Hi,
I am trying to understand this equation and implementing it in julia.
where sigma is second order tensor assuming dimension n1n1. and Nd is shape function matric for scalar value so dimension is 1n2. How this multiplication can be done? I think it would be a third order tensor and if it is right how can I assemble that matric in Julia. I tried to find some examples on that but I couldnt.
Thank you
I assembled before matrices in trial and test spaces with same dimensions. but now i have 2 different dimensions
\bar d is likely a vector of size n2. I can’t make sense of that expression otherwise.
Then the meaning is rather clear: I’ll write it for you with more indices and summation over repeated indices is implied (I also call N_d just N so it’s clearer what the indices are):
\frac{\partial\sigma_{i,j}}{\partial \bar d_k} = -2(1- N_l \bar d_l) \sigma_{i,j} N_k
This object indeed has 3 indices and is thus a 3-tensor.
I am not sure what you mean with “implementing it in Julia” but Julia supports multi-dimensional arrays out of the box. So you could use just a 3 dimensional array to store all the values.
thank you! perfect explanation. so what is the dimension of this 3rd order tensor? n1n1n2?
also for coding part i used gridap FE package like below
function dσfun(ε, ε_in,s_in)
σ_elas = C_mat⊙ε
if tr(ε_in) >= 0
dσ = (2 * s_in)*σ_elas
elseif tr(ε_in) < 0
dσ = (2 * s_in)*P_dev ⊙ σ_elas
end
return dσ
end
function MatrixdAs(pth,uh,sh;fem_params)
dAs_Disp(u,pth,uh,sh,s) = ((p->Em(p))∘pth) * ((dσfun∘(ε(u), ε(uh), sh)) * s)
dAs_mat = assemble_matrix(fem_params.V0_Disp, fem_params.U_PF) do u,s
∫(dAs_Disp(u,pth,uh,sh,s))fem_params.dΩ
end
return dAs_mat
end
but it can not assemble that matrix and getting this error
MethodError: no method matching Float64(::SymTensorValue{2, Float64, 3})
Closest candidates are:
(::Type{T})(::T) where T<:Number at boot.jl:772
(::Type{T})(::AbstractChar) where T<:Union{AbstractChar, Number} at char.jl:50
(::Type{T})(::Base.TwicePrecision) where T<:Number at twiceprecision.jl:266
...
Stacktrace:
[1] convert(#unused#::Type{Float64}, x::SymTensorValue{2, Float64, 3})
@ Base .\number.jl:7
[2] setindex!(A::Vector{Float64}, x::SymTensorValue{2, Float64, 3}, i1::Int64)
@ Base .\array.jl:966
[3] add_entry!(#unused#::typeof(+), a::Gridap.Algebra.InserterCSC{Float64, Int64}, v::SymTensorValue{2, Float64, 3}, i::Int32, j::Int32)
@ Gridap.Algebra C:\Users\marya\.julia\packages\Gridap\971dU\src\Algebra\SparseMatrixCSC.jl:132
[4] _add_entries!
@ C:\Users\marya\.julia\packages\Gridap\971dU\src\Algebra\AlgebraInterfaces.jl:149 [inlined]
[5] add_entries!
