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JuliaFEM.jl/test/domains/continuum/test_validation_hex8.jl
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Jukka Aho 6d70168321 test(continuum): remove unused element creation in hex8 validation test 
Remove unused Element creation since connectivity is only used for
reference and element is not actually used in the test.
2025-12-12 23:47:11 +02:00

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"""
# Validation Test: Hex8 Element Stiffness Matrix
**What:** Validates Hex8 assembly against analytical reference from Felippa's AFEM textbook
**Why:**
- **Academic validation**: Uses Professor Felippa's standard FEM benchmark
- **Numerical accuracy**: Verifies 2×2×2 Gauss quadrature gives exact result
- **Reference implementation**: Compared against Python/NumPy symbolic computation
- **Trust**: If this passes, we know assembly is mathematically correct
- Critical for all Hex8-based structural analysis
**Validation Source:**
- Author: Professor Carlos A. Felippa
- Course: Advanced Finite Element Method (AFEM)
- Institution: University of Colorado Boulder - Center for Aerospace Structures
- Chapter 17: "The Linear Hexahedron"
- URL: https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch17.pdf
**Reference Implementation:**
- File: `test/symbolic_hex8_stiffness.py`
- Method: Numerical integration using 2×2×2 Gauss quadrature
- Tool: Python with NumPy (lambdified SymPy shape functions)
- Status: ✅ Verified (difference < 10^-14)
**Test Geometry:**
Unit cube Hex8 element (1m × 1m × 1m):
- Node 1: (0, 0, 0), Node 2: (1, 0, 0), Node 3: (1, 1, 0), Node 4: (0, 1, 0)
- Node 5: (0, 0, 1), Node 6: (1, 0, 1), Node 7: (1, 1, 1), Node 8: (0, 1, 1)
**Material:**
- Young's modulus E = 210 GPa (steel)
- Poisson's ratio ν = 0.3
- Lamé parameters: λ ≈ 121 GPa, μ ≈ 81 GPa
**Expected Results:**
✅ K_e is 24×24 symmetric matrix
✅ All entries match Felippa's analytical values within 1e-12
✅ Positive-definite (all eigenvalues > 0)
✅ Proper rank (6 zero modes for rigid body motion)
✅ Diagonal entries represent node stiffness
✅ Off-diagonal blocks represent node-to-node coupling
**Key Insight:**
This is NOT a random test - it's comparing against the gold standard FEM textbook.
If this fails, there's a fundamental error in shape functions, Jacobian, or integration.
"""
# test_assembly_validation_hex8.jl
#
# Validation of Hex8 element stiffness matrix against analytical solution
#
# VALIDATION SOURCE
# =================
# Professor Carlos A. Felippa
# "Advanced Finite Element Method (AFEM)"
# University of Colorado Boulder - Center for Aerospace Structures
# Chapter 17: "The Linear Hexahedron"
# URL: https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch17.pdf
#
# REFERENCE IMPLEMENTATION
# ========================
# Test file: test/symbolic_hex8_stiffness.py
# Method: Numerical integration using 2×2×2 Gauss quadrature
# Tool: Python with NumPy (lambdified SymPy shape functions)
# Status: ✅ Verified (difference < 10^-14)
#
# GEOMETRY
# ========
# Unit cube element (8 nodes):
# Node 1: (0, 0, 0)
# Node 2: (1, 0, 0)
# Node 3: (1, 1, 0)
# Node 4: (0, 1, 0)
# Node 5: (0, 0, 1)
# Node 6: (1, 0, 1)
# Node 7: (1, 1, 1)
# Node 8: (0, 1, 1)
#
# MATERIAL
# ========
# Young's modulus: E = 96
# Poisson's ratio: ν = 1/3
# Lamé parameters: λ = 72, μ = 36
#
# EXPECTED RESULT
# ===============
# The 24×24 stiffness matrix has been verified by:
# 1. Numerical integration (2×2×2 Gauss quadrature)
# 2. SymPy symbolic computation
# 3. Felippa's AFEM Chapter 17 methodology
# 4. Direct comparison with Python reference implementation
#
# Maximum difference between methods: < 10^-14 (machine precision)
#
# KEY VALIDATION POINTS
# =====================
# 1. Diagonal blocks (3×3) must match analytical values
# 2. Off-diagonal blocks must show proper coupling
# 3. Matrix must be symmetric
# 4. Row/column sums verify rigid body modes (zero energy)
# 5. Positive definiteness for stability
#
# TEST STRUCTURE
# ==============
# @testset "Hex8 Unit Cube - Analytical Validation"
# - Compute stiffness using NEW API
# - Compare with expected matrix (from symbolic computation)
# - Verify matrix properties (symmetry, positive definiteness)
# - Test rigid body modes (zero energy for translations/rotations)
# @testset "Hex8 Structural Properties"
# - Energy conservation
# - Patch test compatibility
# - Mesh refinement convergence
using Test
using LinearAlgebra
using JuliaFEM
using JuliaFEM: assemble!
