From 8cc891862fd7657b0440a02f7115d392dc10c185 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Wed, 30 Oct 2024 00:00:08 +1100
Subject: [PATCH 01/27] AffineMap is now AffineField. Introduced new AffineMap
 and ConstantMap.

---
 src/Adaptivity/RefinementRules.jl             |  2 +-
 src/Fields/AffineMaps.jl                      | 52 ++++++++++++++-----
 src/Fields/Fields.jl                          |  2 +-
 src/Fields/FieldsInterfaces.jl                | 10 ++++
 src/Geometry/CartesianGrids.jl                |  6 +--
 test/FieldsTests/AffineMapsTests.jl           |  4 +-
 test/FieldsTests/InverseFieldsTests.jl        |  8 +--
 .../BoundaryTriangulationsTests.jl            |  2 +-
 test/GridapTests/issue_879.jl                 |  2 +-
 9 files changed, 61 insertions(+), 27 deletions(-)

diff --git a/src/Adaptivity/RefinementRules.jl b/src/Adaptivity/RefinementRules.jl
index a93d93d20..d617cb091 100644
--- a/src/Adaptivity/RefinementRules.jl
+++ b/src/Adaptivity/RefinementRules.jl
@@ -25,7 +25,7 @@ end
 # - The reason why we are saving both the cell maps and the inverse cell maps is to avoid recomputing
 #   them when needed. This is needed for performance when the RefinementRule is used for MacroFEs.
 #   Also, in the case the ref_grid comes from a CartesianGrid, we save the cell maps as 
-#   AffineMaps, which are more efficient than the default linear_combinations.
+#   AffineFields, which are more efficient than the default linear_combinations.
 # - We cannot parametrise the RefinementRule by all it's fields, because we will have different types of
 #   RefinementRules in a single mesh. It's the same reason why we don't parametrise the ReferenceFE type.
 
diff --git a/src/Fields/AffineMaps.jl b/src/Fields/AffineMaps.jl
index 9ce5725fc..d8f521b3d 100644
--- a/src/Fields/AffineMaps.jl
+++ b/src/Fields/AffineMaps.jl
@@ -1,12 +1,28 @@
 
+struct AffineMap <: Map end
+
+function return_cache(::AffineMap,G::TensorValue{D1,D2,T,L},y0::Point{D2,T},x::Point{D1,T})
+  nothing
+end
+
+function evaluate!(
+  cache,::AffineMap,G::TensorValue{D1,D2},y0::Point{D2},x::Point{D1}
+) where {D1,D2}
+  x⋅G + y0
+end
+
+struct InverseAffineMap <: Map end
+
+
+
 """
 A Field with this form
 y = x⋅G + y0
 """
-struct AffineMap{D1,D2,T,L} <:Field
+struct AffineField{D1,D2,T,L} <: Field
   gradient::TensorValue{D1,D2,T,L}
   origin::Point{D2,T}
-  function AffineMap(
+  function AffineField(
     gradient::TensorValue{D1,D2,T,L},
     origin::Point{D2,T}) where {D1,D2,T,L}
 
@@ -14,21 +30,21 @@ struct AffineMap{D1,D2,T,L} <:Field
   end
 end
 
-affine_map(gradient,origin) = AffineMap(gradient,origin)
+affine_map(gradient,origin) = AffineField(gradient,origin)
 
-function evaluate!(cache,f::AffineMap,x::Point)
+function evaluate!(cache,f::AffineField,x::Point)
   G = f.gradient
   y0 = f.origin
   x⋅G + y0
 end
 
-function return_cache(f::AffineMap,x::AbstractVector{<:Point})
+function return_cache(f::AffineField,x::AbstractVector{<:Point})
   T = return_type(f,testitem(x))
   y = similar(x,T,size(x))
   CachedArray(y)
 end
 
-function evaluate!(cache,f::AffineMap,x::AbstractVector{<:Point})
+function evaluate!(cache,f::AffineField,x::AbstractVector{<:Point})
   setsize!(cache,size(x))
   y = cache.array
   G = f.gradient
@@ -41,11 +57,11 @@ function evaluate!(cache,f::AffineMap,x::AbstractVector{<:Point})
   y
 end
 
-function gradient(h::AffineMap)
+function gradient(h::AffineField)
   ConstantField(h.gradient)
 end
 
-function push_∇∇(∇∇a::Field,ϕ::AffineMap)
+function push_∇∇(∇∇a::Field,ϕ::AffineField)
   # Assuming ϕ is affine map
   Jt = ∇(ϕ)
   Jt_inv = pinvJt(Jt)
@@ -80,7 +96,7 @@ end
 function lazy_map(
   k::Broadcasting{typeof(push_∇∇)},
   cell_∇∇a::AbstractArray,
-  cell_map::AbstractArray{<:AffineMap})
+  cell_map::AbstractArray{<:AffineField})
   cell_Jt = lazy_map(∇,cell_map)
   cell_invJt = lazy_map(Operation(pinvJt),cell_Jt)
   lazy_map(Broadcasting(Operation(push_∇∇)),cell_∇∇a,cell_invJt)
@@ -89,19 +105,19 @@ end
 function lazy_map(
   k::Broadcasting{typeof(push_∇∇)},
   cell_∇∇at::LazyArray{<:Fill{typeof(transpose)}},
-  cell_map::AbstractArray{<:AffineMap})
+  cell_map::AbstractArray{<:AffineField})
   cell_∇∇a = cell_∇∇at.args[1]
   cell_∇∇b = lazy_map(k,cell_∇∇a,cell_map)
   cell_∇∇bt = lazy_map(transpose,cell_∇∇b)
   cell_∇∇bt
 end
 
-function inverse_map(f::AffineMap)
+function inverse_map(f::AffineField)
   Jt = f.gradient
   y0 = f.origin
   invJt = pinvJt(Jt)
   x0 = -y0⋅invJt
-  AffineMap(invJt,x0)
+  AffineField(invJt,x0)
 end
 
 function lazy_map(::typeof(∇),a::LazyArray{<:Fill{typeof(affine_map)}})
@@ -109,8 +125,16 @@ function lazy_map(::typeof(∇),a::LazyArray{<:Fill{typeof(affine_map)}})
   lazy_map(constant_field,gradients)
 end
 
-function Base.zero(::Type{<:AffineMap{D1,D2,T}}) where {D1,D2,T}
+function lazy_map(
+  ::typeof(evaluate),a::LazyArray{<:Fill{typeof(affine_map)}},x::AbstractArray
+)
+  gradients = a.args[1]
+  origins = a.args[2]
+  lazy_map(Broadcasting(AffineMap()),gradients,origins,x)
+end
+
+function Base.zero(::Type{<:AffineField{D1,D2,T}}) where {D1,D2,T}
   gradient = TensorValue{D1,D2}(tfill(zero(T),Val{D1*D2}()))
   origin = Point{D2,T}(tfill(zero(T),Val{D2}()))
-  AffineMap(gradient,origin)
+  AffineField(gradient,origin)
 end
diff --git a/src/Fields/Fields.jl b/src/Fields/Fields.jl
index 05e5e7a56..0dd060b4c 100644
--- a/src/Fields/Fields.jl
+++ b/src/Fields/Fields.jl
@@ -56,7 +56,7 @@ export MockFieldArray
 export Point
 export inverse_map
 
-export AffineMap
+export AffineField
 export affine_map
 
 export gradient
diff --git a/src/Fields/FieldsInterfaces.jl b/src/Fields/FieldsInterfaces.jl
index bdad6a111..799bc96cc 100644
--- a/src/Fields/FieldsInterfaces.jl
+++ b/src/Fields/FieldsInterfaces.jl
@@ -288,6 +288,16 @@ function lazy_map(::Operation{typeof(inv)},a::LazyArray{<:Fill{typeof(constant_f
   lazy_map(constant_field,vinv)
 end
 
+struct ConstantMap <: Map end
+
+return_cache(::ConstantMap,v::Number,x::Point) = nothing
+evaluate!(cache,::ConstantMap,v::Number,x::Point) = v
+
+function lazy_map(::typeof(evaluate),a::LazyArray{<:Fill{typeof(constant_field)}},x::AbstractArray)
+  values = a.args[1]
+  lazy_map(Broadcasting(ConstantMap()),values,x)
+end
+
 ## Make Function behave like Field
 
 return_cache(f::FieldGradient{N,<:Function},x::Point) where N = gradient(f.object,Val(N))
diff --git a/src/Geometry/CartesianGrids.jl b/src/Geometry/CartesianGrids.jl
index ee83694c4..5fd239957 100644
--- a/src/Geometry/CartesianGrids.jl
+++ b/src/Geometry/CartesianGrids.jl
@@ -237,7 +237,7 @@ end
 
 # Cell map
 
-struct CartesianMap{D,T,L} <: AbstractArray{AffineMap{D,D,T,L},D}
+struct CartesianMap{D,T,L} <: AbstractArray{AffineField{D,D,T,L},D}
   data::CartesianDescriptor{D,T,typeof(identity)}
   function CartesianMap(des::CartesianDescriptor{D,T}) where {D,T}
     L = D*D
@@ -256,9 +256,9 @@ function Base.getindex(a::CartesianMap{D,T},I::Vararg{Integer,D}) where {D,T}
   @inbounds for d in 1:D
     p[d] =  x0[d] + (I[d]-1)*dx[d]
   end
-  origin =  Point(p)
+  origin = Point(p)
   grad = diagonal_tensor(VectorValue(dx))
-  AffineMap(grad,origin)
+  AffineField(grad,origin)
 end
 
 function lazy_map(::typeof(∇),a::CartesianMap)
diff --git a/test/FieldsTests/AffineMapsTests.jl b/test/FieldsTests/AffineMapsTests.jl
index 284aaec1d..963e3faf6 100644
--- a/test/FieldsTests/AffineMapsTests.jl
+++ b/test/FieldsTests/AffineMapsTests.jl
@@ -9,7 +9,7 @@ using Test
 origin = Point(1,1)
 g = TensorValue(2,0,0,2)
 
-h = AffineMap(g,origin)
+h = AffineField(g,origin)
 @test isa(∇(h),ConstantField)
 @test isa(Broadcasting(∇)(h),ConstantField)
 
@@ -39,7 +39,7 @@ cell_to_∇hx = lazy_map(evaluate,cell_to_∇h,cell_to_x)
 test_array(cell_to_hx,fill(hx,ncells))
 test_array(cell_to_∇hx,fill(∇hx,ncells))
 
-T = AffineMap{3,3,Int}
+T = AffineField{3,3,Int}
 @test isa(zero(T),T)
 
 #display(cell_to_hx)
diff --git a/test/FieldsTests/InverseFieldsTests.jl b/test/FieldsTests/InverseFieldsTests.jl
index 8cf7bb5d2..f8db6750a 100644
--- a/test/FieldsTests/InverseFieldsTests.jl
+++ b/test/FieldsTests/InverseFieldsTests.jl
@@ -8,22 +8,22 @@ using Test
 
 b0 = Point(0,0)
 m0 = TensorValue(1,0,0,1)
-id = AffineMap(m0,b0)
+id = AffineField(m0,b0)
 
 b1 = Point(1,1)
 m1 = TensorValue(2,0,0,2)
-h1 = AffineMap(m1,b1)
+h1 = AffineField(m1,b1)
 
 b2 = Point(3,3)
 m2 = TensorValue(4,0,0,4)
-h2 = AffineMap(m2,b2)
+h2 = AffineField(m2,b2)
 
 h = h2 ∘ h1
 
 # (4x+3) ∘ (2x+1) = 8x+7
 b3 = Point(7,7)
 m3 = TensorValue(8,0,0,8)
-h3 = AffineMap(m3,b3)
+h3 = AffineField(m3,b3)
 
 h1inv = inverse_map(h1)
 h2inv = inverse_map(h2)
diff --git a/test/GeometryTests/BoundaryTriangulationsTests.jl b/test/GeometryTests/BoundaryTriangulationsTests.jl
index f876d2a86..b775798ef 100644
--- a/test/GeometryTests/BoundaryTriangulationsTests.jl
+++ b/test/GeometryTests/BoundaryTriangulationsTests.jl
@@ -78,7 +78,7 @@ glue = get_glue(btrian,Val(1))
 glue = get_glue(btrian,Val(2))
 @test glue.tface_to_mface === btrian.glue.face_to_cell
 
-@test isa(get_cell_map(btrian)[1],AffineMap)
+@test isa(get_cell_map(btrian)[1],AffineField)
 
 face_s_q = glue.tface_to_mface_map
 
diff --git a/test/GridapTests/issue_879.jl b/test/GridapTests/issue_879.jl
index d0eefea69..bd9613aad 100644
--- a/test/GridapTests/issue_879.jl
+++ b/test/GridapTests/issue_879.jl
@@ -38,6 +38,6 @@ fΓ = interpolate_everywhere(fa, FESpace(Γ,reffe))
 # ERROR: LoadError: DimensionMismatch: matrix is not square: dimensions are (1, 2)
 
 # Corrections
-# Modified src/Fields/AffineMaps.jl
+# Modified src/Fields/AffineFields.jl
 
 end
\ No newline at end of file

From 028897471244ae313d32e649d8d6dfd58f2daee5 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Wed, 30 Oct 2024 00:04:33 +1100
Subject: [PATCH 02/27] Minor fix

---
 src/Fields/AffineMaps.jl | 5 +----
 1 file changed, 1 insertion(+), 4 deletions(-)

diff --git a/src/Fields/AffineMaps.jl b/src/Fields/AffineMaps.jl
index d8f521b3d..e99221107 100644
--- a/src/Fields/AffineMaps.jl
+++ b/src/Fields/AffineMaps.jl
@@ -1,7 +1,7 @@
 
 struct AffineMap <: Map end
 
-function return_cache(::AffineMap,G::TensorValue{D1,D2,T,L},y0::Point{D2,T},x::Point{D1,T})
+function return_cache(::AffineMap,G::TensorValue{D1,D2},y0::Point{D2},x::Point{D1}) where {D1,D2}
   nothing
 end
 
@@ -11,9 +11,6 @@ function evaluate!(
   x⋅G + y0
 end
 
-struct InverseAffineMap <: Map end
-
-
 
 """
 A Field with this form

From dfd7af5e008446ffb34a709f27d62a6991a0dee4 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Wed, 30 Oct 2024 16:31:52 +1100
Subject: [PATCH 03/27] Optimizations for
 lazy_map(evaluate,lazy_map(Reindex(f),ids),x)

---
 src/Arrays/Reindex.jl            | 8 ++++++++
 src/Fields/ApplyOptimizations.jl | 4 ++--
 2 files changed, 10 insertions(+), 2 deletions(-)

diff --git a/src/Arrays/Reindex.jl b/src/Arrays/Reindex.jl
index 7c6442790..429d5b8b3 100644
--- a/src/Arrays/Reindex.jl
+++ b/src/Arrays/Reindex.jl
@@ -129,3 +129,11 @@ end
   i = j_to_i[j]
   i_to_v[i]=v
 end
+
+# This optimization is important when integrating on partial domains (Triangulation views)
+function lazy_map(::typeof(evaluate),a::LazyArray{<:Fill{<:Reindex}},x::AbstractArray)
+  fields = a.maps.value.values
+  ids = a.args[1]
+  vals = lazy_map(evaluate,fields,x)
+  return lazy_map(Reindex(vals),ids)
+end
diff --git a/src/Fields/ApplyOptimizations.jl b/src/Fields/ApplyOptimizations.jl
index 76dd091fd..95df7c730 100644
--- a/src/Fields/ApplyOptimizations.jl
+++ b/src/Fields/ApplyOptimizations.jl
@@ -135,14 +135,14 @@ function lazy_map(
   k::Broadcasting{typeof(∇)}, a::LazyArray{<:Fill{typeof(transpose)}})
 
   i_to_basis = lazy_map(k,a.args[1])
-  lazy_map( transpose, i_to_basis)
+  lazy_map(transpose, i_to_basis)
 end
 
 function lazy_map(
   k::Broadcasting{typeof(∇∇)}, a::LazyArray{<:Fill{typeof(transpose)}})
 
   i_to_basis = lazy_map(k,a.args[1])
-  lazy_map( transpose, i_to_basis)
+  lazy_map(transpose, i_to_basis)
 end
 
 # Gradient rules

From 050f6a8e6f69e40c0ea16c991ef950fc21eebd3e Mon Sep 17 00:00:00 2001
From: Antoine Marteau <antoine.marteau@protonmail.com>
Date: Mon, 11 Nov 2024 11:30:32 +1100
Subject: [PATCH 04/27] small TensorValues docu improvements

---
 NEWS.md                                       |   4 +
 docs/src/TensorValues.md                      | 148 +++++++++++++++++-
 src/TensorValues/MultiValueTypes.jl           |   6 +-
 .../SymTracelessTensorValueTypes.jl           |   3 -
 src/TensorValues/TensorValues.jl              |  72 +--------
 5 files changed, 153 insertions(+), 80 deletions(-)

diff --git a/NEWS.md b/NEWS.md
index f2973a265..dec364e56 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,10 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+### Added
+
+- Small improvements of the documentation of `Gridap.TensorValues`. Since PR[#1051](https://github.com/gridap/Gridap.jl/pull/1051).
+
 ## [0.18.7] - 2024-10-8
 
 ### Added
diff --git a/docs/src/TensorValues.md b/docs/src/TensorValues.md
index 3c0725777..6b04631cf 100644
--- a/docs/src/TensorValues.md
+++ b/docs/src/TensorValues.md
@@ -1,9 +1,151 @@
+# Gridap.TensorValues
+
 ```@meta
 CurrentModule = Gridap.TensorValues
 ```
 
-# Gridap.TensorValues
+This module provides the abstract interface `MultiValue` representing tensors
+that are also `Number`s, along with concrete implementations for the following
+tensors:
+- 1st order [`VectorValue`](@ref),
+- 2nd order [`TensorValue`](@ref),
+- 2nd order and symmetric [`SymTensorValue`](@ref),
+- 2nd order, symmetric and traceless [`SymTracelessTensorValue`](@ref),
+- 3rd order [`ThirdOrderTensorValue`](@ref),
+- 4th order and symmetric [`SymFourthOrderTensorValue`](@ref).
+
+## Generalities
+
+The main feature of this module is that the provided types do not extend from `AbstractArray`, but from `Number`!
+
+This allows one to work with them as if they were scalar values in broadcasted operations on arrays of `VectorValue` objects (also for `TensorValue` or `MultiValue` objects). For instance, one can perform the following manipulations:
+```julia
+# Assign a VectorValue to all the entries of an Array of VectorValues
+A = zeros(VectorValue{2,Int}, (4,5))
+v = VectorValue(12,31)
+A .= v # This is possible since  VectorValue <: Number
 
