From ca34e8da9dac81aed7ae4fc88685ea76a8313090 Mon Sep 17 00:00:00 2001 From: Jukka Aho Date: Sun, 9 Nov 2025 08:29:50 +0200 Subject: [PATCH] chore: Add auto-generated Lagrange basis functions New file: src/basis/lagrange_generated.jl (448 lines, machine-generated) Generated by: julia --project=. src/basis/lagrange_generator.jl Generated at: 2025-11-09 07:28:58 Contains basis functions for 15 element types: - 1D: Seg2Basis, Seg3Basis - 2D triangles: Tri3Basis, Tri6Basis - 2D quads: Quad4Basis, Quad8Basis, Quad9Basis - 3D tets: Tet4Basis, Tet10Basis - 3D hexes: Hex8Basis, Hex20Basis, Hex27Basis - 3D pyramid: Pyr5Basis - 3D wedges: Wedge6Basis, Wedge15Basis All types have "Basis" suffix to avoid conflicts with topology types. DO NOT EDIT MANUALLY - regenerate with generator script. --- src/basis/lagrange_generated.jl | 448 ++++++++++++++++++++++++++++++++ 1 file changed, 448 insertions(+) create mode 100644 src/basis/lagrange_generated.jl diff --git a/src/basis/lagrange_generated.jl b/src/basis/lagrange_generated.jl new file mode 100644 index 0000000..635319e --- /dev/null +++ b/src/basis/lagrange_generated.jl @@ -0,0 +1,448 @@ +# This file is a part of JuliaFEM. +# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE + +# ============================================================================ +# AUTO-GENERATED LAGRANGE BASIS FUNCTIONS +# ============================================================================ +# +# WARNING: DO NOT EDIT THIS FILE MANUALLY! +# +# This file was automatically generated by: +# julia --project=. src/basis/lagrange_generator.jl +# +# To regenerate (e.g., after adding new element types): +# cd /path/to/JuliaFEM.jl +# julia --project=. src/basis/lagrange_generator.jl +# +# Theory: +# See docs/book/lagrange_basis_functions.md +# +# Generator: +# src/basis/lagrange_generator.jl (symbolic engine) +# +# Generated: 2025-11-09 07:28:58 +# ============================================================================ + +# Export all basis types +export Seg2Basis, Seg3Basis, Tri3Basis, Tri6Basis, Quad4Basis, Quad8Basis, Quad9Basis, Tet4Basis, Tet10Basis, Hex8Basis, Hex20Basis, Hex27Basis, Pyr5Basis, Wedge6Basis, Wedge15Basis + +# ────────────────────────────────────────────────────────────────────────────── +# Seg2: 2-node linear segment element +# ────────────────────────────────────────────────────────────────────────────── + + struct Seg2Basis <: AbstractBasis{1} + end + Base.@pure function Base.size(::Type{Seg2Basis}) + return (1, 2) + end + function Base.size(::Type{Seg2Basis}, j::Int) + j == 1 && return 1 + j == 2 && return 2 + end + Base.@pure function Base.length(::Type{Seg2Basis}) + return 2 + end + function get_reference_element_coordinates(::Type{Seg2Basis}) + return Vec{1, Float64}[[-1.0], [1.0]] + end + @inline function eval_basis!(::Type{Seg2Basis}, ::Type{T}, xi::Vec) where T + (u,) = xi + @inbounds return (0.5 + -0.5u, 0.5 + 0.5u) + end + @inline function eval_dbasis!(::Type{Seg2Basis}, xi::Vec) + (u,) = xi + @inbounds return (Vec(float.(tuple(-0.5))), Vec(float.(tuple(0.5)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Seg3: 3-node quadratic segment element +# ────────────────────────────────────────────────────────────────────────────── + + struct Seg3Basis <: AbstractBasis{1} + end + Base.@pure function Base.size(::Type{Seg3Basis}) + return (1, 3) + end + function Base.size(::Type{Seg3Basis}, j::Int) + j == 1 && return 1 + j == 2 && return 3 + end + Base.@pure function Base.length(::Type{Seg3Basis}) + return 3 + end + function get_reference_element_coordinates(::Type{Seg3Basis}) + return Vec{1, Float64}[[-1.0], [1.0], [0.0]] + end + @inline function eval_basis!(::Type{Seg3Basis}, ::Type{T}, xi::Vec) where T + (u,) = xi + @inbounds return (-0.5u + 0.5 * u ^ 2, 0.5u + 0.5 * u ^ 2, 1 + -1.0 * u ^ 2) + end + @inline function eval_dbasis!(::Type{Seg3Basis}, xi::Vec) + (u,) = xi + @inbounds return (Vec(float.(tuple(-0.5 + 0.5 * (2 * u ^ (2 - 1))))), Vec(float.(tuple(0.5 + 0.5 * (2 * u ^ (2 - 1))))), Vec(float.(tuple(-1.0 * (2 * u ^ (2 - 1)))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Tri3: 3-node linear triangular element +# ────────────────────────────────────────────────────────────────────────────── + + struct Tri3Basis <: AbstractBasis{2} + end + Base.@pure function Base.size(::Type{Tri3Basis}) + return (2, 3) + end + function Base.size(::Type{Tri3Basis}, j::Int) + j == 1 && return 2 + j == 2 && return 3 + end + Base.@pure function Base.length(::Type{Tri3Basis}) + return 3 + end + function get_reference_element_coordinates(::Type{Tri3Basis}) + return Vec{2, Float64}[[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]] + end + @inline function eval_basis!(::Type{Tri3Basis}, ::Type{T}, xi::Vec) where T + (u, v) = xi + @inbounds return (1 + -1.0u + -1.0v, +u, +v) + end + @inline function eval_dbasis!(::Type{Tri3Basis}, xi::Vec) + (u, v) = xi + @inbounds return (Vec(float.(tuple(-1.0, -1.0))), Vec(float.(tuple(1, 0))), Vec(float.(tuple(0, 1)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Tri6: 6-node quadratic triangular element +# ────────────────────────────────────────────────────────────────────────────── + + struct Tri6Basis <: AbstractBasis{2} + end + Base.@pure function Base.size(::Type{Tri6Basis}) + return (2, 6) + end + function Base.size(::Type{Tri6Basis}, j::Int) + j == 1 && return 2 + j == 2 && return 6 + end + Base.@pure function Base.length(::Type{Tri6Basis}) + return 6 + end + function get_reference_element_coordinates(::Type{Tri6Basis}) + return Vec{2, Float64}[[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [0.5, 0.0], [0.5, 0.5], [0.0, 0.5]] + end + @inline function eval_basis!(::Type{Tri6Basis}, ::Type{T}, xi::Vec) where T + (u, v) = xi + @inbounds return (1 + -3.0u + -3.0v + 2.0 * u ^ 2 + 4.0 * (u * v) + 2.0 * v ^ 2, -1.0u + 2.0 * u ^ 2, -1.0v + 2.0 * v ^ 2, 4.0u + -4.0 * u ^ 2 + -4.0 * (u * v), +(4.0 * (u * v)), 4.0v + -4.0 * (u * v) + -4.0 * v ^ 2) + end + @inline function eval_dbasis!(::Type{Tri6Basis}, xi::Vec) + (u, v) = xi + @inbounds return (Vec(float.(tuple(-3.0 + 2.0 * (2 * u ^ (2 - 1)) + 4.0v, -3.0 + 4.0u + 2.0 * (2 * v ^ (2 - 1))))), Vec(float.(tuple(-1.0 + 2.0 * (2 * u ^ (2 - 1)), 0))), Vec(float.(tuple(0, -1.0 + 2.0 * (2 * v ^ (2 - 1))))), Vec(float.(tuple(4.0 + -4.0 * (2 * u ^ (2 - 1)) + -4.0v, -4.0u))), Vec(float.(tuple(4.0v, 4.0u))), Vec(float.(tuple(-4.0v, 4.0 + -4.0u + -4.0 * (2 * v ^ (2 - 1)))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Quad4: 4-node bilinear quadrilateral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Quad4Basis <: AbstractBasis{2} + end + Base.@pure function Base.size(::Type{Quad4Basis}) + return (2, 4) + end + function Base.size(::Type{Quad4Basis}, j::Int) + j == 1 && return 2 + j == 2 && return 4 + end + Base.@pure function Base.length(::Type{Quad4Basis}) + return 4 + end + function get_reference_element_coordinates(::Type{Quad4Basis}) + return Vec{2, Float64}[[-1.0, -1.0], [1.0, -1.0], [1.0, 1.0], [-1.0, 1.0]] + end + @inline function eval_basis!(::Type{Quad4Basis}, ::Type{T}, xi::Vec) where T + (u, v) = xi + @inbounds return (0.25 + -0.25u + -0.25v + 0.25 * (u * v), 0.25 + 0.25u + -0.25v + -0.25 * (u * v), 0.25 + 0.25u + 0.25v + 0.25 * (u * v), 0.25 + -0.25u + 0.25v + -0.25 * (u * v)) + end + @inline function eval_dbasis!