3, k=1:3]
@inbounds for (i, j, k) in three_index_components
if i == 1 && j == 1 && k == 1
A[1, 1, 1] = A111;
elseif i == 1 && j == 1 && k == 2
A[1, 1, 2] = A112;
A[1, 2, 1] = A112;
A[2, 1, 1] = A112;
elseif i == 1 && j == 2 && k == 2
A[1, 2, 2] = A122;
A[2, 1, 2] = A122;
A[2, 2, 1] = A122;
elseif i == 1 && j == 1 && k == 3
A[1, 1, 3] = A113;
A[1, 3, 1] = A113;
A[3, 1, 1] = A113;
elseif i == 1 && j == 3 && k == 3
A[1, 3, 3] = A133;
A[3, 1, 3] = A133;
A[3, 3, 1] = A133;
elseif i == 1 && j == 2 && k == 3
A[1, 2, 3] = A123;
A[1, 3, 2] = A123;
A[2, 1, 3] = A123;
A[2, 3, 1] = A123;
A[3, 2, 1] = A123;
A[3, 1, 2] = A123;
elseif i == 2 && j == 3 && k == 3
A[2, 3, 3] = A233;
A[3, 2, 3] = A233;
A[3, 3, 2] = A233;
elseif i == 2 && j == 2 && k == 3
A[2, 2, 3] = A223;
A[2, 3, 2] = A223;
A[3, 2, 2] = A223;
elseif i == 2 && j == 2 && k == 2
A[2, 2, 2] = A222
elseif i == 3 && j == 3 && k == 3
A[3, 3, 3] = A333
end
end
return A
end
# symmetrizes a three-index tensor with its ten independent components already specified
@views function SymmetrizeThreeIndexTensor!(A::AbstractArray)
@inbounds for (i, j, k) in three_index_components
if i == 1 && j == 1 && k == 2
A[1, 2, 1] = A[1, 1, 2];
A[2, 1, 1] = A[1, 1, 2];
elseif i == 1 && j == 2 && k == 2
A[2, 1, 2] = A[1, 2, 2];
A[2, 2, 1] = A[1, 2, 2];
elseif i == 1 && j == 1 && k == 3
A[1, 3, 1] = A[1, 1, 3];
A[3, 1, 1] = A[1, 1, 3];
elseif i == 1 && j == 3 && k == 3
A[3, 1, 3] = A[1, 3, 3];
A[3, 3, 1] = A[1, 3, 3];
elseif i == 1 && j == 2 && k == 3
A[1, 3, 2] = A[1, 2, 3];
A[2, 1, 3] = A[1, 2, 3];
A[2, 3, 1] = A[1, 2, 3];
A[3, 2, 1] = A[1, 2, 3];
A[3, 1, 2] = A[1, 2, 3];
elseif i == 2 && j == 3 && k == 3
A[3, 2, 3] = A[2, 3, 3];
A[3, 3, 2] = A[2, 3, 3];
elseif i == 2 && j == 2 && k == 3
A[2, 3, 2] = A[2, 2, 3];
A[3, 2, 2] = A[2, 2, 3];
end
end
end
# constructs totally symmetric four-index tensor from the specified fifteen independent components
@views function ConstructFourIndexTensor(A1111::AbstractVector{Float64}, A1112::AbstractVector{Float64}, A1122::AbstractVector{Float64}, A1222::AbstractVector{Float64}, A1113::AbstractVector{Float64},
A1133::AbstractVector{Float64}, A1333::AbstractVector{Float64}, A1123::AbstractVector{Float64}, A1223::AbstractVector{Float64}, A1233::AbstractVector{Float64}, A2222::AbstractVector{Float64},
A2223::AbstractVector{Float64}, A2233::AbstractVector{Float64}, A2333::AbstractVector{Float64}, A3333::AbstractVector{Float64})
A = [Float64[] for i=1:3, j=1:3, k=1:3, l=1:3]
@inbounds for (i, j, k, l) in four_index_components
if i == 1 && j == 1 && k == 1 && l == 1
A[1, 1, 1, 1] = A1111
elseif i == 1 && j == 1 && k == 1 && l == 2