-```@autodocs
-Modules = [TensorValues,]
+# Broadcasting of tensor operations in arrays of TensorValues
+t = TensorValue(13,41,53,17) # creates a 2x2 TensorValue
+g = TensorValue(32,41,3,14) # creates another 2x2 TensorValue
+B = fill(t,(1,5))
+C = inner.(g,B) # inner product of g against all TensorValues in the array B
+@show C
+# C = [2494 2494 2494 2494 2494]
 ```
+
+To create a [`::MultiValue`](@ref) tensor from components, these should be given
+as separate arguments or all gathered in a `tuple`. The order of the arguments
+is the order of the linearized Cartesian indices of the corresponding array
+(order of the `Base.LinearIndices` indices):
+```julia
+using StaticArrays
+t = TensorValue( (1, 2, 3, 4) )
+ts= convert(SMatrix{2,2,Int}, t)
+@show ts
+# 2×2 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
+#  1  3
+#  2  4
+t2[1,2] == t[1,2] == 3 # true
+```
+For symmetric tensor types, only the independent components should be given, see
+[`SymTensorValue`](@ref), [`SymTracelessTensorValue`](@ref) and [`SymFourthOrderTensorValue`](@ref).
+
+A `MultiValue` can be created from an `AbstractArray` of the same size. If the
+`MultiValue` type has internal constraints (e.g. symmetries), ONLY the required
+components are picked from the array WITHOUT CHECKING if the given array
+did respect the constraints:
+```julia
+SymTensorValue( [1 2; 3 4] )          # -> SymTensorValue{2, Int64, 3}(1, 2, 4)
+SymTensorValue( SMatrix{2}(1,2,3,4) ) # -> SymTensorValue{2, Int64, 3}(1, 3, 4)
+```
+
+`MultiValue`s can be converted to static and mutable arrays types from
+`StaticArrays.jl` using `convert` and [`mutable`](@ref), respectively.
+
+## Tensor types
+
+The following concrete tensor types are currently implemented:
+
+```@docs
+VectorValue
+TensorValue
+SymTensorValue
+SymTracelessTensorValue
+ThirdOrderTensorValue
+SymFourthOrderTensorValue
+```
+
+### Abstract tensor types
+
+```@docs
+MultiValue
+AbstractSymTensorValue
+```
+
+## Interface
+
+The tensor types implement methods for the following `Base` functions: `getindex`, `length`, `size`, `rand`, `zero`, `real`, `imag` and `conj`.
+
+`one` is also implemented in particular cases: it is defined for second
+and fourth order tensors. For second order, it returns the identity tensor `δij`,
+except `SymTracelessTensorValue` that does not implement `one`. For fourth order symmetric tensors, see [`one`](@ref).
+
+Additionally, the tensor types expose the following interface:
+
+```@docs
+num_components
+mutable
+Mutable
+num_indep_components
+indep_comp_getindex
+indep_components_names
+change_eltype
+
+inner
+dot
+double_contraction
+outer
+```
+
+### Other type specific interfaces
+
+#### For square second order tensors
+
+```@docs
+det
+inv
+symmetric_part
+```
+
+#### For first order tensors
+
+```@docs
+diagonal_tensor
+```
+
+#### For second and third order tensors
+
+```@docs
+tr
+```
+
+#### For first and second order tensors
+
+```@docs
+norm
+meas
+```
+
+#### For `VectorValue` of length 2 and 3
+
+```@docs
+cross
+```
+
+#### For second order non-traceless and symmetric fourth order tensors
+
+```@docs
+one
+```
+
diff --git a/src/TensorValues/MultiValueTypes.jl b/src/TensorValues/MultiValueTypes.jl
index d95047aa7..cc73d31aa 100644
--- a/src/TensorValues/MultiValueTypes.jl
+++ b/src/TensorValues/MultiValueTypes.jl
@@ -11,7 +11,7 @@ Abstract type representing a multi-dimensional number value. The parameters are
 - `N` is the order of the tensor, the length of `S`,
 - `L` is the number of components stored internally.
 
-`MultiValue`s are immutable. See [`TensorValues`](@ref) for more details on usage.
+`MultiValue`s are immutable.
 """
 abstract type MultiValue{S,T,N,L} <: Number end
 
@@ -48,7 +48,7 @@ change_eltype(::Number,::Type{T2}) where {T2} = change_eltype(Number,T2)
     Mutable(T::Type{<:MultiValue}) -> ::Type{<:MArray}
     Mutable(a::MultiValue)
 
-Return the concrete `MArray` type (defined by `StaticArrays.jl`) corresponding
+Return the concrete mutable `MArray` type (defined by `StaticArrays.jl`) corresponding
 to the `MultiValue` type T or array size and type of `a`.
 
 See also [`mutable`](@ref).
@@ -59,7 +59,7 @@ Mutable(::MultiValue) = Mutable(MultiValue)
 """
     mutable(a::MultiValue)
 
-Converts `a` into an array of type `MArray` defined by `StaticArrays.jl`.
+Converts `a` into a mutable array of type `MArray` defined by `StaticArrays.jl`.
 
 See also [`Mutable`](@ref).
 """
diff --git a/src/TensorValues/SymTracelessTensorValueTypes.jl b/src/TensorValues/SymTracelessTensorValueTypes.jl
index 9e3ddfa03..c872641cd 100644
--- a/src/TensorValues/SymTracelessTensorValueTypes.jl
+++ b/src/TensorValues/SymTracelessTensorValueTypes.jl
@@ -34,9 +34,6 @@ end
   Meta.parse("($str)")
 end
 
-"""
-Alias for [`SymTracelessTensorValue`](@ref).
-"""
 const QTensorValue = SymTracelessTensorValue
 
 ###############################################################
diff --git a/src/TensorValues/TensorValues.jl b/src/TensorValues/TensorValues.jl
index 11972a83d..5985195dc 100644
--- a/src/TensorValues/TensorValues.jl
+++ b/src/TensorValues/TensorValues.jl
@@ -1,75 +1,5 @@
 """
-This module provides the abstract interface `MultiValue` representing tensors
-that are also `Number`s, along with concrete implementations for the following
-tensors:
-- 1st order [`VectorValue`](@ref),
-- 2nd order [`TensorValue`](@ref),
-- 2nd order and symmetric [`SymTensorValue`](@ref),
-- 2nd order, symmetric and traceless [`SymTracelessTensorValue`](@ref),
-- 3rd order [`ThirdOrderTensorValue`](@ref),
-- 4th order and symmetric [`SymFourthOrderTensorValue`](@ref)).
-
-## Why
-
-The main feature of this module is that the provided types do not extend from `AbstractArray`, but from `Number`!
-
-This allows one to work with them as if they were scalar values in broadcasted operations on arrays of `VectorValue` objects (also for `TensorValue` or `MultiValue` objects). For instance, one can perform the following manipulations:
-```julia
-# Assign a VectorValue to all the entries of an Array of VectorValues
-A = zeros(VectorValue{2,Int}, (4,5))
-v = VectorValue(12,31)
-A .= v # This is possible since  VectorValue <: Number
-
-# Broadcasting of tensor operations in arrays of TensorValues
-t = TensorValue(13,41,53,17) # creates a 2x2 TensorValue
-g = TensorValue(32,41,3,14) # creates another 2x2 TensorValue
-B = fill(t,(1,5))
-C = inner.(g,B) # inner product of g against all TensorValues in the array B
-@show C
-# C = [2494 2494 2494 2494 2494]
-```
-
-To create a variable of type [`MultiValue`](@ref) from components, these should be given
-as separate arguments or all gathered in a `tuple`. The order of the arguments
-is the order of the linearized Cartesian indices of the corresponding array
-(order of the [`LinearIndices`](@ref) indices):
-```julia
-using StaticArrays
-t = TensorValue( (1, 2, 3, 4) )
-ts= convert(SMatrix{2,2,Int}, t)
-@show ts
-# 2×2 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
-#  1  3
-#  2  4
-t2[1,2] == t[1,2] == 3 # true
-```
-For symetric tensor types, only the independent components should be given, see
-[`SymTensorValue`](@ref), [`SymTracelessTensorValue`](@ref) and [`SymFourthOrderTensorValue`](@ref).
-
-A `MultiValue` can be created from an `AbstractArray` of the same size. If the
-`MultiValue` type has internal constraints (e.g. symmetries), ONLY the required
-components are picked from the array WITHOUT CHECKING if the given array
-did respect the constraints:
-```julia
-SymTensorValue( [1 2; 3 4] )          # -> SymTensorValue{2, Int64, 3}(1, 2, 4)
-SymTensorValue( SMatrix{2}(1,2,3,4) ) # -> SymTensorValue{2, Int64, 3}(1, 3, 4)
-```
-
-`MultiValue`s can be converted to static and mutable arrays types from
-`StaticArrays.jl` using `convert` and [`mutable`](@ref), respectively.
-
-The concrete `MultiValue` types implement methods for the following
-`Base` functions: `length`, `size`, `rand`, `zero`, `real`, `imag` and
-`conj`.
-
-`one` is also implemented in particular cases, it is defined for second
-and fourth order tensors. For second order, it returns the identity tensor `δij`,
-for fourth order, see [`one`](@ref). `SymTracelessTensorValue` does not implement
-`one`.
-
-The exported names are:
-
-$(EXPORTS)
+Immutable tensor types for Gridap.
 """
 module TensorValues
 

From 9a961334f19a8556dfc5bfa6c2dd49a577b10f15 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Tue, 26 Nov 2024 14:19:45 +1100
Subject: [PATCH 05/27] Added Xiao - Gimbutas quadratures

---
 src/ReferenceFEs/ReferenceFEs.jl              |    3 +
 src/ReferenceFEs/XiaoGimbutasQuadratures.jl   | 4379 +++++++++++++++++
 .../XiaoGimbutasQuadraturesTests.jl           |   32 +
 test/ReferenceFEsTests/runtests.jl            |    2 +
 4 files changed, 4416 insertions(+)
 create mode 100644 src/ReferenceFEs/XiaoGimbutasQuadratures.jl
 create mode 100644 test/ReferenceFEsTests/XiaoGimbutasQuadraturesTests.jl

diff --git a/src/ReferenceFEs/ReferenceFEs.jl b/src/ReferenceFEs/ReferenceFEs.jl
index 24cf3d310..428f183ea 100644
--- a/src/ReferenceFEs/ReferenceFEs.jl
+++ b/src/ReferenceFEs/ReferenceFEs.jl
@@ -211,6 +211,7 @@ export test_quadrature
 export tensor_product
 export duffy
 export strang
+export xiao_gimbutas
 
 include("Polytopes.jl")
 
@@ -242,6 +243,8 @@ include("DuffyQuadratures.jl")
 
 include("StrangQuadratures.jl")
 
+include("XiaoGimbutasQuadratures.jl")
+
 include("RaviartThomasRefFEs.jl")
 
 include("BDMRefFEs.jl")
diff --git a/src/ReferenceFEs/XiaoGimbutasQuadratures.jl b/src/ReferenceFEs/XiaoGimbutasQuadratures.jl
new file mode 100644
index 000000000..7fd185d28
--- /dev/null
+++ b/src/ReferenceFEs/XiaoGimbutasQuadratures.jl
@@ -0,0 +1,4379 @@
+
+struct XiaoGimbutas <: QuadratureName end
+const xiao_gimbutas = XiaoGimbutas()
+
+# Reference: 
+#  `A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions`, 
+#   Hong Xiao, Zydrunas Gimbutas, Computers & Mathematics with Applications, (2010)
+#   DOI : https://doi.org/10.1016/j.camwa.2009.10.027
+# Adapted from: 
+#    https://github.com/FEniCS/basix/blob/main/cpp/basix/quadrature.cpp
+function Quadrature(
+  p::Polytope,::XiaoGimbutas,degree::Integer;T::Type{<:AbstractFloat}=Float64
+)
+  msg = """\n
+  `strang` quadrature rule only available for simplices.
+  Use `tensor_product` for n-cubes.
+  """
+  @assert is_simplex(p) msg
+
+  D = num_dims(p)
+  if D == 2
+    x, w = _xiaogimbutas_quad_tri(degree)
+  elseif D == 3
+    x, w = _xiaogimbutas_quad_tet(degree)
+  else
+    msg = """\n
+    `strang` quadrature rule only available for tris and tets.
+    Use `duffy` for other simplices.
+    """
+    @unreachable msg
+  end
+  x_T = convert(Vector{VectorValue{D,T}},x)
+  w_T = convert(Vector{T},w)
+  GenericQuadrature(x_T,w_T,"Xiao-Gimbutas - $(string(p)) - degree $degree")
+end
+
+function _xiaogimbutas_quad_tri(degree)
+  if degree ∉ 1:30
+    msg = """\n
+      `xiaogimbutas` quadrature rule not implemented for degree = $degree on a triangle.
+      Implemented up to degree 30.
+      Use `duffy` instead.
+    """
+    error(msg)
+  end
+
+  if (degree == 1)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333)
+    ]
+    w = [0.5]
+    return x, w
+  elseif (degree == 2)
+    x = [
+      VectorValue(0.16666666666666666, 0.16666666666666666),
+      VectorValue(0.16666666666666666,0.6666666666666667),
+      VectorValue(0.6666666666666667,  0.16666666666666666)
+    ]
+    w = [0.16666666666666666, 0.16666666666666666, 0.16666666666666666]
+    return x, w
+  elseif (degree == 3)
+    x = [
+      VectorValue(0.4459484909159649,  0.4459484909159649), 
+      VectorValue(0.09157621350977085, 0.09157621350977085),
+      VectorValue(0.4459484909159649, 0.10810301816807022),
+      VectorValue(0.09157621350977085, 0.8168475729804583), 
+      VectorValue(0.10810301816807022, 0.4459484909159649),
+      VectorValue(0.8168475729804583, 0.09157621350977085)
+    ]
+    w = [0.11169079483900574, 0.05497587182766094, 0.11169079483900574,
+            0.05497587182766094, 0.11169079483900574, 0.05497587182766094]
+    return x, w
+  elseif (degree == 4)
+    x = [
+      VectorValue(0.4459484909159649, 0.4459484909159649),
+      VectorValue(0.09157621350977085, 0.09157621350977085),
+      VectorValue(0.4459484909159649, 0.10810301816807022),
+      VectorValue(0.09157621350977085, 0.8168475729804583), 
+      VectorValue(0.10810301816807022, 0.4459484909159649),
+      VectorValue(0.8168475729804583, 0.09157621350977085)
+    ]
+    w = [0.11169079483900574, 0.05497587182766094, 0.11169079483900574,
+            0.05497587182766094, 0.11169079483900574, 0.05497587182766094]
+    return x, w
+  elseif (degree == 5)
+    x = [
+      VectorValue(0.3333333333333333,  0.3333333333333333), 
+      VectorValue(0.1012865073234564, 0.1012865073234564),
+      VectorValue(0.47014206410511505, 0.47014206410511505),
+      VectorValue(0.1012865073234564,  0.7974269853530872),  
+      VectorValue(0.47014206410511505,0.05971587178976989), 
+      VectorValue(0.7974269853530872,  0.1012865073234564),
+      VectorValue(0.05971587178976989, 0.47014206410511505)
+    ]
+    w = [
+      0.1125, 0.06296959027241357, 0.0661970763942531, 0.06296959027241357,
+      0.0661970763942531, 0.06296959027241357, 0.0661970763942531
+    ]
+    return x, w
+  elseif (degree == 6)
+    x = [
+      VectorValue(0.21942998254978302,  0.21942998254978302),  
+      VectorValue(0.48013796411221504, 0.48013796411221504),  
+      VectorValue(0.21942998254978302, 0.561140034900434),
+      
+      VectorValue(0.48013796411221504, 0.039724071775569914), 
+      VectorValue(0.561140034900434, 0.21942998254978302),  
+      VectorValue(0.039724071775569914, 0.48013796411221504),
+      
+      VectorValue(0.019371724361240805, 0.14161901592396814),  
+      VectorValue(0.8390092597147911, 0.019371724361240805), 
+      VectorValue(0.14161901592396814, 0.8390092597147911),
+      
+      VectorValue(0.14161901592396814, 0.019371724361240805), 
+      VectorValue(0.8390092597147911, 0.14161901592396814),  
+      VectorValue(0.019371724361240805, 0.8390092597147911)
+    ]
+    w = [0.08566656207649052, 0.04036554479651549, 0.08566656207649052,
+            0.04036554479651549, 0.08566656207649052, 0.04036554479651549,
+            0.02031727989683033, 0.02031727989683033, 0.02031727989683033,
+            0.02031727989683033, 0.02031727989683033, 0.02031727989683033
+        ]
+    return x, w
+  elseif (degree == 7)
+    x = [
+      VectorValue(0.47319565368925104, 0.47319565368925104),  
+      VectorValue(0.057797640054506494,0.057797640054506494), 
+      VectorValue(0.24166360639724743, 0.24166360639724743),
+      VectorValue(0.47319565368925104, 0.05360869262149792),  
+      VectorValue(0.057797640054506494,0.884404719890987),    
+      VectorValue(0.24166360639724743, 0.5166727872055051),
+      VectorValue(0.05360869262149792, 0.47319565368925104),  
+      VectorValue(0.884404719890987,0.057797640054506494), 
+      VectorValue(0.5166727872055051, 0.24166360639724743),
+      VectorValue(0.046971206130085534, 0.2593390118657857),   
+      VectorValue(0.6936897820041288,0.046971206130085534), 
+      VectorValue(0.2593390118657857, 0.6936897820041288),
+      VectorValue(0.2593390118657857, 0.046971206130085534), 
+      VectorValue(0.6936897820041288, 0.2593390118657857),   
+      VectorValue(0.046971206130085534, 0.6936897820041288)
+    ]
+    w = [
+      0.02659041664838023,  0.020459085197028434, 0.06386262428056692,
+      0.02659041664838023,  0.020459085197028434, 0.06386262428056692,
+      0.02659041664838023,  0.020459085197028434, 0.06386262428056692,
+      0.027877270270345547, 0.027877270270345547, 0.027877270270345547,
+      0.027877270270345547, 0.027877270270345547, 0.027877270270345547
+    ]
+    return x, w
+  elseif (degree == 8)
+    x = [
+      VectorValue(0.3333333333333333,   0.3333333333333333),  
+      VectorValue(0.17056930775176027,0.17056930775176027),  
+      VectorValue(0.4592925882927231,  0.4592925882927231),
+      VectorValue(0.05054722831703107,  0.05054722831703107), 
+      VectorValue(0.17056930775176027,0.6588613844964795),   
+      VectorValue(0.4592925882927231,  0.08141482341455375),
+      VectorValue(0.05054722831703107,  0.8989055433659379),  
+      VectorValue(0.6588613844964795,0.17056930775176027),  
+      VectorValue(0.08141482341455375, 0.4592925882927231),
+      VectorValue(0.8989055433659379,   0.05054722831703107), 
+      VectorValue(0.008394777409957675,0.26311282963463806),  
+      VectorValue(0.7284923929554044,  0.008394777409957675),
+      VectorValue(0.26311282963463806,  0.7284923929554044),  
+      VectorValue(0.26311282963463806,0.008394777409957675), 
+      VectorValue(0.7284923929554044,  0.26311282963463806),
+      VectorValue(0.008394777409957675, 0.7284923929554044)
+    ]
+    w = [
+      0.0721578038388936,   0.05160868526735912,  0.04754581713364232,
+      0.01622924881159904,  0.05160868526735912,  0.04754581713364232,
+      0.01622924881159904,  0.05160868526735912,  0.04754581713364232,
+      0.01622924881159904,  0.013615157087217498, 0.013615157087217498,
+      0.013615157087217498, 0.013615157087217498, 0.013615157087217498,
+      0.013615157087217498
+      ]
+    return x, w
+  elseif (degree == 9)
+    x = [
+      VectorValue(0.3333333333333333,  0.3333333333333333),  
+      VectorValue(0.4896825191987376,0.4896825191987376),  
+      VectorValue(0.1882035356190328,  0.1882035356190328),
+      VectorValue(0.43708959149293664, 0.43708959149293664), 
+      VectorValue(0.04472951339445275,0.04472951339445275), 
+      VectorValue(0.4896825191987376,  0.02063496160252476),
+      VectorValue(0.1882035356190328,  0.6235929287619344),  
+      VectorValue(0.43708959149293664,0.12582081701412673), 
+      VectorValue(0.04472951339445275, 0.9105409732110945),
+      VectorValue(0.02063496160252476, 0.4896825191987376),  
+      VectorValue(0.6235929287619344,0.1882035356190328),  
+      VectorValue(0.12582081701412673, 0.43708959149293664),
+      VectorValue(0.9105409732110945,  0.04472951339445275), 
+      VectorValue(0.0368384120547363,0.2219629891607657),  
+      VectorValue(0.741198598784498,   0.0368384120547363),
+      VectorValue(0.2219629891607657,  0.741198598784498),   
+      VectorValue(0.2219629891607657,0.0368384120547363),  
+      VectorValue(0.741198598784498,   0.2219629891607657),
+      VectorValue(0.0368384120547363,  0.741198598784498)
+    ]
+    w = [0.04856789814139942,  0.015667350113569536, 0.03982386946360513,
+            0.03891377050238714,  0.012788837829349017, 0.015667350113569536,
+            0.03982386946360513,  0.03891377050238714,  0.012788837829349017,
+            0.015667350113569536, 0.03982386946360513,  0.03891377050238714,
+            0.012788837829349017, 0.021641769688644688, 0.021641769688644688,
+            0.021641769688644688, 0.021641769688644688, 0.021641769688644688,
+            0.021641769688644688]
+    return x, w
+  elseif (degree == 10)
+    x = [
+      VectorValue(0.3333333333333333,   0.3333333333333333),   
+      VectorValue(0.4951734598011705,0.4951734598011705),   
+      VectorValue(0.019139415242841296, 0.019139415242841296),
+      VectorValue(0.18448501268524653,  0.18448501268524653),  
+      VectorValue(0.42823482094371884,0.42823482094371884),  
+      VectorValue(0.4951734598011705,   0.009653080397658997),
+      VectorValue(0.019139415242841296, 0.9617211695143174),   
+      VectorValue(0.18448501268524653,0.6310299746295069),   
+      VectorValue(0.42823482094371884,  0.14353035811256232),
+      VectorValue(0.009653080397658997, 0.4951734598011705),   
+      VectorValue(0.9617211695143174,0.019139415242841296), 
+      VectorValue(0.6310299746295069,   0.18448501268524653),
+      VectorValue(0.14353035811256232,  0.42823482094371884),  
+      VectorValue(0.03472362048232748,0.13373475510086913),  
+      VectorValue(0.03758272734119169,  0.3266931362813369),
+      VectorValue(0.8315416244168035,   0.03472362048232748),  
+      VectorValue(0.6357241363774714,0.03758272734119169),  
+      VectorValue(0.13373475510086913,  0.8315416244168035),
+      VectorValue(0.3266931362813369,   0.6357241363774714),   
+      VectorValue(0.13373475510086913,0.03472362048232748),  
+      VectorValue(0.3266931362813369,   0.03758272734119169),
+      VectorValue(0.8315416244168035,   0.13373475510086913),  
+      VectorValue(0.6357241363774714,0.3266931362813369),   
+      VectorValue(0.03472362048232748,  0.8315416244168035),
+      VectorValue(0.03758272734119169,  0.6357241363774714)
+      ]
+    w = [0.041807437186986963, 0.004896295249209152, 0.003192679615059327,
+            0.039316884873188636, 0.03762366398427199,  0.004896295249209152,
+            0.003192679615059327, 0.039316884873188636, 0.03762366398427199,
+            0.004896295249209152, 0.003192679615059327, 0.039316884873188636,
+            0.03762366398427199,  0.014481140731628171, 0.019369524543009452,
+            0.014481140731628171, 0.019369524543009452, 0.014481140731628171,
+            0.019369524543009452, 0.014481140731628171, 0.019369524543009452,
+            0.014481140731628171, 0.019369524543009452, 0.014481140731628171,
+            0.019369524543009452];
+    return x, w
+  elseif (degree == 11)
+    x = [
+      VectorValue(0.3333333333333333,    0.3333333333333333),   
+      VectorValue(0.030846895635588123,0.030846895635588123),  
+      VectorValue(0.49878016517846074,  0.49878016517846074),
+      VectorValue(0.11320782728669404,   0.11320782728669404),  
+      VectorValue(0.4366550163931761,0.4366550163931761),    
+      VectorValue(0.21448345861926937,  0.21448345861926937),
+      VectorValue(0.030846895635588123,  0.9383062087288238),   
+      VectorValue(0.49878016517846074,0.0024396696430785125), 
+      VectorValue(0.11320782728669404,  0.7735843454266119),
+      VectorValue(0.4366550163931761,    0.12668996721364778),  
+      VectorValue(0.21448345861926937,0.5710330827614613),    
+      VectorValue(0.9383062087288238,   0.030846895635588123),
+      VectorValue(0.0024396696430785125, 0.49878016517846074),  
+      VectorValue(0.7735843454266119,0.11320782728669404),   
+      VectorValue(0.12668996721364778,  0.4366550163931761),
+      VectorValue(0.5710330827614613,    0.21448345861926937),  
+      VectorValue(0.014366662569555624,0.1593036198376935),    
+      VectorValue(0.04766406697215078,  0.31063121631346313),
+      VectorValue(0.8263297175927509,    0.014366662569555624), 
+      VectorValue(0.6417047167143861,0.04766406697215078),   
+      VectorValue(0.1593036198376935,   0.8263297175927509),
+      VectorValue(0.31063121631346313,   0.6417047167143861),   
+      VectorValue(0.1593036198376935,0.014366662569555624),  
+      VectorValue(0.31063121631346313,  0.04766406697215078),
+      VectorValue(0.8263297175927509,    0.1593036198376935),   
+      VectorValue(0.6417047167143861,0.31063121631346313),   
+      VectorValue(0.014366662569555624, 0.8263297175927509),
+      VectorValue(0.04766406697215078,   0.6417047167143861)
+    ];
+    w = [
+        0.040722567354675644,  0.006124648475353982,  0.0062327459369406904,
+        0.02006462119065416,   0.031547436079949344,  0.033922553871847574,
+        0.006124648475353982,  0.0062327459369406904, 0.02006462119065416,
+        0.031547436079949344,  0.033922553871847574,  0.006124648475353982,
+        0.0062327459369406904, 0.02006462119065416,   0.031547436079949344,
+        0.033922553871847574,  0.007278811668904623,  0.020321424327943236,
+        0.007278811668904623,  0.020321424327943236,  0.007278811668904623,
+        0.020321424327943236,  0.007278811668904623,  0.020321424327943236,
+        0.007278811668904623,  0.020321424327943236,  0.007278811668904623,
+        0.020321424327943236];
+    return x, w
+  elseif (degree == 12)
+    x = [
+      VectorValue(0.27146250701492614,  0.27146250701492614),  
+      VectorValue(0.10925782765935432,0.10925782765935432),  
+      VectorValue(0.4401116486585931,   0.4401116486585931),
+      VectorValue(0.4882037509455415,   0.4882037509455415),   
+      VectorValue(0.02464636343633564,0.02464636343633564),  
+      VectorValue(0.27146250701492614,  0.45707498597014773),
+      VectorValue(0.10925782765935432,  0.7814843446812914),   
+      VectorValue(0.4401116486585931,0.11977670268281382),  
+      VectorValue(0.4882037509455415,   0.02359249810891695),
+      VectorValue(0.02464636343633564,  0.9507072731273287),   
+      VectorValue(0.45707498597014773,0.27146250701492614),  
+      VectorValue(0.7814843446812914,   0.10925782765935432),
+      VectorValue(0.11977670268281382,  0.4401116486585931),   
+      VectorValue(0.02359249810891695,0.4882037509455415),   
+      VectorValue(0.9507072731273287,   0.02464636343633564),
+      VectorValue(0.1162960196779266,   0.25545422863851736),  
+      VectorValue(0.021382490256170623,0.12727971723358936),  
+      VectorValue(0.023034156355267166, 0.29165567973834094),
+      VectorValue(0.6282497516835561,   0.1162960196779266),   
+      VectorValue(0.85133779251024,0.021382490256170623), 
+      VectorValue(0.6853101639063919,   0.023034156355267166),
+      VectorValue(0.25545422863851736,  0.6282497516835561),   
+      VectorValue(0.12727971723358936,0.85133779251024),     
+      VectorValue(0.29165567973834094,  0.6853101639063919),
+      VectorValue(0.25545422863851736,  0.1162960196779266),   
+      VectorValue(0.12727971723358936,0.021382490256170623), 
+      VectorValue(0.29165567973834094,  0.023034156355267166),
+      VectorValue(0.6282497516835561,   0.25545422863851736),  
+      VectorValue(0.85133779251024,0.12727971723358936),  
+      VectorValue(0.6853101639063919,   0.29165567973834094),
+      VectorValue(0.1162960196779266,   0.6282497516835561),   
+      VectorValue(0.021382490256170623,0.85133779251024),     
+      VectorValue(0.023034156355267166, 0.6853101639063919)
+      ];
+    w = [
+        0.03127060659795138,   0.014243026034438775,  0.024959167464030475,
+        0.012133419040726017,  0.0039658212549868194, 0.03127060659795138,
+        0.014243026034438775,  0.024959167464030475,  0.012133419040726017,
+        0.0039658212549868194, 0.03127060659795138,   0.014243026034438775,
+        0.024959167464030475,  0.012133419040726017,  0.0039658212549868194,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378,
+        0.021613681829707104,  0.007541838788255721,  0.01089179251930378];
+    return x, w
+  elseif (degree == 13)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.4961358947410461, 0.4961358947410461),
+      VectorValue(0.4696086896534919, 0.4696086896534919),
+      VectorValue(0.23111028494908226, 0.23111028494908226),
+      VectorValue(0.4144775702790546, 0.4144775702790546),
+      VectorValue(0.11355991257213327, 0.11355991257213327),
+      VectorValue(0.024895931491216494, 0.024895931491216494),
+      VectorValue(0.4961358947410461, 0.007728210517907841),
+      VectorValue(0.4696086896534919, 0.06078262069301621),
+      VectorValue(0.23111028494908226, 0.5377794301018355),
+      VectorValue(0.4144775702790546, 0.17104485944189085),
+      VectorValue(0.11355991257213327, 0.7728801748557335),
+      VectorValue(0.024895931491216494, 0.950208137017567),
+      VectorValue(0.007728210517907841, 0.4961358947410461),
+      VectorValue(0.06078262069301621, 0.4696086896534919),
+      VectorValue(0.5377794301018355, 0.23111028494908226),
+      VectorValue(0.17104485944189085, 0.4144775702790546),
+      VectorValue(0.7728801748557335, 0.11355991257213327),
+      VectorValue(0.950208137017567, 0.024895931491216494),
+      VectorValue(0.01898800438375904, 0.2920786885766364),
+      VectorValue(0.09773603106601653, 0.26674525331035115),
+      VectorValue(0.021966344206529244, 0.1267997757838373),
+      VectorValue(0.6889333070396046, 0.01898800438375904),
+      VectorValue(0.6355187156236324, 0.09773603106601653),
+      VectorValue(0.8512338800096335, 0.021966344206529244),
+      VectorValue(0.2920786885766364, 0.6889333070396046),
+      VectorValue(0.26674525331035115, 0.6355187156236324),
+      VectorValue(0.1267997757838373, 0.8512338800096335),
+      VectorValue(0.2920786885766364, 0.01898800438375904),
+      VectorValue(0.26674525331035115, 0.09773603106601653),
+      VectorValue(0.1267997757838373, 0.021966344206529244),
+      VectorValue(0.6889333070396046, 0.2920786885766364),
+      VectorValue(0.6355187156236324, 0.26674525331035115),
+      VectorValue(0.8512338800096335, 0.1267997757838373),
+      VectorValue(0.01898800438375904, 0.6889333070396046),
+      VectorValue(0.09773603106601653, 0.6355187156236324),
+      VectorValue(0.021966344206529244, 0.8512338800096335),
+    ]
+    w = [0.02581132333214541,   0.004970738180536294, 0.01639062080186149,
+            0.023031204796389124,  0.0234735477710776,   0.015451548987879897,
+            0.0040146998976292115, 0.004970738180536294, 0.01639062080186149,
+            0.023031204796389124,  0.0234735477710776,   0.015451548987879897,
+            0.0040146998976292115, 0.004970738180536294, 0.01639062080186149,
+            0.023031204796389124,  0.0234735477710776,   0.015451548987879897,
+            0.0040146998976292115, 0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089,  0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089,  0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089,  0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089,  0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089,  0.00906274932310044,  0.018605980228630768,
+            0.007696536341891089];
+    return x, w
+  elseif (degree == 14)
+    x = [
+      VectorValue(0.41764471934045394, 0.41764471934045394),
+      VectorValue(0.0617998830908727, 0.0617998830908727),
+      VectorValue(0.2734775283088387, 0.2734775283088387),
+      VectorValue(0.1772055324125435, 0.1772055324125435),
+      VectorValue(0.0193909612487011, 0.0193909612487011),
+      VectorValue(0.4889639103621786, 0.4889639103621786),
+      VectorValue(0.41764471934045394, 0.16471056131909212),
+      VectorValue(0.0617998830908727, 0.8764002338182546),
+      VectorValue(0.2734775283088387, 0.4530449433823226),
+      VectorValue(0.1772055324125435, 0.645588935174913),
+      VectorValue(0.0193909612487011, 0.9612180775025978),
+      VectorValue(0.4889639103621786, 0.022072179275642756),
+      VectorValue(0.16471056131909212, 0.41764471934045394),
+      VectorValue(0.8764002338182546, 0.0617998830908727),
+      VectorValue(0.4530449433823226, 0.2734775283088387),
+      VectorValue(0.645588935174913, 0.1772055324125435),
+      VectorValue(0.9612180775025978, 0.0193909612487011),
+      VectorValue(0.022072179275642756, 0.4889639103621786),
+      VectorValue(0.014646950055654471, 0.29837288213625773),
+      VectorValue(0.09291624935697185, 0.336861459796345),
+      VectorValue(0.05712475740364799, 0.17226668782135557),
+      VectorValue(0.001268330932872076, 0.11897449769695682),
+      VectorValue(0.6869801678080878, 0.014646950055654471),
+      VectorValue(0.5702222908466832, 0.09291624935697185),
+      VectorValue(0.7706085547749965, 0.05712475740364799),
+      VectorValue(0.8797571713701712, 0.001268330932872076),
+      VectorValue(0.29837288213625773, 0.6869801678080878),
+      VectorValue(0.336861459796345, 0.5702222908466832),
+      VectorValue(0.17226668782135557, 0.7706085547749965),
+      VectorValue(0.11897449769695682, 0.8797571713701712),
+      VectorValue(0.29837288213625773, 0.014646950055654471),
+      VectorValue(0.336861459796345, 0.09291624935697185),
+      VectorValue(0.17226668782135557, 0.05712475740364799),
+      VectorValue(0.11897449769695682, 0.001268330932872076),
+      VectorValue(0.6869801678080878, 0.29837288213625773),
+      VectorValue(0.5702222908466832, 0.336861459796345),
+      VectorValue(0.7706085547749965, 0.17226668782135557),
+      VectorValue(0.8797571713701712, 0.11897449769695682),
+      VectorValue(0.014646950055654471, 0.6869801678080878),
+      VectorValue(0.09291624935697185, 0.5702222908466832),
+      VectorValue(0.05712475740364799, 0.7706085547749965),
+      VectorValue(0.001268330932872076, 0.8797571713701712),
+    ]
+    w = [0.016394176772062678, 0.007216849834888334, 0.025887052253645793,
+            0.02108129436849651,  0.002461701801200041, 0.010941790684714446,
+            0.016394176772062678, 0.007216849834888334, 0.025887052253645793,
+            0.02108129436849651,  0.002461701801200041, 0.010941790684714446,
+            0.016394176772062678, 0.007216849834888334, 0.025887052253645793,
+            0.02108129436849651,  0.002461701801200041, 0.010941790684714446,
+            0.007218154056766921, 0.019285755393530342, 0.012332876606281839,
+            0.002505114419250336, 0.007218154056766921, 0.019285755393530342,
+            0.012332876606281839, 0.002505114419250336, 0.007218154056766921,
+            0.019285755393530342, 0.012332876606281839, 0.002505114419250336,
+            0.007218154056766921, 0.019285755393530342, 0.012332876606281839,
+            0.002505114419250336, 0.007218154056766921, 0.019285755393530342,
+            0.012332876606281839, 0.002505114419250336, 0.007218154056766921,
+            0.019285755393530342, 0.012332876606281839, 0.002505114419250336];
+    return x, w
+  elseif (degree == 15)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.1299782299330779, 0.1299782299330779),
+      VectorValue(0.4600769492970597, 0.4600769492970597),
+      VectorValue(0.4916858166302972, 0.4916858166302972),
+      VectorValue(0.22153234079514206, 0.22153234079514206),
+      VectorValue(0.39693373740906057, 0.39693373740906057),
+      VectorValue(0.0563419176961002, 0.0563419176961002),
+      VectorValue(0.1299782299330779, 0.7400435401338442),
+      VectorValue(0.4600769492970597, 0.07984610140588055),
+      VectorValue(0.4916858166302972, 0.016628366739405598),
+      VectorValue(0.22153234079514206, 0.5569353184097159),
+      VectorValue(0.39693373740906057, 0.20613252518187886),
+      VectorValue(0.0563419176961002, 0.8873161646077996),
+      VectorValue(0.7400435401338442, 0.1299782299330779),
+      VectorValue(0.07984610140588055, 0.4600769492970597),
+      VectorValue(0.016628366739405598, 0.4916858166302972),
+      VectorValue(0.5569353184097159, 0.22153234079514206),
+      VectorValue(0.20613252518187886, 0.39693373740906057),
+      VectorValue(0.8873161646077996, 0.0563419176961002),
+      VectorValue(0.08459422148219181, 0.18232178340719132),
+      VectorValue(0.016027089786345473, 0.15020038406523872),
+      VectorValue(0.09765044243024235, 0.32311131516371266),
+      VectorValue(0.018454251904633165, 0.3079476814836729),
+      VectorValue(0.0011135352740137417, 0.03803522930110929),
+      VectorValue(0.733083995110617, 0.08459422148219181),
+      VectorValue(0.8337725261484158, 0.016027089786345473),
+      VectorValue(0.5792382424060449, 0.09765044243024235),
+      VectorValue(0.673598066611694, 0.018454251904633165),
+      VectorValue(0.960851235424877, 0.0011135352740137417),
+      VectorValue(0.18232178340719132, 0.733083995110617),
+      VectorValue(0.15020038406523872, 0.8337725261484158),
+      VectorValue(0.32311131516371266, 0.5792382424060449),
+      VectorValue(0.3079476814836729, 0.673598066611694),
+      VectorValue(0.03803522930110929, 0.960851235424877),
+      VectorValue(0.18232178340719132, 0.08459422148219181),
+      VectorValue(0.15020038406523872, 0.016027089786345473),
+      VectorValue(0.32311131516371266, 0.09765044243024235),
+      VectorValue(0.3079476814836729, 0.018454251904633165),
+      VectorValue(0.03803522930110929, 0.0011135352740137417),
+      VectorValue(0.733083995110617, 0.18232178340719132),
+      VectorValue(0.8337725261484158, 0.15020038406523872),
+      VectorValue(0.5792382424060449, 0.32311131516371266),
+      VectorValue(0.673598066611694, 0.3079476814836729),
+      VectorValue(0.960851235424877, 0.03803522930110929),
+      VectorValue(0.08459422148219181, 0.733083995110617),
+      VectorValue(0.016027089786345473, 0.8337725261484158),
