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test(assemblers): add matrix-free eigensolve smoke tests
This commit is contained in:
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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Matrix-free generalized eigensolve `K φ = λ M φ` tests.
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Locks in the contract of the subspace-iteration eigensolver added in
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A+++:
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1. **Algebraic correctness on assembled matrices**: `lowest_eigenpairs(K, M)`
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matches `eigen(K, M)` to round-off on small SPD test problems.
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2. **Matrix-free path agrees with assembled**: building `op_K` /
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`op_M` from `apply_K!` / `apply_M!` reproduces the same
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eigenvalues as direct factorization of the assembled `K` / `M`
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for both heat and elasticity.
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3. **End-to-end heat-conduction modal analysis**: a 1D bar with
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fixed endpoints (`Δ T = λ T`) recovers the classical
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`λ_k = (k π / L)² / (ρ c)` spectrum to <1% relative error on a
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coarse mesh.
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4. **Elasticity natural frequencies of a clamped-free bar**:
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`ω_k = c · (2k − 1) π / (2L)` (axial modes) recovered to within
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mesh-discretisation error on a small problem.
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5. **`solve_eigenproblem` smoke**: high-level wrapper returns the
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same answer as the low-level `lowest_eigenpairs(op_K, op_M, n)`
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when no constraints are involved (i.e. on a closed/free system).
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"""
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using Test
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using JuliaFEM
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using JuliaFEM: ContinuumFormulation, FullThreeD, Vertex
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using JuliaFEM: @DOFSet, DOF
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using JuliaFEM: LinearElastic, Displacement, ContinuumKernel
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using JuliaFEM: HeatConductivity, HeatKernel, Temperature
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using JuliaFEM: DOFBasedCOOAssembler, DOFBasedCOOCache
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using JuliaFEM: extract_system, apply_K!, apply_M!, assemble_M!
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using JuliaFEM: PenaltyDirichlet, EliminatedDirichlet, apply_constraint!
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using JuliaFEM: matrix_free_op, JacobiPreconditioner
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using JuliaFEM: lowest_eigenpairs, solve_eigenproblem
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using JuliaFEM: create_elements!
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using LinearAlgebra
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using SparseArrays
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using Tensors
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using Random
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# ---------------------------------------------------------------------------
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# Mesh + setup helpers
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# ---------------------------------------------------------------------------
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function _hex8_box(nx::Int, ny::Int, nz::Int;
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Lx::Float64 = 1.0, Ly::Float64 = 1.0, Lz::Float64 = 1.0)
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nodes = Vec{3,Float64}[]
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nidx(i, j, k) = (i - 1) + (j - 1) * (nx + 1) + (k - 1) * (nx + 1) * (ny + 1) + 1
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for k in 1:(nz + 1), j in 1:(ny + 1), i in 1:(nx + 1)
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push!(nodes, Vec{3}((Lx * Float64(i - 1) / nx,
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Ly * Float64(j - 1) / ny,
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Lz * Float64(k - 1) / nz)))
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end
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conns = NTuple{8,UInt32}[]
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for k in 1:nz, j in 1:ny, i in 1:nx
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n1 = nidx(i, j, k)
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n2 = nidx(i + 1, j, k)
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n3 = nidx(i + 1, j + 1, k)
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n4 = nidx(i, j + 1, k)
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n5 = nidx(i, j, k + 1)
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n6 = nidx(i + 1, j, k + 1)
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n7 = nidx(i + 1, j + 1, k + 1)
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n8 = nidx(i, j + 1, k + 1)
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push!(conns, (UInt32(n1), UInt32(n2), UInt32(n3), UInt32(n4),
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UInt32(n5), UInt32(n6), UInt32(n7), UInt32(n8)))
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end
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return Mesh{8,Hexahedron{8}}(nodes, conns)
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end
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function _setup_elasticity(mesh; ρ::Float64 = 7850.0)
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material = LinearElastic(E = 210e9, ν = 0.3)
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kernel = ContinuumKernel(ContinuumFormulation{FullThreeD}(),
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material, Displacement{3}();
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density = ρ)
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S = @DOFSet{u::DOF{Displacement{3}, Vertex}}
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elements, dof_mgr = create_elements!(mesh, Element{Hexahedron{8}, Lagrange{1}, S})
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asm = DOFBasedCOOAssembler()
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cache = DOFBasedCOOCache(elements, dof_mgr, mesh, kernel)
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return cache, asm, kernel, mesh
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end
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function _setup_heat(mesh; k_value::Float64 = 1.0,
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ρcp::Float64 = 1.0)
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material = HeatConductivity(k = k_value)
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kernel = HeatKernel(ContinuumFormulation{FullThreeD}(), material;
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heat_capacity = ρcp)
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S = @DOFSet{T::DOF{Temperature, Vertex}}
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elements, dof_mgr = create_elements!(mesh, Element{Hexahedron{8}, Lagrange{1}, S})
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asm = DOFBasedCOOAssembler()
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cache = DOFBasedCOOCache(elements, dof_mgr, mesh, kernel)
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return cache, asm, kernel, mesh
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end
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# Build assembled K and M from a fresh cache (each `assemble!` /
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# `assemble_M!` overwrites the COO triplets, so we extract between
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# calls).