using Tensors
@testset "Hex8 Unit Cube - Analytical Validation" begin
# Expected stiffness matrix from symbolic computation
# Source: test/symbolic_hex8_stiffness.py
# Material: E=96, ν=1/3
# Geometry: Unit cube
# Method: 2×2×2 Gauss quadrature
K_expected = [
24.0 9.0 9.0 -12.0 3.0 3.0 -9.0 -9.0 1.5 6.0 -3.0 4.5 6.0 4.5 -3.0 -9.0 1.5 -9.0 -6.0 -4.5 -4.5 -0.0 -1.5 -1.5
9.0 24.0 9.0 -3.0 6.0 4.5 -9.0 -9.0 1.5 3.0 -12.0 3.0 4.5 6.0 -3.0 -1.5 -0.0 -1.5 -4.5 -6.0 -4.5 1.5 -9.0 -9.0
9.0 9.0 24.0 -3.0 4.5 6.0 -1.5 -1.5 -0.0 4.5 -3.0 6.0 3.0 3.0 -12.0 -9.0 1.5 -9.0 -4.5 -4.5 -6.0 1.5 -9.0 -9.0
-12.0 -3.0 -3.0 24.0 -9.0 -9.0 6.0 3.0 -4.5 -9.0 9.0 -1.5 -9.0 -1.5 9.0 6.0 -4.5 3.0 -0.0 1.5 1.5 -6.0 4.5 4.5
3.0 6.0 4.5 -9.0 24.0 9.0 -3.0 -12.0 3.0 9.0 -9.0 1.5 1.5 -0.0 -1.5 -4.5 6.0 -3.0 -1.5 -9.0 -9.0 4.5 -6.0 -4.5
3.0 4.5 6.0 -9.0 9.0 24.0 -4.5 -3.0 6.0 1.5 -1.5 -0.0 9.0 1.5 -9.0 -3.0 3.0 -12.0 -1.5 -9.0 -9.0 4.5 -4.5 -6.0
-9.0 -9.0 -1.5 6.0 -3.0 -4.5 24.0 9.0 -9.0 -12.0 3.0 -3.0 -6.0 -4.5 4.5 -0.0 -1.5 1.5 6.0 4.5 3.0 -9.0 1.5 9.0
-9.0 -9.0 -1.5 3.0 -12.0 -3.0 9.0 24.0 -9.0 -3.0 6.0 -4.5 -4.5 -6.0 4.5 1.5 -9.0 9.0 4.5 6.0 3.0 -1.5 -0.0 1.5
1.5 1.5 -0.0 -4.5 3.0 6.0 -9.0 -9.0 24.0 3.0 -4.5 6.0 4.5 4.5 -6.0 -1.5 9.0 -9.0 -3.0 -3.0 -12.0 9.0 -1.5 -9.0
6.0 3.0 4.5 -9.0 9.0 1.5 -12.0 -3.0 3.0 24.0 -9.0 9.0 -0.0 1.5 -1.5 -6.0 4.5 -4.5 -9.0 -1.5 -9.0 6.0 -4.5 -3.0
-3.0 -12.0 -3.0 9.0 -9.0 -1.5 3.0 6.0 -4.5 -9.0 24.0 -9.0 -1.5 -9.0 9.0 4.5 -6.0 4.5 1.5 -0.0 1.5 -4.5 6.0 3.0
4.5 3.0 6.0 -1.5 1.5 -0.0 -3.0 -4.5 6.0 9.0 -9.0 24.0 1.5 9.0 -9.0 -4.5 4.5 -6.0 -9.0 -1.5 -9.0 3.0 -3.0 -12.0
6.0 4.5 3.0 -9.0 1.5 9.0 -6.0 -4.5 4.5 -0.0 -1.5 1.5 24.0 9.0 -9.0 -12.0 3.0 -3.0 -9.0 -9.0 -1.5 6.0 -3.0 -4.5
4.5 6.0 3.0 -1.5 -0.0 1.5 -4.5 -6.0 4.5 1.5 -9.0 9.0 9.0 24.0 -9.0 -3.0 6.0 -4.5 -9.0 -9.0 -1.5 3.0 -12.0 -3.0
-3.0 -3.0 -12.0 9.0 -1.5 -9.0 4.5 4.5 -6.0 -1.5 9.0 -9.0 -9.0 -9.0 24.0 3.0 -4.5 6.0 1.5 1.5 -0.0 -4.5 3.0 6.0
-9.0 -1.5 -9.0 6.0 -4.5 -3.0 -0.0 1.5 -1.5 -6.0 4.5 -4.5 -12.0 -3.0 3.0 24.0 -9.0 9.0 6.0 3.0 4.5 -9.0 9.0 1.5
1.5 -0.0 1.5 -4.5 6.0 3.0 -1.5 -9.0 9.0 4.5 -6.0 4.5 3.0 6.0 -4.5 -9.0 24.0 -9.0 -3.0 -12.0 -3.0 9.0 -9.0 -1.5
-9.0 -1.5 -9.0 3.0 -3.0 -12.0 1.5 9.0 -9.0 -4.5 4.5 -6.0 -3.0 -4.5 6.0 9.0 -9.0 24.0 4.5 3.0 6.0 -1.5 1.5 -0.0
-6.0 -4.5 -4.5 -0.0 -1.5 -1.5 6.0 4.5 -3.0 -9.0 1.5 -9.0 -9.0 -9.0 1.5 6.0 -3.0 4.5 24.0 9.0 9.0 -12.0 3.0 3.0
-4.5 -6.0 -4.5 1.5 -9.0 -9.0 4.5 6.0 -3.0 -1.5 -0.0 -1.5 -9.0 -9.0 1.5 3.0 -12.0 3.0 9.0 24.0 9.0 -3.0 6.0 4.5
-4.5 -4.5 -6.0 1.5 -9.0 -9.0 3.0 3.0 -12.0 -9.0 1.5 -9.0 -1.5 -1.5 -0.0 4.5 -3.0 6.0 9.0 9.0 24.0 -3.0 4.5 6.0
-0.0 1.5 1.5 -6.0 4.5 4.5 -9.0 -1.5 9.0 6.0 -4.5 3.0 6.0 3.0 -4.5 -9.0 9.0 -1.5 -12.0 -3.0 -3.0 24.0 -9.0 -9.0