(::Type{Quad4Basis}, xi::Vec) + (u, v) = xi + @inbounds return (Vec(float.(tuple(-0.25 + 0.25v, -0.25 + 0.25u))), Vec(float.(tuple(0.25 + -0.25v, -0.25 + -0.25u))), Vec(float.(tuple(0.25 + 0.25v, 0.25 + 0.25u))), Vec(float.(tuple(-0.25 + -0.25v, 0.25 + -0.25u)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Quad8: 8-node serendipity quadrilateral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Quad8Basis <: AbstractBasis{2} + end + Base.@pure function Base.size(::Type{Quad8Basis}) + return (2, 8) + end + function Base.size(::Type{Quad8Basis}, j::Int) + j == 1 && return 2 + j == 2 && return 8 + end + Base.@pure function Base.length(::Type{Quad8Basis}) + return 8 + end + function get_reference_element_coordinates(::Type{Quad8Basis}) + return Vec{2, Float64}[[-1.0, -1.0], [1.0, -1.0], [1.0, 1.0], [-1.0, 1.0], [0.0, -1.0], [1.0, 0.0], [0.0, 1.0], [-1.0, 0.0]] + end + @inline function eval_basis!(::Type{Quad8Basis}, ::Type{T}, xi::Vec) where T + (u, v) = xi + @inbounds return (-0.25 + 0.25 * u ^ 2 + 0.25 * (u * v) + 0.25 * v ^ 2 + -0.25 * (u ^ 2 * v) + -0.25 * (u * v ^ 2), -0.25 + 0.25 * u ^ 2 + -0.25 * (u * v) + 0.25 * v ^ 2 + -0.25 * (u ^ 2 * v) + 0.25 * (u * v ^ 2), -0.25 + 0.25 * u ^ 2 + 0.25 * (u * v) + 0.25 * v ^ 2 + 0.25 * (u ^ 2 * v) + 0.25 * (u * v ^ 2), -0.25 + 0.25 * u ^ 2 + -0.25 * (u * v) + 0.25 * v ^ 2 + 0.25 * (u ^ 2 * v) + -0.25 * (u * v ^ 2), 0.5 + -0.5v + -0.5 * u ^ 2 + 0.5 * (u ^ 2 * v), 0.5 + 0.5u + -0.5 * v ^ 2 + -0.5 * (u * v ^ 2), 0.5 + 0.5v + -0.5 * u ^ 2 + -0.5 * (u ^ 2 * v), 0.5 + -0.5u + -0.5 * v ^ 2 + 0.5 * (u * v ^ 2)) + end + @inline function eval_dbasis!(::Type{Quad8Basis}, xi::Vec) + (u, v) = xi + @inbounds return (Vec(float.(tuple(0.25 * (2 * u ^ (2 - 1)) + 0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2, 0.25u + 0.25 * (2 * v ^ (2 - 1)) + -0.25 * u ^ 2 + -0.25 * (u * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.25 * (2 * u ^ (2 - 1)) + -0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2, -0.25u + 0.25 * (2 * v ^ (2 - 1)) + -0.25 * u ^ 2 + 0.25 * (u * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.25 * (2 * u ^ (2 - 1)) + 0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2, 0.25u + 0.25 * (2 * v ^ (2 - 1)) + 0.25 * u ^ 2 + 0.25 * (u * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.25 * (2 * u ^ (2 - 1)) + -0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2, -0.25u + 0.25 * (2 * v ^ (2 - 1)) + 0.25 * u ^ 2 + -0.25 * (u * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.5 * (2 * u ^ (2 - 1)) + 0.5 * ((2 * u ^ (2 - 1)) * v), -0.5 + 0.5 * u ^ 2))), Vec(float.(tuple(0.5 + -0.5 * v ^ 2, -0.5 * (2 * v ^ (2 - 1)) + -0.5 * (u * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.5 * (2 * u ^ (2 - 1)) + -0.5 * ((2 * u ^ (2 - 1)) * v), 0.5 + -0.5 * u ^ 2))), Vec(float.(tuple(-0.5 + 0.5 * v ^ 2, -0.5 * (2 * v ^ (2 - 1)) + 0.5 * (u * (2 * v ^ (2 - 1))))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Quad9: 9-node biquadratic quadrilateral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Quad9Basis <: AbstractBasis{2} + end + Base.@pure function Base.size(::Type{Quad9Basis}) + return (2, 9) + end + function Base.size(::Type{Quad9Basis}, j::Int) + j == 1 && return 2 + j == 2 && return 9 + end + Base.@pure function Base.length(::Type{Quad9Basis}) + return 9 + end + function get_reference_element_coordinates(::Type{Quad9Basis}) + return Vec{2, Float64}[[-1.0, -1.0], [1.0, -1.0], [1.0, 1.0], [-1.0, 1.0], [0.0, -1.0], [1.0, 0.0], [0.0, 1.0], [-1.0, 0.0], [0.0, 0.0]] + end + @inline function eval_basis!(::Type{Quad9Basis}, ::Type{T}, xi::Vec) where T + (u, v) = xi + @inbounds return (0.25 * (u * v) + -0.25 * (u ^ 2 * v) + -0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2), -0.25 * (u * v) + -0.25 * (u ^ 2 * v) + 0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2), 0.25 * (u * v) + 0.25 * (u ^ 2 * v) + 0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2), -0.25 * (u * v) + 0.25 * (u ^ 2 * v) + -0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2), -0.5v + 0.5 * v ^ 2 + 0.5 * (u ^ 2 * v) + -0.5 * (u ^ 2 * v ^ 2), 0.5u + 0.5 * u ^ 2 + -0.5 * (u * v ^ 2) + -0.5 * (u ^ 2 * v ^ 2), 0.5v + 0.5 * v ^ 2 + -0.5 * (u ^ 2 * v) + -0.5 * (u ^ 2 * v ^ 2), -0.5u + 0.5 * u ^ 2 + 0.5 * (u * v ^ 2) + -0.5 * (u ^ 2 * v ^ 2), 1 + -1.0 * u ^ 2 + -1.0 * v ^ 2 + u ^ 2 * v ^ 2) + end + @inline function eval_dbasis!(::Type{Quad9Basis}, xi::Vec) + (u, v) = xi + @inbounds return (Vec(float.(tuple(0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2), 0.25u + -0.25 * u ^ 2 + -0.25 * (u * (2 * v ^ (2 - 1))) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2), -0.25u + -0.25 * u ^ 2 + 0.25 * (u * (2 * v ^ (2 - 1))) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2), 0.25u + 0.25 * u ^ 2 + 0.25 * (u * (2 * v ^ (2 - 1))) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2), -0.25u + 0.25 * u ^ 2 + -0.25 * (u * (2 * v ^ (2 - 1))) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.5 * ((2 * u ^ (2 - 1)) * v) + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2), -0.5 + 0.5 * (2 * v ^ (2 - 1)) + 0.5 * u ^ 2 + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(0.5 + 0.5 * (2 * u ^ (2 - 1)) + -0.5 * v ^ 2 + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2), -0.5 * (u * (2 * v ^ (2 - 1))) + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.5 * ((2 * u ^ (2 - 1)) * v) + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2), 0.5 + 0.5 * (2 * v ^ (2 - 1)) + -0.5 * u ^ 2 + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-0.5 + 0.5 * (2 * u ^ (2 - 1)) + 0.5 * v ^ 2 + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2), 0.5 * (u * (2 * v ^ (2 - 1))) + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1)))))), Vec(float.(tuple(-1.0 * (2 * u ^ (2 - 1)) + (2 * u ^ (2 - 1)) * v ^ 2, -1.0 * (2 * v ^ (2 - 1)) + u ^ 2 * (2 * v ^ (2 - 1)))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Tet4: 4-node linear tetrahedral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Tet4Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Tet4Basis}) + return (3, 4) + end + function Base.size(::Type{Tet4Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 4 + end + Base.@pure function Base.length(::Type{Tet4Basis}) + return 4 + end + function get_reference_element_coordinates(::Type{Tet4Basis}) + return Vec{3, Float64}[[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]] + end + @inline function eval_basis!(::Type{Tet4Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (1 + -1.0u + -1.0v + -1.0w, +u, +v, +w) + end + @inline function eval_dbasis!(::Type{Tet4Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-1.0, -1.0, -1.0))), Vec(float.(tuple(1, 0, 0))), Vec(float.(tuple(0, 1, 0))), Vec(float.(tuple(0, 0, 1)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Tet10: 10-node quadratic tetrahedral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Tet10Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Tet10Basis}) + return (3, 10) + end + function Base.size(::Type{Tet10Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 10 + end + Base.@pure function Base.length(::Type{Tet10Basis}) + return 10 + end + function get_reference_element_coordinates(::Type{Tet10Basis}) + return Vec{3, Float64}[[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.5, 0.0], [0.0, 0.0, 0.5], [0.5, 0.0, 0.5], [0.0, 0.5, 0.5]] + end + @inline function eval_basis!