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Add Lagrange{RefTetrahedron,3} #1255

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Add Lagrange{RefTetrahedron,3}
termi-official Nov 26, 2025
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141 changes: 141 additions & 0 deletions src/interpolations.jl
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Expand Up @@ -28,6 +28,7 @@ The following interpolations are implemented:
* `Lagrange{RefHexahedron, 2}`
* `Lagrange{RefTetrahedron, 1}`
* `Lagrange{RefTetrahedron, 2}`
* `Lagrange{RefTetrahedron, 3}`
* `Lagrange{RefPrism, 1}`
* `Lagrange{RefPrism, 2}`
* `Lagrange{RefPyramid, 1}`
Expand Down Expand Up @@ -932,6 +933,80 @@ function reference_shape_value(ip::Lagrange{RefTetrahedron, 2}, ξ::Vec{3}, i::I
throw(ArgumentError("no shape function $i for interpolation $ip"))
end

###################################
# Lagrange RefTetrahedron order 3 #
###################################
getnbasefunctions(::Lagrange{RefTetrahedron, 3}) = 20
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@AbdAlazezAhmed AbdAlazezAhmed Dec 1, 2025

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maybe

Suggested change
getnbasefunctions(::Lagrange{RefTetrahedron, 3}) = 20
getnbasefunctions(::Lagrange{RefTetrahedron, order}) where order = (order+1)*(order+2)*(order+3)÷6

would be beneficial in the future


facedof_indices(::Lagrange{RefTetrahedron, 3}) = ((1, 3, 2, 10, 9, 8, 7, 6, 5, 17), (1, 2, 4, 5, 6, 13, 14, 11, 12, 18), (2, 3, 4, 7, 8, 15, 16, 13, 14, 19), (1, 4, 3, 11, 12, 15, 16, 9, 10, 20))
facedof_interior_indices(::Lagrange{RefTetrahedron, 3}) = ((17,), (18,), (19,), (20,))
edgedof_indices(::Lagrange{RefTetrahedron, 3}) = ((1, 2, 5, 6), (2, 3, 7, 8), (3, 1, 9, 10), (1, 4, 11, 12), (2, 4, 13, 14), (3, 4, 15, 16))
edgedof_interior_indices(::Lagrange{RefTetrahedron, 3}) = ((5, 6), (7, 8), (9, 10), (11, 12), (13, 14), (15, 16))

function reference_coordinates(::Lagrange{RefTetrahedron, 3})
return [
# Vertices
Vec{3, Float64}((0.0, 0.0, 0.0)),
Vec{3, Float64}((1.0, 0.0, 0.0)),
Vec{3, Float64}((0.0, 1.0, 0.0)),
Vec{3, Float64}((0.0, 0.0, 1.0)),
# Edges
#1
Vec{3, Float64}((1 / 3, 0.0, 0.0)),
Vec{3, Float64}((2 / 3, 0.0, 0.0)),
#2
Vec{3, Float64}((2 / 3, 1 / 3, 0.0)),
Vec{3, Float64}((1 / 3, 2 / 3, 0.0)),
#3
Vec{3, Float64}((0.0, 2 / 3, 0.0)),
Vec{3, Float64}((0.0, 1 / 3, 0.0)),
#4
Vec{3, Float64}((0.0, 0.0, 1 / 3)),
Vec{3, Float64}((0.0, 0.0, 2 / 3)),
#5
Vec{3, Float64}((2 / 3, 0.0, 1 / 3)),
Vec{3, Float64}((1 / 3, 0.0, 2 / 3)),
#6
Vec{3, Float64}((0.0, 2 / 3, 1 / 3)),
Vec{3, Float64}((0.0, 1 / 3, 2 / 3)),
# Faces
Vec{3, Float64}((1 / 3, 1 / 3, 0.0)),
Vec{3, Float64}((1 / 3, 0.0, 1 / 3)),
Vec{3, Float64}((1 / 3, 1 / 3, 1 / 3)),
Vec{3, Float64}((0.0, 1 / 3, 1 / 3)),
]
end