A[1, 1, 1, 2] = A1112;
A[1, 1, 2, 1] = A1112;
A[1, 2, 1, 1] = A1112;
A[2, 1, 1, 1] = A1112;
elseif i == 1 && j == 1 && k == 2 && l == 2
A[1, 1, 2, 2] = A1122;
A[1, 2, 1, 2] = A1122;
A[1, 2, 2, 1] = A1122;
A[2, 1, 1, 2] = A1122;
A[2, 1, 2, 1] = A1122;
A[2, 2, 1, 1] = A1122;
elseif i == 1 && j == 2 && k == 2 && l == 2
A[1, 2, 2, 2] = A1222;
A[2, 1, 2, 2] = A1222;
A[2, 2, 1, 2] = A1222;
A[2, 2, 2, 1] = A1222;
elseif i == 1 && j == 1 && k == 1 && l == 3
A[1, 1, 1, 3] = A1113;
A[1, 1, 3, 1] = A1113;
A[1, 3, 1, 1] = A1113;
A[3, 1, 1, 1] = A1113;
elseif i == 1 && j == 1 && k == 3 && l == 3
A[1, 1, 3, 3] = A1133;
A[1, 3, 1, 3] = A1133;
A[1, 3, 3, 1] = A1133;
A[3, 1, 1, 3] = A1133;
A[3, 1, 3, 1] = A1133;
A[3, 3, 1, 1] = A1133;
elseif i == 1 && j == 3 && k == 3 && l == 3
A[1, 3, 3, 3] = A1333;
A[3, 1, 3, 3] = A1333;
A[3, 3, 1, 3] = A1333;
A[3, 3, 3, 1] = A1333;
elseif i == 1 && j == 1 && k == 2 && l == 3
A[1, 1, 2, 3] = A1123;
A[1, 2, 1, 3] = A1123;
A[1, 2, 3, 1] = A1123;
A[1, 3, 1, 2] = A1123;
A[1, 3, 2, 1] = A1123;
A[2, 1, 1, 3] = A1123;
A[2, 1, 3, 1] = A1123;
A[2, 3, 1, 1] = A1123;
A[3, 1, 1, 2] = A1123;
A[3, 1, 2, 1] = A1123;
A[3, 2, 1, 1] = A1123;
A[1, 1, 3, 2] = A1123;
elseif i == 1 && j == 2 && k == 2 && l == 3
A[1, 2, 2, 3] = A1223;
A[1, 2, 3, 2] = A1223;
A[1, 3, 2, 2] = A1223;
A[2, 2, 1, 3] = A1223;
A[2, 2, 3, 1] = A1223;
A[2, 1, 2, 3] = A1223;
A[2, 1, 3, 2] = A1223;
A[2, 3, 1, 2] = A1223;
A[2, 3, 2, 1] = A1223;
A[3, 2, 2, 1] = A1223;
A[3, 2, 1, 2] = A1223;
A[3, 1, 2, 2] = A1223;
elseif i == 1 && j == 2 && k == 3 && l == 3
A[1, 2, 3, 3] = A1233;
A[1, 3, 2, 3] = A1233;
A[1, 3, 3, 2] = A1233;
A[2, 1, 3, 3] = A1233;
A[2, 3, 1, 3] = A1233;
A[2, 3, 3, 1] = A1233;
A[3, 1, 2, 3] = A1233;
A[3, 1, 3, 2] = A1233;
A[3, 2, 1, 3] = A1233;
A[3, 2, 3, 1] = A1233;
A[3, 3, 1, 2] = A1233;
A[3, 3, 2, 1] = A1233;
elseif i == 2 && j == 2 && k == 2 && l == 2
A[2, 2, 2, 2] = A2222
elseif i == 2 && j == 2 && k == 2 && l == 3
A[2, 2, 2, 3] = A2223;
A[2, 2, 3, 2] = A2223;
A[2, 3, 2, 2] = A2223;
A[3, 2, 2, 2] = A2223;
elseif i == 2 && j == 2 && k == 3 && l == 3
A[2, 2, 3, 3] = A2233;
A[2, 3, 2, 3] = A2233;
A[2, 3, 3, 2] = A2233;
A[3, 2, 2, 3] = A2233;
A[3, 2, 3, 2] = A2233;
A[3, 3, 2, 2] = A2233;
elseif i == 2 && j == 3 && k == 3 && l == 3
A[2, 3, 3, 3] = A2333;
A[3, 2, 3, 3] = A2333;
A[3, 3, 2, 3] = A2333;
A[3, 3, 3, 2] = A2333;
elseif i == 3 && j == 3 && k == 3 && l == 3
A[3, 3, 3, 3] = A3333
end
end
return A
end
# symmetrizes a four-index tensor with its fifteen independent components already specified
@views function SymmetrizeFourIndexTensor!(A::AbstractArray)