+      VectorValue(0.09765044243024235, 0.5792382424060449),
+      VectorValue(0.018454251904633165, 0.673598066611694),
+      VectorValue(0.0011135352740137417, 0.960851235424877),
+    ]
+    w = [
+        0.01486520987403566,   0.00369875203352305,   0.010797043968219226,
+        0.0079161381750109,    0.023143643052599038,  0.023168020695603617,
+        0.007542237123798534,  0.00369875203352305,   0.010797043968219226,
+        0.0079161381750109,    0.023143643052599038,  0.023168020695603617,
+        0.007542237123798534,  0.00369875203352305,   0.010797043968219226,
+        0.0079161381750109,    0.023143643052599038,  0.023168020695603617,
+        0.007542237123798534,  0.012115004391562803,  0.00561425214943903,
+        0.015537610235255475,  0.008218381046413948,  0.0012376330072789582,
+        0.012115004391562803,  0.00561425214943903,   0.015537610235255475,
+        0.008218381046413948,  0.0012376330072789582, 0.012115004391562803,
+        0.00561425214943903,   0.015537610235255475,  0.008218381046413948,
+        0.0012376330072789582, 0.012115004391562803,  0.00561425214943903,
+        0.015537610235255475,  0.008218381046413948,  0.0012376330072789582,
+        0.012115004391562803,  0.00561425214943903,   0.015537610235255475,
+        0.008218381046413948,  0.0012376330072789582, 0.012115004391562803,
+        0.00561425214943903,   0.015537610235255475,  0.008218381046413948,
+        0.0012376330072789582];
+    return x, w
+  elseif (degree == 16)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.06667447224023837, 0.06667447224023837),
+      VectorValue(0.24132168070137838, 0.24132168070137838),
+      VectorValue(0.41279809595522365, 0.41279809595522365),
+      VectorValue(0.15006373658703515, 0.15006373658703515),
+      VectorValue(0.46954803099668496, 0.46954803099668496),
+      VectorValue(0.017041629405718517, 0.017041629405718517),
+      VectorValue(0.06667447224023837, 0.8666510555195233),
+      VectorValue(0.24132168070137838, 0.5173566385972432),
+      VectorValue(0.41279809595522365, 0.1744038080895527),
+      VectorValue(0.15006373658703515, 0.6998725268259297),
+      VectorValue(0.46954803099668496, 0.060903938006630076),
+      VectorValue(0.017041629405718517, 0.965916741188563),
+      VectorValue(0.8666510555195233, 0.06667447224023837),
+      VectorValue(0.5173566385972432, 0.24132168070137838),
+      VectorValue(0.1744038080895527, 0.41279809595522365),
+      VectorValue(0.6998725268259297, 0.15006373658703515),
+      VectorValue(0.060903938006630076, 0.46954803099668496),
+      VectorValue(0.965916741188563, 0.017041629405718517),
+      VectorValue(0.009664954403660254, 0.41376948582708517),
+      VectorValue(0.030305943355186365, 0.30417944822947973),
+      VectorValue(0.010812972776103751, 0.08960908902270585),
+      VectorValue(0.10665316053614844, 0.29661537240038294),
+      VectorValue(0.051354315344013114, 0.16976335515028973),
+      VectorValue(0.0036969427073556124, 0.21404877992584728),
+      VectorValue(0.5765655597692546, 0.009664954403660254),
+      VectorValue(0.6655146084153339, 0.030305943355186365),
+      VectorValue(0.8995779382011905, 0.010812972776103751),
+      VectorValue(0.5967314670634686, 0.10665316053614844),
+      VectorValue(0.7788823295056971, 0.051354315344013114),
+      VectorValue(0.7822542773667971, 0.0036969427073556124),
+      VectorValue(0.41376948582708517, 0.5765655597692546),
+      VectorValue(0.30417944822947973, 0.6655146084153339),
+      VectorValue(0.08960908902270585, 0.8995779382011905),
+      VectorValue(0.29661537240038294, 0.5967314670634686),
+      VectorValue(0.16976335515028973, 0.7788823295056971),
+      VectorValue(0.21404877992584728, 0.7822542773667971),
+      VectorValue(0.41376948582708517, 0.009664954403660254),
+      VectorValue(0.30417944822947973, 0.030305943355186365),
+      VectorValue(0.08960908902270585, 0.010812972776103751),
+      VectorValue(0.29661537240038294, 0.10665316053614844),
+      VectorValue(0.16976335515028973, 0.051354315344013114),
+      VectorValue(0.21404877992584728, 0.0036969427073556124),
+      VectorValue(0.5765655597692546, 0.41376948582708517),
+      VectorValue(0.6655146084153339, 0.30417944822947973),
+      VectorValue(0.8995779382011905, 0.08960908902270585),
+      VectorValue(0.5967314670634686, 0.29661537240038294),
+      VectorValue(0.7788823295056971, 0.16976335515028973),
+      VectorValue(0.7822542773667971, 0.21404877992584728),
+      VectorValue(0.009664954403660254, 0.5765655597692546),
+      VectorValue(0.030305943355186365, 0.6655146084153339),
+      VectorValue(0.010812972776103751, 0.8995779382011905),
+      VectorValue(0.10665316053614844, 0.5967314670634686),
+      VectorValue(0.051354315344013114, 0.7788823295056971),
+      VectorValue(0.0036969427073556124, 0.7822542773667971),
+    ]
+    w = [0.023113955157095672,  0.006212712797780504, 0.020592020534896276,
+            0.020492609893407683,  0.014391748351374455, 0.013546834733855226,
+            0.001894567619132111,  0.006212712797780504, 0.020592020534896276,
+            0.020492609893407683,  0.014391748351374455, 0.013546834733855226,
+            0.001894567619132111,  0.006212712797780504, 0.020592020534896276,
+            0.020492609893407683,  0.014391748351374455, 0.013546834733855226,
+            0.001894567619132111,  0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673, 0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673, 0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673, 0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673, 0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673, 0.004091105276611069, 0.006991803562326784,
+            0.0028759349852485795, 0.015823030840991622, 0.008826540523551642,
+            0.0023073453198645673];
+    return x, w
+  elseif (degree == 17)
+    x = [
+      VectorValue(0.4171034443615992, 0.4171034443615992),
+      VectorValue(0.18035811626637066, 0.18035811626637066),
+      VectorValue(0.2857065024365867, 0.2857065024365867),
+      VectorValue(0.06665406347959701, 0.06665406347959701),
+      VectorValue(0.014755491660754072, 0.014755491660754072),
+      VectorValue(0.46559787161889027, 0.46559787161889027),
+      VectorValue(0.4171034443615992, 0.16579311127680163),
+      VectorValue(0.18035811626637066, 0.6392837674672587),
+      VectorValue(0.2857065024365867, 0.42858699512682663),
+      VectorValue(0.06665406347959701, 0.866691873040806),
+      VectorValue(0.014755491660754072, 0.9704890166784919),
+      VectorValue(0.46559787161889027, 0.06880425676221946),
+      VectorValue(0.16579311127680163, 0.4171034443615992),
+      VectorValue(0.6392837674672587, 0.18035811626637066),
+      VectorValue(0.42858699512682663, 0.2857065024365867),
+      VectorValue(0.866691873040806, 0.06665406347959701),
+      VectorValue(0.9704890166784919, 0.014755491660754072),
+      VectorValue(0.06880425676221946, 0.46559787161889027),
+      VectorValue(0.011575175903180683, 0.07250547079900238),
+      VectorValue(0.013229672760086951, 0.41547545929522905),
+      VectorValue(0.013135870834002753, 0.27179187005535477),
+      VectorValue(0.15750547792686992, 0.29921894247697034),
+      VectorValue(0.06734937786736123, 0.3062815917461865),
+      VectorValue(0.07804234056828245, 0.16872251349525944),
+      VectorValue(0.016017642362119337, 0.15919228747279268),
+      VectorValue(0.9159193532978169, 0.011575175903180683),
+      VectorValue(0.5712948679446841, 0.013229672760086951),
+      VectorValue(0.7150722591106424, 0.013135870834002753),
+      VectorValue(0.5432755795961598, 0.15750547792686992),
+      VectorValue(0.6263690303864522, 0.06734937786736123),
+      VectorValue(0.7532351459364581, 0.07804234056828245),
+      VectorValue(0.824790070165088, 0.016017642362119337),
+      VectorValue(0.07250547079900238, 0.9159193532978169),
+      VectorValue(0.41547545929522905, 0.5712948679446841),
+      VectorValue(0.27179187005535477, 0.7150722591106424),
+      VectorValue(0.29921894247697034, 0.5432755795961598),
+      VectorValue(0.3062815917461865, 0.6263690303864522),
+      VectorValue(0.16872251349525944, 0.7532351459364581),
+      VectorValue(0.15919228747279268, 0.824790070165088),
+      VectorValue(0.07250547079900238, 0.011575175903180683),
+      VectorValue(0.41547545929522905, 0.013229672760086951),
+      VectorValue(0.27179187005535477, 0.013135870834002753),
+      VectorValue(0.29921894247697034, 0.15750547792686992),
+      VectorValue(0.3062815917461865, 0.06734937786736123),
+      VectorValue(0.16872251349525944, 0.07804234056828245),
+      VectorValue(0.15919228747279268, 0.016017642362119337),
+      VectorValue(0.9159193532978169, 0.07250547079900238),
+      VectorValue(0.5712948679446841, 0.41547545929522905),
+      VectorValue(0.7150722591106424, 0.27179187005535477),
+      VectorValue(0.5432755795961598, 0.29921894247697034),
+      VectorValue(0.6263690303864522, 0.3062815917461865),
+      VectorValue(0.7532351459364581, 0.16872251349525944),
+      VectorValue(0.824790070165088, 0.15919228747279268),
+      VectorValue(0.011575175903180683, 0.9159193532978169),
+      VectorValue(0.013229672760086951, 0.5712948679446841),
+      VectorValue(0.013135870834002753, 0.7150722591106424),
+      VectorValue(0.15750547792686992, 0.5432755795961598),
+      VectorValue(0.06734937786736123, 0.6263690303864522),
+      VectorValue(0.07804234056828245, 0.7532351459364581),
+      VectorValue(0.016017642362119337, 0.824790070165088),
+    ]
+    w = [0.013655463264051053, 0.013156315294008993, 0.01885811857639764,
+            0.006229500401152722, 0.001386943788818821, 0.01250972547524868,
+            0.013655463264051053, 0.013156315294008993, 0.01885811857639764,
+            0.006229500401152722, 0.001386943788818821, 0.01250972547524868,
+            0.013655463264051053, 0.013156315294008993, 0.01885811857639764,
+            0.006229500401152722, 0.001386943788818821, 0.01250972547524868,
+            0.002292174200867934, 0.005199219977919768, 0.004346107250500596,
+            0.013085812967668494, 0.011243886273345534, 0.01027894916022726,
+            0.003989150102964797, 0.002292174200867934, 0.005199219977919768,
+            0.004346107250500596, 0.013085812967668494, 0.011243886273345534,
+            0.01027894916022726,  0.003989150102964797, 0.002292174200867934,
+            0.005199219977919768, 0.004346107250500596, 0.013085812967668494,
+            0.011243886273345534, 0.01027894916022726,  0.003989150102964797,
+            0.002292174200867934, 0.005199219977919768, 0.004346107250500596,
+            0.013085812967668494, 0.011243886273345534, 0.01027894916022726,
+            0.003989150102964797, 0.002292174200867934, 0.005199219977919768,
+            0.004346107250500596, 0.013085812967668494, 0.011243886273345534,
+            0.01027894916022726,  0.003989150102964797, 0.002292174200867934,
+            0.005199219977919768, 0.004346107250500596, 0.013085812967668494,
+            0.011243886273345534, 0.01027894916022726,  0.003989150102964797];
+    return x, w
+  elseif (degree == 18)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.4749182113240457, 0.4749182113240457),
+      VectorValue(0.15163850697260495, 0.15163850697260495),
+      VectorValue(0.4110671018759195, 0.4110671018759195),
+      VectorValue(0.2656146099053742, 0.2656146099053742),
+      VectorValue(0.0037589443410684376, 0.0037589443410684376),
+      VectorValue(0.072438705567333, 0.072438705567333),
+      VectorValue(0.4749182113240457, 0.05016357735190857),
+      VectorValue(0.15163850697260495, 0.6967229860547901),
+      VectorValue(0.4110671018759195, 0.177865796248161),
+      VectorValue(0.2656146099053742, 0.46877078018925156),
+      VectorValue(0.0037589443410684376, 0.9924821113178631),
+      VectorValue(0.072438705567333, 0.855122588865334),
+      VectorValue(0.05016357735190857, 0.4749182113240457),
+      VectorValue(0.6967229860547901, 0.15163850697260495),
+      VectorValue(0.177865796248161, 0.4110671018759195),
+      VectorValue(0.46877078018925156, 0.2656146099053742),
+      VectorValue(0.9924821113178631, 0.0037589443410684376),
+      VectorValue(0.855122588865334, 0.072438705567333),
+      VectorValue(0.09042704035434063, 0.3850440344131637),
+      VectorValue(0.012498932483495477, 0.04727614183265175),
+      VectorValue(0.05401173533902428, 0.30206195771287075),
+      VectorValue(0.010505018819241962, 0.2565061597742415),
+      VectorValue(0.06612245802840343, 0.17847912556588763),
+      VectorValue(0.14906691012577386, 0.2685733063960138),
+      VectorValue(0.011691824674667157, 0.41106566867461836),
+      VectorValue(0.014331524778941987, 0.1327788302713893),
+      VectorValue(0.5245289252324957, 0.09042704035434063),
+      VectorValue(0.9402249256838529, 0.012498932483495477),
+      VectorValue(0.6439263069481049, 0.05401173533902428),
+      VectorValue(0.7329888214065166, 0.010505018819241962),
+      VectorValue(0.7553984164057089, 0.06612245802840343),
+      VectorValue(0.5823597834782124, 0.14906691012577386),
+      VectorValue(0.5772425066507145, 0.011691824674667157),
+      VectorValue(0.8528896449496688, 0.014331524778941987),
+      VectorValue(0.3850440344131637, 0.5245289252324957),
+      VectorValue(0.04727614183265175, 0.9402249256838529),
+      VectorValue(0.30206195771287075, 0.6439263069481049),
+      VectorValue(0.2565061597742415, 0.7329888214065166),
+      VectorValue(0.17847912556588763, 0.7553984164057089),
+      VectorValue(0.2685733063960138, 0.5823597834782124),
+      VectorValue(0.41106566867461836, 0.5772425066507145),
+      VectorValue(0.1327788302713893, 0.8528896449496688),
+      VectorValue(0.3850440344131637, 0.09042704035434063),
+      VectorValue(0.04727614183265175, 0.012498932483495477),
+      VectorValue(0.30206195771287075, 0.05401173533902428),
+      VectorValue(0.2565061597742415, 0.010505018819241962),
+      VectorValue(0.17847912556588763, 0.06612245802840343),
+      VectorValue(0.2685733063960138, 0.14906691012577386),
+      VectorValue(0.41106566867461836, 0.011691824674667157),
+      VectorValue(0.1327788302713893, 0.014331524778941987),
+      VectorValue(0.5245289252324957, 0.3850440344131637),
+      VectorValue(0.9402249256838529, 0.04727614183265175),
+      VectorValue(0.6439263069481049, 0.30206195771287075),
+      VectorValue(0.7329888214065166, 0.2565061597742415),
+      VectorValue(0.7553984164057089, 0.17847912556588763),
+      VectorValue(0.5823597834782124, 0.2685733063960138),
+      VectorValue(0.5772425066507145, 0.41106566867461836),
+      VectorValue(0.8528896449496688, 0.1327788302713893),
+      VectorValue(0.09042704035434063, 0.5245289252324957),
+      VectorValue(0.012498932483495477, 0.9402249256838529),
+      VectorValue(0.05401173533902428, 0.6439263069481049),
+      VectorValue(0.010505018819241962, 0.7329888214065166),
+      VectorValue(0.06612245802840343, 0.7553984164057089),
+      VectorValue(0.14906691012577386, 0.5823597834782124),
+      VectorValue(0.011691824674667157, 0.5772425066507145),
+      VectorValue(0.014331524778941987, 0.8528896449496688),
+    ]
+    w = [
+        0.01537426061955793,   0.006553513745869378,  0.0101591694227292,
+        0.01673599702992395,   0.015558198301003067,  0.0002660028084738903,
+        0.006895143302383471,  0.006553513745869378,  0.0101591694227292,
+        0.01673599702992395,   0.015558198301003067,  0.0002660028084738903,
+        0.006895143302383471,  0.006553513745869378,  0.0101591694227292,
+        0.01673599702992395,   0.015558198301003067,  0.0002660028084738903,
+        0.006895143302383471,  0.007664129097276571,  0.0021087583873722216,
+        0.008182954206993283,  0.0038649176400031137, 0.00845582695874004,
+        0.01379644324428974,   0.004793062237180752,  0.0038208524863598183,
+        0.007664129097276571,  0.0021087583873722216, 0.008182954206993283,
+        0.0038649176400031137, 0.00845582695874004,   0.01379644324428974,
+        0.004793062237180752,  0.0038208524863598183, 0.007664129097276571,
+        0.0021087583873722216, 0.008182954206993283,  0.0038649176400031137,
+        0.00845582695874004,   0.01379644324428974,   0.004793062237180752,
+        0.0038208524863598183, 0.007664129097276571,  0.0021087583873722216,
+        0.008182954206993283,  0.0038649176400031137, 0.00845582695874004,
+        0.01379644324428974,   0.004793062237180752,  0.0038208524863598183,
+        0.007664129097276571,  0.0021087583873722216, 0.008182954206993283,
+        0.0038649176400031137, 0.00845582695874004,   0.01379644324428974,
+        0.004793062237180752,  0.0038208524863598183, 0.007664129097276571,
+        0.0021087583873722216, 0.008182954206993283,  0.0038649176400031137,
+        0.00845582695874004,   0.01379644324428974,   0.004793062237180752,
+        0.0038208524863598183];
+    return x, w
+  elseif (degree == 19)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.05252627985410363, 0.05252627985410363),
+      VectorValue(0.11144805571699878, 0.11144805571699878),
+      VectorValue(0.011639027327922657, 0.011639027327922657),
+      VectorValue(0.25516213315312486, 0.25516213315312486),
+      VectorValue(0.4039697179663861, 0.4039697179663861),
+      VectorValue(0.17817100607962755, 0.17817100607962755),
+      VectorValue(0.4591943889568276, 0.4591943889568276),
+      VectorValue(0.4925124498658742, 0.4925124498658742),
+      VectorValue(0.05252627985410363, 0.8949474402917927),
+      VectorValue(0.11144805571699878, 0.7771038885660024),
+      VectorValue(0.011639027327922657, 0.9767219453441547),
+      VectorValue(0.25516213315312486, 0.4896757336937503),
+      VectorValue(0.4039697179663861, 0.19206056406722782),
+      VectorValue(0.17817100607962755, 0.6436579878407449),
+      VectorValue(0.4591943889568276, 0.08161122208634475),
+      VectorValue(0.4925124498658742, 0.014975100268251551),
+      VectorValue(0.8949474402917927, 0.05252627985410363),
+      VectorValue(0.7771038885660024, 0.11144805571699878),
+      VectorValue(0.9767219453441547, 0.011639027327922657),
+      VectorValue(0.4896757336937503, 0.25516213315312486),
+      VectorValue(0.19206056406722782, 0.4039697179663861),
+      VectorValue(0.6436579878407449, 0.17817100607962755),
+      VectorValue(0.08161122208634475, 0.4591943889568276),
+      VectorValue(0.014975100268251551, 0.4925124498658742),
+      VectorValue(0.005005142352350433, 0.1424222825711269),
+      VectorValue(0.009777061438676854, 0.06008389996270236),
+      VectorValue(0.039142449434608845, 0.13070066996053453),
+      VectorValue(0.129312809767979, 0.31131838322398686),
+      VectorValue(0.07456118930435514, 0.22143394188911344),
+      VectorValue(0.04088831446497813, 0.3540259269997119),
+      VectorValue(0.014923638907438481, 0.24189410400689262),
+      VectorValue(0.0020691038491023883, 0.36462041433871),
+      VectorValue(0.8525725750765227, 0.005005142352350433),
+      VectorValue(0.9301390385986208, 0.009777061438676854),
+      VectorValue(0.8301568806048566, 0.039142449434608845),
+      VectorValue(0.5593688070080342, 0.129312809767979),
+      VectorValue(0.7040048688065313, 0.07456118930435514),
+      VectorValue(0.60508575853531, 0.04088831446497813),
+      VectorValue(0.7431822570856689, 0.014923638907438481),
+      VectorValue(0.6333104818121875, 0.0020691038491023883),
+      VectorValue(0.1424222825711269, 0.8525725750765227),
+      VectorValue(0.06008389996270236, 0.9301390385986208),
+      VectorValue(0.13070066996053453, 0.8301568806048566),
+      VectorValue(0.31131838322398686, 0.5593688070080342),
+      VectorValue(0.22143394188911344, 0.7040048688065313),
+      VectorValue(0.3540259269997119, 0.60508575853531),
+      VectorValue(0.24189410400689262, 0.7431822570856689),
+      VectorValue(0.36462041433871, 0.6333104818121875),
+      VectorValue(0.1424222825711269, 0.005005142352350433),
+      VectorValue(0.06008389996270236, 0.009777061438676854),
+      VectorValue(0.13070066996053453, 0.039142449434608845),
+      VectorValue(0.31131838322398686, 0.129312809767979),
+      VectorValue(0.22143394188911344, 0.07456118930435514),
+      VectorValue(0.3540259269997119, 0.04088831446497813),
+      VectorValue(0.24189410400689262, 0.014923638907438481),
+      VectorValue(0.36462041433871, 0.0020691038491023883),
+      VectorValue(0.8525725750765227, 0.1424222825711269),
+      VectorValue(0.9301390385986208, 0.06008389996270236),
+      VectorValue(0.8301568806048566, 0.13070066996053453),
+      VectorValue(0.5593688070080342, 0.31131838322398686),
+      VectorValue(0.7040048688065313, 0.22143394188911344),
+      VectorValue(0.60508575853531, 0.3540259269997119),
+      VectorValue(0.7431822570856689, 0.24189410400689262),
+      VectorValue(0.6333104818121875, 0.36462041433871),
+      VectorValue(0.005005142352350433, 0.8525725750765227),
+      VectorValue(0.009777061438676854, 0.9301390385986208),
+      VectorValue(0.039142449434608845, 0.8301568806048566),
+      VectorValue(0.129312809767979, 0.5593688070080342),
+      VectorValue(0.07456118930435514, 0.7040048688065313),
+      VectorValue(0.04088831446497813, 0.60508575853531),
+      VectorValue(0.014923638907438481, 0.7431822570856689),
+      VectorValue(0.0020691038491023883, 0.6333104818121875),
+    ]
+    w = [
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+        0.0008825962091542701, 0.01587642729376499,   0.01576867932261981,
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+        0.004847759540812101,  0.013173132353722682,  0.009054037295215252,
+        0.008051104730469714,  0.00422796241954674,   0.0016410687574198689,
+        0.0014628462439400358, 0.0016636944202969523, 0.004847759540812101,
+        0.013173132353722682,  0.009054037295215252,  0.008051104730469714,
+        0.00422796241954674,   0.0016410687574198689, 0.0014628462439400358,
+        0.0016636944202969523, 0.004847759540812101,  0.013173132353722682,
+        0.009054037295215252,  0.008051104730469714,  0.00422796241954674,
+        0.0016410687574198689];
+    return x, w
+  elseif (degree == 20)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.18629499774454095, 0.18629499774454095),
+      VectorValue(0.037310880598884766, 0.037310880598884766),
+      VectorValue(0.476245611540499, 0.476245611540499),
+      VectorValue(0.4455510569559248, 0.4455510569559248),
+      VectorValue(0.25457926767333916, 0.25457926767333916),
+      VectorValue(0.39342534781709987, 0.39342534781709987),
+      VectorValue(0.01097614102839789, 0.01097614102839789),
+      VectorValue(0.10938359671171471, 0.10938359671171471),
+      VectorValue(0.18629499774454095, 0.6274100045109181),
+      VectorValue(0.037310880598884766, 0.9253782388022305),
+      VectorValue(0.476245611540499, 0.047508776919002016),
+      VectorValue(0.4455510569559248, 0.10889788608815043),
+      VectorValue(0.25457926767333916, 0.4908414646533217),
+      VectorValue(0.39342534781709987, 0.21314930436580026),
+      VectorValue(0.01097614102839789, 0.9780477179432042),
+      VectorValue(0.10938359671171471, 0.7812328065765706),
+      VectorValue(0.6274100045109181, 0.18629499774454095),
+      VectorValue(0.9253782388022305, 0.037310880598884766),
+      VectorValue(0.047508776919002016, 0.476245611540499),
+      VectorValue(0.10889788608815043, 0.4455510569559248),
+      VectorValue(0.4908414646533217, 0.25457926767333916),
+      VectorValue(0.21314930436580026, 0.39342534781709987),
+      VectorValue(0.9780477179432042, 0.01097614102839789),
+      VectorValue(0.7812328065765706, 0.10938359671171471),
+      VectorValue(0.004854937607623827, 0.06409058560843404),
+      VectorValue(0.10622720472027006, 0.2156070573900944),
+      VectorValue(0.007570780504696579, 0.15913370765706722),
+      VectorValue(0.13980807199179993, 0.317860123835772),
+      VectorValue(0.04656036490766434, 0.19851813222878817),
+      VectorValue(0.038363684775374655, 0.09995229628813862),
+      VectorValue(0.009831548292802588, 0.42002375881622406),
+      VectorValue(0.05498747914298685, 0.33313481730958744),
+      VectorValue(0.01073721285601111, 0.2805814114236652),
+      VectorValue(0.9310544767839422, 0.004854937607623827),
+      VectorValue(0.6781657378896355, 0.10622720472027006),
+      VectorValue(0.8332955118382361, 0.007570780504696579),
+      VectorValue(0.5423318041724281, 0.13980807199179993),
+      VectorValue(0.7549215028635474, 0.04656036490766434),
+      VectorValue(0.8616840189364867, 0.038363684775374655),
+      VectorValue(0.5701446928909732, 0.009831548292802588),
+      VectorValue(0.6118777035474257, 0.05498747914298685),
+      VectorValue(0.7086813757203236, 0.01073721285601111),
+      VectorValue(0.06409058560843404, 0.9310544767839422),
+      VectorValue(0.2156070573900944, 0.6781657378896355),
+      VectorValue(0.15913370765706722, 0.8332955118382361),
+      VectorValue(0.317860123835772, 0.5423318041724281),
+      VectorValue(0.19851813222878817, 0.7549215028635474),
+      VectorValue(0.09995229628813862, 0.8616840189364867),
+      VectorValue(0.42002375881622406, 0.5701446928909732),
+      VectorValue(0.33313481730958744, 0.6118777035474257),
+      VectorValue(0.2805814114236652, 0.7086813757203236),
+      VectorValue(0.06409058560843404, 0.004854937607623827),
+      VectorValue(0.2156070573900944, 0.10622720472027006),
+      VectorValue(0.15913370765706722, 0.007570780504696579),
+      VectorValue(0.317860123835772, 0.13980807199179993),
+      VectorValue(0.19851813222878817, 0.04656036490766434),
+      VectorValue(0.09995229628813862, 0.038363684775374655),
+      VectorValue(0.42002375881622406, 0.009831548292802588),
+      VectorValue(0.33313481730958744, 0.05498747914298685),
+      VectorValue(0.2805814114236652, 0.01073721285601111),
+      VectorValue(0.9310544767839422, 0.06409058560843404),
+      VectorValue(0.6781657378896355, 0.2156070573900944),
+      VectorValue(0.8332955118382361, 0.15913370765706722),
+      VectorValue(0.5423318041724281, 0.317860123835772),
+      VectorValue(0.7549215028635474, 0.19851813222878817),
+      VectorValue(0.8616840189364867, 0.09995229628813862),
+      VectorValue(0.5701446928909732, 0.42002375881622406),
+      VectorValue(0.6118777035474257, 0.33313481730958744),
+      VectorValue(0.7086813757203236, 0.2805814114236652),
+      VectorValue(0.004854937607623827, 0.9310544767839422),
+      VectorValue(0.10622720472027006, 0.6781657378896355),
+      VectorValue(0.007570780504696579, 0.8332955118382361),
+      VectorValue(0.13980807199179993, 0.5423318041724281),
+      VectorValue(0.04656036490766434, 0.7549215028635474),
+      VectorValue(0.038363684775374655, 0.8616840189364867),
+      VectorValue(0.009831548292802588, 0.5701446928909732),
+      VectorValue(0.05498747914298685, 0.6118777035474257),
+      VectorValue(0.01073721285601111, 0.7086813757203236),
+    ]      
+    w = [
+        0.013910110701453116,  0.009173462974252915,  0.0021612754106655778,
+        0.007101825303408441,  0.009452399933232448,  0.014083201307520249,
+        0.013788050629070459,  0.00079884079106662,   0.007830230776074535,
+        0.009173462974252915,  0.0021612754106655778, 0.007101825303408441,
+        0.009452399933232448,  0.014083201307520249,  0.013788050629070459,
+        0.00079884079106662,   0.007830230776074535,  0.009173462974252915,
+        0.0021612754106655778, 0.007101825303408441,  0.009452399933232448,
+        0.014083201307520249,  0.013788050629070459,  0.00079884079106662,
+        0.007830230776074535,  0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856, 0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856, 0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856, 0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856, 0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856, 0.0011298696021258656, 0.007722607822099231,
+        0.002202897418558498,  0.011691745731827735,  0.00598639857895469,
+        0.004145711527613858,  0.003695681500255298,  0.008667225567219335,
+        0.0035782002384576856];
+    return x, w
+  elseif (degree == 21)
+    x = [
+      VectorValue(0.2989362353149826, 0.2989362353149826),
+      VectorValue(0.4970078754686856, 0.4970078754686856),
+      VectorValue(0.40361758654638513, 0.40361758654638513),
+      VectorValue(0.11898857762271953, 0.11898857762271953),
+      VectorValue(0.19028871809127856, 0.19028871809127856),
+      VectorValue(0.4815978686532166, 0.4815978686532166),
+      VectorValue(0.4498127917753624, 0.4498127917753624),
+      VectorValue(0.053627575546145, 0.053627575546145),
+      VectorValue(0.010742456432828507, 0.010742456432828507),
+      VectorValue(0.2989362353149826, 0.4021275293700348),
+      VectorValue(0.4970078754686856, 0.005984249062628844),
+      VectorValue(0.40361758654638513, 0.19276482690722974),
+      VectorValue(0.11898857762271953, 0.762022844754561),
+      VectorValue(0.19028871809127856, 0.6194225638174429),
+      VectorValue(0.4815978686532166, 0.03680426269356685),
+      VectorValue(0.4498127917753624, 0.10037441644927525),
+      VectorValue(0.053627575546145, 0.89274484890771),
+      VectorValue(0.010742456432828507, 0.978515087134343),
+      VectorValue(0.4021275293700348, 0.2989362353149826),
+      VectorValue(0.005984249062628844, 0.4970078754686856),
+      VectorValue(0.19276482690722974, 0.40361758654638513),
+      VectorValue(0.762022844754561, 0.11898857762271953),
+      VectorValue(0.6194225638174429, 0.19028871809127856),
+      VectorValue(0.03680426269356685, 0.4815978686532166),
+      VectorValue(0.10037441644927525, 0.4498127917753624),
+      VectorValue(0.89274484890771, 0.053627575546145),
+      VectorValue(0.978515087134343, 0.010742456432828507),
+      VectorValue(0.20529555933516153, 0.28918949607859473),
+      VectorValue(0.006931809031468116, 0.23787338259799398),
+      VectorValue(0.12377940040549276, 0.31886531079482827),
+      VectorValue(0.03899136262322033, 0.23187362537040096),
+      VectorValue(0.009536247529710598, 0.1331671229413703),
+      VectorValue(0.05305219170121682, 0.34680797980991107),
+      VectorValue(0.10045802007411446, 0.21659962318998252),
+      VectorValue(0.04945106556854055, 0.12882980796205154),
+      VectorValue(0.010254635872924515, 0.3609534080189222),
+      VectorValue(0.010301903643423904, 0.055719565072371954),
+      VectorValue(0.5055149445862437, 0.20529555933516153),
+      VectorValue(0.755194808370538, 0.006931809031468116),
+      VectorValue(0.557355288799679, 0.12377940040549276),
+      VectorValue(0.7291350120063786, 0.03899136262322033),
+      VectorValue(0.857296629528919, 0.009536247529710598),
+      VectorValue(0.6001398284888722, 0.05305219170121682),
+      VectorValue(0.6829423567359031, 0.10045802007411446),
+      VectorValue(0.8217191264694079, 0.04945106556854055),
+      VectorValue(0.6287919561081533, 0.010254635872924515),
+      VectorValue(0.9339785312842042, 0.010301903643423904),
+      VectorValue(0.28918949607859473, 0.5055149445862437),
+      VectorValue(0.23787338259799398, 0.755194808370538),
+      VectorValue(0.31886531079482827, 0.557355288799679),
+      VectorValue(0.23187362537040096, 0.7291350120063786),
+      VectorValue(0.1331671229413703, 0.857296629528919),
+      VectorValue(0.34680797980991107, 0.6001398284888722),
+      VectorValue(0.21659962318998252, 0.6829423567359031),
+      VectorValue(0.12882980796205154, 0.8217191264694079),
+      VectorValue(0.3609534080189222, 0.6287919561081533),
+      VectorValue(0.055719565072371954, 0.9339785312842042),
+      VectorValue(0.28918949607859473, 0.20529555933516153),
+      VectorValue(0.23787338259799398, 0.006931809031468116),
+      VectorValue(0.31886531079482827, 0.12377940040549276),
+      VectorValue(0.23187362537040096, 0.03899136262322033),
+      VectorValue(0.1331671229413703, 0.009536247529710598),
+      VectorValue(0.34680797980991107, 0.05305219170121682),
+      VectorValue(0.21659962318998252, 0.10045802007411446),
+      VectorValue(0.12882980796205154, 0.04945106556854055),
+      VectorValue(0.3609534080189222, 0.010254635872924515),
+      VectorValue(0.055719565072371954, 0.010301903643423904),
+      VectorValue(0.5055149445862437, 0.28918949607859473),
+      VectorValue(0.755194808370538, 0.23787338259799398),
+      VectorValue(0.557355288799679, 0.31886531079482827),
+      VectorValue(0.7291350120063786, 0.23187362537040096),
+      VectorValue(0.857296629528919, 0.1331671229413703),
+      VectorValue(0.6001398284888722, 0.34680797980991107),
+      VectorValue(0.6829423567359031, 0.21659962318998252),
+      VectorValue(0.8217191264694079, 0.12882980796205154),
+      VectorValue(0.6287919561081533, 0.3609534080189222),
+      VectorValue(0.9339785312842042, 0.055719565072371954),
+      VectorValue(0.20529555933516153, 0.5055149445862437),
+      VectorValue(0.006931809031468116, 0.755194808370538),
+      VectorValue(0.12377940040549276, 0.557355288799679),
+      VectorValue(0.03899136262322033, 0.7291350120063786),
+      VectorValue(0.009536247529710598, 0.857296629528919),
+      VectorValue(0.05305219170121682, 0.6001398284888722),
+      VectorValue(0.10045802007411446, 0.6829423567359031),
+      VectorValue(0.04945106556854055, 0.8217191264694079),
+      VectorValue(0.010254635872924515, 0.6287919561081533),
+      VectorValue(0.010301903643423904, 0.9339785312842042),
+    ]      
+    w = [
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+        0.006828016226115099,  0.009727620930375354,  0.006107205081692191,
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+        0.005234952092662423,  0.002240406560950738,  0.007250152959485511,
+        0.007952018352713986,  0.004905985911275205,  0.0034199424289671526,
+        0.0016327142920220426, 0.008747708077881562,  0.002103060144074865,
+        0.009223742423966418,  0.005234952092662423,  0.002240406560950738,
+        0.007250152959485511,  0.007952018352713986,  0.004905985911275205,
+        0.0034199424289671526, 0.0016327142920220426, 0.008747708077881562,
+        0.002103060144074865,  0.009223742423966418,  0.005234952092662423,
+        0.002240406560950738,  0.007250152959485511,  0.007952018352713986,
+        0.004905985911275205,  0.0034199424289671526, 0.0016327142920220426]
+    return x, w
+  elseif (degree == 22)
+    x = [
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+      VectorValue(0.4577694113676721, 0.4577694113676721),
+      VectorValue(0.29455825902995014, 0.29455825902995014),
+      VectorValue(0.18851052363028398, 0.18851052363028398),
+      VectorValue(0.42198188879353493, 0.42198188879353493),
+      VectorValue(0.49616117840970864, 0.49616117840970864),
+      VectorValue(0.029108470670807574, 0.029108470670807574),
+      VectorValue(0.11543153821920499, 0.11543153821920499),
+      VectorValue(0.3851845246273021, 0.22963095074539575),
+      VectorValue(0.4577694113676721, 0.08446117726465585),
+      VectorValue(0.29455825902995014, 0.4108834819400997),
+      VectorValue(0.18851052363028398, 0.622978952739432),
+      VectorValue(0.42198188879353493, 0.15603622241293014),
+      VectorValue(0.49616117840970864, 0.007677643180582727),
+      VectorValue(0.029108470670807574, 0.9417830586583849),
+      VectorValue(0.11543153821920499, 0.76913692356159),
+      VectorValue(0.22963095074539575, 0.3851845246273021),
+      VectorValue(0.08446117726465585, 0.4577694113676721),
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+      VectorValue(0.15603622241293014, 0.42198188879353493),
+      VectorValue(0.007677643180582727, 0.49616117840970864),
+      VectorValue(0.9417830586583849, 0.029108470670807574),
+      VectorValue(0.76913692356159, 0.11543153821920499),
+      VectorValue(0.007876282221582374, 0.06984216946744362),
+      VectorValue(0.04475228434833587, 0.09039883116640775),
+      VectorValue(0.038275234700863824, 0.4113417640205587),
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+      VectorValue(0.19108129796672008, 0.29006682411666884),