(::Type{Tet10Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (1 + -3.0u + -3.0v + -3.0w + 2.0 * u ^ 2 + 2.0 * v ^ 2 + 2.0 * w ^ 2 + 4.0 * (u * v) + 4.0 * (u * w) + 4.0 * (v * w), -1.0u + 2.0 * u ^ 2, -1.0v + 2.0 * v ^ 2, -1.0w + 2.0 * w ^ 2, 4.0u + -4.0 * u ^ 2 + -4.0 * (u * v) + -4.0 * (u * w), +(4.0 * (u * v)), 4.0v + -4.0 * v ^ 2 + -4.0 * (u * v) + -4.0 * (v * w), 4.0w + -4.0 * w ^ 2 + -4.0 * (u * w) + -4.0 * (v * w), +(4.0 * (u * w)), +(4.0 * (v * w))) + end + @inline function eval_dbasis!(::Type{Tet10Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-3.0 + 2.0 * (2 * u ^ (2 - 1)) + 4.0v + 4.0w, -3.0 + 2.0 * (2 * v ^ (2 - 1)) + 4.0u + 4.0w, -3.0 + 2.0 * (2 * w ^ (2 - 1)) + 4.0u + 4.0v))), Vec(float.(tuple(-1.0 + 2.0 * (2 * u ^ (2 - 1)), 0, 0))), Vec(float.(tuple(0, -1.0 + 2.0 * (2 * v ^ (2 - 1)), 0))), Vec(float.(tuple(0, 0, -1.0 + 2.0 * (2 * w ^ (2 - 1))))), Vec(float.(tuple(4.0 + -4.0 * (2 * u ^ (2 - 1)) + -4.0v + -4.0w, -4.0u, -4.0u))), Vec(float.(tuple(4.0v, 4.0u, 0))), Vec(float.(tuple(-4.0v, 4.0 + -4.0 * (2 * v ^ (2 - 1)) + -4.0u + -4.0w, -4.0v))), Vec(float.(tuple(-4.0w, -4.0w, 4.0 + -4.0 * (2 * w ^ (2 - 1)) + -4.0u + -4.0v))), Vec(float.(tuple(4.0w, 0, 4.0u))), Vec(float.(tuple(0, 4.0w, 4.0v)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Hex8: 8-node trilinear hexahedral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Hex8Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Hex8Basis}) + return (3, 8) + end + function Base.size(::Type{Hex8Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 8 + end + Base.@pure function Base.length(::Type{Hex8Basis}) + return 8 + end + function get_reference_element_coordinates(::Type{Hex8Basis}) + return Vec{3, Float64}[[-1.0, -1.0, -1.0], [1.0, -1.0, -1.0], [1.0, 1.0, -1.0], [-1.0, 1.0, -1.0], [-1.0, -1.0, 1.0], [1.0, -1.0, 1.0], [1.0, 1.0, 1.0], [-1.0, 1.0, 1.0]] + end + @inline function eval_basis!(::Type{Hex8Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (0.125 + -0.125u + -0.125v + -0.125w + 0.125 * (u * v) + 0.125 * (u * w) + 0.125 * (v * w) + -0.125 * (u * v * w), 0.125 + 0.125u + -0.125v + -0.125w + -0.125 * (u * v) + -0.125 * (u * w) + 0.125 * (v * w) + 0.125 * (u * v * w), 0.125 + 0.125u + 0.125v + -0.125w + 0.125 * (u * v) + -0.125 * (u * w) + -0.125 * (v * w) + -0.125 * (u * v * w), 0.125 + -0.125u + 0.125v + -0.125w + -0.125 * (u * v) + 0.125 * (u * w) + -0.125 * (v * w) + 0.125 * (u * v * w), 0.125 + -0.125u + -0.125v + 0.125w + 0.125 * (u * v) + -0.125 * (u * w) + -0.125 * (v * w) + 0.125 * (u * v * w), 0.125 + 0.125u + -0.125v + 0.125w + -0.125 * (u * v) + 0.125 * (u * w) + -0.125 * (v * w) + -0.125 * (u * v * w), 0.125 + 0.125u + 0.125v + 0.125w + 0.125 * (u * v) + 0.125 * (u * w) + 0.125 * (v * w) + 0.125 * (u * v * w), 0.125 + -0.125u + 0.125v + 0.125w + -0.125 * (u * v) + -0.125 * (u * w) + 0.125 * (v * w) + -0.125 * (u * v * w)) + end + @inline function eval_dbasis!(::Type{Hex8Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-0.125 + 0.125v + 0.125w + -0.125 * (v * w), -0.125 + 0.125u + 0.125w + -0.125 * (u * w), -0.125 + 0.125u + 0.125v + -0.125 * (u * v)))), Vec(float.(tuple(0.125 + -0.125v + -0.125w + 0.125 * (v * w), -0.125 + -0.125u + 0.125w + 0.125 * (u * w), -0.125 + -0.125u + 0.125v + 0.125 * (u * v)))), Vec(float.(tuple(0.125 + 0.125v + -0.125w + -0.125 * (v * w), 0.125 + 0.125u + -0.125w + -0.125 * (u * w), -0.125 + -0.125u + -0.125v + -0.125 * (u * v)))), Vec(float.(tuple(-0.125 + -0.125v + 0.125w + 0.125 * (v * w), 0.125 + -0.125u + -0.125w + 0.125 * (u * w), -0.125 + 0.125u + -0.125v + 0.125 * (u * v)))), Vec(float.(tuple(-0.125 + 0.125v + -0.125w + 0.125 * (v * w), -0.125 + 0.125u + -0.125w + 0.125 * (u * w), 0.125 + -0.125u + -0.125v + 0.125 * (u * v)))), Vec(float.(tuple(0.125 + -0.125v + 0.125w + -0.125 * (v * w), -0.125 + -0.125u + -0.125w + -0.125 * (u * w), 0.125 + 0.125u + -0.125v + -0.125 * (u * v)))), Vec(float.(tuple(0.125 + 0.125v + 0.125w + 0.125 * (v * w), 0.125 + 0.125u + 0.125w + 0.125 * (u * w), 0.125 + 0.125u + 0.125v + 0.125 * (u * v)))), Vec(float.(tuple(-0.125 + -0.125v + -0.125w + -0.125 * (v * w), 0.125 + -0.125u + 0.125w + -0.125 * (u * w), 0.125 + -0.125u + 0.125v + -0.125 * (u * v))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Hex20: 20-node serendipity hexahedral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Hex20Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Hex20Basis}) + return (3, 20) + end + function Base.size(::Type{Hex20Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 20 + end + Base.@pure function Base.length(::Type{Hex20Basis}) + return 20 + end + function get_reference_element_coordinates(::Type{Hex20Basis}) + return Vec{3, Float64}[[-1.0, -1.0, -1.0], [1.0, -1.0, -1.0], [1.0, 1.0, -1.0], [-1.0, 1.0, -1.0], [-1.0, -1.0, 1.0], [1.0, -1.0, 1.0], [1.0, 1.0, 1.0], [-1.0, 1.0, 1.0], [0.0, -1.0, -1.0], [1.0, 0.0, -1.0], [0.0, 1.0, -1.0], [-1.0, 0.0, -1.0], [-1.0, -1.0, 0.0], [1.0, -1.0, 0.0], [1.0, 1.0, 0.0], [-1.0, 1.0, 0.0], [0.0, -1.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0], [-1.0, 0.0, 1.0]] + end + @inline function eval_basis!(::Type{Hex20Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (-0.25 + 0.125u + 0.125v + 0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (u ^ 2 * v) + -0.125 * (u ^ 2 * w) + -0.125 * (v ^ 2 * u) + -0.125 * (v ^ 2 * w) + -0.125 * (w ^ 2 * u) + -0.125 * (w ^ 2 * v) + -0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2), -0.25 + -0.125u + 0.125v + 0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (u ^ 2 * v) + -0.125 * (u ^ 2 * w) + 0.125 * (v ^ 2 * u) + -0.125 * (v ^ 2 * w) + 0.125 * (w ^ 2 * u) + -0.125 * (w ^ 2 * v) + 0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2), -0.25 + -0.125u + -0.125v + 0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (u ^ 2 * v) + -0.125 * (u ^ 2 * w) + 0.125 * (v ^ 2 * u) + -0.125 * (v ^ 2 * w) + 0.125 * (w ^ 2 * u) + 0.125 * (w ^ 2 * v) + -0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2), -0.25 + 0.125u + -0.125v + 0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (u ^ 2 * v) + -0.125 * (u ^ 2 * w) + -0.125 * (v ^ 2 * u) + -0.125 * (v ^ 2 * w) + -0.125 * (w ^ 2 * u) + 0.125 * (w ^ 2 * v) + 0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2), -0.25 + 0.125u + 0.125v + -0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (u ^ 2 * v) + 0.125 * (u ^ 2 * w) + -0.125 * (v ^ 2 * u) + 0.125 * (v ^ 2 * w) + -0.125 * (w ^ 2 * u) + -0.125 * (w ^ 2 * v) + 0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2), -0.25 + -0.125u + 0.125v + -0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (u ^ 2 * v) + 0.125 * (u ^ 2 * w) + 0.125 * (v ^ 2 * u) + 0.125 * (v ^ 2 * w) + 0.125 * (w ^ 2 * u) + -0.125 * (w ^ 2 * v) + -0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2), -0.25 + -0.125u + -0.125v + -0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (u ^ 2 * v) + 0.125 * (u ^ 2 * w) + 0.125 * (v ^ 2 * u) + 0.125 * (v ^ 2 * w) + 0.125 * (w ^ 2 * u) + 0.125 * (w ^ 2 * v) + 0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2), -0.25 + 0.125u + -0.125v + -0.125w + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (u ^ 2 * v) + 0.125 * (u ^ 2 * w) + -0.125 * (v ^ 2 * u) + 0.125 * (v ^ 2 * w) + -0.125 * (w ^ 2 * u) + 0.125 * (w ^ 2 * v) + -0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2), 0.25 + -0.25v + -0.25w + -0.25 * u ^ 2 + 0.25 * (v * w) + 0.25 * (u ^ 2 * v) + 0.25 * (u ^ 2 * w) + -0.25 * (u ^ 2 * v * w), 0.25 + 0.25u + -0.25w + -0.25 * v ^ 2 + -0.25 * (u * w) + -0.25 * (v ^ 2 * u) + 0.25 * (v ^ 2 * w) + 0.25 * (u * v ^ 2 * w), 0.25 + 0.25v + -0.25w + -0.25 * u ^ 2 + -0.25 * (v * w) + -0.25 * (u ^ 2 * v) + 0.25 * (u ^ 2 * w) + 0.25 * (u ^ 2 * v * w), 0.25 + -0.25u + -0.25w + -0.25 * v ^ 2 + 0.25 * (u * w) + 0.25 * (v ^ 2 * u) + 0.25 * (v ^ 2 * w) + -0.25 * (u * v ^ 2 * w), 0.25 + -0.25u + -0.25v + -0.25 * w ^ 2 + 0.25 * (u * v) + 0.25 * (w ^ 2 * u) + 0.25 * (w ^ 2 * v) + -0.25 * (u * v * w ^ 2), 0.25 + 0.25u + -0.25v + -0.25 * w ^ 2 + -0.25 * (u * v) + -0.25 * (w ^ 2 * u) + 0.25 * (w ^ 2 * v) + 0.25 * (u * v * w ^ 2), 0.25 + 0.25u + 0.25v + -0.25 * w ^ 2 + 0.25 * (u * v) + -0.25 * (w ^ 2 * u) + -0.25 * (w ^ 2 * v) + -0.25 * (u * v * w ^ 2), 0.25 + -0.25u + 0.25v + -0.25 * w ^ 2 + -0.25 * (u * v) + 0.25 * (w ^ 2 * u) + -0.25 * (w ^ 2 * v) + 0.25 * (u * v * w ^ 2), 0.25 + -0.25v + 0.25w + -0.25 * u ^ 2 + -0.25 * (v * w) + 0.25 * (u ^ 2 * v) + -0.25 * (u ^ 2 * w) + 0.25 * (u ^ 2 * v * w), 0.25 + 0.25u + 0.25w + -0.25 * v ^ 2 + 0.25 * (u * w) + -0.25 * (v ^ 2 * u) + -0.25 * (v ^ 2 * w) + -0.25 * (u * v ^ 2 * w), 0.25 + 0.25v + 0.25w + -0.25 * u ^ 2 + 0.25 * (v * w) + -0.25 * (u ^ 2 * v) + -0.25 * (u ^ 2 * w) + -0.25 * (u ^ 2 * v * w), 0.25 + -0.25u + 0.25w + -0.25 * v ^ 2 + -0.25 * (u * w) + 0.25 * (v ^ 2 * u) + -0.25 * (v ^ 2 * w) + 0.25 * (u * v ^ 2 * w)) + end + @inline function eval_dbasis!(::Type{Hex20Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(0.125 + 0.125 * (2 * u ^ (2 - 1)) + -0.125 * ((2 * u ^ (2 - 1)) * v) + -0.125 * ((2 * u ^ (2 - 1)) * w) + -0.125 * v ^ 2 + -0.125 * w ^ 2 + -0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2), 0.125 + 0.125 * (2 * v ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * ((2 * v ^ (2 - 1)) * u) + -0.125 * ((2 * v ^ (2 - 1)) * w) + -0.125 * w ^ 2 + -0.125 * (u * w) + 0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2), 0.125 + 0.125 * (2 * w ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * v ^ 2 + -0.125 * ((2 * w ^ (2 - 1)) * u) + -0.125 * ((2 * w ^ (2 - 1)) * v) + -0.125 * (u * v) + 0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 + 0.125 * (2 * u ^ (2 - 1)) + -0.125 * ((2 * u ^ (2 - 1)) * v) + -0.125 * ((2 * u ^ (2 - 1)) * w) + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2), 0.125 + 0.125 * (2 * v ^ (2 - 1)) + -0.125 * u ^ 2 + 0.125 * ((2 * v ^ (2 - 1)) * u) + -0.125 * ((2 * v ^ (2 - 1)) * w) + -0.125 * w ^ 2 + 0.125 * (u * w) + 0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2), 0.125 + 0.125 * (2 * w ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * v ^ 2 + 0.125 * ((2 * w ^ (2 - 1)) * u) + -0.125 * ((2 * w ^ (2 - 1)) * v) + 0.125 * (u * v) + 0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 + 0.125 * (2 * u ^ (2 - 1)) + 0.125 * ((2 * u ^ (2 - 1)) * v) + -0.125 * ((2 * u ^ (2 - 1)) * w) + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2), -0.125 + 0.125 * (2 * v ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * ((2 * v ^ (2 - 1)) * u) + -0.125 * ((2 * v ^ (2 - 1)) * w) + 0.125 * w ^ 2 + -0.125 * (u * w) + -0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2), 0.125 + 0.125 * (2 * w ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * v ^ 2 + 0.125 * ((2 * w ^ (2 - 1)) * u) + 0.125 * ((2 * w ^ (2 - 1)) * v) + -0.125 * (u * v) + -0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 + 0.125 * (2 * u ^ (2 - 1)) + 0.125 * ((2 * u ^ (2 - 1)) * v) + -0.125 * ((2 * u ^ (2 - 1)) * w) + -0.125 * v ^ 2 + -0.125 * w ^ 2 + 0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2), -0.125 + 0.125 * (2 * v ^ (2 - 1)) + 0.125 * u ^ 2 + -0.125 * ((2 * v ^ (2 - 1)) * u) + -0.125 * ((2 * v ^ (2 - 1)) * w) + 0.125 * w ^ 2 + 0.125 * (u * w) + -0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2), 0.125 + 0.125 * (2 * w ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * v ^ 2 + -0.125 * ((2 * w ^ (2 - 1)) * u) + 0.125 * ((2 * w ^ (2 - 1)) * v) + 0.125 * (u * v) + -0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 + 0.125 * (2 * u ^ (2 - 1)) + -0.125 * ((2 * u ^ (2 - 1)) * v) + 0.125 * ((2 * u ^ (2 - 1)) * w) + -0.125 * v ^ 2 + -0.125 * w ^ 2 + 0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2), 0.125 + 0.125 * (2 * v ^ (2 - 1)) + -0.125 * u ^ 2 + -0.125 * ((2 * v ^ (2 - 1)) * u) + 0.125 * ((2 * v ^ (2 - 1)) * w) + -0.125 * w ^ 2 + 0.125 * (u * w) + -0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2), -0.125 + 0.125 * (2 * w ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * v ^ 2 + -0.125 * ((2 * w ^ (2 - 1)) * u) + -0.125 * ((2 * w ^ (2 - 1)) * v) + 0.125 * (u * v) + -0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 + 0.125 * (2 * u ^ (2 - 1)) + -0.125 * ((2 * u ^ (2 - 1)) * v) + 0.125 * ((2 * u ^ (2 - 1)) * w) + 0.125 * v ^ 2 + 0.125 * w ^ 2 + -0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2), 0.125 + 0.125 * (2 * v ^ (2 - 1)) + -0.125 * u ^ 2 + 0.125 * ((2 * v ^ (2 - 1)) * u) + 0.125 * ((2 * v ^ (2 - 1)) * w) + -0.125 * w ^ 2 + -0.125 * (u * w) + -0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2), -0.125 + 0.125 * (2 * w ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * ((2 * w ^ (2 - 1)) * u) + -0.125 * ((2 * w ^ (2 - 1)) * v) + -0.125 * (u * v) + -0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 + 0.125 * (2 * u ^ (2 - 1)) + 0.125 * ((2 * u ^ (2 - 1)) * v) + 0.125 * ((2 * u ^ (2 - 1)) * w) + 0.125 * v ^ 2 + 0.125 * w ^ 2 + 0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2), -0.125 + 0.125 * (2 * v ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * ((2 * v ^ (2 - 1)) * u) + 0.125 * ((2 * v ^ (2 - 1)) * w) + 0.125 * w ^ 2 + 0.125 * (u * w) + 0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2), -0.125 + 0.125 * (2 * w ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * v ^ 2 + 0.125 * ((2 * w ^ (2 - 1)) * u) + 0.125 * ((2 * w ^ (2 - 1)) * v) + 0.125 * (u * v) + 0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 + 0.125 * (2 * u ^ (2 - 1)) + 0.125 * ((2 * u ^ (2 - 1)) * v) + 0.125 * ((2 * u ^ (2 - 1)) * w) + -0.125 * v ^ 2 + -0.125 * w ^ 2 + -0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2), -0.125 + 0.125 * (2 * v ^ (2 - 1)) + 0.125 * u ^ 2 + -0.125 * ((2 * v ^ (2 - 1)) * u) + 0.125 * ((2 * v ^ (2 - 1)) * w) + 0.125 * w ^ 2 + -0.125 * (u * w) + 0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2), -0.125 + 0.125 * (2 * w ^ (2 - 1)) + 0.125 * u ^ 2 + 0.125 * v ^ 2 + -0.125 * ((2 * w ^ (2 - 1)) * u) + 0.125 * ((2 * w ^ (2 - 1)) * v) + -0.125 * (u * v) + 0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25 * (2 * u ^ (2 - 1)) + 0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * ((2 * u ^ (2 - 1)) * w) + -0.25 * ((2 * u ^ (2 - 1)) * v * w), -0.25 + 0.25w + 0.25 * u ^ 2 + -0.25 * (u ^ 2 * w), -0.25 + 0.25v + 0.25 * u ^ 2 + -0.25 * (u ^ 2 * v)))), Vec(float.(tuple(0.25 + -0.25w + -0.25 * v ^ 2 + 0.25 * (v ^ 2 * w), -0.25 * (2 * v ^ (2 - 1)) + -0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * ((2 * v ^ (2 - 1)) * w) + 0.25 * (u * (2 * v ^ (2 - 1)) * w), -0.25 + -0.25u + 0.25 * v ^ 2 + 0.25 * (u * v ^ 2)))), Vec(float.