@inbounds for (i, j, k, l) in four_index_components
if i == 1 && j == 1 && k == 1 && l == 2
A[1, 1, 2, 1] = A[1, 1, 1, 2];
A[1, 2, 1, 1] = A[1, 1, 1, 2];
A[2, 1, 1, 1] = A[1, 1, 1, 2];
elseif i == 1 && j == 1 && k == 2 && l == 2
A[1, 2, 1, 2] = A[1, 1, 2, 2];
A[1, 2, 2, 1] = A[1, 1, 2, 2];
A[2, 1, 1, 2] = A[1, 1, 2, 2];
A[2, 1, 2, 1] = A[1, 1, 2, 2];
A[2, 2, 1, 1] = A[1, 1, 2, 2];
elseif i == 1 && j == 2 && k == 2 && l == 2
A[2, 1, 2, 2] = A[1, 2, 2, 2];
A[2, 2, 1, 2] = A[1, 2, 2, 2];
A[2, 2, 2, 1] = A[1, 2, 2, 2];
elseif i == 1 && j == 1 && k == 1 && l == 3
A[1, 1, 3, 1] = A[1, 1, 1, 3];
A[1, 3, 1, 1] = A[1, 1, 1, 3];
A[3, 1, 1, 1] = A[1, 1, 1, 3];
elseif i == 1 && j == 1 && k == 3 && l == 3
A[1, 3, 1, 3] = A[1, 1, 3, 3];
A[1, 3, 3, 1] = A[1, 1, 3, 3];
A[3, 1, 1, 3] = A[1, 1, 3, 3];
A[3, 1, 3, 1] = A[1, 1, 3, 3];
A[3, 3, 1, 1] = A[1, 1, 3, 3];
elseif i == 1 && j == 3 && k == 3 && l == 3
A[3, 1, 3, 3] = A[1, 3, 3, 3];
A[3, 3, 1, 3] = A[1, 3, 3, 3];
A[3, 3, 3, 1] = A[1, 3, 3, 3];
elseif i == 1 && j == 1 && k == 2 && l == 3
A[1, 2, 1, 3] = A[1, 1, 2, 3];
A[1, 2, 3, 1] = A[1, 1, 2, 3];
A[1, 3, 1, 2] = A[1, 1, 2, 3];
A[1, 3, 2, 1] = A[1, 1, 2, 3];
A[2, 1, 1, 3] = A[1, 1, 2, 3];
A[2, 1, 3, 1] = A[1, 1, 2, 3];
A[2, 3, 1, 1] = A[1, 1, 2, 3];
A[3, 1, 1, 2] = A[1, 1, 2, 3];
A[3, 1, 2, 1] = A[1, 1, 2, 3];
A[3, 2, 1, 1] = A[1, 1, 2, 3];
A[1, 1, 3, 2] = A[1, 1, 2, 3];
elseif i == 1 && j == 2 && k == 2 && l == 3
A[1, 2, 3, 2] = A[1, 2, 2, 3];
A[1, 3, 2, 2] = A[1, 2, 2, 3];
A[2, 2, 1, 3] = A[1, 2, 2, 3];
A[2, 2, 3, 1] = A[1, 2, 2, 3];
A[2, 1, 2, 3] = A[1, 2, 2, 3];
A[2, 1, 3, 2] = A[1, 2, 2, 3];
A[2, 3, 1, 2] = A[1, 2, 2, 3];
A[2, 3, 2, 1] = A[1, 2, 2, 3];
A[3, 2, 2, 1] = A[1, 2, 2, 3];
A[3, 2, 1, 2] = A[1, 2, 2, 3];
A[3, 1, 2, 2] = A[1, 2, 2, 3];
elseif i == 1 && j == 2 && k == 3 && l == 3
A[1, 3, 2, 3] = A[1, 2, 3, 3];
A[1, 3, 3, 2] = A[1, 2, 3, 3];
A[2, 1, 3, 3] = A[1, 2, 3, 3];
A[2, 3, 1, 3] = A[1, 2, 3, 3];
A[2, 3, 3, 1] = A[1, 2, 3, 3];
A[3, 1, 2, 3] = A[1, 2, 3, 3];
A[3, 1, 3, 2] = A[1, 2, 3, 3];
A[3, 2, 1, 3] = A[1, 2, 3, 3];
A[3, 2, 3, 1] = A[1, 2, 3, 3];
A[3, 3, 1, 2] = A[1, 2, 3, 3];
A[3, 3, 2, 1] = A[1, 2, 3, 3];
elseif i == 2 && j == 2 && k == 2 && l == 3
A[2, 2, 3, 2] = A[2, 2, 2, 3];
A[2, 3, 2, 2] = A[2, 2, 2, 3];
A[3, 2, 2, 2] = A[2, 2, 2, 3];
elseif i == 2 && j == 2 && k == 3 && l == 3
A[2, 3, 2, 3] = A[2, 2, 3, 3];
A[2, 3, 3, 2] = A[2, 2, 3, 3];
A[3, 2, 2, 3] = A[2, 2, 3, 3];
A[3, 2, 3, 2] = A[2, 2, 3, 3];
A[3, 3, 2, 2] = A[2, 2, 3, 3];
elseif i == 2 && j == 3 && k == 3 && l == 3
A[3, 2, 3, 3] = A[2, 3, 3, 3];
A[3, 3, 2, 3] = A[2, 3, 3, 3];
A[3, 3, 3, 2] = A[2, 3, 3, 3];
end
end
end
end