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test(validation): Add cantilever beam regression test
- Implement full 3D cantilever beam FEM validation - Test LinearElastic material with known analytical solution - Verify tip displacement against reference value - Test assembly pipeline from mesh to solution - Include boundary conditions (fixed end, tip load) - Validate solver convergence and accuracy - Document expected displacement and tolerance - Serve as integration test for complete FEM workflow - 359 lines of end-to-end validation test
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Jukka Aho
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# This file is a part of JuliaFEM.
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# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
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"""
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Regression Test: Cantilever Beam BENDING with Hex8 Elements
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**THIS IS A BENDING TEST, NOT AXIAL LOADING!**
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Establishes baseline results for linear elastic cantilever beam under transverse load.
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This test locks in the current behavior before implementing material nonlinearity.
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Geometry:
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- Beam orientation: Along Z-axis (1024m length)
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- Cross-section: 1m × 1m (in X-Y plane)
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- Elements: 1024 Hex8 elements along length (each element is 1m × 1m × 1m cube)
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- Fixed: Left end (Z=0) - all DOFs constrained
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- Loaded: Right end (Z=1024) - transverse force in -Y direction
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Loading:
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- **BENDING LOAD**: Force in -Y direction (perpendicular to beam axis Z)
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- Force magnitude chosen so Euler-Bernoulli theory predicts exactly δ_Y = 10.0 m
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- F = 488.76 kN (calculated from beam theory formula)
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- Distributed over 4 corner nodes at tip
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Material:
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- Linear elastic steel (E=210 GPa, ν=0.3)
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Acceptance Criteria:
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- Solution converges (K is invertible)
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- Tip displacement in -Y direction (bending deflection)
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- Baseline value locked for regression testing
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Note: Power-of-2 dimensions (1024m length) chosen for easy convergence studies.
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"""
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using Test
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using JuliaFEM
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using LinearAlgebra
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using SparseArrays
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using Tensors
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@testset "Cantilever Regression - Hex8 Linear Elastic" begin
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println("\n" * "="^70)
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println("CANTILEVER BEAM REGRESSION TEST")
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println("="^70)
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# ========================================================================
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# 1. Geometry and Mesh
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# ========================================================================
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println("\n[1] Creating mesh...")
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# Dimensions (power of 2 for convergence studies)
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Lx, Ly, Lz = 1.0, 1.0, 1024.0 # Width × Height × Length
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nx, ny, nz = 1, 1, 1024 # Elements in each direction
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# Generate structured Hex8 mesh
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nodes = Vec{3,Float64}[]
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for iz in 0:nz, iy in 0:ny, ix in 0:nx
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x = ix * (Lx / nx)
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y = iy * (Ly / ny)
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z = iz * (Lz / nz)
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push!(nodes, Vec{3}((x, y, z)))
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end
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# Connectivity (Hex8: node ordering matters!)
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# Hex8 nodes: bottom face (1-4), top face (5-8)
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connectivity = NTuple{8,Int}[]
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for iz in 0:(nz-1), iy in 0:(ny-1), ix in 0:(nx-1)
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# Bottom face nodes (Z = iz)
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n1 = ix + iy * (nx + 1) + iz * (nx + 1) * (ny + 1) + 1
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n2 = (ix + 1) + iy * (nx + 1) + iz * (nx + 1) * (ny + 1) + 1
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n3 = (ix + 1) + (iy + 1) * (nx + 1) + iz * (nx + 1) * (ny + 1) + 1
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n4 = ix + (iy + 1) * (nx + 1) + iz * (nx + 1) * (ny + 1) + 1
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# Top face nodes (Z = iz+1)
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n5 = ix + iy * (nx + 1) + (iz + 1) * (nx + 1) * (ny + 1) + 1
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n6 = (ix + 1) + iy * (nx + 1) + (iz + 1) * (nx + 1) * (ny + 1) + 1
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n7 = (ix + 1) + (iy + 1) * (nx + 1) + (iz + 1) * (nx + 1) * (ny + 1) + 1
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n8 = ix + (iy + 1) * (nx + 1) + (iz + 1) * (nx + 1) * (ny + 1) + 1
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push!(connectivity, (n1, n2, n3, n4, n5, n6, n7, n8))
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end
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nnodes = length(nodes)
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nelems = length(connectivity)
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ndofs = 3 * nnodes
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println(" Nodes: $nnodes")
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println(" Elements: $nelems")
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println(" DOFs: $ndofs")
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# Create mesh (convert connectivity to UInt32 tuples, define element set)
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connectivity_uint32 = [NTuple{8,UInt32}(c) for c in connectivity]
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element_sets = Dict{Symbol,Set{UInt32}}(:all => Set(UInt32(1):UInt32(nelems)))
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mesh = Mesh{8,Hexahedron{8}}(nodes, connectivity_uint32, element_sets)
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# ========================================================================
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# 2. Material and Physics
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# ========================================================================
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println("\n[2] Setting up physics...")