@ C:\Users\marya\.julia\packages\Gridap\971dU\src\Algebra\AlgebraInterfaces.jl:127 [inlined]
[6] evaluate!(cache::Nothing, k::Gridap.Arrays.AddEntriesMap{typeof(+)}, A::Gridap.Algebra.InserterCSC{Float64, Int64}, v::Matrix{SymTensorValue{2, Float64, 3}}, i::Vector{Int32}, j::Vector{Int32})
@ Gridap.Arrays C:\Users\marya\.julia\packages\Gridap\971dU\src\Arrays\AlgebraMaps.jl:47
[7] _numeric_loop_matrix!(mat::Gridap.Algebra.InserterCSC{Float64, Int64}, caches::Tuple{Nothing, Tuple{Tuple{Nothing, Tuple{Gridap.Arrays.CachedMatrix{SymTensorValue{2, Float64, 3}, Matrix{SymTensorValue{2, Float64, 3}}}, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedArray{SymTensorValue{2, Float64, 3}, 3, Array{SymTensorValue{2, Float64, 3}, 3}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Gridap.Arrays.CachedVector{Int32, Vector{Int32}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}, Tuple{Tuple{Nothing, Gridap.Arrays.CachedArray{SymTensorValue{2, Float64, 3}, 3, Array{SymTensorValue{2, Float64, 3}, 3}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedArray{SymTensorValue{2, Float64, 3}, 3, Array{SymTensorValue{2, Float64, 3}, 3}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedArray{SymTensorValue{2, Float64, 3}, 3, Array{SymTensorValue{2, Float64, 3}, 3}}, Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedMatrix{TensorValue{2, 2, Float64, 4}, Matrix{TensorValue{2, 2, Float64, 4}}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Tuple{Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{VectorValue{2, Float64}, Vector{VectorValue{2, Float64}}}, Tuple{Gridap.Arrays.CachedVector{Int64, Vector{Int64}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{VectorValue{2, Float64}}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, TensorValue{2, 2, Float64, 4}}}}}, Gridap.Arrays.IndexItemPair{Int64, ConstantField{TensorValue{2, 2, Float64, 4}}}}, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Matrix{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Gridap.Fields.TransposeFieldIndices{Matrix{TensorValue{2, 2, Float64, 4}}, TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Array{SymTensorValue{2, Float64, 3}, 3}}}, Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{SymTensorValue{2, Float64, 3}, Vector{SymTensorValue{2, Float64, 3}}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Tuple{Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{VectorValue{2, Float64}, Vector{VectorValue{2, Float64}}}, Tuple{Gridap.Arrays.CachedVector{Int64, Vector{Int64}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{VectorValue{2, Float64}}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, TensorValue{2, 2, Float64, 4}}}}}, Gridap.Arrays.IndexItemPair{Int64, ConstantField{TensorValue{2, 2, Float64, 4}}}}, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}, Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Gridap.Arrays.CachedVector{Int32, Vector{Int32}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{SymTensorValue{2, Float64, 3}}}}, Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{Float64, Vector{Float64}}, Tuple{Gridap.Arrays.CachedVector{Int32, Vector{Int32}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{Float64}}}}}, Gridap.Arrays.IndexItemPair{Int64, Array{SymTensorValue{2, Float64, 3}, 3}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Array{SymTensorValue{2, Float64, 3}, 3}}}}}, Gridap.Arrays.IndexItemPair{Int64, Array{SymTensorValue{2, Float64, 3}, 3}}}, Nothing, Tuple{Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Nothing, Tuple{Tuple{Tuple{Nothing, Gridap.Arrays.CachedVector{VectorValue{2, Float64}, Vector{VectorValue{2, Float64}}}, Tuple{Gridap.Arrays.CachedVector{Int64, Vector{Int64}}}}, Gridap.Arrays.IndexItemPair{Int64, Vector{VectorValue{2, Float64}}}}, Nothing}}, Gridap.Arrays.IndexItemPair{Int64, TensorValue{2, 2, Float64, 