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+      VectorValue(0.10868994186267199, 0.21678693336494115),
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+      VectorValue(0.8648488844852563, 0.04475228434833587),
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+      VectorValue(0.630023478332822, 0.007400241234710751),
+      VectorValue(0.5188518779166111, 0.19108129796672008),
+      VectorValue(0.6680765517823722, 0.04399164539345585),
+      VectorValue(0.6745231247723869, 0.10868994186267199),
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+      VectorValue(0.06984216946744362, 0.9222815483109741),
+      VectorValue(0.09039883116640775, 0.8648488844852563),
+      VectorValue(0.4113417640205587, 0.5503830012785775),
+      VectorValue(0.3321061050074464, 0.5651468190056222),
+      VectorValue(0.36257628043246726, 0.630023478332822),
+      VectorValue(0.29006682411666884, 0.5188518779166111),
+      VectorValue(0.28793180282417186, 0.6680765517823722),
+      VectorValue(0.21678693336494115, 0.6745231247723869),
+      VectorValue(0.14587371987352518, 0.8449815687515108),
+      VectorValue(0.17629743482450005, 0.7754476410608586),
+      VectorValue(0.24399064603949305, 0.7468454447123217),
+      VectorValue(0.017934321052938986, 0.9802672139581126),
+      VectorValue(0.06984216946744362, 0.007876282221582374),
+      VectorValue(0.09039883116640775, 0.04475228434833587),
+      VectorValue(0.4113417640205587, 0.038275234700863824),
+      VectorValue(0.3321061050074464, 0.10274707598693139),
+      VectorValue(0.36257628043246726, 0.007400241234710751),
+      VectorValue(0.29006682411666884, 0.19108129796672008),
+      VectorValue(0.28793180282417186, 0.04399164539345585),
+      VectorValue(0.21678693336494115, 0.10868994186267199),
+      VectorValue(0.14587371987352518, 0.009144711374964054),
+      VectorValue(0.17629743482450005, 0.048254924114641384),
+      VectorValue(0.24399064603949305, 0.009163909248185229),
+      VectorValue(0.017934321052938986, 0.0017984649889483744),
+      VectorValue(0.9222815483109741, 0.06984216946744362),
+      VectorValue(0.8648488844852563, 0.09039883116640775),
+      VectorValue(0.5503830012785775, 0.4113417640205587),
+      VectorValue(0.5651468190056222, 0.3321061050074464),
+      VectorValue(0.630023478332822, 0.36257628043246726),
+      VectorValue(0.5188518779166111, 0.29006682411666884),
+      VectorValue(0.6680765517823722, 0.28793180282417186),
+      VectorValue(0.6745231247723869, 0.21678693336494115),
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+      VectorValue(0.7754476410608586, 0.17629743482450005),
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+      VectorValue(0.007876282221582374, 0.9222815483109741),
+      VectorValue(0.04475228434833587, 0.8648488844852563),
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+      VectorValue(0.10274707598693139, 0.5651468190056222),
+      VectorValue(0.007400241234710751, 0.630023478332822),
+      VectorValue(0.19108129796672008, 0.5188518779166111),
+      VectorValue(0.04399164539345585, 0.6680765517823722),
+      VectorValue(0.10868994186267199, 0.6745231247723869),
+      VectorValue(0.009144711374964054, 0.8449815687515108),
+      VectorValue(0.048254924114641384, 0.7754476410608586),
+      VectorValue(0.009163909248185229, 0.7468454447123217),
+      VectorValue(0.0017984649889483744, 0.9802672139581126),
+    ]      
+    w = [
+        0.006746541941805331,  0.006930699762117096,  0.010537881978726092,
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+        0.005281792483873449,  0.0025270384487923007, 0.0003202142655857129];
+    return x, w
+  elseif (degree == 23)
+    x = [
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+      VectorValue(0.08684104820763322, 0.8263179035847336),
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+      VectorValue(0.11410136032236454, 0.005189821760844536),
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+      VectorValue(0.3163043076538381, 0.1548301554055162),
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+      VectorValue(0.27879416981410227, 0.03299370819253279),
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+      VectorValue(0.8717190926395587, 0.0955398781717349),
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+      VectorValue(0.007162539910244482, 0.9455758306400301),
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+      VectorValue(0.10172832932728422, 0.6577888986377031),
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+      VectorValue(0.1548301554055162, 0.5288655369406456),
+      VectorValue(0.014758969729945169, 0.5874824534670472),
+      VectorValue(0.03299370819253279, 0.688212121993365),
+    ]      
+    w = [
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+        0.004479958512756771,  0.011837304231564011,  0.011903931443749882,
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+        0.0052351282465650335, 0.010422197929484405,  0.0035488894172609124,
+        0.005087787328353519];
+    return x, w
+  elseif (degree == 24)
+    x = [
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+      VectorValue(0.4188909749106028, 0.4188909749106028),
+      VectorValue(0.16236063371692644, 0.16236063371692644),
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+      VectorValue(0.5881529574639623, 0.17036728246244368),
+      VectorValue(0.5012643952150376, 0.169759795860736),
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+      VectorValue(0.8172747318363464, 0.007987921880847964),
+      VectorValue(0.9546336170785, 0.008074910870208776),
+      VectorValue(0.241479760073594, 0.5881529574639623),
+      VectorValue(0.3289758089242264, 0.5012643952150376),
+      VectorValue(0.09316740977988115, 0.8685143643990995),
+      VectorValue(0.39452027980019433, 0.5128232386790481),
+      VectorValue(0.16267741639447741, 0.7961338693570472),
+      VectorValue(0.25358901421887947, 0.7068400808109625),
+      VectorValue(0.36225224131779127, 0.5991550585073127),
+      VectorValue(0.28162257770616084, 0.6238424605572401),
+      VectorValue(0.3832726649926592, 0.6093393403750466),
+      VectorValue(0.2737503525162605, 0.7187036443214267),
+      VectorValue(0.09412134279736603, 0.8986440987448518),
+      VectorValue(0.18039615188676572, 0.724037578585869),
+      VectorValue(0.17473734628280568, 0.8172747318363464),
+      VectorValue(0.03729147205129122, 0.9546336170785),
+      VectorValue(0.241479760073594, 0.17036728246244368),
+      VectorValue(0.3289758089242264, 0.169759795860736),
+      VectorValue(0.09316740977988115, 0.03831822582101938),
+      VectorValue(0.39452027980019433, 0.09265648152075752),
+      VectorValue(0.16267741639447741, 0.041188714248475373),
+      VectorValue(0.25358901421887947, 0.03957090497015804),
+      VectorValue(0.36225224131779127, 0.038592700174896126),
+      VectorValue(0.28162257770616084, 0.09453496173659899),
+      VectorValue(0.3832726649926592, 0.007387994632294238),
+      VectorValue(0.2737503525162605, 0.007546003162312815),
+      VectorValue(0.09412134279736603, 0.007234558457782137),
+      VectorValue(0.18039615188676572, 0.09556626952736523),
+      VectorValue(0.17473734628280568, 0.007987921880847964),
+      VectorValue(0.03729147205129122, 0.008074910870208776),
+      VectorValue(0.5881529574639623, 0.241479760073594),
+      VectorValue(0.5012643952150376, 0.3289758089242264),
+      VectorValue(0.8685143643990995, 0.09316740977988115),
+      VectorValue(0.5128232386790481, 0.39452027980019433),
+      VectorValue(0.7961338693570472, 0.16267741639447741),
+      VectorValue(0.7068400808109625, 0.25358901421887947),
+      VectorValue(0.5991550585073127, 0.36225224131779127),
+      VectorValue(0.6238424605572401, 0.28162257770616084),
+      VectorValue(0.6093393403750466, 0.3832726649926592),
+      VectorValue(0.7187036443214267, 0.2737503525162605),
+      VectorValue(0.8986440987448518, 0.09412134279736603),
+      VectorValue(0.724037578585869, 0.18039615188676572),
+      VectorValue(0.8172747318363464, 0.17473734628280568),
+      VectorValue(0.9546336170785, 0.03729147205129122),
+      VectorValue(0.17036728246244368, 0.5881529574639623),
+      VectorValue(0.169759795860736, 0.5012643952150376),
+      VectorValue(0.03831822582101938, 0.8685143643990995),
+      VectorValue(0.09265648152075752, 0.5128232386790481),
+      VectorValue(0.041188714248475373, 0.7961338693570472),
+      VectorValue(0.03957090497015804, 0.7068400808109625),
+      VectorValue(0.038592700174896126, 0.5991550585073127),
+      VectorValue(0.09453496173659899, 0.6238424605572401),
+      VectorValue(0.007387994632294238, 0.6093393403750466),
+      VectorValue(0.007546003162312815, 0.7187036443214267),
+      VectorValue(0.007234558457782137, 0.8986440987448518),
+      VectorValue(0.09556626952736523, 0.724037578585869),
+      VectorValue(0.007987921880847964, 0.8172747318363464),
+      VectorValue(0.008074910870208776, 0.9546336170785),
+    ]      
+    w = [
+        0.00627284492280016,   0.006555266350942619,  0.0051895080282000966,
+        0.0019168498654645917, 0.0003086272527483216, 0.002171623361085349,
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+        0.010260004335754922,  0.005176247385426301,  0.005013696533694453,
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+        0.002683135727083884,  0.0075159271748706695, 0.0036020670873987137,
+        0.004452438464081782,  0.004973625937841209,  0.007176175789078727,
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+        0.0018536133821231541, 0.0008984737927232883, 0.007072522903242423,
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+        0.0036020670873987137, 0.004452438464081782,  0.004973625937841209,
+        0.007176175789078727,  0.0021210746334018845, 0.0020406375385582255,
+        0.0012946061911989924, 0.005921781071271555,  0.0018536133821231541,
+        0.0008984737927232883];
+    return x, w
+  elseif (degree == 25)
+    x = [
+      VectorValue(0.3876420304045634, 0.3876420304045634),
+      VectorValue(0.21100450806149668, 0.21100450806149668),
+      VectorValue(0.2994923158045085, 0.2994923158045085),
+      VectorValue(0.03722292599244087, 0.03722292599244087),
+      VectorValue(0.1451092435745004, 0.1451092435745004),
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+      VectorValue(0.4622087087487061, 0.4622087087487061),
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+      VectorValue(0.48903936966039546, 0.48903936966039546),
+      VectorValue(0.3876420304045634, 0.22471593919087318),
+      VectorValue(0.21100450806149668, 0.5779909838770066),
+      VectorValue(0.2994923158045085, 0.40101536839098295),
+      VectorValue(0.03722292599244087, 0.9255541480151183),
+      VectorValue(0.1451092435745004, 0.7097815128509992),
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+      VectorValue(0.09294970170076994, 0.8141005965984601),
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+      VectorValue(0.48903936966039546, 0.021921260679209076),
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+      VectorValue(0.40101536839098295, 0.2994923158045085),
+      VectorValue(0.9255541480151183, 0.03722292599244087),
+      VectorValue(0.7097815128509992, 0.1451092435745004),
+      VectorValue(0.1504813909188505, 0.42475930454057476),
+      VectorValue(0.07558258250258776, 0.4622087087487061),
+      VectorValue(0.8141005965984601, 0.09294970170076994),
+      VectorValue(0.9843293114347923, 0.007835344282603851),
+      VectorValue(0.021921260679209076, 0.48903936966039546),
+      VectorValue(0.0018188666342743875, 0.4404169274793433),
+      VectorValue(0.03696014157967147, 0.15900790619732788),
+      VectorValue(0.07885806800563527, 0.1773537967572529),
+      VectorValue(0.06884752943149791, 0.2700667358209594),
+      VectorValue(0.11599980764096017, 0.34139103302114987),
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+      VectorValue(0.007128314501257424, 0.09913306334168219),
+      VectorValue(0.20369291058425096, 0.29950641862967453),
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+      VectorValue(0.012913883250032529, 0.362068801895972),
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+      VectorValue(0.02454006024752439, 0.2565954097090198),
+      VectorValue(0.007188828261693038, 0.041068819111784644),
+      VectorValue(0.0008914643174981278, 0.2794161886492607),
+      VectorValue(0.5577642058863823, 0.0018188666342743875),
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+      VectorValue(0.7437881352371118, 0.07885806800563527),
+      VectorValue(0.6610857347475427, 0.06884752943149791),
+      VectorValue(0.5426091593378899, 0.11599980764096017),
+      VectorValue(0.5777445859930387, 0.04831743428737695),
+      VectorValue(0.8937386221570605, 0.007128314501257424),
+      VectorValue(0.4968006707860745, 0.20369291058425096),
+      VectorValue(0.8141339896484356, 0.007236161747948156),
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+      VectorValue(0.8735191347263744, 0.037687949784259066),
+      VectorValue(0.6293704957712138, 0.13700669408707095),
+      VectorValue(0.7188645300434557, 0.02454006024752439),
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+      VectorValue(0.29950641862967453, 0.4968006707860745),
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+      VectorValue(0.362068801895972, 0.6250173148539955),
+      VectorValue(0.08879291548936656, 0.8735191347263744),
+      VectorValue(0.23362281014171524, 0.6293704957712138),
+      VectorValue(0.2565954097090198, 0.7188645300434557),
+      VectorValue(0.041068819111784644, 0.9517423526265223),
+      VectorValue(0.2794161886492607, 0.7196923470332413),
+      VectorValue(0.4404169274793433, 0.0018188666342743875),
+      VectorValue(0.15900790619732788, 0.03696014157967147),
+      VectorValue(0.1773537967572529, 0.07885806800563527),
+      VectorValue(0.2700667358209594, 0.06884752943149791),
+      VectorValue(0.34139103302114987, 0.11599980764096017),
+      VectorValue(0.3739379797195844, 0.04831743428737695),
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+      VectorValue(0.29950641862967453, 0.20369291058425096),
+      VectorValue(0.17862984860361625, 0.007236161747948156),
+      VectorValue(0.362068801895972, 0.012913883250032529),
+      VectorValue(0.08879291548936656, 0.037687949784259066),
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+      VectorValue(0.2794161886492607, 0.0008914643174981278),
+      VectorValue(0.5577642058863823, 0.4404169274793433),
+      VectorValue(0.8040319522230007, 0.15900790619732788),
+      VectorValue(0.7437881352371118, 0.1773537967572529),
+      VectorValue(0.6610857347475427, 0.2700667358209594),
+      VectorValue(0.5426091593378899, 0.34139103302114987),
+      VectorValue(0.5777445859930387, 0.3739379797195844),
+      VectorValue(0.8937386221570605, 0.09913306334168219),
+      VectorValue(0.4968006707860745, 0.29950641862967453),
+      VectorValue(0.8141339896484356, 0.17862984860361625),
+      VectorValue(0.6250173148539955, 0.362068801895972),
+      VectorValue(0.8735191347263744, 0.08879291548936656),
+      VectorValue(0.6293704957712138, 0.23362281014171524),
+      VectorValue(0.7188645300434557, 0.2565954097090198),
+      VectorValue(0.9517423526265223, 0.041068819111784644),
+      VectorValue(0.7196923470332413, 0.2794161886492607),
+      VectorValue(0.0018188666342743875, 0.5577642058863823),
+      VectorValue(0.03696014157967147, 0.8040319522230007),
+      VectorValue(0.07885806800563527, 0.7437881352371118),
+      VectorValue(0.06884752943149791, 0.6610857347475427),
+      VectorValue(0.11599980764096017, 0.5426091593378899),
+      VectorValue(0.04831743428737695, 0.5777445859930387),
+      VectorValue(0.007128314501257424, 0.8937386221570605),
+      VectorValue(0.20369291058425096, 0.4968006707860745),
+      VectorValue(0.007236161747948156, 0.8141339896484356),
+      VectorValue(0.012913883250032529, 0.6250173148539955),
+      VectorValue(0.037687949784259066, 0.8735191347263744),
+      VectorValue(0.13700669408707095, 0.6293704957712138),
+      VectorValue(0.02454006024752439, 0.7188645300434557),
+      VectorValue(0.007188828261693038, 0.9517423526265223),
+      VectorValue(0.0008914643174981278, 0.7196923470332413),
+    ]      
+    w = [
+        0.006844925774136122,  0.005793631618005297,  0.009008820350850738,
+        0.001698648860952368,  0.005745762931282399,  0.00795565506872921,
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+        0.004222042973260539,  0.006844925774136122,  0.005793631618005297,
+        0.009008820350850738,  0.001698648860952368,  0.005745762931282399,
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+        0.0004032551441623084, 0.004222042973260539,  0.006844925774136122,
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+        0.0027273191839872145, 0.00263628096071471,   0.006870041296011276,
+        0.0036571704539664234, 0.0008464918170636662, 0.0007558510392294402,
+        0.0008374089159673526, 0.003155739012379637,  0.004757510783727886,
+        0.00544219680621846,   0.007920176143949218,  0.0053200853477543926,
+        0.0012726358126745072, 0.00895691044613803,   0.0016318698410246215,
+        0.0027273191839872145, 0.00263628096071471,   0.006870041296011276,
+        0.0036571704539664234, 0.0008464918170636662, 0.0007558510392294402];
+    return x, w
+  elseif (degree == 26)
+    x = [
+      VectorValue(0.3333333333333333, 0.3333333333333333),
+      VectorValue(0.06673712257646625, 0.06673712257646625),
+      VectorValue(0.0063401164920769415, 0.0063401164920769415),
+      VectorValue(0.4937530328963848, 0.4937530328963848),
+      VectorValue(0.388787497107594, 0.388787497107594),
+      VectorValue(0.2731471009290788, 0.2731471009290788),
+      VectorValue(0.471828563321166, 0.471828563321166),
+      VectorValue(0.1542014303645443, 0.1542014303645443),
+      VectorValue(0.21204316330220568, 0.21204316330220568),
+      VectorValue(0.4359854193843832, 0.4359854193843832),
+      VectorValue(0.06673712257646625, 0.8665257548470675),
+      VectorValue(0.0063401164920769415, 0.9873197670158461),
+      VectorValue(0.4937530328963848, 0.01249393420723044),
+      VectorValue(0.388787497107594, 0.22242500578481195),
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+      VectorValue(0.007332472549040455, 0.7629478741931164),
+      VectorValue(0.0004903284434629743, 0.8518541504366672),
+    ]
+    w = [
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+        0.0026063109364009383, 0.006735657699024688,  0.007873982890681327,
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+        0.002311193555890555,  0.0016971268694035136, 0.00042330306788192527];
+    return x, w
+  elseif (degree == 28)
+    x = [
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+    
+    w = [0.007181233150323068,  0.00015556760434074816,
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+        0.0029589418861769453, 0.003130267321484276,
+        0.0015700888217440182, 0.0063935692114779,
+        0.0016024580953965175, 0.0003625672974930415];
+    return x, w
+  elseif (degree == 29)
+    x = [
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+      VectorValue(0.4343804267617306, 0.4343804267617306),
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+      VectorValue(0.395227177569743, 0.3023864112151285),
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+      VectorValue(0.6607456749400253, 0.0238589269426556),
+      VectorValue(0.7364820152304077, 0.04029533454477179),
+      VectorValue(0.6437257357698205, 0.06787840431144707),
+      VectorValue(0.5930980425461418, 0.13953560718108263),
+      VectorValue(0.5861680189969418, 0.008066585704166612),
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+      VectorValue(0.2883958599187324, 0.6437257357698205),
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+      VectorValue(0.058942108840229206, 0.002728743247921069),
+      VectorValue(0.34978801000933185, 0.15717769986719343),
+      VectorValue(0.32300182354355017, 0.0021009666448275587),
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+      VectorValue(0.029549468261353372, 0.010830958603609348),
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+      VectorValue(0.0755221785129958, 0.02128689624073325),
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+      VectorValue(0.39221011498043484, 0.04152706126882266),
+      VectorValue(0.3597112755099759, 0.0938490411451324),
+      VectorValue(0.24587469948287544, 0.008686029804384135),
+      VectorValue(0.16700273817492314, 0.01758912404404562),
+      VectorValue(0.11500859863194643, 0.005523524512212553),
+      VectorValue(0.31539539811731915, 0.0238589269426556),
+      VectorValue(0.2232226502248206, 0.04029533454477179),
+      VectorValue(0.2883958599187324, 0.06787840431144707),
+      VectorValue(0.2673663502727756, 0.13953560718108263),
+      VectorValue(0.40576539529889155, 0.008066585704166612),
+      VectorValue(0.18632072767535954, 0.00012344681228740494),
+      VectorValue(0.9383291479118497, 0.058942108840229206),
+      VectorValue(0.4930342901234747, 0.34978801000933185),
+      VectorValue(0.6748972098116224, 0.32300182354355017),
+      VectorValue(0.7736883369367374, 0.15814585424951613),
+      VectorValue(0.9596195731350372, 0.029549468261353372),
+      VectorValue(0.4892484845932904, 0.29181917342653707),
+      VectorValue(0.903190925246271, 0.0755221785129958),
+      VectorValue(0.8420361139149987, 0.11711666950889889),
+      VectorValue(0.987230285395787, 0.011166218108169191),
+      VectorValue(0.6904083654940789, 0.20804564927908714),
+      VectorValue(0.5662628237507425, 0.39221011498043484),
+      VectorValue(0.5464396833448917, 0.3597112755099759),
+      VectorValue(0.7454392707127404, 0.24587469948287544),
+      VectorValue(0.8154081377810312, 0.16700273817492314),
+      VectorValue(0.8794678768558409, 0.11500859863194643),
+      VectorValue(0.6607456749400253, 0.31539539811731915),
+      VectorValue(0.7364820152304077, 0.2232226502248206),
+      VectorValue(0.6437257357698205, 0.2883958599187324),
+      VectorValue(0.5930980425461418, 0.2673663502727756),
+      VectorValue(0.5861680189969418, 0.40576539529889155),
+      VectorValue(0.8135558255123531, 0.18632072767535954),
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+      VectorValue(0.04152706126882266, 0.5662628237507425),
+      VectorValue(0.0938490411451324, 0.5464396833448917),
+      VectorValue(0.008686029804384135, 0.7454392707127404),
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+      VectorValue(0.005523524512212553, 0.8794678768558409),
+      VectorValue(0.0238589269426556, 0.6607456749400253),
+      VectorValue(0.04029533454477179, 0.7364820152304077),
+      VectorValue(0.06787840431144707, 0.6437257357698205),
+      VectorValue(0.13953560718108263, 0.5930980425461418),
+      VectorValue(0.008066585704166612, 0.5861680189969418),
+      VectorValue(0.00012344681228740494, 0.8135558255123531),
+    ]      
+    w = [
+        0.0007582310515784511,  0.005585550147583929,  0.0014302457530784435,
+        0.006269601721311614,   0.005285708929113789,  0.0030672005823594553,
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+        0.004086439220613566,   0.00515655831762925,   0.002804423566415652,
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+        0.006259183249417213,   0.0018206702056403684, 0.0003443363125208805];
+    return x, w
+  elseif (degree == 30)
+    x = [
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+      VectorValue(0.7356278532534755, 0.022758384295000066),
+      VectorValue(0.6234067740413924, 0.1238119787706746),
+      VectorValue(0.6992119263799823, 0.11588196723610056),
+      VectorValue(0.7397781780644783, 0.0665174447818816),
+      VectorValue(0.9191599701835511, 0.00443878137706136),
+      VectorValue(0.7862883331546218, 0.004663579392688625),
+      VectorValue(0.6967533241762803, 0.004703681764477044),
+      VectorValue(0.6404388932621002, 0.025182066703868706),
+      VectorValue(0.8109926237945619, 0.06577657382474286),
+      VectorValue(0.5353617425851649, 0.12612409498498942),
+      VectorValue(0.4507254468957941, 0.19487095092351842),
+      VectorValue(0.5459302776699086, 0.19100142457228309),
+      VectorValue(0.5379004199107804, 0.027533406124549888),
+      VectorValue(0.8078650909487487, 0.028063921981372968),
+      VectorValue(0.9414155840249848, 0.015902416268934703),
+      VectorValue(0.8787459746951743, 0.027294230652095765),
+      VectorValue(0.8588999350346495, 0.005691211445416102),
+      VectorValue(0.5986161382437196, 0.005162347016621321),
+      VectorValue(0.9699822487416316, 0.000533708660694491),
+      VectorValue(0.25905388452106737, 0.6931109923381601),
+      VectorValue(0.39157218829125634, 0.5287682847771431),
+      VectorValue(0.35614028329623354, 0.5861663154298922),
+      VectorValue(0.2830124973495888, 0.6397260650735271),
+      VectorValue(0.2416137624515244, 0.7356278532534755),
+      VectorValue(0.25278124718793293, 0.6234067740413924),
+      VectorValue(0.18490610638391713, 0.6992119263799823),
+      VectorValue(0.19370437715364014, 0.7397781780644783),
+      VectorValue(0.07640124843938755, 0.9191599701835511),
+      VectorValue(0.20904808745268963, 0.7862883331546218),
+      VectorValue(0.2985429940592427, 0.6967533241762803),
+      VectorValue(0.33437904003403107, 0.6404388932621002),
+      VectorValue(0.12323080238069513, 0.8109926237945619),
+      VectorValue(0.3385141624298457, 0.5353617425851649),
+      VectorValue(0.35440360218068745, 0.4507254468957941),
+      VectorValue(0.2630682977578083, 0.5459302776699086),
+      VectorValue(0.4345661739646696, 0.5379004199107804),
+      VectorValue(0.16407098706987833, 0.8078650909487487),
+      VectorValue(0.0426819997060804, 0.9414155840249848),
+      VectorValue(0.09395979465272987, 0.8787459746951743),
+      VectorValue(0.13540885351993445, 0.8588999350346495),
+      VectorValue(0.3962215147396591, 0.5986161382437196),
+      VectorValue(0.02948404259767394, 0.9699822487416316),
+      VectorValue(0.25905388452106737, 0.047835123140772554),
+      VectorValue(0.39157218829125634, 0.07965952693160062),
+      VectorValue(0.35614028329623354, 0.05769340127387423),
+      VectorValue(0.2830124973495888, 0.0772614375768841),
+      VectorValue(0.2416137624515244, 0.022758384295000066),
+      VectorValue(0.25278124718793293, 0.1238119787706746),
+      VectorValue(0.18490610638391713, 0.11588196723610056),
+      VectorValue(0.19370437715364014, 0.0665174447818816),
+      VectorValue(0.07640124843938755, 0.00443878137706136),
+      VectorValue(0.20904808745268963, 0.004663579392688625),
+      VectorValue(0.2985429940592427, 0.004703681764477044),
+      VectorValue(0.33437904003403107, 0.025182066703868706),
+      VectorValue(0.12323080238069513, 0.06577657382474286),
+      VectorValue(0.3385141624298457, 0.12612409498498942),
+      VectorValue(0.35440360218068745, 0.19487095092351842),
+      VectorValue(0.2630682977578083, 0.19100142457228309),
+      VectorValue(0.4345661739646696, 0.027533406124549888),
+      VectorValue(0.16407098706987833, 0.028063921981372968),
+      VectorValue(0.0426819997060804, 0.015902416268934703),
+      VectorValue(0.09395979465272987, 0.027294230652095765),
+      VectorValue(0.13540885351993445, 0.005691211445416102),
+      VectorValue(0.3962215147396591, 0.005162347016621321),
+      VectorValue(0.02948404259767394, 0.000533708660694491),
+      VectorValue(0.6931109923381601, 0.25905388452106737),
+      VectorValue(0.5287682847771431, 0.39157218829125634),
+      VectorValue(0.5861663154298922, 0.35614028329623354),
+      VectorValue(0.6397260650735271, 0.2830124973495888),
+      VectorValue(0.7356278532534755, 0.2416137624515244),
+      VectorValue(0.6234067740413924, 0.25278124718793293),
+      VectorValue(0.6992119263799823, 0.18490610638391713),
+      VectorValue(0.7397781780644783, 0.19370437715364014),
+      VectorValue(0.9191599701835511, 0.07640124843938755),
+      VectorValue(0.7862883331546218, 0.20904808745268963),
+      VectorValue(0.6967533241762803, 0.2985429940592427),
+      VectorValue(0.6404388932621002, 0.33437904003403107),
+      VectorValue(0.8109926237945619, 0.12323080238069513),
+      VectorValue(0.5353617425851649, 0.3385141624298457),
+      VectorValue(0.4507254468957941, 0.35440360218068745),
+      VectorValue(0.5459302776699086, 0.2630682977578083),
+      VectorValue(0.5379004199107804, 0.4345661739646696),
+      VectorValue(0.8078650909487487, 0.16407098706987833),
+      VectorValue(0.9414155840249848, 0.0426819997060804),
+      VectorValue(0.8787459746951743, 0.09395979465272987),
+      VectorValue(0.8588999350346495, 0.13540885351993445),
+      VectorValue(0.5986161382437196, 0.3962215147396591),
+      VectorValue(0.9699822487416316, 0.02948404259767394),
+      VectorValue(0.047835123140772554, 0.6931109923381601),
+      VectorValue(0.07965952693160062, 0.5287682847771431),
+      VectorValue(0.05769340127387423, 0.5861663154298922),
+      VectorValue(0.0772614375768841, 0.6397260650735271),
+      VectorValue(0.022758384295000066, 0.7356278532534755),
+      VectorValue(0.1238119787706746, 0.6234067740413924),
+      VectorValue(0.11588196723610056, 0.6992119263799823),
+      VectorValue(0.0665174447818816, 0.7397781780644783),
+      VectorValue(0.00443878137706136, 0.9191599701835511),
+      VectorValue(0.004663579392688625, 0.7862883331546218),
+      VectorValue(0.004703681764477044, 0.6967533241762803),
+      VectorValue(0.025182066703868706, 0.6404388932621002),
+      VectorValue(0.06577657382474286, 0.8109926237945619),
+      VectorValue(0.12612409498498942, 0.5353617425851649),
+      VectorValue(0.19487095092351842, 0.4507254468957941),
+      VectorValue(0.19100142457228309, 0.5459302776699086),
+      VectorValue(0.027533406124549888, 0.5379004199107804),
+      VectorValue(0.028063921981372968, 0.8078650909487487),
+      VectorValue(0.015902416268934703, 0.9414155840249848),
+      VectorValue(0.027294230652095765, 0.8787459746951743),
+      VectorValue(0.005691211445416102, 0.8588999350346495),
+      VectorValue(0.005162347016621321, 0.5986161382437196),
+      VectorValue(0.000533708660694491, 0.9699822487416316),
+    ]      
+    w = [8.586495068552472e-05,  0.001900966334207759,
+        0.0014222168338437478,  0.0038920643661654237,
+        0.0004232750726644445,  0.0036934559174571445,
+        0.005376425482716406,   0.007798045662912898,
+        0.006202479514073004,   0.00745727538134288,
+        0.0013956590598703843,  8.586495068552472e-05,
+        0.001900966334207759,   0.0014222168338437478,
+        0.0038920643661654237,  0.0004232750726644445,
+        0.0036934559174571445,  0.005376425482716406,
+        0.007798045662912898,   0.006202479514073004,
+        0.00745727538134288,    0.0013956590598703843,
+        8.586495068552472e-05,  0.001900966334207759,
+        0.0014222168338437478,  0.0038920643661654237,
+        0.0004232750726644445,  0.0036934559174571445,
+        0.005376425482716406,   0.007798045662912898,
+        0.006202479514073004,   0.00745727538134288,
+        0.0013956590598703843,  0.002107379231956217,
+        0.0029624613725460075,  0.0029721100922122435,
+        0.003438592193767963,   0.0020462494841735996,
+        0.0044363272530086725,  0.003767161464773241,
+        0.003402003378050937,   0.000603348284271059,
+        0.0009794742889466218,  0.0011372229504993392,
+        0.002682342423093155,   0.0029350349015327634,
+        0.005651511771468744,   0.007158242634833795,
+        0.006463011725059148,   0.003140798290445832,
+        0.0023509684970529836,  0.0009152534380744139,
+        0.0019109402735475102,  0.0009599402787344589,
+        0.0013212839615816257,  0.00016781085573320113,
+        0.002107379231956217,   0.0029624613725460075,
+        0.0029721100922122435,  0.003438592193767963,
+        0.0020462494841735996,  0.0044363272530086725,
+        0.003767161464773241,   0.003402003378050937,
+        0.000603348284271059,   0.0009794742889466218,
+        0.0011372229504993392,  0.002682342423093155,
+        0.0029350349015327634,  0.005651511771468744,
+        0.007158242634833795,   0.006463011725059148,
+        0.003140798290445832,   0.0023509684970529836,
+        0.0009152534380744139,  0.0019109402735475102,
+        0.0009599402787344589,  0.0013212839615816257,
+        0.00016781085573320113, 0.002107379231956217,
+        0.0029624613725460075,  0.0029721100922122435,
+        0.003438592193767963,   0.0020462494841735996,
+        0.0044363272530086725,  0.003767161464773241,
+        0.003402003378050937,   0.000603348284271059,
+        0.0009794742889466218,  0.0011372229504993392,
+        0.002682342423093155,   0.0029350349015327634,
+        0.005651511771468744,   0.007158242634833795,
+        0.006463011725059148,   0.003140798290445832,
+        0.0023509684970529836,  0.0009152534380744139,
+        0.0019109402735475102,  0.0009599402787344589,
+        0.0013212839615816257,  0.00016781085573320113,
+        0.002107379231956217,   0.0029624613725460075,
+        0.0029721100922122435,  0.003438592193767963,
+        0.0020462494841735996,  0.0044363272530086725,
+        0.003767161464773241,   0.003402003378050937,
+        0.000603348284271059,   0.0009794742889466218,
+        0.0011372229504993392,  0.002682342423093155,
+        0.0029350349015327634,  0.005651511771468744,
+        0.007158242634833795,   0.006463011725059148,
+        0.003140798290445832,   0.0023509684970529836,
+        0.0009152534380744139,  0.0019109402735475102,
+        0.0009599402787344589,  0.0013212839615816257,
+        0.00016781085573320113, 0.002107379231956217,
+        0.0029624613725460075,  0.0029721100922122435,
+        0.003438592193767963,   0.0020462494841735996,
+        0.0044363272530086725,  0.003767161464773241,
+        0.003402003378050937,   0.000603348284271059,
+        0.0009794742889466218,  0.0011372229504993392,
+        0.002682342423093155,   0.0029350349015327634,
+        0.005651511771468744,   0.007158242634833795,
+        0.006463011725059148,   0.003140798290445832,
+        0.0023509684970529836,  0.0009152534380744139,
+        0.0019109402735475102,  0.0009599402787344589,
+        0.0013212839615816257,  0.00016781085573320113,
+        0.002107379231956217,   0.0029624613725460075,
+        0.0029721100922122435,  0.003438592193767963,
+        0.0020462494841735996,  0.0044363272530086725,
+        0.003767161464773241,   0.003402003378050937,
+        0.000603348284271059,   0.0009794742889466218,
+        0.0011372229504993392,  0.002682342423093155,
+        0.0029350349015327634,  0.005651511771468744,
+        0.007158242634833795,   0.006463011725059148,
+        0.003140798290445832,   0.0023509684970529836,
+        0.0009152534380744139,  0.0019109402735475102,
+        0.0009599402787344589,  0.0013212839615816257,
+        0.00016781085573320113];
+    return x, w
+  end
+end
+
+##########################################################################################
+##########################################################################################
+##########################################################################################
+##########################################################################################
+##########################################################################################
+##########################################################################################
+##########################################################################################
+##########################################################################################
+
+function _xiaogimbutas_quad_tet(degree)
+  if degree ∉ 1:15
+    msg = """\n
+      `xiaogimbutas` quadrature rule not implemented for degree = $degree on a triangle.