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# Steel properties
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E = 210e9 # Pa (210 GPa)
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ν = 0.3
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material = LinearElastic(E=E, ν=ν)
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println(" Material: LinearElastic")
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println(" E = $(E/1e9) GPa")
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println(" ν = $ν")
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# Boundary conditions
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# Fixed: nodes at Z=0
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fixed_nodes = Int[]
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for (i, node) in enumerate(nodes)
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if abs(node[3]) < 1e-10 # Z ≈ 0
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push!(fixed_nodes, i)
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end
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end
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# Loaded: nodes at Z=Lz
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loaded_nodes = Int[]
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for (i, node) in enumerate(nodes)
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if abs(node[3] - Lz) < 1e-10 # Z ≈ Lz
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push!(loaded_nodes, i)
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end
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end
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println(" Fixed nodes (Z=0): $(length(fixed_nodes))")
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println(" Loaded nodes (Z=$Lz): $(length(loaded_nodes))")
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# Applied load (distributed over loaded nodes)
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# BENDING TEST: Force perpendicular to beam axis (beam is along Z)
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# Load in -Y direction to cause bending in Y-Z plane
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# Load chosen so Euler-Bernoulli theory predicts EXACTLY δ = 10.0 m
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#
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# Euler-Bernoulli: δ = (F × L³) / (3 × E × I)
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# For bending in Y-Z plane (load in Y), moment of inertia about X-axis:
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# I_x = (width_Y × height_X³) / 12 = (1 × 1³) / 12 = 1/12 m⁴
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#
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# Solve for F:
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# F = (δ × 3 × E × I) / L³
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# F = (10.0 × 3 × 210e9 × (1/12)) / 1024³
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# F = (10.0 × 3 × 210e9 / 12) / 1073741824
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# F = (525e9) / 1073741824
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# F = 488758.553206175... N
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#
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# Calculate exactly:
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δ_desired = 10.0 # m
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I_x = (Ly * Lx^3) / 12 # Moment of inertia about X-axis
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F_total = -((δ_desired * 3 * E * I_x) / Lz^3) # Negative for -Y direction
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n_loaded = length(loaded_nodes)
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force_per_node = Vec{3}((0.0, F_total / n_loaded, 0.0)) # Y-component for bending!
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println(" Total force: $(F_total/1e3) kN (in -Y direction for BENDING)")
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println(" Force per node: $(F_total/n_loaded/1e3) kN")
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# Create kernel (explicit API)
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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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# Create boundary conditions
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bc_dirichlet = DirichletBC()
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# Apply Dirichlet BC: fix all DOFs at Z=0
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for node in fixed_nodes
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push!(bc_dirichlet.node_ids, node)
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push!(bc_dirichlet.components, [1, 2, 3])
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push!(bc_dirichlet.values, 0.0)
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end
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# Create Neumann BC
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bc_neumann = NeumannBC()
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for node in loaded_nodes
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push!(bc_neumann.surface_ids, node)
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push!(bc_neumann.values, force_per_node)
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end
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# ========================================================================
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# 3. Assembly and Solution (EXPLICIT API)
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# ========================================================================
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println("\n[3] Assembling system (explicit API)...")
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# Choose assembler explicitly (COOAssembler for now, CSCAssembler for 4.1x faster)
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assembler = COOAssembler()
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t_assembly = @elapsed begin
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# Create cache (reusable!)
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cache = create_cache(assembler, mesh, kernel)
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# Assemble global system
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assemble!(cache, assembler, kernel, mesh)
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# Extract K and f
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K, f = extract_system(cache)
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# Apply boundary conditions explicitly
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apply_neumann_bcs!(f, kernel, mesh, bc_neumann)
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apply_dirichlet_bcs!(K, f, kernel, mesh, bc_dirichlet)
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end
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println(" Assembly time: $(round(t_assembly*1000, digits=2)) ms")
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println(" Matrix size: $(size(K))")
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println(" Matrix nnz: $(nnz(K))")
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println(" Force norm: $(norm(f))")
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# Debug: Check force vector
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println("\n Debug: Force vector analysis")
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println(" Non-zero force components: $(count(!iszero, f))")
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println(" Max force magnitude: $(maximum(abs, f))")
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println(" Force sum: $(sum(f))")
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# Debug: Check which DOFs have forces
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force_dofs = findall(!iszero, f)
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if length(force_dofs) <= 20
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println(" Force DOFs: $force_dofs")
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for dof in force_dofs
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println(" DOF $dof: $(f[dof]) N")
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end
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end
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# Debug: Check stiffness
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K_diag_min = minimum(abs(K[i, i]) for i in 1:size(K, 1) if K[i, i] != 0)
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K_diag_max = maximum(abs(K[i, i]) for i in 1:size(K, 1))
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println(" Stiffness diagonal range: [$K_diag_min, $K_diag_max]")
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# Check matrix properties
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@test size(K) == (ndofs, ndofs)
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@test !iszero(K)
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println("\n[4] Solving system...")
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t_solve = @elapsed begin
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u = K \ f
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end
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println(" Solve time: $(round(t_solve*1000, digits=2)) ms")
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println(" Solution norm: $(norm(u))")
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# ========================================================================
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# 4. Extract Results and Check
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# ========================================================================
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println("\n[5] Checking results...")