4}}}}}, Gridap.Arrays.IndexItemPair{Int64, ConstantField{TensorValue{2, 2, Float64, 4}}}}, Gridap.Arrays.CachedVector{TensorValue{2, 2, Float64, 4}, Vector{TensorValue{2, 2, Float64, 4}}}, Tuple{Nothing}}, Gridap.Arrays.IndexItemPair{Int64, Vector{TensorValue{2, 2, Float64, 4}}}}}}, Gridap.Arrays.IndexItemPair{Int64, Matrix{SymTensorValue{2, Float64, 3}}}}, Gridap.Arrays.CachedVector{Int32, Vector{Int32}}, Gridap.Arrays.CachedVector{Int32, Vector{Int32}}}, cell_vals::Gridap.Arrays.LazyArray{FillArrays.Fill{IntegrationMap, 1, Tuple{Base.OneTo{Int64}}}, Matrix{SymTensorValue{2, Float64, 3}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(*)}, 1, Tuple{Base.OneTo{Int64}}}, Array{SymTensorValue{2, Float64, 3}, 3}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{var"#78#81"}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{var"#31#32"}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Gridap.Arrays.PosNegReindex{Vector{Float64}, Vector{Float64}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.Table{Int32, Vector{Int32}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Matrix{Float64}, 1, Vector{Matrix{Float64}}, Vector{Int8}}}}}}}}, Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(*)}, 1, Tuple{Base.OneTo{Int64}}}, Array{SymTensorValue{2, Float64, 3}, 3}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(dσfun)}, 1, Tuple{Base.OneTo{Int64}}}, Array{SymTensorValue{2, Float64, 3}, 3}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(symmetric_part)}, 1, Tuple{Base.OneTo{Int64}}}, Array{SymTensorValue{2, Float64, 3}, 3}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.TransposeMap, 1, Tuple{Base.OneTo{Int64}}}, Gridap.Fields.TransposeFieldIndices{Matrix{TensorValue{2, 2, Float64, 4}}, TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(dot)}, 1, Tuple{Base.OneTo{Int64}}}, Matrix{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{typeof(pinvJt)}, 1, Tuple{Base.OneTo{Int64}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{Gridap.Arrays.LazyArray{FillArrays.Fill{typeof(constant_field), 1, Tuple{Base.OneTo{Int64}}}, ConstantField{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, TensorValue{2, 2, Float64, 4}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Reindex{Vector{VectorValue{2, Float64}}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{VectorValue{2, Float64}}, 1, Tuple{Gridap.Arrays.Table{Int64, Vector{Int64}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, Gridap.Arrays.CompressedArray{Matrix{TensorValue{2, 2, Float64, 4}}, 1, Vector{Matrix{TensorValue{2, 2, Float64, 4}}}, Vector{Int8}}}}}}}}, Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(symmetric_part)}, 1, Tuple{Base.OneTo{Int64}}}, Vector{SymTensorValue{2, Float64, 3}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.BroadcastingFieldOpMap{typeof(dot)}, 1, Tuple{Base.OneTo{Int64}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{typeof(pinvJt)}, 1, Tuple{Base.OneTo{Int64}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{Gridap.Arrays.LazyArray{FillArrays.Fill{typeof(constant_field), 1, Tuple{Base.OneTo{Int64}}}, ConstantField{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, TensorValue{2, 2, Float64, 4}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Reindex{Vector{VectorValue{2, Float64}}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{VectorValue{2, Float64}}, 1, Tuple{Gridap.Arrays.Table{Int64, Vector{Int64}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Gridap.Arrays.PosNegReindex{Vector{Float64}, Vector{Float64}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.Table{Int32, Vector{Int32}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Matrix{TensorValue{2, 2, Float64, 4}}, 1, Vector{Matrix{TensorValue{2, 2, Float64, 4}}}, Vector{Int8}}}}}}}}, Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Gridap.Arrays.PosNegReindex{Vector{Float64}, Vector{Float64}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{Float64}, 