# http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch09.d/AFEM.Ch09.pdf
# http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch10.d/AFEM.Ch10.pdf
function reference_shape_value(ip::Lagrange{RefTetrahedron, 3}, ξ::Vec{3, T}, i::Int) where {T}
ξ_x = ξ[1]
ξ_y = ξ[2]
ξ_z = ξ[3]
# TODO implement as tensor product form after https://github.com/Ferrite-FEM/Ferrite.jl/pull/1188 is merged
i == 1 && return T(1.0) + T(-5.5) * ξ_x + T(9.0) * ξ_x^2 + T(-4.5) * ξ_x^3 + T(-5.5) * ξ_y + T(18.0) * ξ_x * ξ_y + T(-13.5) * ξ_x^2 * ξ_y + T(9.0) * ξ_y^2 + T(-13.5) * ξ_x * ξ_y^2 + T(-4.5) * ξ_y^3 + T(-5.5) * ξ_z + T(18.0) * ξ_x * ξ_z + T(-13.5) * ξ_x^2 * ξ_z + T(18.0) * ξ_y * ξ_z + T(-27.0) * ξ_x * ξ_y * ξ_z + T(-13.5) * ξ_y^2 * ξ_z + T(9.0) * ξ_z^2 + T(-13.5) * ξ_x * ξ_z^2 + T(-13.5) * ξ_y * ξ_z^2 + T(-4.5) * ξ_z^3
i == 2 && return ξ_x + T(-4.5) * ξ_x^2 + T(4.5) * ξ_x^3
i == 3 && return ξ_y + T(-4.5) * ξ_y^2 + T(4.5) * ξ_y^3
i == 4 && return ξ_z + T(-4.5) * ξ_z^2 + T(4.5) * ξ_z^3
i == 5 && return T(9.0) * ξ_x + T(-22.5) * ξ_x^2 + T(13.5) * ξ_x^3 + T(-22.5) * ξ_x * ξ_y + T(27.0) * ξ_x^2 * ξ_y + T(13.5) * ξ_x * ξ_y^2 + T(-22.5) * ξ_x * ξ_z + T(27.0) * ξ_x^2 * ξ_z + T(27.0) * ξ_x * ξ_y * ξ_z + T(13.5) * ξ_x * ξ_z^2
i == 6 && return T(-4.5) * ξ_x + T(18.0) * ξ_x^2 + T(-13.5) * ξ_x^3 + T(4.5) * ξ_x * ξ_y + T(-13.5) * ξ_x^2 * ξ_y + T(4.5) * ξ_x * ξ_z + T(-13.5) * ξ_x^2 * ξ_z
i == 7 && return T(-4.5) * ξ_x * ξ_y + T(13.5) * ξ_x^2 * ξ_y
i == 8 && return T(-4.5) * ξ_x * ξ_y + T(13.5) * ξ_x * ξ_y^2
i == 9 && return T(-4.5) * ξ_y + T(4.5) * ξ_x * ξ_y + T(18) * ξ_y^2 + T(-13.5) * ξ_x * ξ_y^2 + T(-13.5) * ξ_y^3 + T(4.5) * ξ_y * ξ_z + T(-13.5) * ξ_y^2 * ξ_z
i == 10 && return T(9.0) * ξ_y + T(-22.5) * ξ_x * ξ_y + T(13.5) * ξ_x^2 * ξ_y + T(-22.5) * ξ_y^2 + T(27.0) * ξ_x * ξ_y^2 + T(13.5) * ξ_y^3 + T(-22.5) * ξ_y * ξ_z + T(27.0) * ξ_x * ξ_y * ξ_z + T(27.0) * ξ_y^2 * ξ_z + T(13.5) * ξ_y * ξ_z^2
i == 11 && return T(9.0) * ξ_z + T(-22.5) * ξ_x * ξ_z + T(13.5) * ξ_x^2 * ξ_z + T(-22.5) * ξ_y * ξ_z + T(27.0) * ξ_x * ξ_y * ξ_z + T(13.5) * ξ_y^2 * ξ_z + T(-22.5) * ξ_z^2 + T(27.0) * ξ_x * ξ_z^2 + T(27.0) * ξ_y * ξ_z^2 + T(13.5) * ξ_z^3
i == 12 && return T(-4.5) * ξ_z + T(4.5) * ξ_x * ξ_z + T(4.5) * ξ_y * ξ_z + T(18.0) * ξ_z^2 + T(-13.5) * ξ_x * ξ_z^2 + T(-13.5) * ξ_y * ξ_z^2 + T(-13.5) * ξ_z^3
i == 13 && return T(-4.5) * ξ_x * ξ_z + T(13.5) * ξ_x^2 * ξ_z
i == 14 && return T(-4.5) * ξ_x * ξ_z + T(13.5) * ξ_x * ξ_z^2
i == 15 && return T(-4.5) * ξ_y * ξ_z + T(13.5) * ξ_y^2 * ξ_z
i == 16 && return T(-4.5) * ξ_y * ξ_z + T(13.5) * ξ_y * ξ_z^2
i == 17 && return T(27.0) * ξ_x * ξ_y + T(-27.0) * ξ_x^2 * ξ_y + T(-27.0) * ξ_x * ξ_y^2 + T(-27.0) * ξ_x * ξ_y * ξ_z
i == 18 && return T(27.0) * ξ_x * ξ_z + T(-27.0) * ξ_x^2 * ξ_z + T(-27.0) * ξ_x * ξ_y * ξ_z + T(-27.0) * ξ_x * ξ_z^2
i == 19 && return T(27.0) * ξ_x * ξ_y * ξ_z
i == 20 && return T(27.0) * ξ_y * ξ_z + T(-27.0) * ξ_x * ξ_y * ξ_z + T(-27.0) * ξ_y^2 * ξ_z + T(-27.0) * ξ_y * ξ_z^2
throw(ArgumentError("no shape function $i for interpolation $ip"))
end