(tuple(-0.25 * (2 * u ^ (2 - 1)) + -0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * ((2 * u ^ (2 - 1)) * w) + 0.25 * ((2 * u ^ (2 - 1)) * v * w), 0.25 + -0.25w + -0.25 * u ^ 2 + 0.25 * (u ^ 2 * w), -0.25 + -0.25v + 0.25 * u ^ 2 + 0.25 * (u ^ 2 * v)))), Vec(float.(tuple(-0.25 + 0.25w + 0.25 * v ^ 2 + -0.25 * (v ^ 2 * w), -0.25 * (2 * v ^ (2 - 1)) + 0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * ((2 * v ^ (2 - 1)) * w) + -0.25 * (u * (2 * v ^ (2 - 1)) * w), -0.25 + 0.25u + 0.25 * v ^ 2 + -0.25 * (u * v ^ 2)))), Vec(float.(tuple(-0.25 + 0.25v + 0.25 * w ^ 2 + -0.25 * (v * w ^ 2), -0.25 + 0.25u + 0.25 * w ^ 2 + -0.25 * (u * w ^ 2), -0.25 * (2 * w ^ (2 - 1)) + 0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * ((2 * w ^ (2 - 1)) * v) + -0.25 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25 + -0.25v + -0.25 * w ^ 2 + 0.25 * (v * w ^ 2), -0.25 + -0.25u + 0.25 * w ^ 2 + 0.25 * (u * w ^ 2), -0.25 * (2 * w ^ (2 - 1)) + -0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25 + 0.25v + -0.25 * w ^ 2 + -0.25 * (v * w ^ 2), 0.25 + 0.25u + -0.25 * w ^ 2 + -0.25 * (u * w ^ 2), -0.25 * (2 * w ^ (2 - 1)) + -0.25 * ((2 * w ^ (2 - 1)) * u) + -0.25 * ((2 * w ^ (2 - 1)) * v) + -0.25 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25 + -0.25v + 0.25 * w ^ 2 + 0.25 * (v * w ^ 2), 0.25 + -0.25u + -0.25 * w ^ 2 + 0.25 * (u * w ^ 2), -0.25 * (2 * w ^ (2 - 1)) + 0.25 * ((2 * w ^ (2 - 1)) * u) + -0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (u * v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25 * (2 * u ^ (2 - 1)) + 0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * ((2 * u ^ (2 - 1)) * w) + 0.25 * ((2 * u ^ (2 - 1)) * v * w), -0.25 + -0.25w + 0.25 * u ^ 2 + 0.25 * (u ^ 2 * w), 0.25 + -0.25v + -0.25 * u ^ 2 + 0.25 * (u ^ 2 * v)))), Vec(float.(tuple(0.25 + 0.25w + -0.25 * v ^ 2 + -0.25 * (v ^ 2 * w), -0.25 * (2 * v ^ (2 - 1)) + -0.25 * ((2 * v ^ (2 - 1)) * u) + -0.25 * ((2 * v ^ (2 - 1)) * w) + -0.25 * (u * (2 * v ^ (2 - 1)) * w), 0.25 + 0.25u + -0.25 * v ^ 2 + -0.25 * (u * v ^ 2)))), Vec(float.(tuple(-0.25 * (2 * u ^ (2 - 1)) + -0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * ((2 * u ^ (2 - 1)) * w) + -0.25 * ((2 * u ^ (2 - 1)) * v * w), 0.25 + 0.25w + -0.25 * u ^ 2 + -0.25 * (u ^ 2 * w), 0.25 + 0.25v + -0.25 * u ^ 2 + -0.25 * (u ^ 2 * v)))), Vec(float.(tuple(-0.25 + -0.25w + 0.25 * v ^ 2 + 0.25 * (v ^ 2 * w), -0.25 * (2 * v ^ (2 - 1)) + 0.25 * ((2 * v ^ (2 - 1)) * u) + -0.25 * ((2 * v ^ (2 - 1)) * w) + 0.25 * (u * (2 * v ^ (2 - 1)) * w), 0.25 + -0.25u + -0.25 * v ^ 2 + 0.25 * (u * v ^ 2))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Hex27: 27-node triquadratic hexahedral element +# ────────────────────────────────────────────────────────────────────────────── + + struct Hex27Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Hex27Basis}) + return (3, 27) + end + function Base.size(::Type{Hex27Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 27 + end + Base.@pure function Base.length(::Type{Hex27Basis}) + return 27 + end + function get_reference_element_coordinates(::Type{Hex27Basis}) + return Vec{3, Float64}[[-1.0, -1.0, -1.0], [1.0, -1.0, -1.0], [1.0, 1.0, -1.0], [-1.0, 1.0, -1.0], [-1.0, -1.0, 1.0], [1.0, -1.0, 1.0], [1.0, 1.0, 1.0], [-1.0, 1.0, 1.0], [0.0, -1.0, -1.0], [1.0, 0.0, -1.0], [0.0, 1.0, -1.0], [-1.0, 0.0, -1.0], [-1.0, -1.0, 0.0], [1.0, -1.0, 0.0], [1.0, 1.0, 0.0], [-1.0, 1.0, 0.0], [0.0, -1.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0], [-1.0, 0.0, 1.0], [0.0, 0.0, -1.0], [0.0, 0.0, 1.0], [0.0, -1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [-1.0, 0.0, 0.0], [0.0, 0.0, 0.0]] + end + @inline function eval_basis!(::Type{Hex27Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (-0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2) + -0.125 * (u ^ 2 * v ^ 2 * w) + -0.125 * (u ^ 2 * v * w ^ 2) + -0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), 0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2) + -0.125 * (u ^ 2 * v ^ 2 * w) + -0.125 * (u ^ 2 * v * w ^ 2) + 0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), -0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2) + -0.125 * (u ^ 2 * v ^ 2 * w) + 0.125 * (u ^ 2 * v * w ^ 2) + 0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), 0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2) + -0.125 * (u ^ 2 * v ^ 2 * w) + 0.125 * (u ^ 2 * v * w ^ 2) + -0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), 0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w) + -0.125 * (u ^ 2 * v * w ^ 2) + -0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), -0.125 * (u * v * w) + -0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w) + -0.125 * (u ^ 2 * v * w ^ 2) + 0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), 0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + 0.125 * (u * v ^ 2 * w) + 0.125 * (u * v * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w) + 0.125 * (u ^ 2 * v * w ^ 2) + 0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), -0.125 * (u * v * w) + 0.125 * (u ^ 2 * v * w) + -0.125 * (u * v ^ 2 * w) + -0.125 * (u * v * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w) + 0.125 * (u ^ 2 * v * w ^ 2) + -0.125 * (u * v ^ 2 * w ^ 2) + 0.125 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (v * w) + -0.25 * (v ^ 2 * w) + -0.25 * (w ^ 2 * v) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v * w) + 0.25 * (u ^ 2 * v ^ 2 * w) + 0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (u * w) + -0.25 * (u ^ 2 * w) + 0.25 * (w ^ 2 * u) + 0.25 * (u ^ 2 * w ^ 2) + 0.25 * (u * v ^ 2 * w) + 0.25 * (u ^ 2 * v ^ 2 * w) + -0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (v * w) + -0.25 * (v ^ 2 * w) + 0.25 * (w ^ 2 * v) + 0.25 * (v ^ 2 * w ^ 2) + 0.25 * (u ^ 2 * v * w) + 0.25 * (u ^ 2 * v ^ 2 * w) + -0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (u * w) + -0.25 * (u ^ 2 * w) + -0.25 * (w ^ 2 * u) + 0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u * v ^ 2 * w) + 0.25 * (u ^ 2 * v ^ 2 * w) + 0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (u * v) + -0.25 * (u ^ 2 * v) + -0.25 * (v ^ 2 * u) + 0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u * v * w ^ 2) + 0.25 * (u ^ 2 * v * w ^ 2) + 0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (u * v) + -0.25 * (u ^ 2 * v) + 0.25 * (v ^ 2 * u) + 0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u * v * w ^ 2) + 0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (u * v) + 0.25 * (u ^ 2 * v) + 0.25 * (v ^ 2 * u) + 0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u * v * w ^ 2) + -0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (u * v) + 0.25 * (u ^ 2 * v) + -0.25 * (v ^ 2 * u) + 0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u * v * w ^ 2) + -0.25 * (u ^ 2 * v * w ^ 2) + 0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (v * w) + 0.25 * (v ^ 2 * w) + -0.25 * (w ^ 2 * v) + 0.25 * (v ^ 2 * w ^ 2) + 0.25 * (u ^ 2 * v * w) + -0.25 * (u ^ 2 * v ^ 2 * w) + 0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (u * w) + 0.25 * (u ^ 2 * w) + 0.25 * (w ^ 2 * u) + 0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u * v ^ 2 * w) + -0.25 * (u ^ 2 * v ^ 2 * w) + -0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), 