+      Implemented up to degree 30.
+      Use `duffy` instead.
+    """
+    error(msg)
+  end
+  
+  if (degree == 1)
+    x = [
+      VectorValue(0.25, 0.25, 0.25)
+    ]
+    w = [0.16666666666666666]
+    return x, w
+  elseif (degree == 2)
+    x = [
+      VectorValue(0.1236668003284584, 0.8215725409676198, 0.03993304864149842),
+      VectorValue(0.4574615870855955, 0.155933120499186, 0.3817653560693467),
+      VectorValue(0.3653145188146345, 0.1800296935103654, 0.006923235573627467),
+      VectorValue(0.0003755150287292757, 0.2160764291848478, 0.4307017070778361),
+      ]
+    w = [0.016934591412496782, 0.04646292944776137,
+        0.050086823222829334, 0.05318232258357918];
+    return x, w
+  elseif (degree == 3)
+    x = [
+      VectorValue(0.6414297914956963, 0.1620014916985245, 0.1838503504920977),
+      VectorValue(0.3454441557197307, 0.01090521221118924, 0.2815238021235462),
+      VectorValue(0.439858947649275, 0.1901170024392839, 0.01140332944455717),
+      VectorValue(0.03787163178235702, 0.170816925164989, 0.1528181430909273),
+      VectorValue(0.1248048621652472, 0.1586851632274406, 0.5856628056552158),
+      VectorValue(0.1414827519695045, 0.5712260521491151, 0.1469183900871696),
+      ]      
+    w = [0.020387000459557516, 0.021344402118457815, 0.022094671190740867,
+            0.0234374016100672,   0.0374025278195929,   0.042000663468250383];
+    return x, w
+  elseif (degree == 4)
+    x = [
+      VectorValue(0.1746940586972306, 0.04049050672759043, 0.01356070187980288),
+      VectorValue(0.08140491840285925, 0.752508507009655, 0.06809937093820666),
+      VectorValue(0.7412288820936226, 0.0672232948933834, 0.03518392977359872),
+      VectorValue(0.05334123953574518, 0.419266313879513, 0.04778143555908666),
+      VectorValue(0.4329534904813556, 0.4507658760912768, 0.05945661629943383),
+      VectorValue(0.5380072039161857, 0.1294113737889104, 0.3301904148374645),
+      VectorValue(0.00899126009333578, 0.1215419913339278, 0.3064939884296903),
+      VectorValue(0.1066041725619936, 0.09720464458758327, 0.68439041545304),
+      VectorValue(0.3292329597426469, 0.02956949520647961, 0.3179035602133946),
+      VectorValue(0.1038441164109932, 0.4327102390477686, 0.3538232392092971),
+      VectorValue(0.3044484024344968, 0.2402766649280726, 0.126801725915392),
+    ]
+    w = [0.006541848487473326, 0.009212228192656149, 0.009232299811929395,
+            0.009988864191093254, 0.011578327656272562, 0.012693785874259726,
+            0.013237780011337552, 0.01774467235924835,  0.018372372071416284,
+            0.02582935266937435,  0.03223513534160575];
+    return x, w
+  elseif (degree == 5)
+    x = [
+      VectorValue(0.4544962958743503, 0.4544962958743504, 0.04550370412564962),
+      VectorValue(0.04550370412564967, 0.4544962958743504, 0.4544962958743504),
+      VectorValue(0.04550370412564973, 0.4544962958743503, 0.04550370412564969),
+      VectorValue(0.4544962958743503, 0.04550370412564966, 0.4544962958743504),
+      VectorValue(0.4544962958743503, 0.04550370412564968, 0.04550370412564962),
+      VectorValue(0.0455037041256497, 0.04550370412564966, 0.4544962958743504),
+      VectorValue(0.09273525031089128, 0.7217942490673263, 0.09273525031089122),
+      VectorValue(0.721794249067326, 0.09273525031089128, 0.09273525031089129),
+      VectorValue(0.09273525031089132, 0.09273525031089114, 0.09273525031089129),
+      VectorValue(0.0927352503108913, 0.0927352503108913, 0.7217942490673263),
+      VectorValue(0.3108859192633006, 0.06734224221009831, 0.3108859192633006),
+      VectorValue(0.06734224221009824, 0.3108859192633006, 0.3108859192633007),
+      VectorValue(0.3108859192633006, 0.3108859192633007, 0.3108859192633006),
+      VectorValue(0.3108859192633006, 0.3108859192633007, 0.06734224221009814),
+    ] 
+    w = [0.0070910034628469025, 0.007091003462846909, 0.007091003462846909,
+            0.007091003462846912,  0.007091003462846912, 0.0070910034628469155,
+            0.012248840519393652,  0.012248840519393652, 0.012248840519393655,
+            0.012248840519393659,  0.018781320953002632, 0.018781320953002632,
+            0.018781320953002632,  0.01878132095300265];
+    return x, w
+  elseif (degree == 6)
+    x = [
+      VectorValue(0.02431897424814286, 0.03883608434488445, 0.9029287990136113),
+      VectorValue(0.02286582381402311, 0.9037700013321819, 0.02933572108317866),
+      VectorValue(0.008781957777518898, 0.0405760510668179, 0.08860035046891021),
+      VectorValue(0.8411389516623184, 0.05132520616520296, 0.0372647521383555),
+      VectorValue(0.2112976585815863, 0.007354523838069352, 0.2511844952775297),
+      VectorValue(0.02346779557305456, 0.06477516044710505, 0.3908620506710118),
+      VectorValue(0.2130411832361856, 0.06001058302026912, 0.02584268626070331),
+      VectorValue(0.2678441981835756, 0.06476943693005288, 0.6367675085585139),
+      VectorValue(0.05399614083591447, 0.2757863004698506, 0.06001614916616868),
+      VectorValue(0.3293797185491983, 0.3251196585770252, 0.3268335046190458),
+      VectorValue(0.6243213635534294, 0.06592492316000995, 0.2535936747432003),
+      VectorValue(0.06319998094256953, 0.6174557201472688, 0.2584491489839256),
+      VectorValue(0.248449540118895, 0.6265402017088824, 0.06211553318359875),
+      VectorValue(0.06373289529499766, 0.277903669330078, 0.5949096890217955),
+      VectorValue(0.06517799276337043, 0.5947173018757956, 0.06660329800760315),
+      VectorValue(0.08367881406005505, 0.06609866241468051, 0.6300545551109896),
+      VectorValue(0.5773457813897267, 0.2877250948264642, 0.06462063807336853),
+      VectorValue(0.038288670738245, 0.3283881712312217, 0.3202874336976925),
+      VectorValue(0.3519391973347045, 0.05509902249072568, 0.3810843089063102),
+      VectorValue(0.5300632754810166, 0.06678959978173812, 0.07699271710096725),
+      VectorValue(0.1521038113099309, 0.1246499636374863, 0.201234567364421),
+      VectorValue(0.3041692653497818, 0.3191942803489312, 0.04438334435720821),
+      VectorValue(0.2558207842649862, 0.2794200529459882, 0.269569929633272),
+      ]      
+    w = [
+        0.0011826324752765881, 0.001206879481977829,  0.0017372226206159916,
+        0.0026542465308339587, 0.0037609445463571384, 0.0040385478129073915,
+        0.00425072071117374,   0.005251568313784406,  0.006619016274847046,
+        0.0072065494492455666, 0.007265066343438196,  0.007768855687763452,
+        0.007858005078710203,  0.008148345983740361,  0.008294771681919054,
+        0.00883888731802823,   0.008989168438051998,  0.009970224610238195,
+        0.010435745880218545,  0.010511060314253423,  0.010722336995514588,
+        0.011189302702092839,  0.018766567415678];
+    return x, w
+  elseif (degree == 7)
+    x = [
+      VectorValue(0.001996825818299818, 0.01920799348858535, 0.6513348958482376),
+      VectorValue(0.06092218458545083, 0.3234568417895977, 0.6151709883118704),
+      VectorValue(0.0005004334442718418, 0.6355215105837613, 0.0598944722319085),
+      VectorValue(0.6279832293585974, 0.293770036523707, 0.02748237819283441),
+      VectorValue(0.05213668905801093, 0.06201109193664409, 0.05718215451677883),
+      VectorValue(0.8245440666953954, 0.05989419506998693, 0.05677586668994691),
+      VectorValue(0.062815072845237, 0.8207453007415948, 0.05891041282560915),
+      VectorValue(0.6315484739180046, 0.02583316773173042, 0.280405238101906),
+      VectorValue(0.001613532619990097, 0.1991105720528834, 0.2637473753385648),
+      VectorValue(0.3196583760970118, 0.5991009436200256, 0.03521379529745457),
+      VectorValue(0.5508889781422127, 0.2234702025301428, 0.2255285376972644),
+      VectorValue(0.3505284068372833, 0.003929651487087849, 0.2418446130585829),
+      VectorValue(0.2307849002376704, 0.01403308447330531, 0.5661630745306973),
+      VectorValue(0.063969430325799, 0.06247402252315021, 0.812872555571),
+      VectorValue(0.3239480709891824, 0.0624444127129091, 0.5863212858301218),
+      VectorValue(0.2343964973359623, 0.528711306413653, 0.2169982993458658),
+      VectorValue(0.3501153045071709, 0.2615087691765827, 0.01197587688915757),
+      VectorValue(0.2753098106871322, 0.05300013833454678, 0.04497766312688006),
+      VectorValue(0.0762414702839689, 0.03316569983103569, 0.2748015348979936),
+      VectorValue(0.02253298349383202, 0.2604205879982621, 0.5646501024676913),
+      VectorValue(0.04463026790663657, 0.6018235776318118, 0.2899940343655131),
+      VectorValue(0.5990192398798975, 0.0528776293788545, 0.05497958289551413),
+      VectorValue(0.0680111048992614, 0.2977852343835241, 0.04764944310089234),
+      VectorValue(0.1572418559860032, 0.5504559416248597, 0.04374347016073189),
+      VectorValue(0.5003786698498154, 0.2582805473674438, 0.08266909560739215),
+      VectorValue(0.2898612819086906, 0.3971744029949173, 0.1710488778187786),
+      VectorValue(0.1112511334269427, 0.1074624307831534, 0.4758491617393153),
+      VectorValue(0.07400870213911578, 0.4103580959949396, 0.2447616509193416),
+      VectorValue(0.2175544442163533, 0.2521305306293954, 0.4487392553835752),
+      VectorValue(0.4372480897645487, 0.1098959763270211, 0.2765716827388576),
+      VectorValue(0.2188126225475045, 0.176438223014948, 0.1757661102664513),
+      ]
+    w = [
+        0.0012846968603334146, 0.002000632031369977,  0.002085684575720105,
+        0.002783666843940815,  0.0030095129140263084, 0.0032004686326964665,
+        0.0034317247720467565, 0.003452013693960475,  0.0034787841936317,
+        0.0035154609736464167, 0.0036967198625352964, 0.003724778580430695,
+        0.0037352275658985514, 0.003869468221310365,  0.0038818311595533,
+        0.00409373831432887,   0.004554985719607738,  0.0046768630523221786,
+        0.00471879793532209,   0.004799380599205763,  0.005235187909249938,
+        0.005632644217125163,  0.0061559681852397475, 0.006973088601266729,
+        0.007165193660663451,  0.008796944275592277,  0.010050252598534952,
+        0.010698139822576646,  0.011118534527621958,  0.011123248236813444,
+        0.01372302813009497];
+    return x, w
+  elseif (degree == 8)
+    x = [
+      VectorValue(0.0009718783690150193, 0.9573816260583027, 0.01600798423129822),
+      VectorValue(0.01760715612687848, 0.008855594278705747, 0.9485639297990048),
+      VectorValue(0.9408082028316022, 0.0300522477759322, 0.006719678822153188),
+      VectorValue(0.004571184374515704, 0.04868900463902199, 0.0221620710966624),
+      VectorValue(0.4618117251682627, 0.4653151604130365, 0.005422183307178087),
+      VectorValue(0.03576257087872006, 0.5003296720329762, 0.008112050755706058),
+      VectorValue(0.2307434967946494, 0.6113365336918841, 0.005575773030438238),
+      VectorValue(0.02580684887253231, 0.02777675799356541, 0.4665405858909191),
+      VectorValue(0.4139212834388201, 0.4982549617822043, 0.07975787096690573),
+      VectorValue(0.1999067290329215, 0.4794066533528659, 0.3200956067503981),
+      VectorValue(0.1643869309848081, 0.03985667792342303, 0.0344820754789425),
+      VectorValue(0.03678064973474819, 0.7205111657178718, 0.2043689239073254),
+      VectorValue(0.1890744688700332, 0.03620467192851837, 0.7313673600360894),
+      VectorValue(0.1506377117814253, 0.7623678849544528, 0.05243838992870327),
+      VectorValue(0.2496370830578753, 0.1955460819226021, 0.5417931824618095),
+      VectorValue(0.2579166731543092, 0.2209774649988177, 0.01239592336019548),
+      VectorValue(0.05045500693172469, 0.03938021868450564, 0.1731956086833511),
+      VectorValue(0.7485745607457628, 0.04543453035210961, 0.1632993432993541),
+      VectorValue(0.04648612941109276, 0.1564866850400663, 0.7514965361518376),
+      VectorValue(0.723709495041749, 0.04620058214866413, 0.04349881445195029),
+      VectorValue(0.04908534432749626, 0.04149415901827835, 0.71717735709395),
+      VectorValue(0.7016567760643901, 0.2044918137472231, 0.0468686532808731),
+      VectorValue(0.04478021549603177, 0.719573330071695, 0.05599064468210205),
+      VectorValue(0.4487780077524824, 0.04105029814548231, 0.04568536574611817),
+      VectorValue(0.04312238337787981, 0.4150765842699887, 0.4947991693199913),
+      VectorValue(0.03095052921613287, 0.4246500540795894, 0.1362062412251983),
+      VectorValue(0.2504719219173209, 0.02261090963666046, 0.2224346424927786),
+      VectorValue(0.05438301560623344, 0.2162641488906512, 0.05125376513387382),
+      VectorValue(0.4721332195496732, 0.04544737426285883, 0.4356669945456628),
+      VectorValue(0.03082247209993105, 0.1812621911709453, 0.2593463891596102),
+      VectorValue(0.5111060719135769, 0.02882725890093539, 0.2295407130726556),
+      VectorValue(0.1239784893278184, 0.06224439946102048, 0.4036547750100588),
+      VectorValue(0.3828805011657271, 0.397275457461443, 0.08173362820479114),
+      VectorValue(0.4935900808340203, 0.2161396926300791, 0.03919057319753563),
+      VectorValue(0.03383673809722395, 0.2125192261175291, 0.5056419509803703),
+      VectorValue(0.04243030114562282, 0.4862252458661596, 0.2752360272999065),
+      VectorValue(0.2502274462930384, 0.04529074988858361, 0.4931447679590647),
+      VectorValue(0.2280801840550788, 0.1425146299088945, 0.1281437397666177),
+      VectorValue(0.4835182802910101, 0.2295406150635747, 0.2420175573174804),
+      VectorValue(0.1822460492233608, 0.4284077707360187, 0.06194696154684466),
+      VectorValue(0.2096423028706461, 0.4853135576380692, 0.1986199776692312),
+      VectorValue(0.1646955059003284, 0.2495207660799524, 0.249905550725755),
+      VectorValue(0.4118962036484124, 0.1582590933458902, 0.2073917351406235),
+      VectorValue(0.2003307068778031, 0.2274620499867547, 0.4489545286636978),
+      ]      
+    w = [
+        0.0002523242325191773, 0.000290863922550942,  0.00034651409450314665,
+        0.0004244834198904958, 0.0011495713860626099, 0.0012535250586434693,
+        0.0016460861324811202, 0.0016881535417914368, 0.00197757011617887,
+        0.0021332305621032666, 0.0022892954263809117, 0.002453317862571215,
+        0.00251062824751718,   0.002564740902068485,  0.002602731533963437,
+        0.0027548944667587136, 0.0028292048763357883, 0.0028982004270852453,
+        0.002935422697522245,  0.002983133967882152,  0.0029981483163099916,
+        0.0033577851411315802, 0.0034241727240937898, 0.0036540831010869316,
+        0.0037507609946701936, 0.0037598731255837213, 0.0037906201057506216,
+        0.003916487256448844,  0.00402639065463561,   0.00418984767843144,
+        0.004519589747580942,  0.004542489322412185,  0.004697614316667847,
+        0.0053735422239879,    0.005382166773474393,  0.0054545457922569145,
+        0.006091681784605513,  0.006510451658158449,  0.006683134190221676,
+        0.006927915069245575,  0.008629937323168083,  0.008810488117670844,
+        0.008892300407975902,  0.009298747966287926];
+    return x, w
+  elseif (degree == 9)
+    x = [
+      VectorValue(0.006891366839295159, 0.7203807203359198, 0.2440786635406753),
+      VectorValue(0.1555789083027259, 0.7830238380676567, 0.03722731406074435),
+      VectorValue(0.8847735952623145, 0.03746715582104113, 0.04313984945003826),
+      VectorValue(0.04038577127605769, 0.2233476678136009, 0.723724094807067),
+      VectorValue(0.515285061557277, 0.4405214048473231, 0.03307435692542832),
+      VectorValue(0.03474022441842899, 0.8739842844692827, 0.04802973388725062),
+      VectorValue(0.2737549691522871, 0.5866194063678511, 0.1331737132533926),
+      VectorValue(0.03138783669262657, 0.02302476689782933, 0.2010151099960878),
+      VectorValue(0.04195473200768755, 0.04206648540987851, 0.8751960265284056),
+      VectorValue(0.2133537804622216, 0.01492382397847686, 0.72761122983064),
+      VectorValue(0.05137368203860852, 0.737898449375068, 0.01052487006572961),
+      VectorValue(0.01167918776735336, 0.03967712051591504, 0.47317164143189),
+      VectorValue(0.6123408021820864, 0.1432143403978234, 0.0006662179087437572),
+      VectorValue(0.04585901947602267, 0.04589832979249515, 0.03930065125704704),
+      VectorValue(0.1030115138063945, 0.2313142488715318, 0.002864577988396536),
+      VectorValue(0.00837326895113094, 0.218303550548395, 0.0638376081133778),
+      VectorValue(0.7518893444883036, 0.03020971497018418, 0.03931328269724702),
+      VectorValue(0.2307731120717099, 0.03171156153538288, 0.02914913102984441),
+      VectorValue(0.005807381258551988, 0.4350661767994387, 0.4471841503422259),
+      VectorValue(0.3807370349415273, 0.1946216802654709, 0.4205051742028665),
+      VectorValue(0.2760774398234713, 0.6335347557774519, 0.02029485866160777),
+      VectorValue(0.06107660031735783, 0.4443452478104636, 0.474856882867235),
+      VectorValue(0.04011137824044635, 0.03313166695541811, 0.713691124061697),
+      VectorValue(0.02545934399020729, 0.1809521324516702, 0.6820586129684939),
+      VectorValue(0.6044262999606023, 0.01758681325508173, 0.2037168634602605),
+      VectorValue(0.7098027195586388, 0.04483105915111014, 0.211317992359171),
+      VectorValue(0.7304989942620459, 0.1871472921145495, 0.03706986589769196),
+      VectorValue(0.5007701133228333, 0.02448223828136365, 0.04933213970351709),
+      VectorValue(0.03332720320375657, 0.4837323966974713, 0.0372285314523825),
+      VectorValue(0.3567521368987188, 0.1328834330395908, 0.02051457140178872),
+      VectorValue(0.1838549978176258, 0.02094427630108402, 0.1704389178072653),
+      VectorValue(0.2268373199643008, 0.4167820833252522, 0.007798392818596213),
+      VectorValue(0.1923717952701477, 0.1242856487305183, 0.6561382235043771),
+      VectorValue(0.3619508813743887, 0.01552011599218286, 0.3924336805270703),
+      VectorValue(0.01965056042084037, 0.6443583098389682, 0.1340945354491865),
+      VectorValue(0.458244669513536, 0.0366754648901354, 0.4635857264004116),
+      VectorValue(0.01538959981066759, 0.3394322940952808, 0.2418695119980843),
+      VectorValue(0.1242356229882021, 0.02743271521385679, 0.4125431242926855),
+      VectorValue(0.5655163520825289, 0.2158455335267193, 0.1861909388269384),
+      VectorValue(0.1031571468282741, 0.6454494634370023, 0.2025517153954042),
+      VectorValue(0.04580873781449243, 0.1306656875755306, 0.2354771046476909),
+      VectorValue(0.2222475737162643, 0.3744969790769868, 0.3639253527626089),
+      VectorValue(0.4587456757955503, 0.360156713211527, 0.03689983110212465),
+      VectorValue(0.3633566801123643, 0.4147619222990371, 0.1625263637506013),
+      VectorValue(0.1839853909173693, 0.05281258992885371, 0.5791440724832208),
+      VectorValue(0.03912033521689179, 0.1799045015970503, 0.4765094010040439),
+      VectorValue(0.1524683171735235, 0.1476870681014711, 0.09407076219108002),
+      VectorValue(0.1668749728706788, 0.5729563685802239, 0.07774426835124648),
+      VectorValue(0.1417161598745218, 0.2271598693150327, 0.5175989795285014),
+      VectorValue(0.3516091771883741, 0.05503739382968048, 0.1969410619376471),
+      VectorValue(0.3364345271215077, 0.2289840090706933, 0.0827335053473291),
+      VectorValue(0.10961245832859, 0.3537003584345384, 0.1179344061919236),
+      VectorValue(0.5571603219749478, 0.1325043854190557, 0.1224035951881698),
+      VectorValue(0.08342378852063631, 0.4222286016719213, 0.3031418645744979),
+      VectorValue(0.2709662131081065, 0.3336191905680228, 0.1998230301014543),
+      VectorValue(0.3787443941211687, 0.143482922419707, 0.3469408062547597),
+      VectorValue(0.1819772234774806, 0.1650628409073057, 0.3092464405790883),
+      ]       
+    w = [
+        0.0007162016044045418, 0.0009077771144629987, 0.0009168420416400242,
+        0.0009750266138313078, 0.0011001900925731798, 0.0011235667369375677,
+        0.001171941882444346,  0.0011744621157262513, 0.0011822813558184364,
+        0.001184296682706247,  0.0011900066193666035, 0.0011999643701743605,
+        0.001313356829704151,  0.0013368684225426673, 0.0013663833806215103,
+        0.0013827059604750302, 0.0015244284474830849, 0.001527532399215677,
+        0.0016299004290249847, 0.0016375601841337004, 0.0017185809772670582,
+        0.00193749218890168,   0.0019604227565410766, 0.0020954845204123134,
+        0.0021625430372239733, 0.0021645885229682614, 0.00218097825718004,
+        0.002205183817607957,  0.0022123193411248848, 0.0022143750651612668,
+        0.0022795392347588314, 0.0023777410142237997, 0.002482591391785385,
+        0.00251979649860075,   0.0025639554658188667, 0.0026195628789069518,
+        0.00274589846301462,   0.00331480176776181,   0.0035590499516530514,
+        0.00373011850287772,   0.0038349148702950617, 0.004215437802807641,
+        0.0042320852343912235, 0.004275313663630222,  0.0043800821139580864,
+        0.00439593140299428,   0.00468786977428942,   0.005307550768351452,
+        0.005439951471336866,  0.0055201790703796666, 0.005870060438901905,
+        0.005879894123846905,  0.005952755449705822,  0.006370180985658904,
+        0.006979224618881529,  0.007535230513202575,  0.008183687426958101];
+    return x, w
+  elseif (degree == 10)
+    x = [
+      VectorValue(0.004844889527768171, 0.004844889527573282, 0.9854653314167374),
+      VectorValue(0.946719848292354, 0.006667080448351054, 0.00806750556190488),
+      VectorValue(0.006667080448658992, 0.03854556569737443, 0.008067505561779886),
+      VectorValue(0.0385455656974801, 0.9467198482918522, 0.008067505562173938),
+      VectorValue(0.00382689193302969, 0.8740251327651677, 0.01855431505435714),
+      VectorValue(0.1035936602480028, 0.003826891933043923, 0.01855431505433685),
+      VectorValue(0.8740251327649323, 0.1035936602475548, 0.01855431505434555),
+      VectorValue(0.01735404744568466, 0.6202583012900509, 0.0324034637009163),
+      VectorValue(0.6202583012898868, 0.3299841875634282, 0.03240346370091293),
+      VectorValue(0.3299841875637114, 0.01735404744578351, 0.03240346370092632),
+      VectorValue(0.03118131875220204, 0.1266394908785819, 0.8114772294984899),
+      VectorValue(0.1266394908788713, 0.03070196087069638, 0.8114772294981942),
+      VectorValue(0.03070196087074504, 0.03118131875221086, 0.8114772294979709),
+      VectorValue(0.7805722458243883, 0.03518958226876812, 0.03307359982554328),
+      VectorValue(0.03518958226870847, 0.1511645720816399, 0.03307359982557704),
+      VectorValue(0.1511645720815177, 0.7805722458241509, 0.03307359982557292),
+      VectorValue(0.8149022829407979, 0.03351563550542005, 0.1193767951360695),
+      VectorValue(0.03351563550541749, 0.03220528641772838, 0.1193767951358615),
+      VectorValue(0.03220528641771444, 0.8149022829405032, 0.1193767951363346),
+      VectorValue(0.5596600888554596, 0.1807602495702272, 0.007438795731167321),
+      VectorValue(0.1807602495700562, 0.2521408658433209, 0.007438795731199615),
+      VectorValue(0.2521408658430335, 0.5596600888557026, 0.007438795731214367),
+      VectorValue(0.03029393775770431, 0.3510309266619137, 0.03266058005326467),
+      VectorValue(0.3510309266617719, 0.5860145555272518, 0.03266058005328976),
+      VectorValue(0.5860145555274775, 0.03029393775772133, 0.03266058005327201),
+      VectorValue(0.03114461249008678, 0.5992738011392268, 0.3365975263830276),
+      VectorValue(0.03298405998767925, 0.03114461249007869, 0.3365975263825117),
+      VectorValue(0.5992738011395161, 0.03298405998767671, 0.3365975263827067),
+      VectorValue(0.01267643164854561, 0.1645472158899154, 0.4090752240119478),
+      VectorValue(0.1645472158900111, 0.4137011284494345, 0.4090752240120063),
+      VectorValue(0.4137011284495361, 0.01267643164855134, 0.4090752240118714),
+      VectorValue(0.03573908441154557, 0.03377912559111834, 0.592532870170714),
+      VectorValue(0.3379489198264275, 0.03573908441154142, 0.5925328701708977),
+      VectorValue(0.03377912559113959, 0.3379489198260952, 0.5925328701712224),
+      VectorValue(0.1336599495323923, 0.1336599495326111, 0.5990201514025275),
+      VectorValue(0.1669597693172063, 0.1669597693175803, 0.4991206920481123),
+      VectorValue(0.0122804285624344, 0.4126008361791699, 0.3979043621857173),
+      VectorValue(0.1772143730727422, 0.01228042856244396, 0.3979043621855937),
+      VectorValue(0.4126008361792021, 0.1772143730727065, 0.3979043621855659),
+      VectorValue(0.3696126448561478, 0.1145379863160246, 0.02495402206716512),
+      VectorValue(0.1145379863158415, 0.4908953467609971, 0.02495402206723796),
+      VectorValue(0.4908953467605529, 0.369612644856217, 0.0249540220672398),
+      VectorValue(0.08302494688196357, 0.7282094809974271, 0.03849678186899388),
+      VectorValue(0.7282094809970688, 0.1502687902517552, 0.03849678186899534),
+      VectorValue(0.1502687902517038, 0.08302494688216776, 0.03849678186896475),
+      VectorValue(0.01680959146959896, 0.4043770144883662, 0.1613371421710751),
+      VectorValue(0.417476251870908, 0.01680959146964861, 0.1613371421710812),
+      VectorValue(0.4043770144883684, 0.4174762518708103, 0.1613371421711606),
+      VectorValue(0.1696138475604023, 0.6339280343903018, 0.1733608592737587),
+      VectorValue(0.02309725877555648, 0.1696138475603947, 0.1733608592737318),
+      VectorValue(0.6339280343903569, 0.0230972587755699, 0.1733608592736602),
+      VectorValue(0.02434102832026448, 0.6335875653142609, 0.1650530553465075),
+      VectorValue(0.1770183510189467, 0.02434102832027353, 0.1650530553463749),
+      VectorValue(0.6335875653143981, 0.1770183510189177, 0.1650530553463994),
+      VectorValue(0.02935511686037216, 0.1740037654545786, 0.6208415104357218),
+      VectorValue(0.1740037654546278, 0.1757996072492151, 0.6208415104357764),
+      VectorValue(0.1757996072493023, 0.02935511686041814, 0.6208415104356415),
+      VectorValue(0.1121243864649791, 0.5325889236500517, 0.2362419588457611),
+      VectorValue(0.1190447310392825, 0.112124386465023, 0.2362419588456988),
+      VectorValue(0.5325889236499357, 0.119044731039276, 0.2362419588456608),
+      VectorValue(0.1193225934917416, 0.4534859243589937, 0.1285563923599583),
+      VectorValue(0.2986350897893351, 0.119322593491917, 0.1285563923597633),
+      VectorValue(0.4534859243587895, 0.298635089789364, 0.1285563923599528),
+      VectorValue(0.3020019073266451, 0.3283455214738413, 0.3030261289420949),
+      VectorValue(0.3283455214738281, 0.0666264422574022, 0.3030261289418549),
+      VectorValue(0.06662644225731254, 0.3020019073267441, 0.3030261289420782),
+      VectorValue(0.5159168166142745, 0.1324670107447715, 0.1074464615850585),
+      VectorValue(0.1324670107446421, 0.244169711055904, 0.1074464615851541),
+      VectorValue(0.2441697110556646, 0.5159168166144414, 0.1074464615851919),
+      VectorValue(0.318206200820324, 0.3182062008205924, 0.04538139753862875),
+      VectorValue(0.1207131116358087, 0.3395130626211381, 0.4110926574948671),
+      VectorValue(0.3395130626209017, 0.1286811682481305, 0.4110926574949583),
+      VectorValue(0.1286811682481935, 0.1207131116358556, 0.4110926574950111),
+      VectorValue(0.2570719807624278, 0.2570719807625507, 0.2287840577124626),
+      ]      
+    w = [6.365271938318e-05,     0.00013168318579095435,
+                        0.00013168318579197532, 0.00013168318579277535,
+                        0.0002820943065241142,  0.0002820943065245265,
+                        0.0002820943065259753,  0.0010231923733042463,
+                        0.0010231923733075212,  0.0010231923733079599,
+                        0.0011033909837475048,  0.0011033909837480274,
+                        0.0011033909837487538,  0.00116416016749959,
+                        0.0011641601675003968,  0.0011641601675006052,
+                        0.001191461909106609,   0.0011914619091069783,
+                        0.0011914619091077897,  0.001368674554135069,
+                        0.0013686745541369087,  0.0013686745541372192,
+                        0.001522996656033314,   0.00152299665603406,
+                        0.0015229966560349054,  0.0016958310152239383,
+                        0.0016958310152242648,  0.00169583101522512,
+                        0.0018736985584823533,  0.0018736985584836066,
+                        0.0018736985584844332,  0.0018987030975030218,
+                        0.0018987030975037966,  0.0018987030975040383,
+                        0.0019028288945829631,  0.0019591843153447786,
+                        0.001969493864818562,   0.001969493864819973,
+                        0.001969493864825328,   0.002064660777657845,
+                        0.0020646607776616814,  0.0020646607776622417,
+                        0.002083913166714165,   0.002083913166715285,
+                        0.0020839131667166597,  0.002094685984369893,
+                        0.002094685984373115,   0.002094685984375275,
+                        0.00219141969443908,    0.0021914196944394283,
+                        0.002191419694440678,   0.002284450780288395,
+                        0.0022844507802891865,  0.002284450780289795,
+                        0.0027774049091755302,  0.00277740490917618,
+                        0.0027774049091794984,  0.003445098889335028,
+                        0.0034450988893415164,  0.0034450988893420715,
+                        0.004234512641858648,   0.004234512641859507,
+                        0.004234512641860643,   0.004471157472950426,
+                        0.004471157472951478,   0.004471157472952013,
+                        0.0045538831991611866,  0.004553883199161758,
+                        0.004553883199163369,   0.004651927948037353,
+                        0.004682093742486137,   0.004682093742490215,
+                        0.004682093742491071,   0.0077630869974032015];
+    return x, w
+  elseif (degree == 11)
+    x = [
+      VectorValue(0.02174356161974667, 0.02174356162123492, 0.9347693151393625),
+      VectorValue(0.01214887718924207, 0.4733927262830146, 0.5068373557636653),
+      VectorValue(0.473392726282396, 0.007621040764793446, 0.5068373557636205),
+      VectorValue(0.007621040764693851, 0.01214887718929203, 0.5068373557640636),
+      VectorValue(0.8283411311394426, 0.01186743374839475, 0.03349202862194989),
+      VectorValue(0.1262994064902889, 0.8283411311393326, 0.03349202862186627),
+      VectorValue(0.01186743374856977, 0.1262994064902864, 0.03349202862190801),
+      VectorValue(0.03884286462495556, 0.0266245121640002, 0.02791260212206558),
+      VectorValue(0.906620021088922, 0.03884286462496065, 0.02791260212209652),
+      VectorValue(0.02662451216400466, 0.9066200210888388, 0.02791260212224616),
+      VectorValue(0.01779838979231115, 0.02570671787901103, 0.1452959108855631),
+      VectorValue(0.8111989814430212, 0.01779838979236296, 0.1452959108856611),
+      VectorValue(0.02570671787902293, 0.8111989814422269, 0.1452959108864052),
+      VectorValue(0.4336893678654693, 0.03413212384063684, 0.004124974507952623),
+      VectorValue(0.5280535337860727, 0.4336893678650325, 0.0041249745082621),
+      VectorValue(0.03413212384056524, 0.528053533786363, 0.004124974508272195),
+      VectorValue(0.02723844389731109, 0.02928619896418311, 0.8221371579049448),
+      VectorValue(0.1213381992336525, 0.02723844389723012, 0.8221371579047962),
+      VectorValue(0.0292861989645277, 0.1213381992371994, 0.8221371579007392),
+      VectorValue(0.1055219304212634, 0.4058583387143239, 0.4825471524757858),
+      VectorValue(0.40585833871604, 0.006072578389145572, 0.4825471524736373),
+      VectorValue(0.00607257838916173, 0.1055219304213615, 0.482547152473874),
+      VectorValue(0.7375158260909277, 0.09328386025279926, 0.01187574741300857),
+      VectorValue(0.09328386025323457, 0.1573245662432324, 0.01187574741303624),
+      VectorValue(0.1573245662429647, 0.7375158260902062, 0.01187574741367739),
+      VectorValue(0.7570556764755156, 0.1851613866917053, 0.02607843400608577),
+      VectorValue(0.1851613866916737, 0.03170450282669195, 0.02607843400609957),
+      VectorValue(0.0317045028266906, 0.7570556764755849, 0.02607843400619944),
+      VectorValue(0.3085734208212338, 0.6353194772067751, 0.03266098603990054),
+      VectorValue(0.635319477206719, 0.02344611593205012, 0.03266098604008431),
+      VectorValue(0.02344611593211183, 0.3085734208211992, 0.03266098603998926),
+      VectorValue(0.02841949702328586, 0.2801664445485925, 0.6575245517573061),
+      VectorValue(0.2801664445453317, 0.03388950667075014, 0.6575245517602716),
+      VectorValue(0.03388950667083485, 0.02841949702363653, 0.6575245517604836),
+      VectorValue(0.7288732494846533, 0.1160230818544248, 0.1355923645754941),
+      VectorValue(0.1160230818543114, 0.01951130408545599, 0.1355923645754723),
+      VectorValue(0.01951130408539934, 0.7288732494841443, 0.135592364576106),
+      VectorValue(0.144054962296286, 0.680860745134503, 0.1577793967317463),
+      VectorValue(0.01730489583748775, 0.1440549622963796, 0.1577793967318802),
+      VectorValue(0.6808607451345954, 0.01730489583762223, 0.157779396731644),
+      VectorValue(0.1579550425986422, 0.01216008923516082, 0.6749562807643449),
+      VectorValue(0.1549285874017353, 0.1579550425979384, 0.6749562807650454),
+      VectorValue(0.01216008923534246, 0.1549285874013449, 0.6749562807649698),
+      VectorValue(0.01001242225693271, 0.4226210458987284, 0.4504683351507885),
+      VectorValue(0.1168981966933887, 0.01001242225711663, 0.4504683351504025),
+      VectorValue(0.4226210458990461, 0.1168981966933672, 0.4504683351504458),
+      VectorValue(0.5566694034765993, 0.3465053726509192, 0.07740138872102775),
+      VectorValue(0.01942383515147987, 0.5566694034766593, 0.07740138872115238),
+      VectorValue(0.3465053726511215, 0.01942383515155061, 0.07740138872070887),
+      VectorValue(0.613355585987273, 0.03707747991970328, 0.3147249103897943),
+      VectorValue(0.037077479919721, 0.03484202370319439, 0.3147249103898168),
+      VectorValue(0.0348420237032176, 0.6133555859856407, 0.314724910391352),
+      VectorValue(0.3389609703820722, 0.01688864018974739, 0.280802863730256),
+      VectorValue(0.01688864018976847, 0.3633475256969008, 0.280802863730816),
+      VectorValue(0.3633475256976239, 0.3389609703821428, 0.2808028637303945),
+      VectorValue(0.2689933017087768, 0.1524656156298647, 0.02218921323001911),
+      VectorValue(0.5563518694315872, 0.2689933017084892, 0.0221892132299926),
+      VectorValue(0.1524656156298783, 0.5563518694313629, 0.02218921323010947),
+      VectorValue(0.3277711127242067, 0.3277711127240469, 0.01668666182758015),
+      VectorValue(0.4945020621274046, 0.02951774250327373, 0.2969980096590332),
+      VectorValue(0.02951774250352841, 0.1789821857098627, 0.2969980096594547),
+      VectorValue(0.1789821857092536, 0.4945020621265333, 0.2969980096607454),
+      VectorValue(0.1978395470663011, 0.02960272586703677, 0.2278827225106128),
+      VectorValue(0.02960272586707343, 0.544675004555141, 0.2278827225112254),
+      VectorValue(0.5446750045559754, 0.1978395470660405, 0.2278827225108745),
+      VectorValue(0.03229881331828368, 0.3371690700776922, 0.1320500662715863),
+      VectorValue(0.4984820503323908, 0.03229881331833241, 0.1320500662716444),
+      VectorValue(0.3371690700771687, 0.4984820503329197, 0.1320500662716443),
+      VectorValue(0.1207512536386751, 0.6778351364529377, 0.1011781353097973),
+      VectorValue(0.6778351364538195, 0.1002354745986857, 0.1011781353085159),
+      VectorValue(0.1002354745985345, 0.1207512536390304, 0.1011781353086743),
+      VectorValue(0.138106941917413, 0.3354607968315433, 0.03273663417213666),
+      VectorValue(0.4936956270790593, 0.1381069419173125, 0.03273663417222188),
+      VectorValue(0.3354607968314335, 0.4936956270789277, 0.03273663417220138),
+      VectorValue(0.2500532412376334, 0.2507616099201166, 0.4585809066512544),
+      VectorValue(0.2507616099209062, 0.04060424219100611, 0.4585809066503929),
+      VectorValue(0.04060424219112386, 0.2500532412374494, 0.4585809066512888),
+      VectorValue(0.09261674502310314, 0.3090936802353301, 0.5079960391158374),
+      VectorValue(0.3090936802365318, 0.0902935356263497, 0.5079960391138479),
+      VectorValue(0.09029353562658468, 0.09261674502301917, 0.5079960391137064),