##################################
# Lagrange RefHexahedron order 1 #
##################################
Expand Down Expand Up @@ -2132,3 +2207,69 @@ adjust_dofs_during_distribution(::Nedelec{RefHexahedron, 1}) = false
function get_direction(::Nedelec{RefHexahedron, 1}, shape_nr, cell)
return get_edge_direction(cell, shape_nr) # shape_nr = edge_nr
end

# TODO where to put this? I am a lazy man and I won't compute this by hand.
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@KnutAM KnutAM Dec 1, 2025

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Seems like this should be a macro or generated function?

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@termi-official termi-official Dec 1, 2025

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Not 100% sure yet. This is for now just a template to get things started on higher order element generators, as we can simply generate these in a systematic way.

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@AbdAlazezAhmed AbdAlazezAhmed Dec 1, 2025

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Maybe

@generated function do_magic(::Lagrange{RefTetrahedron, order}, ξ::Vec{3, Float64}, i::Int) where {order}

    N = 0
    for p₃ in 0:order
        for p₂ in 0:(order - p₃)
            for p₁ in 0:(order - p₃ - p₂)
                N += 1
            end
        end
    end

    @assert N == 20
    xs = reference_coordinates(Lagrange{RefTetrahedron, order})
    V = zeros(N, N)
    j = 0
    for p₃ in 0:order
        for p₂ in 0:(order - p₃)
            for p₁ in 0:(order - p₃ - p₂)
                j += 1
                for i in 1:N
                    x = xs[i]
                    V[i, j] = x[1]^p₁ * x[2]^p₂ * x[3]^p₃
                end
            end
        end
    end
    C = inv(V)' # These are our coefficients
    ex = :(    ξ_x = ξ[1];
    ξ_y = ξ[2];
    ξ_z = ξ[3];)
    for i in 1:N
        j = 0
        x = xs[i]
        ex_i = :(0.0)
        for p₃ in 0:order
            for p₂ in 0:(order - p₃)
                for p₁ in 0:(order - p₃ - p₂)
                    j += 1
                    ex_i = :($ex_i + $(C[i, j]) * ξ_x^$p₁ * ξ_y^$p₂ * ξ_z^$p₃)
                end
            end
        end
        ex = :($ex; i == $i && return $ex_i)
    end

    return ex
end
julia> @benchmark Ferrite.do_magic($ip, $(Vec(-0.0,-0.0,-0.0)),$(1))
BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
 Range (min … max):  4.882 ns … 20.719 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     5.008 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   5.030 ns ±  0.389 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

   ▅     █▂                                                   
  ▅█▆▃▂▂▄██▄▂▂▂▂▁▂▂▂▁▂▄▄▃▂▂▂▂▂▂▂▂▁▁▂▂▂▂▁▁▁▂▂▂▂▁▁▂▂▁▁▁▁▁▂▂▁▁▂ ▃
  4.88 ns        Histogram: frequency by time        5.86 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark Ferrite.reference_shape_value($ip, $(Vec(-0.0,-0.0,-0.0)),$(1))
BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
 Range (min … max):  4.581 ns … 9.742 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     4.619 ns             ┊ GC (median):    0.00%
 Time  (mean ± σ):   4.673 ns ± 0.209 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