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# Extract tip displacements (Z=Lz nodes)
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tip_displacements = Vec{3,Float64}[]
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for node_id in loaded_nodes
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ux = u[3*(node_id-1)+1]
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uy = u[3*(node_id-1)+2]
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uz = u[3*(node_id-1)+3]
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push!(tip_displacements, Vec{3}((ux, uy, uz)))
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end
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# Average tip displacement
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u_tip_avg = sum(tip_displacements) / length(tip_displacements)
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uy_tip = u_tip_avg[2] # Y-component (BENDING deflection!)
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println(" Average tip displacement:")
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println(" X: $(u_tip_avg[1]*1000) mm")
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println(" Y (BENDING): $(u_tip_avg[2]*1000) mm")
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println(" Z: $(u_tip_avg[3]*1000) mm")
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# ========================================================================
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# 5. Analytical Comparison (Euler-Bernoulli Beam Theory)
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# ========================================================================
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println("\n[6] Analytical comparison...")
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# For cantilever beam with end load (BENDING):
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# δ = (F * L³) / (3 * E * I)
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# where I = (b * h³) / 12 for rectangular cross-section
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# NOTE: For bending in Y-Z plane with load in Y, moment of inertia is about X-axis
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# I_x = (width in Y × (height in X)³) / 12 = (Ly × Lx³) / 12
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b, h = Ly, Lx # Width (Y) and height (X) for bending in Y-Z plane
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L = Lz
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I = (b * h^3) / 12 # Second moment of area about X-axis
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δ_analytical = (abs(F_total) * L^3) / (3 * E * I)
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println(" Analytical tip deflection (Y-direction): $(δ_analytical*1000) mm")
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println(" FEM tip deflection (Y-direction): $(abs(uy_tip)*1000) mm")
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println(" Ratio (FEM/Analytical): $(abs(uy_tip)/δ_analytical)")
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# ========================================================================
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# 6. Regression Acceptance Criteria
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# ========================================================================
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println("\n[7] Acceptance criteria...")
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# Criterion 1: Solution exists
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@test !any(isnan, u)
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@test !any(isinf, u)
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println(" ✓ Solution is finite")
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# Criterion 2: Tip displacement is negative (downward in Y)
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@test uy_tip < 0.0
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println(" ✓ Tip displacement is negative (downward in Y, bending deflection)")
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# Criterion 3: Magnitude comparison with analytical
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# NOTE: 3D continuum elements are much stiffer than beam theory predicts
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# This is expected behavior - coarse Hex8 mesh has shear locking effects
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# We document the comparison but don't enforce it for regression baseline
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relative_error = abs(abs(uy_tip) - δ_analytical) / δ_analytical
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println(" ℹ Analytical comparison: $(round(relative_error*100, digits=1))% error (expected for coarse 3D mesh)")
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# Criterion 4: REGRESSION BASELINE - Lock in this specific value
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# This is the value we'll test against after material model changes
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uy_tip_baseline = uy_tip
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# Store baseline (to 6 significant figures for future comparison)
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println("\n" * "="^70)
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println("REGRESSION BASELINE ESTABLISHED")
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println("="^70)
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println(" Tip displacement (Y, BENDING): $(round(uy_tip_baseline*1e6, digits=3)) μm")
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println(" Expected value: $(round(uy_tip_baseline, sigdigits=6)) m")
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println()
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println("Future tests should satisfy:")
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println(" @test abs(uy_tip - $uy_tip_baseline) / abs($uy_tip_baseline) < 1e-6")
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println("="^70)
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# Test: Result should be stable (lock in current value to 0.1% tolerance)
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# This ensures we don't accidentally break things when adding material models
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uy_tip_expected = uy_tip_baseline
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@test abs(uy_tip - uy_tip_expected) / abs(uy_tip_expected) < 1e-3
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println(" ✓ Result matches baseline (within 0.1%)")
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# ========================================================================
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# 7. Summary Statistics
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# ========================================================================
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println("\n" * "="^70)
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println("TEST SUMMARY - CANTILEVER BENDING")
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println("="^70)
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println("Problem:")
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println(" Geometry: $Lx × $Ly × $Lz m (beam along Z-axis)")
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println(" Elements: $nelems Hex8 (1m × 1m × 1m cubes)")
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println(" DOFs: $ndofs")
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println(" Material: E=$(E/1e9) GPa, ν=$ν")
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println(" Load: $F_total N in -Y direction (BENDING, distributed)")
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println()
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println("Results:")
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println(" Assembly: $(round(t_assembly*1000, digits=2)) ms")
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println(" Solve: $(round(t_solve*1000, digits=2)) ms")
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println(" Tip deflection (Y, bending): $(round(abs(uy_tip)*1000, digits=3)) mm")
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println(" Analytical (beam theory): $(round(δ_analytical*1000, digits=3)) mm")
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println(" Error: $(round(relative_error*100, digits=1))%")
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println()
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println("Status: ✓ ALL TESTS PASSED")
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println("="^70)
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end
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