1, Tuple{Gridap.Arrays.Table{Int32, Vector{Int32}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Matrix{Float64}, 1, Vector{Matrix{Float64}}, Vector{Int8}}}}}}, Gridap.Arrays.CompressedArray{Matrix{Float64}, 1, Vector{Matrix{Float64}}, Vector{Int8}}}}}}, Gridap.Arrays.CompressedArray{Vector{Float64}, 1, Vector{Vector{Float64}}, Vector{Int8}}, Gridap.Arrays.LazyArray{Gridap.Arrays.LazyArray{FillArrays.Fill{typeof(constant_field), 1, Tuple{Base.OneTo{Int64}}}, ConstantField{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Gridap.Fields.LinearCombinationMap{Colon}, 1, Tuple{Base.OneTo{Int64}}}, TensorValue{2, 2, Float64, 4}, 1, Tuple{Gridap.Arrays.LazyArray{FillArrays.Fill{Broadcasting{Reindex{Vector{VectorValue{2, Float64}}}}, 1, Tuple{Base.OneTo{Int64}}}, Vector{VectorValue{2, Float64}}, 1, Tuple{Gridap.Arrays.Table{Int64, Vector{Int64}, Vector{Int32}}}}, Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, Vector{TensorValue{2, 2, Float64, 4}}, 1, Tuple{Gridap.Arrays.CompressedArray{Vector{VectorValue{2, Float64}}, 1, Vector{Vector{VectorValue{2, Float64}}}, Vector{Int8}}}}}}, cell_rows::Gridap.Arrays.Table{Int32, Vector{Int32}, Vector{Int32}}, cell_cols::Gridap.Arrays.Table{Int32, Vector{Int32}, Vector{Int32}})
@ Gridap.FESpaces C:\Users\marya\.julia\packages\Gridap\971dU\src\FESpaces\SparseMatrixAssemblers.jl:261
[8] numeric_loop_matrix!(A::Gridap.Algebra.InserterCSC{Float64, Int64}, a::Gridap.FESpaces.GenericSparseMatrixAssembler, matdata::Tuple{Vector{Any}, Vector{Any}, Vector{Any}})
@ Gridap.FESpaces C:\Users\marya\.julia\packages\Gridap\971dU\src\FESpaces\SparseMatrixAssemblers.jl:248
[9] assemble_matrix(a::Gridap.FESpaces.GenericSparseMatrixAssembler, matdata::Tuple{Vector{Any}, Vector{Any}, Vector{Any}})
@ Gridap.FESpaces C:\Users\marya\.julia\packages\Gridap\971dU\src\FESpaces\SparseMatrixAssemblers.jl:107
[10] assemble_matrix(f::var"#79#82"{NamedTuple{(:V0_Disp, :U_PF, :V0_PF, :Q, :P, :Qf, :Pf, :np, :Ω, :dΩ, :n_Γ_Load, :dΓ_Load), Tuple{Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{VectorValue{2, Int32}}}, TrialFESpace{Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{Int32}}}, Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{Int32}}, Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Nothing}, TrialFESpace{Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Nothing}}, Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{Int32}}, TrialFESpace{Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{Int32}}}, Int64, BodyFittedTriangulation{2, 2, UnstructuredDiscreteModel{2, 2, Float64, Oriented}, UnstructuredGrid{2, 2, Float64, Oriented, Nothing}, Gridap.Arrays.IdentityVector{Int64}}, Gridap.CellData.GenericMeasure, GenericCellField{ReferenceDomain}, Gridap.CellData.GenericMeasure}}, Gridap.CellData.OperationCellField{ReferenceDomain}, Gridap.FESpaces.SingleFieldFEFunction{GenericCellField{ReferenceDomain}}, Gridap.FESpaces.SingleFieldFEFunction{GenericCellField{ReferenceDomain}}, var"#dAs_Disp#80"}, a::Gridap.FESpaces.GenericSparseMatrixAssembler, U::Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{VectorValue{2, Int32}}}, V::TrialFESpace{Gridap.FESpaces.UnconstrainedFESpace{Vector{Float64}, Gridap.FESpaces.NodeToDofGlue{Int32}}})
@ Gridap.FESpaces C:\Users\marya\.julia\packages\Gridap\971dU\src\FESpaces\Assemblers.jl:285
[11] assemble_matrix
@ C:\Users\marya\.julia\packages\Gridap\971dU\src\FESpaces\Assemblers.jl:342 [inlined]
[12] #MatrixdAs#77
@ .\In[48]:9 [inlined]
[13] top-level scope
@ In[48]:14
You can choose how you want to arrange the dimensions. Choose whatever will be practical in the following code.
I am sorry, but I am not familiar with this package. So someone else needs to chime in and comment on that 
thank you the indices notation helps a lot