   ▅█                                                        
  ▂██▂▂▂▂▄▄▂▂▂▂▂▂▂▂▂▂▂▄▂▂▂▂▂▂▂▁▂▂▂▁▂▂▂▂▁▁▁▁▁▁▁▂▁▂▂▂▂▂▁▁▁▁▁▂ ▂
  4.58 ns        Histogram: frequency by time       5.53 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

compiler vroom

function tabulate_lagrange(interpolation::Lagrange{RefTetrahedron, order}; tol = 1.0e-8, with_T = false) where {order}
N = 0
for p₃ in 0:order
for p₂ in 0:(order - p₃)
for p₁ in 0:(order - p₃ - p₂)
N += 1
end
end
end

@assert N == getnbasefunctions(interpolation)
xs = reference_coordinates(interpolation)
V = zeros(N, N)
j = 0
for p₃ in 0:order
for p₂ in 0:(order - p₃)
for p₁ in 0:(order - p₃ - p₂)
j += 1
for i in 1:N
x = xs[i]
V[i, j] = x[1]^p₁ * x[2]^p₂ * x[3]^p₃
end
end
end
end
C = inv(V)' # These are our coefficients

for i in 1:N
j = 0
x = xs[i]
isfirst = true
for p₃ in 0:order
for p₂ in 0:(order - p₃)
for p₁ in 0:(order - p₃ - p₂)
j += 1
(abs(C[i, j]) < tol) && continue
if isfirst
print("i == $i && return")
else
print(" +")
end
isfirst = false
if with_T
print(" T($(C[i, j]))")
else
print(" $(C[i, j])")
end
if p₁ > 0
p₁ > 1 ? print(" * ξ_x^$p₁") : print(" * ξ_x")
end
if p₂ > 0
p₂ > 1 ? print(" * ξ_y^$p₂") : print(" * ξ_y")
end
if p₃ > 0
p₃ > 1 ? print(" * ξ_z^$p₃") : print(" * ξ_z")
end
end
end
end
println()
end

return C
end
3 changes: 2 additions & 1 deletion test/integration/test_simple_scalar_convergence.jl
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Expand Up @@ -15,7 +15,7 @@ module ConvergenceTestHelper
get_geometry(::Ferrite.Interpolation{RefPyramid}) = Pyramid

get_quadrature_order(::Lagrange{shape, order}) where {shape, order} = max(2 * order - 1, 2)
get_quadrature_order(::Lagrange{RefTriangle, 5}) where {shape, order} = 8
get_quadrature_order(::Lagrange{RefTriangle, 5}) = 8
get_quadrature_order(::Lagrange{RefPrism, order}) where {order} = 2 * order # Don't know why
get_quadrature_order(::Serendipity{shape, order}) where {shape, order} = max(2 * order - 1, 2)
get_quadrature_order(::CrouzeixRaviart{shape, order}) where {shape, order} = max(2 * order - 1, 2)
Expand Down Expand Up @@ -195,6 +195,7 @@ end
Lagrange{RefTriangle, 2}(),
Lagrange{RefHexahedron, 2}(),
Lagrange{RefTetrahedron, 2}(),
Lagrange{RefTetrahedron, 3}(),
Lagrange{RefPrism, 2}(),
CrouzeixRaviart{RefTriangle, 1}(),
CrouzeixRaviart{RefTetrahedron, 1}(),
Expand Down
6 changes: 4 additions & 2 deletions test/test_interpolations.jl
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Expand Up @@ -157,6 +157,7 @@ end
Serendipity{RefHexahedron, 2}(),
Lagrange{RefTetrahedron, 1}(),
Lagrange{RefTetrahedron, 2}(),
Lagrange{RefTetrahedron, 3}(),
Lagrange{RefPrism, 1}(),
Lagrange{RefPrism, 2}(),
Lagrange{RefPyramid, 1}(),
Expand Down Expand Up @@ -219,14 +220,15 @@ end
end
end

# Check for dirac delta property of interpolation
# Check for dirac delta property of interpolatio
@testset "dirac delta property of dof $dof" for dof in 1:n_basefuncs
for k in 1:n_basefuncs
N_dof = reference_shape_value(interpolation, coords[dof], k)
if k == dof
@test N_dof ≈ 1.0
else
factor = interpolation isa Lagrange{RefQuadrilateral, 3} ? 200 : 4
# High order elements are right now not implemented in factorized form, so there is some small numerical noise in the evaluation.
factor = Ferrite.getorder(interpolation) > 2 ? 200 : 4
@test N_dof ≈ 0.0 atol = factor * eps(typeof(N_dof))
end
end
Expand Down
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