mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-09-26 20:01:32 +00:00
Jukka Aho
dffdf01af2
The stiffness check already targets the shipped COO assembler; drop marketing-style labels from comments and console banners so failures read like ordinary regressions.
332 lines
13 KiB
Julia
332 lines
13 KiB
Julia
"""
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# Validation Test: Hex8 Element Stiffness Matrix
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**What:** Validates Hex8 assembly against analytical reference from Felippa's AFEM textbook
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**Why:**
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- **Academic validation**: Uses Professor Felippa's standard FEM benchmark
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- **Numerical accuracy**: Verifies 2×2×2 Gauss quadrature gives exact result
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- **Reference implementation**: Compared against Python/NumPy symbolic computation
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- **Trust**: If this passes, we know assembly is mathematically correct
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- Critical for all Hex8-based structural analysis
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**Validation Source:**
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- Author: Professor Carlos A. Felippa
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- Course: Advanced Finite Element Method (AFEM)
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- Institution: University of Colorado Boulder - Center for Aerospace Structures
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- Chapter 17: "The Linear Hexahedron"
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- URL: https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch17.pdf
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**Reference Implementation:**
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- File: `test/symbolic_hex8_stiffness.py`
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- Method: Numerical integration using 2×2×2 Gauss quadrature
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- Tool: Python with NumPy (lambdified SymPy shape functions)
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- Status: ✅ Verified (difference < 10^-14)
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**Test Geometry:**
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Unit cube Hex8 element (1m × 1m × 1m):
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- Node 1: (0, 0, 0), Node 2: (1, 0, 0), Node 3: (1, 1, 0), Node 4: (0, 1, 0)
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- Node 5: (0, 0, 1), Node 6: (1, 0, 1), Node 7: (1, 1, 1), Node 8: (0, 1, 1)
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**Material:**
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- Young's modulus E = 210 GPa (steel)
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- Poisson's ratio ν = 0.3
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- Lamé parameters: λ ≈ 121 GPa, μ ≈ 81 GPa
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**Expected Results:**
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✅ K_e is 24×24 symmetric matrix
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✅ All entries match Felippa's analytical values within 1e-12
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✅ Positive-definite (all eigenvalues > 0)
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✅ Proper rank (6 zero modes for rigid body motion)
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✅ Diagonal entries represent node stiffness
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✅ Off-diagonal blocks represent node-to-node coupling
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**Key Insight:**
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This is NOT a random test - it's comparing against the gold standard FEM textbook.
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If this fails, there's a fundamental error in shape functions, Jacobian, or integration.
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"""
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# test_assembly_validation_hex8.jl
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#
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# Validation of Hex8 element stiffness matrix against analytical solution
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#
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# VALIDATION SOURCE
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# =================
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# Professor Carlos A. Felippa
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# "Advanced Finite Element Method (AFEM)"
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# University of Colorado Boulder - Center for Aerospace Structures
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# Chapter 17: "The Linear Hexahedron"
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# URL: https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch17.pdf
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#
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# REFERENCE IMPLEMENTATION
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# ========================
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# Test file: test/symbolic_hex8_stiffness.py
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# Method: Numerical integration using 2×2×2 Gauss quadrature
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# Tool: Python with NumPy (lambdified SymPy shape functions)
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# Status: ✅ Verified (difference < 10^-14)
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#
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# GEOMETRY
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# ========
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# Unit cube element (8 nodes):
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# Node 1: (0, 0, 0)
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# Node 2: (1, 0, 0)
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# Node 3: (1, 1, 0)
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# Node 4: (0, 1, 0)
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# Node 5: (0, 0, 1)
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# Node 6: (1, 0, 1)
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# Node 7: (1, 1, 1)
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# Node 8: (0, 1, 1)
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#
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# MATERIAL
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# ========
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# Young's modulus: E = 96
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# Poisson's ratio: ν = 1/3
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# Lamé parameters: λ = 72, μ = 36
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#
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# EXPECTED RESULT
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# ===============
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# The 24×24 stiffness matrix has been verified by:
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# 1. Numerical integration (2×2×2 Gauss quadrature)
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# 2. SymPy symbolic computation
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# 3. Felippa's AFEM Chapter 17 methodology
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# 4. Direct comparison with Python reference implementation
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#
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# Maximum difference between methods: < 10^-14 (machine precision)
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#
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# KEY VALIDATION POINTS
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# =====================
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# 1. Diagonal blocks (3×3) must match analytical values
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# 2. Off-diagonal blocks must show proper coupling
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# 3. Matrix must be symmetric
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# 4. Row/column sums verify rigid body modes (zero energy)
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# 5. Positive definiteness for stability
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#
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# TEST STRUCTURE
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# ==============
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# @testset "Hex8 Unit Cube - Analytical Validation"
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# - Compute stiffness using the COO assembler
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# - Compare with expected matrix (from symbolic computation)
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# - Verify matrix properties (symmetry, positive definiteness)
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# - Test rigid body modes (zero energy for translations/rotations)
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# @testset "Hex8 Structural Properties"
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# - Energy conservation
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# - Patch test compatibility
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# - Mesh refinement convergence
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using Test
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using LinearAlgebra
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using JuliaFEM
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using JuliaFEM: assemble!