+      VectorValue(0.1139695348123683, 0.1139695348116085, 0.6580913955636913),
+      VectorValue(0.1429946401664338, 0.2384743646845937, 0.1443488665029351),
+      VectorValue(0.4741821286458542, 0.1429946401665482, 0.1443488665032992),
+      VectorValue(0.2384743646842464, 0.4741821286455784, 0.1443488665038495),
+      VectorValue(0.2864949113772423, 0.1115967035331357, 0.1108203324715355),
+      VectorValue(0.1115967035330135, 0.4910880526179628, 0.1108203324717666),
+      VectorValue(0.4910880526178663, 0.286494911377532, 0.1108203324716498),
+      VectorValue(0.125536113167282, 0.1148821967817834, 0.2994703853758982),
+      VectorValue(0.4601113046750214, 0.1255361131668398, 0.2994703853761758),
+      VectorValue(0.1148821967814947, 0.4601113046733321, 0.299470385377665),
+      VectorValue(0.1968268645044536, 0.1968268645044899, 0.4095194064864269),
+      VectorValue(0.1275456273615645, 0.3107744236691848, 0.2609901451138755),
+      VectorValue(0.3107744236696469, 0.3006898038543989, 0.2609901451144841),
+      VectorValue(0.3006898038549698, 0.1275456273614465, 0.260990145113821),
+      VectorValue(0.296008469913471, 0.2960084699134847, 0.1119745902596869),
+      ]      
+    w = [0.00018520507764804083, 0.00024392157838708066,
+                        0.000243921578397049,   0.000243921578397797,
+                        0.0004426544798853663,  0.00044265447988711795,
+                        0.000442654479888878,   0.00046562748965273685,
+                        0.00046562748965351384, 0.00046562748965519116,
+                        0.000527992042630873,   0.000527992042630885,
+                        0.000527992042634342,   0.0005450543594569904,
+                        0.000545054359463543,   0.0005450543594640881,
+                        0.0007232685232691291,  0.0007232685232705268,
+                        0.0007232685232867013,  0.000852258543339523,
+                        0.0008522585433631999,  0.0008522585433649738,
+                        0.0008570829478892283,  0.0008570829478907689,
+                        0.0008570829479115868,  0.001012842250592951,
+                        0.001012842250593049,   0.0010128422505964068,
+                        0.0010128681282489697,  0.0010128681282520911,
+                        0.001012868128252403,   0.0010556629969665655,
+                        0.001055662996989173,   0.001055662996989902,
+                        0.0010621542172256188,  0.0010621542172265183,
+                        0.0010621542172285483,  0.0011697598043711175,
+                        0.0011697598043717676,  0.0011697598043740865,
+                        0.0012002940734976602,  0.001200294073497819,
+                        0.0012002940735026382,  0.001204257126245662,
+                        0.0012042571262564923,  0.0012042571262568921,
+                        0.001384148614025617,   0.001384148614025973,
+                        0.001384148614030434,   0.0015051557681943553,
+                        0.0015051557681955198,  0.0015051557682016624,
+                        0.0016274985904939964,  0.0016274985904981987,
+                        0.0016274985905059345,  0.0018737072129849982,
+                        0.00187370721298583,    0.0018737072129913232,
+                        0.00193958466403963,    0.0019496326390221017,
+                        0.0019496326390307033,  0.0019496326390484552,
+                        0.00207150412472298,    0.0020715041247248048,
+                        0.002071504124725327,   0.0021498066057806515,
+                        0.002149806605781448,   0.0021498066057846865,
+                        0.00253409922583179,    0.00253409922583231,
+                        0.002534099225833395,   0.0025971204712147886,
+                        0.002597120471219267,   0.002597120471221267,
+                        0.0028928877153485666,  0.002892887715353863,
+                        0.00289288771535987,    0.0029018056586740813,
+                        0.0029018056586777047,  0.002901805658680232,
+                        0.0029993712524892485,  0.003281967182103737,
+                        0.0032819671821101765,  0.0032819671821297963,
+                        0.0033417252382084,     0.0033417252382101736,
+                        0.0033417252382157863,  0.0036081871202086548,
+                        0.0036081871202103036,  0.0036081871202163014,
+                        0.00450476478426666,    0.004597380383431189,
+                        0.0045973803834320915,  0.00459738038343333,
+                        0.004960765552103523];
+    return x, w
+  elseif (degree == 12)
+    x = [
+      VectorValue(0.005336077019643312, 0.00533607701965122, 0.9839917689410563),
+      VectorValue(0.02289609338296681, 0.9546184848667337, 0.01790586600742269),
+      VectorValue(0.004579555742875841, 0.0228960933829682, 0.0179058660074233),
+      VectorValue(0.9546184848667292, 0.004579555742878448, 0.0179058660074248),
+      VectorValue(0.003634038039874281, 0.8589044391845247, 0.01981617795956102),
+      VectorValue(0.1176453448160352, 0.003634038039875694, 0.01981617795956089),
+      VectorValue(0.858904439184522, 0.1176453448160376, 0.01981617795956242),
+      VectorValue(0.03542041822023295, 0.5224715513554079, 0.01083320231284623),
+      VectorValue(0.5224715513554018, 0.4312748281115071, 0.01083320231285078),
+      VectorValue(0.43127482811151, 0.03542041822023723, 0.01083320231285082),
+      VectorValue(0.7939888499122254, 0.003557095377268929, 0.03305477917755645),
+      VectorValue(0.003557095377268694, 0.1693992755329503, 0.0330547791775573),
+      VectorValue(0.1693992755329477, 0.7939888499122263, 0.0330547791775572),
+      VectorValue(0.03490045658756769, 0.002190912398338879, 0.6722668687771673),
+      VectorValue(0.2906417622369189, 0.03490045658756785, 0.6722668687771728),
+      VectorValue(0.002190912398345305, 0.2906417622369231, 0.6722668687771658),
+      VectorValue(0.2341448840740503, 0.692501417588051, 0.0005245712210146334),
+      VectorValue(0.07282912711688336, 0.2341448840740487, 0.000524571221016469),
+      VectorValue(0.6925014175880473, 0.07282912711688572, 0.0005245712210174548),
+      VectorValue(0.3050609665772057, 0.01338905088310755, 0.02647035150983993),
+      VectorValue(0.6550796310298418, 0.3050609665772103, 0.02647035150984013),
+      VectorValue(0.01338905088311035, 0.6550796310298342, 0.02647035150984075),
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+    w = [3.336665703934045e-05,  8.429047994003496e-05,
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+                        0.0036587068547407633,  0.003658706854740793,
+                        0.004252858800216373,   0.0049174945629996405];
+    return x, w
+  elseif (degree == 13)
+    x = [
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+    w = [4.134324458614672e-05,  0.00010841532703717533,
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+                        0.0033033665682094884,  0.004303940769897278];
+    return x, w
+  elseif (degree == 14)
+    x = [
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+    w = [2.2105078592571967e-05, 5.104037863193469e-05,
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+                        0.00298344565801923,    0.0029834456580193486,
+                        0.0031781597181588447];
+    return x, w
+  elseif (degree == 15)
+    x = [
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+      VectorValue(0.08504170597569295, 0.01871950766556529, 0.8174647095402776),
+      VectorValue(0.2289622580755599, 0.7059180690514001, 0.01703231872933662),
+      VectorValue(0.7059180690513981, 0.04808735414370217, 0.01703231872933689),
+      VectorValue(0.04808735414368957, 0.2289622580755674, 0.01703231872933699),
+      VectorValue(0.01343952012215375, 0.188978322150464, 0.07671766832162848),
+      VectorValue(0.1889783221504602, 0.7208644894057589, 0.07671766832162524),
+      VectorValue(0.7208644894057502, 0.01343952012215593, 0.07671766832162946),
+      VectorValue(0.1856695714205305, 0.00940743474102812, 0.498715106027506),
+      VectorValue(0.3062078878109278, 0.1856695714205285, 0.4987151060275142),
+      VectorValue(0.009407434741032658, 0.306207887810909, 0.4987151060275297),
+      VectorValue(0.07134313304985408, 0.7168988880207371, 0.01586351574919136),
+      VectorValue(0.7168988880207331, 0.1958944631802114, 0.01586351574919187),
+      VectorValue(0.1958944631802093, 0.07134313304985869, 0.01586351574919585),
+      VectorValue(0.6081230921397938, 0.01132540328384632, 0.2960917880883974),
+      VectorValue(0.01132540328384961, 0.08445971648796122, 0.2960917880884005),
+      VectorValue(0.08445971648796277, 0.6081230921397877, 0.2960917880884),
+      VectorValue(0.7831901908047053, 0.08946628745362917, 0.1093980943900156),
+      VectorValue(0.01794542735164876, 0.7831901908046981, 0.1093980943900213),
+      VectorValue(0.08946628745363563, 0.01794542735165081, 0.1093980943900156),
+      VectorValue(0.5244117804550043, 0.2084577994081171, 0.01133447823052404),
+      VectorValue(0.2084577994081089, 0.255795941906351, 0.01133447823052516),
+      VectorValue(0.2557959419063432, 0.5244117804550297, 0.01133447823052761),
+      VectorValue(0.01588261437502311, 0.3863456880386881, 0.5325804036187812),
+      VectorValue(0.06519129396751876, 0.0158826143750247, 0.5325804036187707),
+      VectorValue(0.3863456880386817, 0.0651912939675148, 0.5325804036187769),
+      VectorValue(0.03236911071658531, 0.6667349326998179, 0.2320665678040404),
+      VectorValue(0.06882938877957372, 0.03236911071658261, 0.2320665678040363),
+      VectorValue(0.6667349326998023, 0.06882938877957215, 0.2320665678040363),
+      VectorValue(0.2154208621405496, 0.01783785666517354, 0.07794412055941285),
+      VectorValue(0.01783785666517584, 0.6887971606348672, 0.07794412055941696),
+      VectorValue(0.6887971606348626, 0.2154208621405385, 0.07794412055942193),
+      VectorValue(0.4031765079784496, 0.01499093171087654, 0.5021066311640049),
+      VectorValue(0.01499093171087712, 0.07972592914667045, 0.5021066311640021),
+      VectorValue(0.07972592914666891, 0.4031765079784464, 0.5021066311640048),
+      VectorValue(0.1848636811678036, 0.6499516355495718, 0.0182999648331871),
+      VectorValue(0.6499516355495523, 0.1468847184494371, 0.01829996483318704),
+      VectorValue(0.1468847184494266, 0.184863681167814, 0.01829996483318721),
+      VectorValue(0.1804453441648688, 0.451820102412822, 0.3540241023636928),
+      VectorValue(0.01371045105861957, 0.1804453441648663, 0.3540241023636804),
+      VectorValue(0.4518201024128161, 0.01371045105861805, 0.3540241023636947),
+      VectorValue(0.08105206237388583, 0.3766532156247404, 0.01518945086644218),
+      VectorValue(0.5271052711349223, 0.08105206237388458, 0.01518945086644177),
+      VectorValue(0.3766532156247387, 0.5271052711349294, 0.01518945086644269),
+      VectorValue(0.331408497762104, 0.1126435977537889, 0.0134868064234796),
+      VectorValue(0.5424610980606107, 0.3314084977621181, 0.01348680642347998),
+      VectorValue(0.1126435977537936, 0.5424610980606109, 0.01348680642348059),
+      VectorValue(0.5607935139776807, 0.01909773271486895, 0.06348929065765115),
+      VectorValue(0.01909773271486825, 0.3566194626497967, 0.06348929065765373),
+      VectorValue(0.3566194626497901, 0.5607935139776836, 0.0634892906576574),
+      VectorValue(0.07227368768344591, 0.2738988990502447, 0.6053416400059596),
+      VectorValue(0.04848577326034448, 0.07227368768344192, 0.6053416400059548),
+      VectorValue(0.2738988990502546, 0.04848577326034462, 0.6053416400059433),
+      VectorValue(0.3706144545187708, 0.01602307963396246, 0.1897016601820003),
+      VectorValue(0.01602307963396413, 0.4236608056652554, 0.1897016601819962),
+      VectorValue(0.4236608056652498, 0.3706144545187751, 0.1897016601820099),
+      VectorValue(0.07313934036770678, 0.06778253195192757, 0.7047038738820789),
+      VectorValue(0.1543742537982787, 0.0731393403677088, 0.7047038738820692),
+      VectorValue(0.06778253195194402, 0.1543742537982604, 0.7047038738820758),
+      VectorValue(0.2255550687888375, 0.3828261323216449, 0.01264429188566553),
+      VectorValue(0.3828261323216364, 0.3789745070038535, 0.01264429188566583),
+      VectorValue(0.3789745070038492, 0.2255550687888252, 0.01264429188566767),
+      VectorValue(0.01610664762850615, 0.1955976722408662, 0.504888949609758),
+      VectorValue(0.1955976722408699, 0.2834067305208563, 0.5048889496097685),
+      VectorValue(0.2834067305208526, 0.01610664762850647, 0.5048889496097625),
+      VectorValue(0.07394396840975409, 0.08972209215481677, 0.06088720165051376),
+      VectorValue(0.7754467377849116, 0.07394396840975978, 0.06088720165051331),
+      VectorValue(0.08972209215481254, 0.7754467377849096, 0.06088720165051947),
+      VectorValue(0.02198550415554825, 0.4522698873067332, 0.3656528451019294),
+      VectorValue(0.4522698873067705, 0.1600917634357737, 0.3656528451019036),
+      VectorValue(0.1600917634358059, 0.02198550415555332, 0.3656528451018906),
+      VectorValue(0.6425340162574237, 0.03023340240085394, 0.1777342490770398),
+      VectorValue(0.1494983322647011, 0.6425340162574126, 0.1777342490770339),
+      VectorValue(0.03023340240085156, 0.1494983322646933, 0.1777342490770401),
+      VectorValue(0.2982386203137134, 0.3492984375384283, 0.3282766894848435),
+      VectorValue(0.02418625266301134, 0.2982386203137263, 0.3282766894848561),
+      VectorValue(0.3492984375384166, 0.02418625266301722, 0.3282766894848335),
+      VectorValue(0.02492578585848872, 0.5613484323473249, 0.1898351044830521),
+      VectorValue(0.2238906773111161, 0.02492578585848644, 0.1898351044830608),
+      VectorValue(0.5613484323473145, 0.22389067731113, 0.1898351044830645),
+      VectorValue(0.1769461529053036, 0.05090872345953187, 0.6068946490861404),
+      VectorValue(0.1652504745490214, 0.1769461529053149, 0.6068946490861201),
+      VectorValue(0.05090872345953583, 0.1652504745490102, 0.6068946490861318),
+      VectorValue(0.267326785868413, 0.09253312310838845, 0.162728234374953),
+      VectorValue(0.09253312310841946, 0.4774118566482484, 0.1627282343749322),
+      VectorValue(0.4774118566482272, 0.2673267858684251, 0.162728234374927),
+      VectorValue(0.6324086041104882, 0.194821777558372, 0.07976498347271095),
+      VectorValue(0.09300463485842678, 0.6324086041104942, 0.0797649834727117),
+      VectorValue(0.1948217775583615, 0.09300463485842231, 0.07976498347272819),
+      VectorValue(0.1286081121536503, 0.06206882668877994, 0.5069361346327252),
+      VectorValue(0.3023869265248393, 0.1286081121536543, 0.5069361346327149),
+      VectorValue(0.06206882668879442, 0.3023869265248427, 0.5069361346327274),
+      VectorValue(0.0670060501118351, 0.07446918750056565, 0.362874044201665),
+      VectorValue(0.4956507181859352, 0.06700605011182939, 0.3628740442016639),
+      VectorValue(0.07446918750057789, 0.4956507181859199, 0.3628740442016637),
+      VectorValue(0.04935190074328522, 0.3139246227522204, 0.1497252367121568),
+      VectorValue(0.3139246227522261, 0.4869982397923315, 0.1497252367121615),
+      VectorValue(0.4869982397923459, 0.04935190074327669, 0.149725236712158),
+      VectorValue(0.1083529025389752, 0.08656298623704123, 0.1758982241285937),
+      VectorValue(0.629185887095386, 0.1083529025389925, 0.1758982241285794),
+      VectorValue(0.08656298623703901, 0.629185887095383, 0.1758982241285948),
+      VectorValue(0.4947915054658847, 0.3668061114674768, 0.0797762381765776),
+      VectorValue(0.05862614489007139, 0.4947915054658749, 0.07977623817658046),
+      VectorValue(0.3668061114674683, 0.05862614489007745, 0.07977623817658072),
+      VectorValue(0.6107564898848346, 0.09155687274905353, 0.08682921283620239),
+      VectorValue(0.2108574245299135, 0.6107564898848316, 0.0868292128362),
+      VectorValue(0.09155687274905454, 0.2108574245299094, 0.08682921283620657),
+      VectorValue(0.27189840912706, 0.2718984091270191, 0.1843047726189553),
+      VectorValue(0.08364528610473303, 0.4598959913496589, 0.2836943220226789),
+      VectorValue(0.4598959913496499, 0.1727644005229201, 0.2836943220226703),
+      VectorValue(0.1727644005229245, 0.0836452861047611, 0.2836943220226696),
+      VectorValue(0.220307180915872, 0.2203071809158613, 0.3390784572525623),
+      VectorValue(0.4498492745599813, 0.1402964159526019, 0.06810558755187382),
+      VectorValue(0.3417487219355564, 0.4498492745599653, 0.0681055875518764),
+      VectorValue(0.1402964159526053, 0.3417487219355464, 0.06810558755187403),
+      VectorValue(0.4796245561760771, 0.06988987460087778, 0.2572931400794461),
+      VectorValue(0.1931924291436032, 0.4796245561760668, 0.2572931400794509),
+      VectorValue(0.06988987460089365, 0.1931924291436036, 0.2572931400794577),
+      VectorValue(0.4819852013641325, 0.2605590256904803, 0.06519355736441289),
+      VectorValue(0.260559025690468, 0.1922622155809665, 0.06519355736441458),
+      VectorValue(0.1922622155809799, 0.481985201364128, 0.06519355736441629),
+      VectorValue(0.09109001795840514, 0.343030573977829, 0.2354338687541656),
+      VectorValue(0.3304455393096081, 0.09109001795841783, 0.2354338687541547),
+      VectorValue(0.343030573977809, 0.3304455393095925, 0.2354338687541694),
+      VectorValue(0.08624240167701186, 0.1525520261883117, 0.4323311285083857),
+      VectorValue(0.3288744436262873, 0.0862424016770105, 0.4323311285083789),
+      VectorValue(0.1525520261883389, 0.3288744436262735, 0.4323311285083771),
+      VectorValue(0.3337847727363109, 0.3372700741462638, 0.1536568898623008),
+      VectorValue(0.3372700741462386, 0.1752882632551174, 0.1536568898623003),
+      VectorValue(0.1752882632551262, 0.333784772736308, 0.1536568898623202),
+      VectorValue(0.3113177461109946, 0.3113177461109871, 0.06604676166701309),
+      VectorValue(0.2898527003164425, 0.2359060680311884, 0.3891009454304513),
+      VectorValue(0.2359060680311902, 0.08514028622192746, 0.3891009454304601),
+      VectorValue(0.0851402862219274, 0.2898527003164147, 0.38910094543047),
+      VectorValue(0.1915586123296988, 0.4738597830753307, 0.1624688668332693),
+      VectorValue(0.4738597830753175, 0.1721127377617093, 0.1624688668332666),
+      VectorValue(0.1721127377617219, 0.1915586123296906, 0.1624688668332857),
+      VectorValue(0.343066772448071, 0.1725513660667974, 0.2910280425505992),
+      VectorValue(0.1933538189344859, 0.3430667724481031, 0.2910280425505916),
+      VectorValue(0.1725513660668472, 0.1933538189345398, 0.2910280425506466),
+      VectorValue(0.1627284320161738, 0.1627284320161655, 0.5118147039514781),
+      ]
+    w = [5.0949499461256536e-05, 5.094949946129215e-05,
+                        5.0949499461363764e-05, 8.091578025333847e-05,
+                        8.09157802534108e-05,   8.091578025341743e-05,
+                        9.003858468151016e-05,  9.003858468155147e-05,
+                        9.003858468155359e-05,  0.00011096788435521842,
+                        0.00011096788435523033, 0.00011096788435523135,
+                        0.00014301129642297394, 0.0001430112964229811,
+                        0.00014301129642299053, 0.0001435686140277277,
+                        0.00014356861402773594, 0.00014356861402778517,
+                        0.00017047619231215117, 0.00017047619231217833,
+                        0.00017047619231222533, 0.00017795546384389533,
+                        0.00017795546384394634, 0.00017795546384398616,
+                        0.00018791251008007368, 0.00018791251008008316,
+                        0.00018791251008011434, 0.00018853254351190317,
+                        0.00018853254351192117, 0.0001885325435119355,
+                        0.000203420107255822,   0.00020342010725582285,
+                        0.00020342010725585117, 0.00023831202538122532,
+                        0.000238312025381286,   0.00023831202538146515,
+                        0.00024702490709429967, 0.0002470249070943653,
+                        0.00024702490709436765, 0.00025848719013847165,
+                        0.00025848719013854435, 0.00025848719013861867,
+                        0.000265583230932468,   0.00026558323093246863,
+                        0.0002655832309324773,  0.00029768749691803264,
+                        0.0002976874969180593,  0.000297687496918183,
+                        0.0002989438206765175,  0.0002989438206765232,
+                        0.0002989438206765322,  0.0003205499531253637,
+                        0.0003205499531254577,  0.0003205499531255287,
+                        0.0003351988396668535,  0.00033519883966689985,
+                        0.00033519883966691964, 0.000357511187619173,
+                        0.000357511187619211,   0.00035751118761923035,
+                        0.0003766593641388355,  0.00037665936413884397,
+                        0.00037665936413898253, 0.0004170560045706985,
+                        0.00041705600457073996, 0.00041705600457075487,
+                        0.0004206908093603285,  0.0004206908093604563,
+                        0.00042069080936051436, 0.0004403404439190608,
+                        0.0004403404439191172,  0.00044034044391935267,
+                        0.0004510176089945963,  0.00045101760899459884,
+                        0.0004510176089946125,  0.00048092536064853767,
+                        0.0004809253606485387,  0.0004809253606485603,
+                        0.0004921683513968802,  0.000492168351396902,
+                        0.0004921683513969398,  0.0004972651138408006,
+                        0.0004972651138408805,  0.0004972651138410044,
+                        0.0005046209614997817,  0.0005046209614998177,
+                        0.0005046209614999028,  0.0005053606662589742,
+                        0.0005053606662590495,  0.0005053606662590833,
+                        0.000507488410074817,   0.000507488410074818,
+                        0.0005074884100748625,  0.0005296283626829228,
+                        0.0005296283626829545,  0.0005296283626830274,
+                        0.0005961391122951584,  0.0005961391122951768,
+                        0.0005961391122952683,  0.0006063199394915311,
+                        0.0006063199394916362,  0.0006063199394917882,
+                        0.0006305106887630419,  0.0006305106887630918,
+                        0.0006305106887631449,  0.0006490437338464095,
+                        0.00064904373384643,    0.0006490437338465401,
+                        0.0006784672244327731,  0.0006784672244328733,
+                        0.0006784672244328974,  0.0007018738744954993,
+                        0.0007018738744955992,  0.0007018738744956027,
+                        0.0007044998612670034,  0.0007044998612670082,
+                        0.0007044998612670204,  0.0007123210856255668,
+                        0.0007123210856256313,  0.0007123210856256491,
+                        0.0007180840505148808,  0.0007180840505148981,
+                        0.0007180840505149077,  0.0007320095837554124,
+                        0.000732009583755433,   0.0007320095837555596,
+                        0.0007487529433902918,  0.0007487529433903294,
+                        0.0007487529433904035,  0.0007633124179686057,
+                        0.0007633124179687115,  0.0007633124179687585,
+                        0.0007665117920190594,  0.0007665117920191146,
+                        0.0007665117920191253,  0.000782684926528277,
+                        0.0007826849265283078,  0.0007826849265283831,
+                        0.0008216462765614071,  0.0008216462765614102,
+                        0.0008216462765614202,  0.0008401904995806664,
+                        0.0008401904995807659,  0.000840190499580874,
+                        0.0008649691231126786,  0.0008649691231126842,
+                        0.0008649691231126886,  0.0009771082729624124,
+                        0.0009771082729625377,  0.0009771082729625865,
+                        0.001003226140444279,   0.0010032261404442897,
+                        0.001003226140444467,   0.0011107068782117186,
+                        0.0011107068782118005,  0.001110706878211849,
+                        0.0011541194419159395,  0.0011541194419163337,
+                        0.0011541194419164846,  0.0011888869781639686,
+                        0.0011888869781640197,  0.0011888869781641177,
+                        0.0012039315081215961,  0.0012039315081217672,
+                        0.0012039315081217863,  0.0012677548100906834,
+                        0.0012677548100907092,  0.0012677548100908191,
+                        0.0013075346711990234,  0.001307534671199086,
+                        0.0013075346711991051,  0.0013386099794468343,
+                        0.0013386099794469022,  0.0013386099794470629,
+                        0.0013454689764230332,  0.0013454689764231366,
+                        0.0013454689764231468,  0.001398358195255936,
+                        0.0013983581952560142,  0.0013983581952561012,
+                        0.001511722175784935,   0.001536829401278278,
+                        0.0015368294012785917,  0.0015368294012786117,
+                        0.001568496765944744,   0.0016371705077032532,
+                        0.0016371705077033274,  0.0016371705077033551,
+                        0.0016680098648530085,  0.0016680098648530152,
+                        0.0016680098648534517,  0.0016801415184265068,
+                        0.0016801415184265085,  0.0016801415184265866,
+                        0.0016912148002698566,  0.0016912148002698867,
+                        0.0016912148002702135,  0.0018093524420684734,
+                        0.0018093524420685033,  0.001809352442068565,
+                        0.0018270943485695949,  0.0018270943485701834,
+                        0.0018270943485707415,  0.0018747167751995899,
+                        0.0018892190654910483,  0.0018892190654914516,
+                        0.0018892190654915483,  0.0019096117089971417,
+                        0.0019096117089973984,  0.0019096117089977185,
+                        0.002049458570185687,   0.00204945857018569,
+                        0.002049458570187277,   0.0024074895531075533];
+    return x, w
+  end
+end
diff --git a/test/ReferenceFEsTests/XiaoGimbutasQuadraturesTests.jl b/test/ReferenceFEsTests/XiaoGimbutasQuadraturesTests.jl
new file mode 100644
index 000000000..ab1f570e4
--- /dev/null
+++ b/test/ReferenceFEsTests/XiaoGimbutasQuadraturesTests.jl
@@ -0,0 +1,32 @@
+module XiaoGimbutasQuadraturesTests
+
+using Test
+using Gridap.ReferenceFEs, Gridap.Fields
+
+for degree in 1:30
+  quad = Quadrature(TRI,xiao_gimbutas,degree)
+  @test sum(get_weights(quad)) ≈ 0.5
+
+  ref_quad = Quadrature(TRI,duffy,degree)
+  f = get_shapefuns(ReferenceFE(TRI,lagrangian,Float64,degree))
+  f1 = integrate(f,get_coordinates(quad),get_weights(quad))
+  f2 = integrate(f,get_coordinates(ref_quad),get_weights(ref_quad))
+  err = maximum(abs.(f1 .- f2))/maximum(abs.(f1))
+  # println("degree = ",degree," err = ",maximum(abs.(f1 .- f2)))
+  @test err < 1.0e-7
+end
+
+for degree in 1:15
+  quad = Quadrature(TET,xiao_gimbutas,degree)
+  @test sum(get_weights(quad)) ≈ 0.5*1/3
+
+  ref_quad = Quadrature(TET,duffy,degree)
+  f = get_shapefuns(ReferenceFE(TET,lagrangian,Float64,degree))
+  f1 = integrate(f,get_coordinates(quad),get_weights(quad))
+  f2 = integrate(f,get_coordinates(ref_quad),get_weights(ref_quad))
+  err = maximum(abs.(f1 .- f2))/maximum(abs.(f1))
+  println("degree = ",degree," err = ",maximum(abs.(f1 .- f2)))
+  @test err < 1.0e-7
+end
+
+end # module
\ No newline at end of file
diff --git a/test/ReferenceFEsTests/runtests.jl b/test/ReferenceFEsTests/runtests.jl
index 47cd088da..70564edc8 100644
--- a/test/ReferenceFEsTests/runtests.jl
+++ b/test/ReferenceFEsTests/runtests.jl
@@ -32,6 +32,8 @@ using Test
 
 @testset "StrangQuadratures" begin include("StrangQuadraturesTests.jl") end
 
+@testset "XiaoGimbutasQuadratures" begin include("XiaoGimbutasQuadraturesTests.jl") end
+
 @testset "RaviartThomasRefFEs" begin include("RaviartThomasRefFEsTests.jl") end
 
 @testset "NedelecRefFEs" begin include("NedelecRefFEsTests.jl") end

From 24b7629028626101fc5db4a3608e463e809e5c1c Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Tue, 26 Nov 2024 14:21:54 +1100
Subject: [PATCH 06/27] Updated NEWS

---
 NEWS.md | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/NEWS.md b/NEWS.md
index 96f07aee0..f78603c09 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -7,6 +7,10 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+### Added
+
+- Added Xiao-Gimbutas quadratures for simplices. Since PR[#1058](https://github.com/gridap/Gridap.jl/pull/1058).
+
 ### Fixed
 
 - Fixed #974, an error when weak form is real but unknown vector is complex. Since PR[#1050](https://github.com/gridap/Gridap.jl/pull/1050).

From c4ab198fd50690620cd337f77dd8033fea590f13 Mon Sep 17 00:00:00 2001
From: Antoine Marteau <antoine.marteau@protonmail.com>
Date: Wed, 27 Nov 2024 15:38:32 +1100
Subject: [PATCH 07/27] optimized MonomialBases evaluations

added associated benchmark
---
 NEWS.md                           |   3 +
 benchmark/Project.toml            |   1 +
 benchmark/benchmarks.jl           |   1 +
 benchmark/bm/bm_monomial_basis.jl | 190 ++++++++++++++++++++++++++++++
 src/Polynomials/ModalC0Bases.jl   |  49 ++++----
 src/Polynomials/MonomialBases.jl  |  92 +++++++++------
 6 files changed, 279 insertions(+), 57 deletions(-)
 create mode 100644 benchmark/bm/bm_monomial_basis.jl

diff --git a/NEWS.md b/NEWS.md
index f78603c09..db7cb91dd 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -16,6 +16,9 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - Fixed #974, an error when weak form is real but unknown vector is complex. Since PR[#1050](https://github.com/gridap/Gridap.jl/pull/1050).
 - Fixed issue where barycentric refinement rule in 3D would not produce oriented meshes. Since PR[#1055](https://github.com/gridap/Gridap.jl/pull/1055).
 
+### Changed
+- Optimized MonomialBasis low-level functions. Since PR[#1059](https://github.com/gridap/Gridap.jl/pull/1059).
+
 ## [0.18.7] - 2024-10-8
 
 ### Added
diff --git a/benchmark/Project.toml b/benchmark/Project.toml
index 03356cbf5..132ed8442 100644
--- a/benchmark/Project.toml
+++ b/benchmark/Project.toml
@@ -2,3 +2,4 @@
 BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
 Gridap = "56d4f2e9-7ea1-5844-9cf6-b9c51ca7ce8e"
 PkgBenchmark = "32113eaa-f34f-5b0d-bd6c-c81e245fc73d"
+StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
diff --git a/benchmark/benchmarks.jl b/benchmark/benchmarks.jl
index ad8aa0aa6..fa037dd8a 100644
--- a/benchmark/benchmarks.jl
+++ b/benchmark/benchmarks.jl
@@ -12,3 +12,4 @@ end
 const SUITE = BenchmarkGroup()
 
 @include_bm SUITE "bm_assembly"
+@include_bm SUITE "bm_monomial_basis"
diff --git a/benchmark/bm/bm_monomial_basis.jl b/benchmark/bm/bm_monomial_basis.jl
new file mode 100644
index 000000000..ba7094ba4
--- /dev/null
+++ b/benchmark/bm/bm_monomial_basis.jl
@@ -0,0 +1,190 @@
+module bm_monomial_basis
+
+using PkgBenchmark, BenchmarkTools
+using Gridap
+using Gridap.Polynomials
+using Gridap.TensorValues
+using StaticArrays
+
+################################################
+# src/Polynomials/MonomialBasis.jl: _set_value_!
+################################################
+
+gradient_type = Gridap.Fields.gradient_type
+
+_set_value! = Gridap.Polynomials._set_value!
+
+function set_value_driver(f,T,D,x,n)
+  k = 1
+  s = one(T)
+  for i in 1:n
+    k = f(x,s,k)
+  end
+end
+
+function set_value_benchmarkable(D, T, V, n)
+  C = num_indep_components(V)
+  x = zeros(V,n*C)
+  return @benchmarkable set_value_driver($_set_value!,$T,$D,$x,$n)
+end
+
+##################################################
+# src/Polynomials/ModalC0Bases.jl: _set_value_mc0!
+##################################################
+
+_set_value_mc0! = Gridap.Polynomials._set_value_mc0!
+
+function set_value_mc0_driver(f,T,D,x,n)
+  k = 1
+  s = one(T)
+  for i in 1:n
+    k = f(x,s,k,2)
+  end
+end
+
+function set_value_mc0_benchmarkable(D, T, V, n)
+  C = num_indep_components(V)
+  x = zeros(V,2*n*C)
+  return @benchmarkable set_value_mc0_driver($_set_value_mc0!,$T,$D,$x,$n)
+end
+
+###################################################
+# src/Polynomials/MonomialBasis.jl: _set_gradient!
+###################################################
+
+ _set_gradient! = Gridap.Polynomials. _set_gradient!
+
+function set_gradient_driver(f,T,D,V,x,n)
+  k = 1
+  s = VectorValue{D,T}(ntuple(_->one(T),D))
+  for i in 1:n
+    k = f(x,s,k,V)
+  end
+end
+
+function set_gradient_benchmarkable(D, T, V, n)
+  C = num_indep_components(V)
+  G = gradient_type(V, zero(Point{D,T}))
+  x = zeros(G,n*C);
+  return @benchmarkable set_gradient_driver($_set_gradient!,$T,$D,$V,$x,$n)
+end
+
+#####################################################
+# src/Polynomials/ModalC0Bases.jl: _set_gradient_mc0!
+#####################################################
+
+ _set_gradient_mc0! = Gridap.Polynomials. _set_gradient_mc0!
+
+function set_gradient_mc0_driver(f,T,D,V,x,n)
+  k = 1
+  s = VectorValue{D,T}(ntuple(_->one(T),D))
+  for i in 1:n
+    k = f(x,s,k,1,V)
+  end
+end
+
+function set_gradient_mc0_benchmarkable(D, T, V, n)
+  C = num_indep_components(V)
+  G = gradient_type(V, zero(Point{D,T}))
+  x = zeros(G,n*C);
+  return @benchmarkable set_gradient_mc0_driver($_set_gradient_mc0!,$T,$D,$V,$x,$n)
+end
+
+#################################################
+# src/Polynomials/MonomialBasis.jl: _evaluate_1d!
+#################################################
+
+_evaluate_1d! = Gridap.Polynomials._evaluate_1d!
+
+function evaluate_1d_driver(f,order,D,v,x_vec)
+  for x in x_vec
+    f(v,x,order,D)
+  end
+end
+
+function evaluate_1d_benchmarkable(D, T, V, n)
+  n = Integer(n/50)
+  order = num_indep_components(V)
+  v = zeros(D,order+1);
+  x = rand(MVector{n,T})
+  return @benchmarkable evaluate_1d_driver($_evaluate_1d!,$order,$D,$v,$x)
+end
+
+################################################
+# src/Polynomials/MonomialBasis.jl:_gradient_1d!
+################################################
+
+_gradient_1d! = Gridap.Polynomials._gradient_1d!
+
+function gradient_1d_driver(f,order,D,v,x_vec)
+  for x in x_vec
+    f(v,x,order,D)
+  end
+end
+
+function gradient_1d_benchmarkable(D, T, V, n)
+  n = Integer(n/10)
+  order = num_indep_components(V)
+  v = zeros(D,order+1);
+  x = rand(MVector{n,T})
+  return @benchmarkable gradient_1d_driver($_gradient_1d!,$order,$D,$v,$x)
+end
+
+################################################
+# src/Polynomials/MonomialBasis.jl:_hessian_1d!
+################################################
+
+_hessian_1d! = Gridap.Polynomials._hessian_1d!
+
+function hessian_1d_driver(f,order,D,v,x_vec)
+  for x in x_vec
+    f(v,x,order,D)
+  end
+end
+
+function hessian_1d_benchmarkable(D, T, V, n)
+  n = Integer(n/10)
+  order = num_indep_components(V)
+  v = zeros(D,order+1);
+  x = rand(MVector{n,T})
+  return @benchmarkable hessian_1d_driver($_hessian_1d!,$order,$D,$v,$x)
+end
+
+#####################
+# benchmarkable suite
+#####################
+
+const SUITE = BenchmarkGroup()
+
+const benchmarkables = (
+  set_value_benchmarkable,
+  set_value_mc0_benchmarkable,
+  set_gradient_benchmarkable,
+  set_gradient_mc0_benchmarkable,
+  evaluate_1d_benchmarkable,
+  gradient_1d_benchmarkable,
+  hessian_1d_benchmarkable
+)
+
+const dims=(1, 2, 3, 5, 8)
+const n = 3000
+const T = Float64
+
+for benchable in benchmarkables
+  for D in dims
+    TV = [
+      VectorValue{D,T},
+      TensorValue{D,D,T,D*D},
+      SymTensorValue{D,T,Integer(D*(D+1)/2)},
+      SymTracelessTensorValue{D,T,Integer(D*(D+1)/2)}
+    ]
+
+    for V in TV
+      if V == SymTracelessTensorValue{1,T,1} continue end # no dofs
+      name = "monomial_basis_$(D)D_$(V)_$(benchable)"
+      SUITE[name] = benchable(D, T, V, n)
+    end
+  end
+end
+
+end # module
diff --git a/src/Polynomials/ModalC0Bases.jl b/src/Polynomials/ModalC0Bases.jl
index b98b6515f..3fbe918d2 100644