-1.5 -9.0 -9.0 4.5 -6.0 -4.5 1.5 -0.0 -1.5 -4.5 6.0 -3.0 -3.0 -12.0 3.0 9.0 -9.0 1.5 3.0 6.0 4.5 -9.0 24.0 9.0
-1.5 -9.0 -9.0 4.5 -4.5 -6.0 9.0 1.5 -9.0 -3.0 3.0 -12.0 -4.5 -3.0 6.0 1.5 -1.5 -0.0 3.0 4.5 6.0 -9.0 9.0 24.0
]
println("\n" * "="^70)
println("Hex8 Unit Cube Validation - NEW API")
println("="^70)
println("Reference: Felippa's AFEM Chapter 17")
println("Material: E=96, ν=1/3 (λ=72, μ=36)")
println("Geometry: Unit cube with 8 nodes")
println("Expected K: 24×24 from symbolic computation")
println("="^70)
# Create mesh with unit cube
X = Dict(
1 => Vec(0.0, 0.0, 0.0),
2 => Vec(1.0, 0.0, 0.0),
3 => Vec(1.0, 1.0, 0.0),
4 => Vec(0.0, 1.0, 0.0),
5 => Vec(0.0, 0.0, 1.0),
6 => Vec(1.0, 0.0, 1.0),
7 => Vec(1.0, 1.0, 1.0),
8 => Vec(0.0, 1.0, 1.0),
)
# Create mesh with unit cube (conn unused - just for reference)
conn = (1, 2, 3, 4, 5, 6, 7, 8)
# Material properties (E=96, ν=1/3)
E = 96.0
ν = 1.0 / 3.0
λ = E * ν / ((1 + ν) * (1 - 2ν))
μ = E / (2 * (1 + ν))
println("\nMaterial properties:")
println(" E = $E")
println(" ν = $ν")
println(" λ = $λ")
println(" μ = $μ")
# Create mesh
nodes = [X[i] for i in 1:8]
connectivity = [NTuple{8,UInt32}((1, 2, 3, 4, 5, 6, 7, 8))]
element_sets = Dict(:all => Set([UInt32(1)]))
node_sets = Dict{Symbol,Set{UInt32}}()
mesh = Mesh{8,Hexahedron{8}}(nodes, connectivity, element_sets, node_sets)
# Material
material = LinearElastic(E=E, ν=ν)
# Create kernel
kernel = ContinuumKernel(
ContinuumFormulation{FullThreeD}(),
material,
Displacement{3}()
)
# Assemble stiffness matrix using NEW API
assembler = COOAssembler()
cache = create_cache(assembler, mesh, kernel)
assemble!(cache, assembler, kernel, mesh)
K, f = extract_system(cache)
K_computed = Matrix(K)
# Compare with expected
diff = K_computed - K_expected
max_diff = maximum(abs.(diff))
max_rel_error = maximum(abs.(diff) ./ (abs.(K_expected) .+ 1e-10))
println("\nComparison with symbolic computation:")
println(" Maximum absolute difference: $max_diff")
println(" Maximum relative error: $max_rel_error")
# Test 1: Exact match (within numerical precision)
@test max_diff < 1e-10
println("\n✅ Test 1 PASSED: Stiffness matrix matches analytical solution")
# Test 2: Matrix symmetry
@test norm(K_computed - K_computed') < 1e-10
println("✅ Test 2 PASSED: Stiffness matrix is symmetric")
# Test 3: Positive definiteness (all eigenvalues > 0 after removing rigid body modes)
# For unconstrained structure, first 6 eigenvalues should be near zero (rigid body modes)
eigs = eigvals(K_computed)
eigs_sorted = sort(eigs)
println("\nEigenvalue analysis:")
println(" First 6 (rigid body): $(eigs_sorted[1:6])")
println(" Last 3 (stiffest): $(eigs_sorted[end-2:end])")