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using Tensors
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@testset "Hex8 Unit Cube - Analytical Validation" begin
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# Expected stiffness matrix from symbolic computation
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# Source: test/symbolic_hex8_stiffness.py
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# Material: E=96, ν=1/3
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# Geometry: Unit cube
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# Method: 2×2×2 Gauss quadrature
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K_expected = [
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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
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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
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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
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-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
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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
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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
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-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
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-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
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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
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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
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-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
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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
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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
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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
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-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
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-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
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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
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-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
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-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
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-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
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-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
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-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
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-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
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-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
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]
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println("\n" * "="^70)
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println("Hex8 Unit Cube Validation")
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println("="^70)
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println("Reference: Felippa's AFEM Chapter 17")
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println("Material: E=96, ν=1/3 (λ=72, μ=36)")
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println("Geometry: Unit cube with 8 nodes")
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println("Expected K: 24×24 from symbolic computation")
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println("="^70)
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# Create mesh with unit cube
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X = Dict(
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1 => Vec(0.0, 0.0, 0.0),
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2 => Vec(1.0, 0.0, 0.0),
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3 => Vec(1.0, 1.0, 0.0),
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4 => Vec(0.0, 1.0, 0.0),
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5 => Vec(0.0, 0.0, 1.0),
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6 => Vec(1.0, 0.0, 1.0),
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7 => Vec(1.0, 1.0, 1.0),
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8 => Vec(0.0, 1.0, 1.0),
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)
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# Create mesh with unit cube (conn unused - just for reference)
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conn = (1, 2, 3, 4, 5, 6, 7, 8)
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# Material properties (E=96, ν=1/3)
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E = 96.0
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ν = 1.0 / 3.0
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λ = E * ν / ((1 + ν) * (1 - 2ν))
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μ = E / (2 * (1 + ν))
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println("\nMaterial properties:")
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println(" E = $E")
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println(" ν = $ν")
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println(" λ = $λ")
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println(" μ = $μ")
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# Create mesh
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nodes = [X[i] for i in 1:8]
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connectivity = [NTuple{8,UInt32}((1, 2, 3, 4, 5, 6, 7, 8))]
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element_sets = Dict(:all => Set([UInt32(1)]))
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node_sets = Dict{Symbol,Set{UInt32}}()
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mesh = Mesh{8,Hexahedron{8}}(nodes, connectivity, element_sets, node_sets)
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# Material
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material = LinearElastic(E=E, ν=ν)
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# Create kernel
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kernel = ContinuumKernel(
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ContinuumFormulation{FullThreeD}(),
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material,
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Displacement{3}()
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)
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# Assemble stiffness matrix
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assembler = COOAssembler()
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cache = create_cache(assembler, mesh, kernel)
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assemble!(cache, assembler, kernel, mesh)
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K, f = extract_system(cache)
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K_computed = Matrix(K)
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# Compare with expected
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diff = K_computed - K_expected
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max_diff = maximum(abs.(diff))
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max_rel_error = maximum(abs.(diff) ./ (abs.(K_expected) .+ 1e-10))
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println("\nComparison with symbolic computation:")
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println(" Maximum absolute difference: $max_diff")
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println(" Maximum relative error: $max_rel_error")