0.25 * (v * w) + 0.25 * (v ^ 2 * w) + 0.25 * (w ^ 2 * v) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v * w) + -0.25 * (u ^ 2 * v ^ 2 * w) + -0.25 * (u ^ 2 * v * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.25 * (u * w) + 0.25 * (u ^ 2 * w) + -0.25 * (w ^ 2 * u) + 0.25 * (u ^ 2 * w ^ 2) + 0.25 * (u * v ^ 2 * w) + -0.25 * (u ^ 2 * v ^ 2 * w) + 0.25 * (u * v ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * v ^ 2 * w ^ 2), -0.5w + 0.5 * w ^ 2 + 0.5 * (u ^ 2 * w) + 0.5 * (v ^ 2 * w) + -0.5 * (u ^ 2 * w ^ 2) + -0.5 * (v ^ 2 * w ^ 2) + -0.5 * (u ^ 2 * v ^ 2 * w) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), 0.5w + 0.5 * w ^ 2 + -0.5 * (u ^ 2 * w) + -0.5 * (v ^ 2 * w) + -0.5 * (u ^ 2 * w ^ 2) + -0.5 * (v ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * w) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), -0.5v + 0.5 * v ^ 2 + 0.5 * (u ^ 2 * v) + 0.5 * (w ^ 2 * v) + -0.5 * (u ^ 2 * v ^ 2) + -0.5 * (v ^ 2 * w ^ 2) + -0.5 * (u ^ 2 * v * w ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), 0.5u + 0.5 * u ^ 2 + -0.5 * (v ^ 2 * u) + -0.5 * (w ^ 2 * u) + -0.5 * (u ^ 2 * v ^ 2) + -0.5 * (u ^ 2 * w ^ 2) + 0.5 * (u * v ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), 0.5v + 0.5 * v ^ 2 + -0.5 * (u ^ 2 * v) + -0.5 * (w ^ 2 * v) + -0.5 * (u ^ 2 * v ^ 2) + -0.5 * (v ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * v * w ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), -0.5u + 0.5 * u ^ 2 + 0.5 * (v ^ 2 * u) + 0.5 * (w ^ 2 * u) + -0.5 * (u ^ 2 * v ^ 2) + -0.5 * (u ^ 2 * w ^ 2) + -0.5 * (u * v ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * w ^ 2), 1 + -1.0 * u ^ 2 + -1.0 * v ^ 2 + -1.0 * w ^ 2 + u ^ 2 * v ^ 2 + u ^ 2 * w ^ 2 + v ^ 2 * w ^ 2 + -1.0 * (u ^ 2 * v ^ 2 * w ^ 2)) + end + @inline function eval_dbasis!(::Type{Hex27Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2) + -0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.125 * (u * w) + 0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2) + -0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.125 * (u ^ 2 * w ^ 2) + -0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.125 * (u * v) + 0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1))) + -0.125 * (u ^ 2 * v ^ 2) + -0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2) + -0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.125 * (u * w) + 0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2) + -0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.125 * (u ^ 2 * w ^ 2) + 0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.125 * (u * v) + 0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1))) + -0.125 * (u ^ 2 * v ^ 2) + -0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2) + -0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.125 * (u * w) + -0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2) + -0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.125 * (u ^ 2 * w ^ 2) + 0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.125 * (u * v) + -0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1))) + -0.125 * (u ^ 2 * v ^ 2) + 0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2) + -0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.125 * (u * w) + -0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2) + -0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.125 * (u ^ 2 * w ^ 2) + -0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.125 * (u * v) + -0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1))) + -0.125 * (u ^ 2 * v ^ 2) + 0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.125 * (u * w) + -0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.125 * (u ^ 2 * w ^ 2) + -0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.125 * (u * v) + -0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2) + -0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 * (v * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.125 * (u * w) + -0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.125 * (u ^ 2 * w ^ 2) + 0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.125 * (u * v) + -0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2) + -0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + 0.125 * (v ^ 2 * w) + 0.125 * (v * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.125 * (u * w) + 0.125 * (u ^ 2 * w) + 0.125 * (u * (2 * v ^ (2 - 1)) * w) + 0.125 * (u * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.125 * (u ^ 2 * w ^ 2) + 0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.125 * (u * v) + 0.125 * (u ^ 2 * v) + 0.125 * (u * v ^ 2) + 0.125 * (u * v * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2) + 0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.125 * (v * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w) + -0.125 * (v ^ 2 * w) + -0.125 * (v * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.125 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.125 * (v ^ 2 * w ^ 2) + 0.125 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.125 * (u * w) + 0.125 * (u ^ 2 * w) + -0.125 * (u * (2 * v ^ (2 - 1)) * w) + -0.125 * (u * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.125 * (u ^ 2 * w ^ 2) + -0.125 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.125 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.125 * (u * v) + 0.125 * (u ^ 2 * v) + -0.125 * (u * v ^ 2) + -0.125 * (u * v * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2) + 0.125 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.125 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.125 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25 * ((2 * u ^ (2 - 1)) * v * w) + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25w + -0.25 * ((2 * v ^ (2 - 1)) * w) + -0.25 * w ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * w) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25v + -0.25 * v ^ 2 + -0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v) + 0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25w + -0.25 * ((2 * u ^ (2 - 1)) * w) + 0.25 * w ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * w ^ 2) + 0.25 * (v ^ 2 * w) + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25 * (u * (2 * v ^ (2 - 1)) * w) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25u + -0.25 * u ^ 2 + 0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * w ^ (2 - 1))) + 0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25 * ((2 * u ^ (2 - 1)) * v * w) + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25w + -0.25 * ((2 * v ^ (2 - 1)) * w) + 0.25 * w ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * w ^ 2) + 0.25 * (u ^ 2 * w) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25v + -0.25 * v ^ 2 + 0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (v ^ 2 * (2 * w ^ (2 - 1))) + 0.25 * (u ^ 2 * v) + 0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25w + -0.25 * ((2 * u ^ (2 - 1)) * w) + -0.25 * w ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * w ^ 2) + -0.25 * (v ^ 2 * w) + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25 * (u * (2 * v ^ (2 - 1)) * w) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25u + -0.25 * u ^ 2 + -0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u * v ^ 2) + 0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2) + -0.25 * (v * w ^ 2) + 0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25u + -0.25 * u ^ 2 + -0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1))) + -0.25 * (u * w ^ 2) + 0.25 * (u ^ 2 * w ^ 2) + 0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25 * (u * v * (2 * w ^ (2 - 1))) + 0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25v + -0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2) + 0.25 * (v * w ^ 2) + 0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25u + -0.25 * u ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1))) + 0.25 * (u * w ^ 2) + 0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25 * (u * v * (2 * w ^ (2 - 1))) + 0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + 0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2) + -0.25 * (v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25u + 0.25 * u ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1))) + -0.25 * (u * w ^ 2) + -0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25 * (u * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25v + 0.25 * ((2 * u ^ (2 - 1)) * v) + -0.25 * v ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * v ^ 2) + 0.25 * (v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25u + 0.25 * u ^ 2 + -0.25 * ((2 * v ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * v ^ (2 - 1))) + 0.25 * (u * w ^ 2) + -0.25 * (u ^ 2 * w ^ 2) + 0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25 * (u * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25 * ((2 * u ^ (2 - 1)) * v * w) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25w + 0.25 * ((2 * v ^ (2 - 1)) * w) + -0.25 * w ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * w ^ 2) + 0.25 * (u ^ 2 * w) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25v + 0.25 * v ^ 2 + -0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (v ^ 2 * (2 * w ^ (2 - 1))) + 0.25 * (u ^ 2 * v) + -0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.25w + 0.25 * ((2 * u ^ (2 - 1)) * w) + 0.25 * w ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * w ^ 2) + -0.25 * (v ^ 2 * w) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.25 * (u * (2 * v ^ (2 - 1)) * w) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25u + 0.25 * u ^ 2 + 0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u * v ^ 2) + -0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25 * ((2 * u ^ (2 - 1)) * v * w) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + -0.25 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25w + 0.25 * ((2 * v ^ (2 - 1)) * w) + 0.25 * w ^ 2 + 0.25 * ((2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * w) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + -0.25 * (u ^ 2 * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.25v + 0.25 * v ^ 2 + 0.25 * ((2 * w ^ (2 - 1)) * v) + 0.25 * (v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v) + -0.25 * (u ^ 2 * v ^ 2) + -0.25 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.25w + 0.25 * ((2 * u ^ (2 - 1)) * w) + -0.25 * w ^ 2 + 0.25 * ((2 * u ^ (2 - 1)) * w ^ 2) + 0.25 * (v ^ 2 * w) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.25 * (v ^ 2 * w ^ 2) + -0.25 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.25 * (u * (2 * v ^ (2 - 1)) * w) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.25 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + -0.25 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.25u + 0.25 * u ^ 2 + -0.25 * ((2 * w ^ (2 - 1)) * u) + 0.25 * (u ^ 2 * (2 * w ^ (2 - 1))) + 0.25 * (u * v ^ 2) + -0.25 * (u ^ 2 * v ^ 2) + 0.25 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + -0.25 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.5 * ((2 * u ^ (2 - 1)) * w) + -0.5 * ((2 * u ^ (2 - 1)) * w ^ 2) + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.5 * ((2 * v ^ (2 - 1)) * w) + -0.5 * ((2 * v ^ (2 - 1)) * w ^ 2) + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.5 + 0.5 * (2 * w ^ (2 - 1)) + 0.5 * u ^ 2 + 0.5 * v ^ 2 + -0.5 * (u ^ 2 * (2 * w ^ (2 - 1))) + -0.5 * (v ^ 2 * (2 * w ^ (2 - 1))) + -0.5 * (u ^ 2 * v ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.5 * ((2 * u ^ (2 - 1)) * w) + -0.5 * ((2 * u ^ (2 - 1)) * w ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.5 * ((2 * v ^ (2 - 1)) * w) + -0.5 * ((2 * v ^ (2 - 1)) * w ^ 2) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.5 + 0.5 * (2 * w ^ (2 - 1)) + -0.5 * u ^ 2 + -0.5 * v ^ 2 + -0.5 * (u ^ 2 * (2 * w ^ (2 - 1))) + -0.5 * (v ^ 2 * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v ^ 2) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.5 * ((2 * u ^ (2 - 1)) * v) + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2) + -0.5 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.5 + 0.5 * (2 * v ^ (2 - 1)) + 0.5 * u ^ 2 + 0.5 * w ^ 2 + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1))) + -0.5 * ((2 * v ^ (2 - 1)) * w ^ 2) + -0.5 * (u ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.5 * ((2 * w ^ (2 - 1)) * v) + -0.5 * (v ^ 2 * (2 * w ^ (2 - 1))) + -0.5 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0.5 + 0.5 * (2 * u ^ (2 - 1)) + -0.5 * v ^ 2 + -0.5 * w ^ 2 + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2) + -0.5 * ((2 * u ^ (2 - 1)) * w ^ 2) + 0.5 * (v ^ 2 * w ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -0.5 * ((2 * v ^ (2 - 1)) * u) + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1))) + 0.5 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.5 * ((2 * w ^ (2 - 1)) * u) + -0.5 * (u ^ 2 * (2 * w ^ (2 - 1))) + 0.5 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.5 * ((2 * u ^ (2 - 1)) * v) + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v * w ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.5 + 0.5 * (2 * v ^ (2 - 1)) + -0.5 * u ^ 2 + -0.5 * w ^ 2 + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1))) + -0.5 * ((2 * v ^ (2 - 1)) * w ^ 2) + 0.5 * (u ^ 2 * w ^ 2) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -0.5 * ((2 * w ^ (2 - 1)) * v) + -0.5 * (v ^ 2 * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-0.5 + 0.5 * (2 * u ^ (2 - 1)) + 0.5 * v ^ 2 + 0.5 * w ^ 2 + -0.5 * ((2 * u ^ (2 - 1)) * v ^ 2) + -0.5 * ((2 * u ^ (2 - 1)) * w ^ 2) + -0.5 * (v ^ 2 * w ^ 2) + 0.5 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), 0.5 * ((2 * v ^ (2 - 1)) * u) + -0.5 * (u ^ 2 * (2 * v ^ (2 - 1))) + -0.5 * (u * (2 * v ^ (2 - 1)) * w ^ 2) + 0.5 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), 0.5 * ((2 * w ^ (2 - 1)) * u) + -0.5 * (u ^ 2 * (2 * w ^ (2 - 1))) + -0.5 * (u * v ^ 2 * (2 * w ^ (2 - 1))) + 0.5 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-1.0 * (2 * u ^ (2 - 1)) + (2 * u ^ (2 - 1)) * v ^ 2 + (2 * u ^ (2 - 1)) * w ^ 2 + -1.0 * ((2 * u ^ (2 - 1)) * v ^ 2 * w ^ 2), -1.0 * (2 * v ^ (2 - 1)) + u ^ 2 * (2 * v ^ (2 - 1)) + (2 * v ^ (2 - 1)) * w ^ 2 + -1.0 * (u ^ 2 * (2 * v ^ (2 - 1)) * w ^ 2), -1.0 * (2 * w ^ (2 - 1)) + u ^ 2 * (2 * w ^ (2 - 1)) + v ^ 2 * (2 * w ^ (2 - 1)) + -1.0 * (u ^ 2 * v ^ 2 * (2 * w ^ (2 - 1))))))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Pyr5: 5-node linear pyramid element +# ────────────────────────────────────────────────────────────────────────────── + + struct Pyr5Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Pyr5Basis}) + return (3, 5) + end + function Base.size(::Type{Pyr5Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 5 + end + Base.