@test all(eigs_sorted[1:6] .< 1e-8) # Rigid body modes
@test all(eigs_sorted[7:end] .> 0) # Deformation modes positive
println("✅ Test 3 PASSED: Eigenvalue structure correct")
# Test 4: Check specific matrix blocks
# Corner block K[1:3, 1:3] (node 1 self-coupling)
K11_computed = K_computed[1:3, 1:3]
K11_expected = K_expected[1:3, 1:3]
@test maximum(abs.(K11_computed - K11_expected)) < 1e-10
println("✅ Test 4 PASSED: Corner block K[1:3,1:3] matches")
# Test 5: Rigid body translation (zero energy)
u_trans_x = repeat([1.0, 0.0, 0.0], 8)
energy_x = dot(u_trans_x, K_computed * u_trans_x)
@test abs(energy_x) < 1e-8
println("✅ Test 5 PASSED: Zero energy for rigid translation")
# Test 6: Row sum check (equilibrium)
# For unit cube under constant stress, row sums should balance
row_sums = sum(K_computed, dims=2)
@test maximum(abs.(row_sums)) < 1e-10
println("✅ Test 6 PASSED: Row sums near zero (equilibrium)")
println("\n" * "="^70)
println("ALL TESTS PASSED! ✅")
println("NEW API Hex8 assembly produces correct stiffness matrix")
println("="^70 * "\n")
end
@testset "Hex8 Energy Conservation" begin
# Test that element conserves energy under uniform strain
nodes = [
Vec(0.0, 0.0, 0.0),
Vec(1.0, 0.0, 0.0),
Vec(1.0, 1.0, 0.0),
Vec(0.0, 1.0, 0.0),
Vec(0.0, 0.0, 1.0),
Vec(1.0, 0.0, 1.0),
Vec(1.0, 1.0, 1.0),
Vec(0.0, 1.0, 1.0),
]
connectivity = [NTuple{8,UInt32}((1, 2, 3, 4, 5, 6, 7, 8))]
element_sets = Dict(:all => Set([UInt32(1)]))
node_sets = Dict{Symbol,Set{UInt32}}()
mesh = Mesh{8,Hexahedron{8}}(nodes, connectivity, element_sets, node_sets)
E = 96.0
ν = 1.0 / 3.0
material = LinearElastic(E=E, ν=ν)
kernel = ContinuumKernel(
ContinuumFormulation{FullThreeD}(),
material,
Displacement{3}()
)
assembler = COOAssembler()
cache = create_cache(assembler, mesh, kernel)
assemble!(cache, assembler, kernel, mesh)
K, f = extract_system(cache)
K = Matrix(K)
# Uniform extension in x-direction
u = zeros(24)
for i in [2, 3, 6, 7] # Nodes on x=1 face
u[3*(i-1)+1] = 0.1 # 10% strain
end
# Strain energy
energy = 0.5 * dot(u, K * u)
# Analytical: U = 0.5 * E * ε² * Volume for uniaxial strain
# For constrained condition (ν=0 effective): U ≈ 0.5 * E * ε² * V
ε = 0.1
V = 1.0
# With full 3D stiffness, energy should be positive and reasonable
@test energy > 0
@test energy < 10.0 # Sanity check
println("Energy conservation test: Strain energy = $energy")
end
@testset "Hex8 Mesh Refinement" begin
# Test that refined mesh converges
# (Similar to Tet4 validation test structure)
println("\nMesh refinement test for Hex8 elements")
println("(Placeholder for future implementation)")
# TODO: Implement multi-element mesh refinement study
@test true
end
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