--- a/src/Polynomials/ModalC0Bases.jl
+++ b/src/Polynomials/ModalC0Bases.jl
@@ -393,16 +393,10 @@ end
 
 @inline function _set_value_mc0!(v::AbstractVector{V},s::T,k,l) where {V,T}
   ncomp = num_indep_components(V)
-  m = zero(MVector{ncomp,T})
   z = zero(T)
-  js = 1:ncomp
-  for j in js
-    for i in js
-      @inbounds m[i] = z
-    end
-    @inbounds m[j] = s
-    i = k+l*(j-1)
-    @inbounds v[i] = Tuple(m)
+  for j in 1:ncomp
+    m = k+l*(j-1)
+    @inbounds v[m] = ntuple(i -> ifelse(i == j, s, z),Val(ncomp))
   end
   k+1
 end
@@ -466,25 +460,36 @@ end
 # Indexing and m definition should be fixed if G contains symmetries, that is
 # if the code is  optimized for symmetric tensor V valued FESpaces
 # (if gradient_type(V) returned a symmetric higher order tensor type G)
-@inline function _set_gradient_mc0!(
+@inline @generated function _set_gradient_mc0!(
   v::AbstractVector{G},s,k,l,::Type{V}) where {V,G}
+  # Git blame me for readable non-generated version
   @notimplementedif num_indep_components(G) != num_components(G) "Not implemented for symmetric Jacobian or Hessian"
-
-  T = eltype(s)
-  m = zero(Mutable(G))
-  w = zero(V)
-  z = zero(T)
-  for (ij,j) in enumerate(CartesianIndices(w))
+  
+  m = Array{String}(undef, size(G))
+  N_val_dims = length(size(V))
+  s_size = size(G)[1:end-N_val_dims]
+
+  body = "T = eltype(s); z = zero(T);"
+  for ci in CartesianIndices(s_size)
+    id = join(Tuple(ci))
+    body *= "@inbounds s$id = s[$ci];"
+  end
+  
+  V_size = size(V)
+  for (ij,j) in enumerate(CartesianIndices(V_size))
     for i in CartesianIndices(m)
-      @inbounds m[i] = z
+      m[i] = "z"
     end
-    for i in CartesianIndices(s)
-      @inbounds m[i,j] = s[i]
+    for ci in CartesianIndices(s_size)
+      id = join(Tuple(ci))
+      m[ci,j] = "s$id"
     end
-    i = k+l*(ij-1)
-    @inbounds v[i] = m
+    body *= "i = k + l*($ij-1);"
+    body *= "@inbounds v[i] = ($(join(tuple(m...), ", ")));"
   end
-  k+1
+
+  body = Meta.parse(string("begin ",body," end"))
+  return Expr(:block, body ,:(return k+1))
 end
 
 function _hessian_nd_mc0!(
diff --git a/src/Polynomials/MonomialBases.jl b/src/Polynomials/MonomialBases.jl
index 25ccd9a36..f1452f34b 100644
--- a/src/Polynomials/MonomialBases.jl
+++ b/src/Polynomials/MonomialBases.jl
@@ -351,8 +351,11 @@ function _evaluate_1d!(v::AbstractMatrix{T},x,order,d) where T
   n = order + 1
   z = one(T)
   @inbounds v[d,1] = z
+  @inbounds xd = x[d]
+  xn = xd
   for i in 2:n
-    @inbounds v[d,i] = x[d]^(i-1)
+    @inbounds v[d,i] = xn
+    xn *= xd
   end
 end
 
@@ -360,8 +363,11 @@ function _gradient_1d!(v::AbstractMatrix{T},x,order,d) where T
   n = order + 1
   z = zero(T)
   @inbounds v[d,1] = z
+  @inbounds xd = x[d]
+  xn = one(T)
   for i in 2:n
-    @inbounds v[d,i] = (i-1)*x[d]^(i-2)
+    @inbounds v[d,i] = (i-1)*xn
+    xn *= xd
   end
 end
 
@@ -372,8 +378,11 @@ function _hessian_1d!(v::AbstractMatrix{T},x,order,d) where T
   if n>1
     @inbounds v[d,2] = z
   end
+  @inbounds xd = x[d]
+  xn = one(T)
   for i in 3:n
-    @inbounds v[d,i] = (i-1)*(i-2)*x[d]^(i-3)
+    @inbounds v[d,i] = (i-1)*(i-2)*xn
+    xn *= xd
   end
 end
 
@@ -406,16 +415,10 @@ function _evaluate_nd!(
 end
 
 function _set_value!(v::AbstractVector{V},s::T,k) where {V,T}
-  ncomp::Int = num_indep_components(V)
-  m = zero(MVector{ncomp,T})
+  ncomp = num_indep_components(V)
   z = zero(T)
-  js = SOneTo(ncomp)#1:ncomp
-  for j in js
-    for i in js
-      @inbounds m[i] = z
-    end
-    m[j] = s
-    v[k] = Tuple(m)
+  @inbounds for j in 1:ncomp
+    v[k] = ntuple(i -> ifelse(i == j, s, z),Val(ncomp))
     k += 1
   end
   k
@@ -474,58 +477,77 @@ function _set_gradient!(
   k+1
 end
 
-function _set_gradient!(
+@generated function  _set_gradient!(
   v::AbstractVector{G},s,k,::Type{V}) where {V,G}
+  # Git blame me for readable non-generated version
 
-  T = eltype(s)
-  m = zero(Mutable(G))
   w = zero(V)
-  z = zero(T)
+  m = Array{String}(undef, size(G))
+  N_val_dims = length(size(V))
+  s_size = size(G)[1:end-N_val_dims]
+
+  body = "T = eltype(s); z = zero(T);"
+  for ci in CartesianIndices(s_size)
+    id = join(Tuple(ci))
+    body *= "@inbounds s$id = s[$ci];"
+  end
+
   for j in CartesianIndices(w)
     for i in CartesianIndices(m)
-     @inbounds m[i] = z
+      m[i] = "z"
     end
-    for i in CartesianIndices(s)
-      @inbounds m[i,j] = s[i]
+    for ci in CartesianIndices(s_size)
+      id = join(Tuple(ci))
+      m[ci,j] = "s$id"
     end
-    @inbounds v[k] = m
-    k += 1
+    body *= "@inbounds v[k] = ($(join(tuple(m...), ", ")));"
+    body *= "k = k + 1;"
   end
-  k
+
+  body = Meta.parse(string("begin ",body," end"))
+  return Expr(:block, body ,:(return k))
 end
 
 # Specialization for SymTensorValue and SymTracelessTensorValue,
 # necessary as long as outer(Point, V<:AbstractSymTensorValue)::G does not
 # return a tensor type that implements the appropriate symmetries of the
 # gradient (and hessian)
-function _set_gradient!(
+@generated function _set_gradient!(
   v::AbstractVector{G},s,k,::Type{V}) where {V<:AbstractSymTensorValue{D},G} where D
-
+  # Git blame me for readable non-generated version
+  
   T = eltype(s)
-  m = zero(Mutable(G))
-  z = zero(T)
+  m = Array{String}(undef, size(G))
+  s_length = size(G)[1]
 
   is_traceless = V <: SymTracelessTensorValue
   skip_last_diagval = is_traceless ? 1 : 0    # Skid V_DD if traceless
 
+  body = "z = $(zero(T));"
+  for i in 1:s_length 
+    body *= "@inbounds s$i = s[$i];"
+  end
+  
   for c in 1:(D-skip_last_diagval) # Go over cols
     for r in c:D                   # Go over lower triangle, current col
-      for i in CartesianIndices(m)
-        @inbounds m[i] = z
+      for i in eachindex(m)
+        m[i] = "z"
       end
-      for i in CartesianIndices(s)
-        @inbounds m[i,r,c] = s[i]
+      for i in 1:s_length # indices of the Vector s
+        m[i,r,c] = "s$i"
         if (r!=c)
-          @inbounds m[i,c,r] = s[i]
+          m[i,c,r] = "s$i"
         elseif is_traceless # V_rr contributes negatively to V_DD (tracelessness)
-          @inbounds m[i,D,D] = -s[i]
+          m[i,D,D] = "-s$i"
         end
       end
-      @inbounds v[k] = m
-      k += 1
+      body *= "@inbounds v[k] = ($(join(tuple(m...), ", ")));"
+      body *= "k = k + 1;"
     end
   end
-  k
+
+  body = Meta.parse(string("begin ",body," end"))
+  return Expr(:block, body ,:(return k))
 end
 
 function _hessian_nd!(

From 9e6bb7ff4ff247108e898b12789dc4bd7e9a2157 Mon Sep 17 00:00:00 2001
From: Antoine Marteau <antoine.marteau@protonmail.com>
Date: Thu, 28 Nov 2024 15:47:09 +1100
Subject: [PATCH 08/27] Improve `Gridap.TensorValues`' docstring

---
 src/TensorValues/TensorValues.jl | 35 +++++++++++++++++++++++++++++++-
 1 file changed, 34 insertions(+), 1 deletion(-)

diff --git a/src/TensorValues/TensorValues.jl b/src/TensorValues/TensorValues.jl
index 5985195dc..5d491f997 100644
--- a/src/TensorValues/TensorValues.jl
+++ b/src/TensorValues/TensorValues.jl
@@ -1,5 +1,38 @@
 """
-Immutable tensor types for Gridap.
+Immutable tensor types for Gridap. The currently implemented tensor types are
+- 1st order [`VectorValue`](@ref),
+- 2nd order [`TensorValue`](@ref),
+- 2nd order and symmetric [`SymTensorValue`](@ref),
+- 2nd order, symmetric and traceless [`SymTracelessTensorValue`](@ref),
+- 3rd order [`ThirdOrderTensorValue`](@ref),
+- 4th order and symmetric [`SymFourthOrderTensorValue`](@ref).
+
+Example usage:
+```julia
+# create a 2D vector from components
+v = VectorValue(12,31)
+
+# Assign a VectorValue to all the entries of an Array of VectorValues
+A = zeros(VectorValue{2,Int}, (4,5))
+A .= v # This is possible since  VectorValue <: Number
+
+using StaticArrays
+# create 2x2 tensor from component tuple
+t = TensorValue( (1, 2, 3, 4) )
+# conversion to StaticArrays type
+ts= convert(SMatrix{2,2,Int}, t)
+@show ts
+# 2×2 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
+#  1  3
+#  2  4
+t2[1,2] == t[1,2] == 3 # true
+
+# conversion from Array or StaticArray types, symmetric tensor types only store required components
+SymTensorValue( [1 2; 3 4] )          # SymTensorValue{2, Int64, 3}(1, 2, 4)
+SymTensorValue( SMatrix{2}(1,2,3,4) ) # SymTensorValue{2, Int64, 3}(1, 3, 4)
+```
+
+See the official documentation for more details.
 """
 module TensorValues
 

From 67320c416ca17f8c2c8889178194c0b187a71ec1 Mon Sep 17 00:00:00 2001
From: "dependabot[bot]" <49699333+dependabot[bot]@users.noreply.github.com>
Date: Sun, 1 Dec 2024 23:51:11 +0000
Subject: [PATCH 09/27] Bump codecov/codecov-action from 4 to 5

Bumps [codecov/codecov-action](https://github.com/codecov/codecov-action) from 4 to 5.
- [Release notes](https://github.com/codecov/codecov-action/releases)
- [Changelog](https://github.com/codecov/codecov-action/blob/main/CHANGELOG.md)
- [Commits](https://github.com/codecov/codecov-action/compare/v4...v5)

---
updated-dependencies:
- dependency-name: codecov/codecov-action
  dependency-type: direct:production
  update-type: version-update:semver-major
...

Signed-off-by: dependabot[bot] <support@github.com>
---
 .github/workflows/ci.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index d609848c8..294cceaea 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -42,7 +42,7 @@ jobs:
       - uses: julia-actions/julia-buildpkg@v1
       - uses: julia-actions/julia-runtest@v1
       - uses: julia-actions/julia-processcoverage@v1
-      - uses: codecov/codecov-action@v4
+      - uses: codecov/codecov-action@v5
         with:
           file: lcov.info
           verbose: true

From ff4eb470bd009f552531b3958be1d00208d619cc Mon Sep 17 00:00:00 2001
From: Jordi Manyer Fuertes <jordi.manyer@monash.edu>
Date: Mon, 2 Dec 2024 14:05:09 +1100
Subject: [PATCH 10/27] Update NEWS.md

---
 NEWS.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/NEWS.md b/NEWS.md
index eb476063c..01e873efb 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,7 +5,7 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
-## [Unreleased]
+## [0.18.18] - 2024-12-2
 
 ### Added
 

From 4da8586e1d59cc1cbc75e8b27c555dc84e4d7b42 Mon Sep 17 00:00:00 2001
From: Jordi Manyer Fuertes <jordi.manyer@monash.edu>
Date: Mon, 2 Dec 2024 14:05:36 +1100
Subject: [PATCH 11/27] Bump version to 0.18.8

---
 Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project.toml b/Project.toml
index 14540d3da..e54caab82 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "Gridap"
 uuid = "56d4f2e9-7ea1-5844-9cf6-b9c51ca7ce8e"
 authors = ["Santiago Badia <santiago.badia@monash.edu>", "Francesc Verdugo <f.verdugo.rojano@vu.nl>", "Alberto F. Martin <alberto.f.martin@anu.edu.au>"]
-version = "0.18.7"
+version = "0.18.8"
 
 [deps]
 AbstractTrees = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"

From b05bebc7f26792935485509e03d1b9ad3baf9238 Mon Sep 17 00:00:00 2001
From: Jordi Manyer Fuertes <jordi.manyer@monash.edu>
Date: Mon, 2 Dec 2024 14:05:50 +1100
Subject: [PATCH 12/27] Update NEWS.md

---
 NEWS.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/NEWS.md b/NEWS.md
index 01e873efb..7cbffd2b2 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,7 +5,7 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
-## [0.18.18] - 2024-12-2
+## [0.18.8] - 2024-12-2
 
 ### Added
 

From b09aa85e5b4750aaa44261dfab98def15083709a Mon Sep 17 00:00:00 2001
From: janmodderman <janmodderman@gmail.com>
Date: Mon, 2 Dec 2024 10:21:14 +0100
Subject: [PATCH 13/27] add get_dof_value_type to FESpacesWithLinearConstraints

---
 src/FESpaces/FESpacesWithLinearConstraints.jl | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/FESpaces/FESpacesWithLinearConstraints.jl b/src/FESpaces/FESpacesWithLinearConstraints.jl
index d89a98926..a1a5771fd 100644
--- a/src/FESpaces/FESpacesWithLinearConstraints.jl
+++ b/src/FESpaces/FESpacesWithLinearConstraints.jl
@@ -297,6 +297,8 @@ function get_fe_dof_basis(f::FESpaceWithLinearConstraints)
   get_fe_dof_basis(f.space)
 end
 
+get_dof_value_type(f::FESpaceWithLinearConstraints) = get_dof_value_type(f.space)
+
 get_dirichlet_dof_ids(f::FESpaceWithLinearConstraints) = Base.OneTo(length(f.mDOF_to_DOF) - f.n_fmdofs)
 
 num_dirichlet_tags(f::FESpaceWithLinearConstraints) = num_dirichlet_tags(f.space)

From 7b5c91756977fd0e6f0742076b36e1c9ef972843 Mon Sep 17 00:00:00 2001
From: janmodderman <janmodderman@gmail.com>
Date: Mon, 2 Dec 2024 14:26:43 +0100
Subject: [PATCH 14/27] added reproduction file

---
 test/repro.jl | 28 ++++++++++++++++++++++++++++
 1 file changed, 28 insertions(+)
 create mode 100644 test/repro.jl

diff --git a/test/repro.jl b/test/repro.jl
new file mode 100644
index 000000000..5b2822099
--- /dev/null
+++ b/test/repro.jl
@@ -0,0 +1,28 @@
+module reproduction_file
+using Gridap
+using Gridap.Geometry
+using Gridap.TensorValues
+using Gridap.Fields
+using GridapEmbedded
+using Gridap.Arrays
+using Gridap.FESpaces
+
+model = CartesianDiscreteModel(Point(0.0, 0.0), Point(1.0, 1.0), (5,5))
+Ω = Interior(model)
+dΩ = Measure(Ω,2)
+geo = disk(0.1, x0=Point(0.5,0.5))
+cutgeo = cut(model, !geo)
+
+W1 = FESpace(Ω, ReferenceFE(lagrangian, Float64, 1), vector_type=Vector{ComplexF64})
+@show get_dof_value_type(W1)
+
+W2 = AgFEMSpace(W1,aggregate(AggregateCutCellsByThreshold(1.0), cutgeo, geo, OUT))  # AgFEMSpace returns a FESpaceWithLinearConstraints
+@show get_dof_value_type(W2)
+Φ = TrialFESpace(W2)
+@show get_dof_value_type(Φ)
+
+a_wϕ(ϕ, w) = ∫( im*∇(ϕ)⋅∇(w) )dΩ        # if we do NOT include the imaginary operator im here, then the code will pass, but the FE Spaces will be of type Float64
+A_wϕ1 = assemble_matrix(a_wϕ, Φ, W1)    # function will NOT fail here, due to get_dof_value_type function being implemented
+A_wϕ2 = assemble_matrix(a_wϕ, Φ, W2)    # function will fail here, because W2 should be ComplexF64, but is Float64 due to missing get_dof_value_type function
+
+end # module
\ No newline at end of file

From 7a8e05dd21dd01613674989d863b956a52cc8b97 Mon Sep 17 00:00:00 2001
From: janmodderman <janmodderman@gmail.com>
Date: Mon, 2 Dec 2024 16:36:58 +0100
Subject: [PATCH 15/27] clean up for PR

---
 NEWS.md                                       |  1 +
 .../FESpacesWithLinearConstraintsTests.jl     | 11 ++++++++
 test/repro.jl                                 | 28 -------------------
 3 files changed, 12 insertions(+), 28 deletions(-)
 delete mode 100644 test/repro.jl

diff --git a/NEWS.md b/NEWS.md
index 7cbffd2b2..e9287dbdc 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -9,6 +9,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ### Added
 
+- Added get_dof_value_type for FESpacesWithLinearConstraints. Since PR[#1062](https://github.com/gridap/Gridap.jl/pull/1062).
 - Added Xiao-Gimbutas quadratures for simplices. Since PR[#1058](https://github.com/gridap/Gridap.jl/pull/1058).
 - Small improvements of the documentation of `Gridap.TensorValues`. Since PR[#1051](https://github.com/gridap/Gridap.jl/pull/1051).
 
diff --git a/test/FESpacesTests/FESpacesWithLinearConstraintsTests.jl b/test/FESpacesTests/FESpacesWithLinearConstraintsTests.jl
index 662e5f72b..fc5f6222d 100644
--- a/test/FESpacesTests/FESpacesWithLinearConstraintsTests.jl
+++ b/test/FESpacesTests/FESpacesWithLinearConstraintsTests.jl
@@ -90,4 +90,15 @@ tol = 1.e-9
 @test e_l2 < tol
 @test e_h1 < tol
 
+V2 = FESpace(
+  model,ReferenceFE(lagrangian,Float64,1), conformity=:H1, dirichlet_tags="dirichlet", vector_type=ComplexF64)
+
+Vc2 = FESpaceWithLinearConstraints(
+  sDOF_to_dof,
+  sDOF_to_dofs,
+  sDOF_to_coeffs,
+  V2)
+
+@test get_dof_value_type(Vc2) <: ComplexF64
+
 end # module
diff --git a/test/repro.jl b/test/repro.jl
deleted file mode 100644
index 5b2822099..000000000
--- a/test/repro.jl
+++ /dev/null
@@ -1,28 +0,0 @@
-module reproduction_file
-using Gridap
-using Gridap.Geometry
-using Gridap.TensorValues
-using Gridap.Fields
-using GridapEmbedded
-using Gridap.Arrays
-using Gridap.FESpaces
-
-model = CartesianDiscreteModel(Point(0.0, 0.0), Point(1.0, 1.0), (5,5))
-Ω = Interior(model)
-dΩ = Measure(Ω,2)
-geo = disk(0.1, x0=Point(0.5,0.5))
-cutgeo = cut(model, !geo)
-
-W1 = FESpace(Ω, ReferenceFE(lagrangian, Float64, 1), vector_type=Vector{ComplexF64})
-@show get_dof_value_type(W1)
-
-W2 = AgFEMSpace(W1,aggregate(AggregateCutCellsByThreshold(1.0), cutgeo, geo, OUT))  # AgFEMSpace returns a FESpaceWithLinearConstraints
-@show get_dof_value_type(W2)
-Φ = TrialFESpace(W2)
-@show get_dof_value_type(Φ)
-
-a_wϕ(ϕ, w) = ∫( im*∇(ϕ)⋅∇(w) )dΩ        # if we do NOT include the imaginary operator im here, then the code will pass, but the FE Spaces will be of type Float64
-A_wϕ1 = assemble_matrix(a_wϕ, Φ, W1)    # function will NOT fail here, due to get_dof_value_type function being implemented
-A_wϕ2 = assemble_matrix(a_wϕ, Φ, W2)    # function will fail here, because W2 should be ComplexF64, but is Float64 due to missing get_dof_value_type function
-
-end # module
\ No newline at end of file

From c7eeb9477667f3f5a88538c57aff58e5802c9e30 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Tue, 3 Dec 2024 09:45:29 +1100
Subject: [PATCH 16/27] Some optimisations to NVB

---
 src/Adaptivity/EdgeBasedRefinement.jl | 69 +++++++++++++++------------
 test/AdaptivityTests/estimators.jl    | 18 +++++++
 2 files changed, 56 insertions(+), 31 deletions(-)
 create mode 100644 test/AdaptivityTests/estimators.jl

diff --git a/src/Adaptivity/EdgeBasedRefinement.jl b/src/Adaptivity/EdgeBasedRefinement.jl
index 724f74fd4..fc50d45ea 100644
--- a/src/Adaptivity/EdgeBasedRefinement.jl
+++ b/src/Adaptivity/EdgeBasedRefinement.jl
@@ -78,7 +78,6 @@ function refine_edge_based_topology(
   c2n_map_new = get_refined_cell_to_vertex_map(topo,rrules,faces_list)
   polys_new, cell_type_new = _get_cell_polytopes(rrules)
   orientation = NonOriented()
-
   return UnstructuredGridTopology(coords_new,c2n_map_new,cell_type_new,polys_new,orientation)
 end
 
@@ -109,17 +108,28 @@ The new vertices are ordered by parent dimension and face id (in that order).
 function get_new_coordinates_from_faces(p::Union{Polytope{D},GridTopology{D}},faces_list::Tuple) where {D}
   @check length(faces_list) == D+1
 
-  nN_new     = sum(x->length(x),faces_list)
+  nN_new     = sum(length,faces_list)
   coords_old = get_vertex_coordinates(p)
   coords_new = Vector{eltype(coords_old)}(undef,nN_new)
 
   n = 1
-  for (d,dfaces) in enumerate(faces_list)
-    if length(dfaces) > 0
-      nf = length(dfaces)
-      d2n_map = get_faces(p,d-1,0)
-      coords_new[n:n+nf-1] .= map(f -> sum(coords_old[d2n_map[f]])/length(d2n_map[f]), dfaces)
-      n += nf
+  # Nodes
+  if !isempty(faces_list[1])
+    for node in faces_list[1]
+      coords_new[n] = coords_old[node]
+      n += 1
+    end
+  end
+  # Faces (d > 0)
+  for (d,dfaces) in enumerate(faces_list[2:end])
+    if !isempty(dfaces)
+      d2n_map = get_faces(p,d,0)
+      cache = array_cache(d2n_map)
+      for face in dfaces
+        face_nodes = getindex!(cache,d2n_map,face)
+        coords_new[n] = sum(coords_old[face_nodes])/length(face_nodes)
+        n += 1
+      end
     end
   end
 
@@ -255,12 +265,13 @@ function setup_edge_based_rrules(method::NVBRefinement, topo::UnstructuredGridTo
   setup_edge_based_rrules(method, topo, collect(1:num_faces(topo,Dc)))
 end
 
-function setup_edge_based_rrules(method::NVBRefinement, topo::UnstructuredGridTopology{Dc},cells_to_refine::AbstractArray{<:Integer}) where Dc
+function setup_edge_based_rrules(
+  method::NVBRefinement, topo::UnstructuredGridTopology{Dc}, cells_to_refine::AbstractArray{<:Integer}
+) where Dc
   nE = num_faces(topo,1)
-  c2e_map       = get_faces(topo,Dc,1)
-  c2e_map_cache = array_cache(c2e_map)
-  e2c_map       = get_faces(topo,1,Dc)
-  polys       = topo.polytopes
+  c2e_map = get_faces(topo,Dc,1)
+  e2c_map = get_faces(topo,1,Dc)
+  polys   = topo.polytopes
   cell_types  = topo.cell_type
   cell_color  = copy(cell_types) # WHITE
   # Hardcoded for TRI
@@ -274,43 +285,39 @@ function setup_edge_based_rrules(method::NVBRefinement, topo::UnstructuredGridTo
   c_to_longest_edge_lid = method.cell_to_longest_edge_lid
   is_refined = falses(nE)
   # Loop over cells and mark edges to refine i.e. is_refined
-  # The reason to not loop directly on c is that we need to change c within
-  # a given iteration of the for loop
-  for i in 1:length(cells_to_refine)
-    c = cells_to_refine[i]
-    e_longest = c_to_longest_edge_gid[c]
+  for i in eachindex(cells_to_refine)
     # Has to terminate because an edge is marked each iteration or we skip an
     # iteration due to a boundary cell
-    while !is_refined[e_longest]
-      is_refined[e_longest] = true
-      c_nbor_lid = findfirst(c′ -> c′ != c, e2c_map[e_longest])
-      if isnothing(c_nbor_lid) # We've reach the boundary
+    c = cells_to_refine[i]
+    e = c_to_longest_edge_gid[c]
+    while !is_refined[e]
+      is_refined[e] = true
+      e_cells = view(e2c_map,e)
+      if length(e_cells) == 1 # We've reach the boundary
         continue
       else
-        # Get the longest edge of the neighbor
-        c_nbor_gid = e2c_map[e_longest][c_nbor_lid]
-        e_longest = c_to_longest_edge_gid[c_nbor_gid]
-        # Set the current cell gid to that of the neighbor
-        c = c_nbor_gid
+        # Propagate to neighboring cell
+        c = ifelse(e_cells[1] == c, e_cells[2], e_cells[1])
+        e = c_to_longest_edge_gid[c]
       end
     end
   end
   # Loop over cells and refine based on marked edges
   for c in 1:length(c2e_map)
-    c_edges = getindex!(c2e_map_cache, c2e_map, c)
+    c_edges = view(c2e_map,c)
     refined_edge_lids = findall(is_refined[c_edges])
-    # GREEN refinement because only one edge should be bisected
     if length(refined_edge_lids) == 1
+      # GREEN refinement because only one edge should be bisected
       ref_edge = refined_edge_lids[1]
       cell_color[c] = GREEN + Int8(ref_edge-1)
-      # BLUE refinement: two bisected
     elseif length(refined_edge_lids) == 2
+      # BLUE refinement: two bisected
       long_ref_edge_lid = c_to_longest_edge_lid[c]
       short_ref_edge_lid = setdiff(refined_edge_lids, long_ref_edge_lid)[1]
       blue_idx = BLUE_dict[(long_ref_edge_lid, short_ref_edge_lid)]
       cell_color[c] = BLUE + Int8(blue_idx - 1)
-      # DOUBLE BLUE refinement: three bisected edges (somewhat rare)
     elseif length(refined_edge_lids) == 3
+      # DOUBLE BLUE refinement: three bisected edges (somewhat rare)
       long_ref_edge_lid = c_to_longest_edge_lid[c]
       cell_color[c] = BLUE_DOUBLE + Int(long_ref_edge_lid - 1)
     end
diff --git a/test/AdaptivityTests/estimators.jl b/test/AdaptivityTests/estimators.jl
new file mode 100644
index 000000000..f15a605d3
--- /dev/null
+++ b/test/AdaptivityTests/estimators.jl
@@ -0,0 +1,18 @@
+
+using Gridap, Gridap.Geometry, Gridap.Adaptivity
+
+function LShapedModel(n)
+  model = CartesianDiscreteModel((0,1,0,1),(n,n))
+  cell_coords = map(mean,get_cell_coordinates(model))
+  l_shape_filter(x) = (x[1] < 0.5) || (x[2] < 0.5)
+  mask = map(l_shape_filter,cell_coords)
+  return simplexify(DiscreteModelPortion(model,mask))
+end
+
+model = LShapedModel(10)
+
+method = Adaptivity.NVBRefinement(model)
+cells_to_refine = [collect(1:10)...,collect(20:30)...]
+fmodel = refine(method,model;cells_to_refine)
+
+writevtk(Triangulation(fmodel),"tmp/fmodel";append=false)

From c73e24f0549ff53220dbc97a04346d9ccb51ad46 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 16:15:28 +1100
Subject: [PATCH 17/27] Added DorflerMarking

---
 src/Adaptivity/AdaptiveMeshRefinement.jl      | 97 +++++++++++++++++++
 src/Adaptivity/Adaptivity.jl                  |  3 +
 ...tors.jl => AdaptiveMeshRefinementTests.jl} | 25 ++++-
 3 files changed, 124 insertions(+), 1 deletion(-)
 create mode 100644 src/Adaptivity/AdaptiveMeshRefinement.jl
 rename test/AdaptivityTests/{estimators.jl => AdaptiveMeshRefinementTests.jl} (52%)

diff --git a/src/Adaptivity/AdaptiveMeshRefinement.jl b/src/Adaptivity/AdaptiveMeshRefinement.jl
new file mode 100644
index 000000000..5b57d24d8
--- /dev/null
+++ b/src/Adaptivity/AdaptiveMeshRefinement.jl
@@ -0,0 +1,97 @@
+
+struct DorflerMarking
+  θ :: Float64
+  ν :: Float64
+  strategy :: Symbol
+  function DorflerMarking(
+    θ::Float64;
+    ν::Float64 = 0.5,
+    strategy::Symbol = :sort
+  )
+    @assert 0 < θ < 1
+    @assert strategy ∈ (:sort,:binsort,:quickmark) "Strategy not recognized. Available values are (:sort)"
+    new(θ,ν,strategy)
+  end
+end
+
+mark(m::DorflerMarking, η::Vector{<:Real}) = mark(Val(m.strategy), m, η)
+
+function mark(::Val{:sort}, m::DorflerMarking, η::Vector{<:Real})
+  target = m.θ * sum(η)
+  perm = sortperm(η, rev=true, alg=QuickSort)
+  s = zero(eltype(η))
+  k = 0
+  while s < target
+    k += 1
+    s += η[perm[k]]
+  end
+  return perm[1:k]
+end
+
+function mark(::Val{:binsort}, m::DorflerMarking, η::Vector{<:Real})
+  target = m.θ * sum(η)
+  M = maximum(η)
+  N = length(η)
+  
+  # Find minimal K such that
+  #   νᴷ⁺¹ M ≤ (1 - θ) / (θ * N) * target
+  K = 0
+  a = M*m.ν
+  b = (1 - m.θ) / (m.θ * N * M) * target
+  while a > b
+    K += 1
+    a *= m.ν
+  end
+
+  # Sort into bins Bk such that ηi ∈ Bk if
+  #    νᴷ⁺¹ M ≤ ηi < νᴷ M
+  bins = zeros(Int,N)
+  sums = zeros(Float64,K+1)
+  lbs = zeros(Float64,K+1)
+  lbs[1] = M * m.ν
+  for i in 2:K
+    lbs[i] = lbs[i-1] * m.ν
+  end
+  lbs[end] = 0.0
+
+  for (i,ηi) in enumerate(η)
+    k = 1
+    while ηi < lbs[k]
+      k += 1
+    end
+    bins[i] = k
+    sums[k] += ηi
+  end
+
+  # Find minimal set of bins that gets over target
+  k = 0
+  s = 0.0
+  while s < target
+    k += 1
+    s += sums[k]
+  end
+  
+  return findall(i -> i <= k, bins)
+end
+
+function mark(::Val{:quickmark}, m::DorflerMarking, η::Vector{<:Real})
+  function quickmark!(η, perm, l, u, target) :: Int
+    m = (u - l) ÷ 2
+    sort!(view(perm,l:u), by=i->η[i], rev=true, alg=PartialQuickSort(m))
+
+    p = l + m
+    t = η[perm[p]]
+    σ = sum(η[perm[l:p-1]])
+    
+    (σ >= target) && return quickmark!(η, perm, l, p, target)
+    (σ + t >= target) && return p
+    return quickmark!(η, perm, p + 1, u, target - σ - t)
+  end
+
+  N = length(η)
+  l, u = 1, N
+  perm = collect(1:N)
+  target = m.θ * sum(η)
+  m = quickmark!(η, perm, l, u, target)
+  return perm[1:m]
+end
diff --git a/src/Adaptivity/Adaptivity.jl b/src/Adaptivity/Adaptivity.jl
index d80eb0c3e..5b85e0460 100644
--- a/src/Adaptivity/Adaptivity.jl
+++ b/src/Adaptivity/Adaptivity.jl
@@ -40,6 +40,8 @@ export AdaptedTriangulation
 export Triangulation, is_change_possible, best_target, get_adapted_model
 export change_domain, move_contributions
 
+export DorflerMarking, mark
+
 include("RefinementRules.jl")
 include("FineToCoarseFields.jl")
 include("OldToNewFields.jl")
@@ -51,5 +53,6 @@ include("MacroFEs.jl")
 include("CompositeQuadratures.jl")
 include("EdgeBasedRefinement.jl")
 include("SimplexifyRefinement.jl")
+include("AdaptiveMeshRefinement.jl")
 
 end # module
diff --git a/test/AdaptivityTests/estimators.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
similarity index 52%
rename from test/AdaptivityTests/estimators.jl
rename to test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index f15a605d3..da240a6a7 100644
--- a/test/AdaptivityTests/estimators.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -1,5 +1,27 @@
-
+using Test
 using Gridap, Gridap.Geometry, Gridap.Adaptivity
+using DataStructures
+
+# Marking tests
+
+function test_marking()
+  for strategy in (:sort,:binsort,:quickmark)
+    for n in (1000,10000)
+      for θ in (0.3,0.5)
+        η = abs.(randn(n))
+        m = DorflerMarking(θ;strategy)
+        I = Adaptivity.mark(m,η)
+        @test sum(η[I]) > θ * sum(η)
+        println("Strategy: $strategy, θ: $θ, n: $n")
+        println("  > N marked = $(length(I)), val marked = $(sum(η[I]) / sum(η))")
+      end
+    end
+  end
+end
+
+test_marking()
+
+# AMR tests
 
 function LShapedModel(n)
   model = CartesianDiscreteModel((0,1,0,1),(n,n))
@@ -16,3 +38,4 @@ cells_to_refine = [collect(1:10)...,collect(20:30)...]
 fmodel = refine(method,model;cells_to_refine)
 
 writevtk(Triangulation(fmodel),"tmp/fmodel";append=false)
+

From 31175b04805faa78faff176c68b1da380ab9ec7a Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 16:30:47 +1100
Subject: [PATCH 18/27] Added documentation

---
 src/Adaptivity/AdaptiveMeshRefinement.jl      | 44 ++++++++++++++++++-
 .../AdaptiveMeshRefinementTests.jl            |  4 +-
 2 files changed, 45 insertions(+), 3 deletions(-)

diff --git a/src/Adaptivity/AdaptiveMeshRefinement.jl b/src/Adaptivity/AdaptiveMeshRefinement.jl
index 5b57d24d8..2f5d83f34 100644
--- a/src/Adaptivity/AdaptiveMeshRefinement.jl
+++ b/src/Adaptivity/AdaptiveMeshRefinement.jl
@@ -1,4 +1,46 @@
 
+"""
+  struct DorflerMarking
+    θ :: Float64
+    ν :: Float64
+    strategy :: Symbol
+  end
+
+  DorflerMarking(θ::Float64; ν::Float64 = 0.5, strategy::Symbol = :quickmark)
+
+Implements the Dorfler marking strategy. Given a vector `η` of real positive numbers, 
+the marking strategy find a subset of indices `I` such that 
+
+  sum(η[I]) > θ * sum(η)
+
+where `0 < θ < 1` is a threshold parameter. 
+
+For more details, see the following reference: 
+
+`Dörfler marking with minimal cardinality is a linear complexity problem`, Pfeiler et al. (2020)
+
+The marking algorithm is controlled by the `strategy` parameter, which can take 
+the following values: 
+
+- `:sort`: Optimal cardinality, O(N log N) complexity. See Algorithm 2 in the reference.
+- `:binsort`: Quasi-optimal cardinality, O(N) complexity. See Algorithm 7 in the reference.
+- `:quickmark`: Optimal cardinality, O(N) complexity.  See Algorithm 10 in the reference.
+
+# Arguments
+
+- `θ::Float64`: The threshold parameter. Between 0 and 1.
+- `ν::Float64`: Extra parameter for `:binsort`. Default is 0.5.
+- `strategy::Symbol`: The marking strategy. Default is `:quickmark`.
+
+# Usage
+
+```julia
+η = abs.(randn(1000))
+m = DorflerMarking(0.5)
+I = mark(m,η)
+```
+
+"""
 struct DorflerMarking
   θ :: Float64
   ν :: Float64
@@ -6,7 +48,7 @@ struct DorflerMarking
   function DorflerMarking(
     θ::Float64;
     ν::Float64 = 0.5,
-    strategy::Symbol = :sort
+    strategy::Symbol = :quickmark
   )
     @assert 0 < θ < 1
     @assert strategy ∈ (:sort,:binsort,:quickmark) "Strategy not recognized. Available values are (:sort)"
diff --git a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index da240a6a7..40776de45 100644
--- a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -4,7 +4,7 @@ using DataStructures
 
 # Marking tests
 
-function test_marking()
+function test_dorfler_marking()
   for strategy in (:sort,:binsort,:quickmark)
     for n in (1000,10000)
       for θ in (0.3,0.5)
@@ -19,7 +19,7 @@ function test_marking()
   end
 end
 
-test_marking()
+@testset "Dorfler marking" test_dorfler_marking()
 
 # AMR tests
 

From 60be5128409aebbdf5ced02e8e3c714068ae4f12 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 17:27:19 +1100
Subject: [PATCH 19/27] Added amr driver

---
 src/Adaptivity/AdaptiveMeshRefinement.jl      | 42 +++++++++++++++
 src/Adaptivity/Adaptivity.jl                  |  2 +-
 .../AdaptiveMeshRefinementTests.jl            | 51 +++++++++++++++++--
 3 files changed, 89 insertions(+), 6 deletions(-)

diff --git a/src/Adaptivity/AdaptiveMeshRefinement.jl b/src/Adaptivity/AdaptiveMeshRefinement.jl
index 2f5d83f34..0dfc9173d 100644
--- a/src/Adaptivity/AdaptiveMeshRefinement.jl
+++ b/src/Adaptivity/AdaptiveMeshRefinement.jl
@@ -1,4 +1,6 @@
 
+# Marking strategies
+
 """
   struct DorflerMarking
     θ :: Float64
@@ -56,6 +58,12 @@ struct DorflerMarking
   end
 end
 
+"""
+    mark(m::DorflerMarking, η::Vector{<:Real}) -> Vector{Int}
+
+Given a vector `η` of real positive numbers, returns a subset of indices `I` such that 
+satisfying the Dorfler marking condition.
+"""
 mark(m::DorflerMarking, η::Vector{<:Real}) = mark(Val(m.strategy), m, η)
 
 function mark(::Val{:sort}, m::DorflerMarking, η::Vector{<:Real})
@@ -137,3 +145,37 @@ function mark(::Val{:quickmark}, m::DorflerMarking, η::Vector{<:Real})
   m = quickmark!(η, perm, l, u, target)
   return perm[1:m]
 end
+
+# Estimators
+
+function estimate(f::Function, uh)
+  collect_estimator(f(uh))
+end
+
+function collect_estimator(c::DomainContribution)
+  trians = get_domains(c)
+  bgmodel = get_background_model(first(trians))
+  msg = "Estimator not implemented for mixed background models"
+  @notimplementedif !all([bgmodel == get_background_model(trian) for trian in trians]) msg
+
+  Dc = num_cell_dims(bgmodel)
+  η = zeros(Float64,num_cells(bgmodel))
+  for trian in trians
+    glue = get_glue(trian,Val(Dc))
+    collect_estimator!(η,glue,get_contribution(c,trian))
+  end
+
+  return η
+end
+
+function collect_estimator!(η, glue::FaceToFaceGlue, c)
+  cache = array_cache(c)
+  for (face,bgcell) in enumerate(glue.tface_to_mface)
+    η[bgcell] += getindex!(cache,c,face)
+  end
+end
+
+function collect_estimator!(η, glue::SkeletonPair, c)
+  collect_estimator!(η,glue.plus,c)
+  collect_estimator!(η,glue.minus,c)
+end
diff --git a/src/Adaptivity/Adaptivity.jl b/src/Adaptivity/Adaptivity.jl
index 5b85e0460..32a5d1594 100644
--- a/src/Adaptivity/Adaptivity.jl
+++ b/src/Adaptivity/Adaptivity.jl
@@ -40,7 +40,7 @@ export AdaptedTriangulation
 export Triangulation, is_change_possible, best_target, get_adapted_model
 export change_domain, move_contributions
 
-export DorflerMarking, mark
+export DorflerMarking, mark, estimate
 
 include("RefinementRules.jl")
 include("FineToCoarseFields.jl")
diff --git a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index 40776de45..6a0165941 100644
--- a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -31,11 +31,52 @@ function LShapedModel(n)
   return simplexify(DiscreteModelPortion(model,mask))
 end
 
-model = LShapedModel(10)
+function amr_step(model)
+  order = 1
+  reffe = ReferenceFE(lagrangian,Float64,order)
+  V = TestFESpace(model,reffe)
+  
+  Ω = Triangulation(model)
+  Γ = Boundary(model)
+  Λ = Skeleton(model)
+  
+  dΩ = Measure(Ω,2*order)
+  dΓ = Measure(Γ,2*order)
+  dΛ = Measure(Λ,2*order)
+  
+  hK = CellField(sqrt.(collect(get_array(∫(1)dΩ))),Ω)
+  
+  f(x) = 1.0 / ((x[1]-0.5)^2 + (x[2]-0.5)^2)^(1/2)
+  a(u,v) = ∫(∇(u)⋅∇(v))dΩ
+  l(v)   = ∫(f*v)dΩ
+  ηh(u)  = ∫(hK*f)dΩ + ∫(hK*∇(u)⋅∇(u))dΓ + ∫(hK*jump(∇(u))⋅jump(∇(u)))dΛ
+  
+  op = AffineFEOperator(a,l,V,V)
+  uh = solve(op)
+  η = estimate(ηh,uh)
+  
+  m = DorflerMarking(0.5)
+  I = Adaptivity.mark(m,η)
+  
+  method = Adaptivity.NVBRefinement(model)
+  fmodel = Adaptivity.get_model(refine(method,model;cells_to_refine=I))
 
-method = Adaptivity.NVBRefinement(model)
-cells_to_refine = [collect(1:10)...,collect(20:30)...]
-fmodel = refine(method,model;cells_to_refine)
+  return fmodel, uh, η, I
+end
 
-writevtk(Triangulation(fmodel),"tmp/fmodel";append=false)
+model = LShapedModel(10)
 
+for i in 1:10
+  fmodel, uh, η, I = amr_step(model)
+  is_refined = map(i -> ifelse(i ∈ I, 1, -1), 1:num_cells(model))
+  Ω = Triangulation(model)
+  writevtk(
+    Ω,"tmp/model_$(i-1)",append=false,
+    cellfields = [
+      "uh" => uh,
+      "η" => CellField(η,Ω),
+      "is_refined" => CellField(is_refined,Ω)
+    ],
+  )
+  model = fmodel
+end

From 66d44a326b9c3b9fc00d5000268dc390de4b8397 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 17:30:59 +1100
Subject: [PATCH 20/27] Minor

---
 test/AdaptivityTests/AdaptiveMeshRefinementTests.jl | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index 6a0165941..81fcc3dcc 100644
--- a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -64,9 +64,10 @@ function amr_step(model)
   return fmodel, uh, η, I
 end
 
+nsteps = 10
 model = LShapedModel(10)
 
-for i in 1:10
+for i in 1:nsteps
   fmodel, uh, η, I = amr_step(model)
   is_refined = map(i -> ifelse(i ∈ I, 1, -1), 1:num_cells(model))
   Ω = Triangulation(model)

From 2ba6583774c9faeb5c8f9a4a31d3afd364766dc2 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 17:43:46 +1100
Subject: [PATCH 21/27] More docs

---
 docs/src/Adaptivity.md                   | 12 ++++++++++++
 src/Adaptivity/AdaptiveMeshRefinement.jl |  7 +++++++
 2 files changed, 19 insertions(+)

diff --git a/docs/src/Adaptivity.md b/docs/src/Adaptivity.md
index 377cd5d23..5a698528a 100644
--- a/docs/src/Adaptivity.md
+++ b/docs/src/Adaptivity.md
@@ -80,6 +80,18 @@ The API is given by the following methods:
   MacroReferenceFE
 ```
 