@pure function Base.length(::Type{Pyr5Basis}) + return 5 + end + function get_reference_element_coordinates(::Type{Pyr5Basis}) + return Vec{3, Float64}[[-1.0, -1.0, 0.0], [1.0, -1.0, 0.0], [1.0, 1.0, 0.0], [-1.0, 1.0, 0.0], [0.0, 0.0, 1.0]] + end + @inline function eval_basis!(::Type{Pyr5Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (0.25 + -0.25u + -0.25v + -0.25w + 0.25 * (u * v), 0.25 + 0.25u + -0.25v + -0.25w + -0.25 * (u * v), 0.25 + 0.25u + 0.25v + -0.25w + 0.25 * (u * v), 0.25 + -0.25u + 0.25v + -0.25w + -0.25 * (u * v), +w) + end + @inline function eval_dbasis!(::Type{Pyr5Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-0.25 + 0.25v, -0.25 + 0.25u, -0.25))), Vec(float.(tuple(0.25 + -0.25v, -0.25 + -0.25u, -0.25))), Vec(float.(tuple(0.25 + 0.25v, 0.25 + 0.25u, -0.25))), Vec(float.(tuple(-0.25 + -0.25v, 0.25 + -0.25u, -0.25))), Vec(float.(tuple(0, 0, 1)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Wedge6: 6-node linear wedge element (triangular prism) +# ────────────────────────────────────────────────────────────────────────────── + + struct Wedge6Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Wedge6Basis}) + return (3, 6) + end + function Base.size(::Type{Wedge6Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 6 + end + Base.@pure function Base.length(::Type{Wedge6Basis}) + return 6 + end + function get_reference_element_coordinates(::Type{Wedge6Basis}) + return Vec{3, Float64}[[0.0, 0.0, -1.0], [1.0, 0.0, -1.0], [0.0, 1.0, -1.0], [0.0, 0.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0]] + end + @inline function eval_basis!(::Type{Wedge6Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (0.5 + -0.5u + -0.5v + -0.5w + 0.5 * (u * w) + 0.5 * (v * w), 0.5u + -0.5 * (u * w), 0.5v + -0.5 * (v * w), 0.5 + -0.5u + -0.5v + 0.5w + -0.5 * (u * w) + -0.5 * (v * w), 0.5u + 0.5 * (u * w), 0.5v + 0.5 * (v * w)) + end + @inline function eval_dbasis!(::Type{Wedge6Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-0.5 + 0.5w, -0.5 + 0.5w, -0.5 + 0.5u + 0.5v))), Vec(float.(tuple(0.5 + -0.5w, 0, -0.5u))), Vec(float.(tuple(0, 0.5 + -0.5w, -0.5v))), Vec(float.(tuple(-0.5 + -0.5w, -0.5 + -0.5w, 0.5 + -0.5u + -0.5v))), Vec(float.(tuple(0.5 + 0.5w, 0, 0.5u))), Vec(float.(tuple(0, 0.5 + 0.5w, 0.5v)))) + end + +# ────────────────────────────────────────────────────────────────────────────── +# Wedge15: 15-node quadratic wedge element +# ────────────────────────────────────────────────────────────────────────────── + + struct Wedge15Basis <: AbstractBasis{3} + end + Base.@pure function Base.size(::Type{Wedge15Basis}) + return (3, 15) + end + function Base.size(::Type{Wedge15Basis}, j::Int) + j == 1 && return 3 + j == 2 && return 15 + end + Base.@pure function Base.length(::Type{Wedge15Basis}) + return 15 + end + function get_reference_element_coordinates(::Type{Wedge15Basis}) + return Vec{3, Float64}[[0.0, 0.0, -1.0], [1.0, 0.0, -1.0], [0.0, 1.0, -1.0], [0.0, 0.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0], [0.5, 0.0, -1.0], [0.5, 0.5, -1.0], [0.0, 0.5, -1.0], [0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.5, 0.0, 1.0], [0.5, 0.5, 1.0], [0.0, 0.5, 1.0]] + end + @inline function eval_basis!(::Type{Wedge15Basis}, ::Type{T}, xi::Vec) where T + (u, v, w) = xi + @inbounds return (-1.0u + -1.0v + -0.5w + u ^ 2 + v ^ 2 + 0.5 * w ^ 2 + 2.0 * (u * v) + 1.5 * (u * w) + 1.5 * (v * w) + -1.0 * (u ^ 2 * w) + -1.0 * (v ^ 2 * w) + -2.0 * (u * v * w) + -0.5 * (u * w ^ 2) + -0.5 * (v * w ^ 2), -1.0u + u ^ 2 + 0.5 * (u * w) + -1.0 * (u ^ 2 * w) + 0.5 * (u * w ^ 2), -1.0v + v ^ 2 + 0.5 * (v * w) + -1.0 * (v ^ 2 * w) + 0.5 * (v * w ^ 2), -1.0u + -1.0v + 0.5w + u ^ 2 + v ^ 2 + 0.5 * w ^ 2 + 2.0 * (u * v) + -1.5 * (u * w) + -1.5 * (v * w) + u ^ 2 * w + v ^ 2 * w + 2.0 * (u * v * w) + -0.5 * (u * w ^ 2) + -0.5 * (v * w ^ 2), -1.0u + u ^ 2 + -0.5 * (u * w) + u ^ 2 * w + 0.5 * (u * w ^ 2), -1.0v + v ^ 2 + -0.5 * (v * w) + v ^ 2 * w + 0.5 * (v * w ^ 2), 2.0u + -2.0 * u ^ 2 + -2.0 * (u * v) + -2.0 * (u * w) + 2.0 * (u ^ 2 * w) + 2.0 * (u * v * w), 2.0 * (u * v) + -2.0 * (u * v * w), 2.0v + -2.0 * v ^ 2 + -2.0 * (u * v) + -2.0 * (v * w) + 2.0 * (v ^ 2 * w) + 2.0 * (u * v * w), 1 + -1.0u + -1.0v + -1.0 * w ^ 2 + u * w ^ 2 + v * w ^ 2, u + -1.0 * (u * w ^ 2), v + -1.0 * (v * w ^ 2), 2.0u + -2.0 * u ^ 2 + -2.0 * (u * v) + 2.0 * (u * w) + -2.0 * (u ^ 2 * w) + -2.0 * (u * v * w), 2.0 * (u * v) + 2.0 * (u * v * w), 2.0v + -2.0 * v ^ 2 + -2.0 * (u * v) + 2.0 * (v * w) + -2.0 * (v ^ 2 * w) + -2.0 * (u * v * w)) + end + @inline function eval_dbasis!(::Type{Wedge15Basis}, xi::Vec) + (u, v, w) = xi + @inbounds return (Vec(float.(tuple(-1.0 + 2 * u ^ (2 - 1) + 2.0v + 1.5w + -1.0 * ((2 * u ^ (2 - 1)) * w) + -2.0 * (v * w) + -0.5 * w ^ 2, -1.0 + 2 * v ^ (2 - 1) + 2.0u + 1.5w + -1.0 * ((2 * v ^ (2 - 1)) * w) + -2.0 * (u * w) + -0.5 * w ^ 2, -0.5 + 0.5 * (2 * w ^ (2 - 1)) + 1.5u + 1.5v + -1.0 * u ^ 2 + -1.0 * v ^ 2 + -2.0 * (u * v) + -0.5 * (u * (2 * w ^ (2 - 1))) + -0.5 * (v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-1.0 + 2 * u ^ (2 - 1) + 0.5w + -1.0 * ((2 * u ^ (2 - 1)) * w) + 0.5 * w ^ 2, 0, 0.5u + -1.0 * u ^ 2 + 0.5 * (u * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0, -1.0 + 2 * v ^ (2 - 1) + 0.5w + -1.0 * ((2 * v ^ (2 - 1)) * w) + 0.5 * w ^ 2, 0.5v + -1.0 * v ^ 2 + 0.5 * (v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-1.0 + 2 * u ^ (2 - 1) + 2.0v + -1.5w + (2 * u ^ (2 - 1)) * w + 2.0 * (v * w) + -0.5 * w ^ 2, -1.0 + 2 * v ^ (2 - 1) + 2.0u + -1.5w + (2 * v ^ (2 - 1)) * w + 2.0 * (u * w) + -0.5 * w ^ 2, 0.5 + 0.5 * (2 * w ^ (2 - 1)) + -1.5u + -1.5v + u ^ 2 + v ^ 2 + 2.0 * (u * v) + -0.5 * (u * (2 * w ^ (2 - 1))) + -0.5 * (v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(-1.0 + 2 * u ^ (2 - 1) + -0.5w + (2 * u ^ (2 - 1)) * w + 0.5 * w ^ 2, 0, -0.5u + u ^ 2 + 0.5 * (u * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0, -1.0 + 2 * v ^ (2 - 1) + -0.5w + (2 * v ^ (2 - 1)) * w + 0.5 * w ^ 2, -0.5v + v ^ 2 + 0.5 * (v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(2.0 + -2.0 * (2 * u ^ (2 - 1)) + -2.0v + -2.0w + 2.0 * ((2 * u ^ (2 - 1)) * w) + 2.0 * (v * w), -2.0u + 2.0 * (u * w), -2.0u + 2.0 * u ^ 2 + 2.0 * (u * v)))), Vec(float.(tuple(2.0v + -2.0 * (v * w), 2.0u + -2.0 * (u * w), -2.0 * (u * v)))), Vec(float.(tuple(-2.0v + 2.0 * (v * w), 2.0 + -2.0 * (2 * v ^ (2 - 1)) + -2.0u + -2.0w + 2.0 * ((2 * v ^ (2 - 1)) * w) + 2.0 * (u * w), -2.0v + 2.0 * v ^ 2 + 2.0 * (u * v)))), Vec(float.(tuple(-1.0 + w ^ 2, -1.0 + w ^ 2, -1.0 * (2 * w ^ (2 - 1)) + u * (2 * w ^ (2 - 1)) + v * (2 * w ^ (2 - 1))))), Vec(float.(tuple(1 + -1.0 * w ^ 2, 0, -1.0 * (u * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(0, 1 + -1.0 * w ^ 2, -1.0 * (v * (2 * w ^ (2 - 1)))))), Vec(float.(tuple(2.0 + -2.0 * (2 * u ^ (2 - 1)) + -2.0v + 2.0w + -2.0 * ((2 * u ^ (2 - 1)) * w) + -2.0 * (v * w), -2.0u + -2.0 * (u * w), 2.0u + -2.0 * u ^ 2 + -2.0 * (u * v)))), Vec(float.(tuple(2.0v + 2.0 * (v * w), 2.0u + 2.0 * (u * w), 2.0 * (u * v)))), Vec(float.(tuple(-2.0v + -2.0 * (v * w), 2.0 + -2.0 * (2 * v ^ (2 - 1)) + -2.0u + 2.0w + -2.0 * ((2 * v ^ (2 - 1)) * w) + -2.0 * (u * w), 2.0v + -2.0 * v ^ 2 + -2.0 * (u * v))))) + end +