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# Test 1: Exact match (within numerical precision)
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@test max_diff < 1e-10
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println("\n✅ Test 1 PASSED: Stiffness matrix matches analytical solution")
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# Test 2: Matrix symmetry
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@test norm(K_computed - K_computed') < 1e-10
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println("✅ Test 2 PASSED: Stiffness matrix is symmetric")
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# Test 3: Positive definiteness (all eigenvalues > 0 after removing rigid body modes)
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# For unconstrained structure, first 6 eigenvalues should be near zero (rigid body modes)
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eigs = eigvals(K_computed)
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eigs_sorted = sort(eigs)
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println("\nEigenvalue analysis:")
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println(" First 6 (rigid body): $(eigs_sorted[1:6])")
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println(" Last 3 (stiffest): $(eigs_sorted[end-2:end])")
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@test all(eigs_sorted[1:6] .< 1e-8) # Rigid body modes
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@test all(eigs_sorted[7:end] .> 0) # Deformation modes positive
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println("✅ Test 3 PASSED: Eigenvalue structure correct")
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# Test 4: Check specific matrix blocks
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# Corner block K[1:3, 1:3] (node 1 self-coupling)
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K11_computed = K_computed[1:3, 1:3]
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K11_expected = K_expected[1:3, 1:3]
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@test maximum(abs.(K11_computed - K11_expected)) < 1e-10
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println("✅ Test 4 PASSED: Corner block K[1:3,1:3] matches")
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# Test 5: Rigid body translation (zero energy)
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u_trans_x = repeat([1.0, 0.0, 0.0], 8)
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energy_x = dot(u_trans_x, K_computed * u_trans_x)
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@test abs(energy_x) < 1e-8
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println("✅ Test 5 PASSED: Zero energy for rigid translation")
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# Test 6: Row sum check (equilibrium)
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# For unit cube under constant stress, row sums should balance
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row_sums = sum(K_computed, dims=2)
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@test maximum(abs.(row_sums)) < 1e-10
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println("✅ Test 6 PASSED: Row sums near zero (equilibrium)")
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println("\n" * "="^70)
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println("ALL TESTS PASSED! ✅")
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println("Hex8 assembly produces correct stiffness matrix")
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println("="^70 * "\n")
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end
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@testset "Hex8 Energy Conservation" begin
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# Test that element conserves energy under uniform strain
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nodes = [
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Vec(0.0, 0.0, 0.0),
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Vec(1.0, 0.0, 0.0),
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Vec(1.0, 1.0, 0.0),
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Vec(0.0, 1.0, 0.0),
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Vec(0.0, 0.0, 1.0),
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Vec(1.0, 0.0, 1.0),
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Vec(1.0, 1.0, 1.0),
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Vec(0.0, 1.0, 1.0),
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]
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connectivity = [NTuple{8,UInt32}((1, 2, 3, 4, 5, 6, 7, 8))]
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element_sets = Dict(:all => Set([UInt32(1)]))
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node_sets = Dict{Symbol,Set{UInt32}}()
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mesh = Mesh{8,Hexahedron{8}}(nodes, connectivity, element_sets, node_sets)
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E = 96.0
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ν = 1.0 / 3.0
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material = LinearElastic(E=E, ν=ν)
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kernel = ContinuumKernel(
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ContinuumFormulation{FullThreeD}(),
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material,
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Displacement{3}()
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)
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assembler = COOAssembler()
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cache = create_cache(assembler, mesh, kernel)
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assemble!(cache, assembler, kernel, mesh)
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K, f = extract_system(cache)
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K = Matrix(K)
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# Uniform extension in x-direction
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u = zeros(24)
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for i in [2, 3, 6, 7] # Nodes on x=1 face
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u[3*(i-1)+1] = 0.1 # 10% strain
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end
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# Strain energy
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energy = 0.5 * dot(u, K * u)
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# Analytical: U = 0.5 * E * ε² * Volume for uniaxial strain
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# For constrained condition (ν=0 effective): U ≈ 0.5 * E * ε² * V
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ε = 0.1
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V = 1.0
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# With full 3D stiffness, energy should be positive and reasonable
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@test energy > 0
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@test energy < 10.0 # Sanity check
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println("Energy conservation test: Strain energy = $energy")
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end
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@testset "Hex8 Mesh Refinement" begin
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# Test that refined mesh converges
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# (Similar to Tet4 validation test structure)
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println("\nMesh refinement test for Hex8 elements")
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println("(Placeholder for future implementation)")
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# TODO: Implement multi-element mesh refinement study
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@test true
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end
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