+## Adaptive Mesh Refinement
+
+One of the main uses of mesh refinement is Adaptive Mesh Refinement, where the mesh is refined only in regions of interest.
+
+The typical AMR workflow is the so-called `solve-estimate-mark-refine` loop. Since estimators will generally be problem-dependent, we only aim to provide some generic tools that can be combined by the user:
+
+```@docs
+  DorflerMarking
+  mark
+  estimate
+```
+
 ## Notes for users
 
 Most of the tools provided by this module are showcased in the tests of the module itself, as well as the following tutorial (coming soon).
diff --git a/src/Adaptivity/AdaptiveMeshRefinement.jl b/src/Adaptivity/AdaptiveMeshRefinement.jl
index 0dfc9173d..78ca189f0 100644
--- a/src/Adaptivity/AdaptiveMeshRefinement.jl
+++ b/src/Adaptivity/AdaptiveMeshRefinement.jl
@@ -148,6 +148,13 @@ end
 
 # Estimators
 
+"""
+    estimate(f::Function, uh::Function) -> Vector{Float64}
+
+Given a functional `f` and a function `uh`, such that `f(uh)` produces a 
+scalar-valued `DomainContribution`, collects the estimator values for 
+each cell in the background model.
+"""
 function estimate(f::Function, uh)
   collect_estimator(f(uh))
 end

From ce22f58316751f01d62e1804f8b26bc7ae183535 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Fri, 6 Dec 2024 17:49:38 +1100
Subject: [PATCH 22/27] Minor

---
 src/Adaptivity/AdaptiveMeshRefinement.jl | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/Adaptivity/AdaptiveMeshRefinement.jl b/src/Adaptivity/AdaptiveMeshRefinement.jl
index 78ca189f0..40f9eea42 100644
--- a/src/Adaptivity/AdaptiveMeshRefinement.jl
+++ b/src/Adaptivity/AdaptiveMeshRefinement.jl
@@ -2,13 +2,13 @@
 # Marking strategies
 
 """
-  struct DorflerMarking
-    θ :: Float64
-    ν :: Float64
-    strategy :: Symbol
-  end
+    struct DorflerMarking
+      θ :: Float64
+      ν :: Float64
+      strategy :: Symbol
+    end
 
-  DorflerMarking(θ::Float64; ν::Float64 = 0.5, strategy::Symbol = :quickmark)
+    DorflerMarking(θ::Float64; ν::Float64 = 0.5, strategy::Symbol = :quickmark)
 
 Implements the Dorfler marking strategy. Given a vector `η` of real positive numbers, 
 the marking strategy find a subset of indices `I` such that 
@@ -19,7 +19,7 @@ where `0 < θ < 1` is a threshold parameter.
 
 For more details, see the following reference: 
 
-`Dörfler marking with minimal cardinality is a linear complexity problem`, Pfeiler et al. (2020)
+"Dörfler marking with minimal cardinality is a linear complexity problem", Pfeiler et al. (2020)
 
 The marking algorithm is controlled by the `strategy` parameter, which can take 
 the following values: 

From 9cf8f080ee01e29127575c185ce6663353f4d265 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Mon, 9 Dec 2024 12:55:16 +1100
Subject: [PATCH 23/27] Added tests

---
 .../AdaptiveMeshRefinementTests.jl            | 83 +++++++++++++------
 test/AdaptivityTests/runtests.jl              |  4 +
 2 files changed, 60 insertions(+), 27 deletions(-)

diff --git a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index 81fcc3dcc..c512e11ae 100644
--- a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -1,3 +1,5 @@
+module AdaptiveMeshRefinementTests
+
 using Test
 using Gridap, Gridap.Geometry, Gridap.Adaptivity
 using DataStructures
@@ -19,8 +21,6 @@ function test_dorfler_marking()
   end
 end
 
-@testset "Dorfler marking" test_dorfler_marking()
-
 # AMR tests
 
 function LShapedModel(n)
@@ -31,53 +31,82 @@ function LShapedModel(n)
   return simplexify(DiscreteModelPortion(model,mask))
 end
 
-function amr_step(model)
-  order = 1
+l2_norm(he,xh,dΩ) = ∫(he*(xh*xh))*dΩ
+l2_norm(xh,dΩ) = ∫(xh*xh)*dΩ
+
+function amr_step(model,u_exact;order=1)
   reffe = ReferenceFE(lagrangian,Float64,order)
-  V = TestFESpace(model,reffe)
+  V = TestFESpace(model,reffe;dirichlet_tags=["boundary"])
+  U = TrialFESpace(V,u_exact)
   
   Ω = Triangulation(model)
   Γ = Boundary(model)
   Λ = Skeleton(model)
   
-  dΩ = Measure(Ω,2*order)
+  dΩ = Measure(Ω,4*order)
   dΓ = Measure(Γ,2*order)
   dΛ = Measure(Λ,2*order)
   
   hK = CellField(sqrt.(collect(get_array(∫(1)dΩ))),Ω)
-  
-  f(x) = 1.0 / ((x[1]-0.5)^2 + (x[2]-0.5)^2)^(1/2)
+
+  nΓ = get_normal_vector(Γ)
+  nΛ = get_normal_vector(Λ)
+
+  ∇u(x)  = ∇(u_exact)(x)
+  f(x)   = -Δ(u_exact)(x)
   a(u,v) = ∫(∇(u)⋅∇(v))dΩ
   l(v)   = ∫(f*v)dΩ
-  ηh(u)  = ∫(hK*f)dΩ + ∫(hK*∇(u)⋅∇(u))dΓ + ∫(hK*jump(∇(u))⋅jump(∇(u)))dΛ
+  ηh(u)  = l2_norm(hK,(f + Δ(u)),dΩ) + l2_norm(hK,(∇(u) - ∇u)⋅nΓ,dΓ) + l2_norm(hK,jump(∇(u)⋅nΛ),dΛ)
   
-  op = AffineFEOperator(a,l,V,V)
+  op = AffineFEOperator(a,l,U,V)
   uh = solve(op)
   η = estimate(ηh,uh)
   
-  m = DorflerMarking(0.5)
+  m = DorflerMarking(0.8)
   I = Adaptivity.mark(m,η)
   
   method = Adaptivity.NVBRefinement(model)
   fmodel = Adaptivity.get_model(refine(method,model;cells_to_refine=I))
 
-  return fmodel, uh, η, I
+  error = sum(l2_norm(uh - u_exact,dΩ))
+  return fmodel, uh, η, I, error
 end
 
-nsteps = 10
-model = LShapedModel(10)
+function test_amr(nsteps,order)
+  model = LShapedModel(10)
 
-for i in 1:nsteps
-  fmodel, uh, η, I = amr_step(model)
-  is_refined = map(i -> ifelse(i ∈ I, 1, -1), 1:num_cells(model))
-  Ω = Triangulation(model)
-  writevtk(
-    Ω,"tmp/model_$(i-1)",append=false,
-    cellfields = [
-      "uh" => uh,
-      "η" => CellField(η,Ω),
-      "is_refined" => CellField(is_refined,Ω)
-    ],
-  )
-  model = fmodel
+  ϵ = 1e-2
+  r(x) = ((x[1]-0.5)^2 + (x[2]-0.5)^2)^(1/2)
+  u_exact(x) = 1.0 / (ϵ + r(x))
+
+  vtk = false
+  last_error = Inf
+  for i in 1:nsteps
+    fmodel, uh, η, I, error = amr_step(model,u_exact;order)
+    if vtk
+      is_refined = map(i -> ifelse(i ∈ I, 1, -1), 1:num_cells(model))
+      Ω = Triangulation(model)
+      writevtk(
+        Ω,"tmp/model_$(i-1)",append=false,
+        cellfields = [
+          "uh" => uh,
+          "η" => CellField(η,Ω),
+          "is_refined" => CellField(is_refined,Ω),
+          "u_exact" => CellField(u_exact,Ω),
+        ],
+      )
+    end
+
+    println("Error: $error, Error η: $(sum(η))")
+    @test (i < 3) || (error < last_error)
+    last_error = error
+    model = fmodel
+  end
 end
+
+############################################################################################
+
+@testset "Dorfler marking" test_dorfler_marking()
+@testset "AMR - Poisson" test_amr(20,2)
+
+end # module
\ No newline at end of file
diff --git a/test/AdaptivityTests/runtests.jl b/test/AdaptivityTests/runtests.jl
index 87437afc3..2d2a569da 100644
--- a/test/AdaptivityTests/runtests.jl
+++ b/test/AdaptivityTests/runtests.jl
@@ -23,4 +23,8 @@ end
   include("MacroFEStokesTests.jl")
 end
 
+@testset "AMR" begin
+  include("AdaptiveMeshRefinementTests.jl")
+end
+
 end # module
\ No newline at end of file

From c5969723c5125eda6fe4bec050c412e6b6285652 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Mon, 9 Dec 2024 15:23:22 +1100
Subject: [PATCH 24/27] Minor

---
 test/AdaptivityTests/AdaptiveMeshRefinementTests.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
index c512e11ae..6f0915d0e 100644
--- a/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
+++ b/test/AdaptivityTests/AdaptiveMeshRefinementTests.jl
@@ -56,7 +56,7 @@ function amr_step(model,u_exact;order=1)
   f(x)   = -Δ(u_exact)(x)
   a(u,v) = ∫(∇(u)⋅∇(v))dΩ
   l(v)   = ∫(f*v)dΩ
-  ηh(u)  = l2_norm(hK,(f + Δ(u)),dΩ) + l2_norm(hK,(∇(u) - ∇u)⋅nΓ,dΓ) + l2_norm(hK,jump(∇(u)⋅nΛ),dΛ)
+  ηh(u)  = l2_norm(hK*(f + Δ(u)),dΩ) + l2_norm(hK*(∇(u) - ∇u)⋅nΓ,dΓ) + l2_norm(jump(hK*∇(u)⋅nΛ),dΛ)
   
   op = AffineFEOperator(a,l,U,V)
   uh = solve(op)

From bcfba317cb0494482672e17420c00d942345122a Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Mon, 9 Dec 2024 16:57:18 +1100
Subject: [PATCH 25/27] Updated NEWS

---
 NEWS.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/NEWS.md b/NEWS.md
index e9287dbdc..524a03951 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,12 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [Unreleased]
+
+### Added
+
+- Added AMR-related methods `mark` and `estimate` to `Adaptivity` module. Implemented Dorfler marking strategy. Since PR[#1063](https://github.com/gridap/Gridap.jl/pull/1063).
+
 ## [0.18.8] - 2024-12-2
 
 ### Added

From 509830bdf49de2e128c0349fb0ec68b592ce328c Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Sun, 15 Dec 2024 00:06:59 +1100
Subject: [PATCH 26/27] Reverted some changes

---
 src/Arrays/Reindex.jl | 8 --------
 1 file changed, 8 deletions(-)

diff --git a/src/Arrays/Reindex.jl b/src/Arrays/Reindex.jl
index 429d5b8b3..7c6442790 100644
--- a/src/Arrays/Reindex.jl
+++ b/src/Arrays/Reindex.jl
@@ -129,11 +129,3 @@ end
   i = j_to_i[j]
   i_to_v[i]=v
 end
-
-# This optimization is important when integrating on partial domains (Triangulation views)
-function lazy_map(::typeof(evaluate),a::LazyArray{<:Fill{<:Reindex}},x::AbstractArray)
-  fields = a.maps.value.values
-  ids = a.args[1]
-  vals = lazy_map(evaluate,fields,x)
-  return lazy_map(Reindex(vals),ids)
-end

From e279910cab2c55ed2d4ccf6f69c3d056768e8348 Mon Sep 17 00:00:00 2001
From: JordiManyer <jordi.manyer@monash.edu>
Date: Sun, 15 Dec 2024 11:16:22 +1100
Subject: [PATCH 27/27] Update NEWS

---
 NEWS.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/NEWS.md b/NEWS.md
index 524a03951..9645ab6ee 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -11,6 +11,10 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 - Added AMR-related methods `mark` and `estimate` to `Adaptivity` module. Implemented Dorfler marking strategy. Since PR[#1063](https://github.com/gridap/Gridap.jl/pull/1063).
 
+### Changed
+
+- Low level optimisations to reduce allocations. `AffineMap` renamed to `AffineField`. New `AffineMap <: Map`, doing the same as `AffineField` without struct allocation. New `ConstantMap <: Map`, doing the same as `ConstantField` without struct allocation. Since PR[#1043](https://github.com/gridap/Gridap.jl/pull/1043).
+
 ## [0.18.8] - 2024-12-2
 
 ### Added
@@ -25,6 +29,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - Fixed issue where barycentric refinement rule in 3D would not produce oriented meshes. Since PR[#1055](https://github.com/gridap/Gridap.jl/pull/1055).
 
 ### Changed
+
 - Optimized MonomialBasis low-level functions. Since PR[#1059](https://github.com/gridap/Gridap.jl/pull/1059).
 
 ## [0.18.7] - 2024-10-8