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function _assemble_KM(cache, asm, kernel, mesh)
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assemble!(cache, asm, kernel, mesh); K, _ = extract_system(cache)
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assemble_M!(cache, asm, kernel, mesh); M, _ = extract_system(cache)
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return K, M
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end
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# Reduce a system by deleting fixed DOF rows/cols (clean elimination
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# for the eigenproblem; avoids polluting the spectrum with `λ = K_dd / M_dd`
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# spurious eigenvalues from penalty / identity-row tricks).
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function _reduce_KM(K::AbstractMatrix, M::AbstractMatrix, fixed::Vector{Int})
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n = size(K, 1)
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free = setdiff(1:n, fixed)
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return K[free, free], M[free, free], free
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end
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# ---------------------------------------------------------------------------
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# 1. Algebraic correctness on small SPD problems
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# ---------------------------------------------------------------------------
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@testset "lowest_eigenpairs: dense SPD (assembled K, M)" begin
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println("\n" * "=" ^ 70)
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println("lowest_eigenpairs — algebraic correctness on dense SPD")
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println("=" ^ 70)
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Random.seed!(20260508)
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@testset "Tridiagonal Laplacian K, identity M" begin
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for n in (20, 50)
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K = Matrix(SymTridiagonal(2.0 * ones(n), -1.0 * ones(n - 1)))
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M = Matrix(LinearAlgebra.I(n) * 1.0)
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λ_ref = sort(eigvals(K, M))[1:5]
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λ_mf, V_mf = lowest_eigenpairs(K, M; nev = 5, tol = 1e-10,
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maxiter = 100)
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relerr = maximum(abs.(λ_mf - λ_ref) ./ abs.(λ_ref))
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@test relerr < 1e-8
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# M-orthonormality: V' M V = I
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ortho_err = norm(V_mf' * M * V_mf - LinearAlgebra.I(5))
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@test ortho_err < 1e-6
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# Eigenpair residuals: ‖K v_k − λ_k M v_k‖ / ‖λ_k M v_k‖
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resid_err = 0.0
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for k in 1:5
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vk = V_mf[:, k]
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resid = norm(K * vk - λ_mf[k] * (M * vk)) /
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max(abs(λ_mf[k]) * norm(M * vk), 1.0)
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resid_err = max(resid_err, resid)
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end
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@test resid_err < 1e-5
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println(" n=$n λ-relerr=$(round(relerr; sigdigits=3)) " *
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"M-ortho-err=$(round(ortho_err; sigdigits=3)) " *
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"resid=$(round(resid_err; sigdigits=3))")
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end
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end
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@testset "Random dense SPD (K = AAᵀ + I, M = BBᵀ + I)" begin
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for n in (15, 30)
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A = randn(n, n); K = Matrix(A * A' + 1.0 * LinearAlgebra.I)
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B = randn(n, n); M = Matrix(B * B' + 1.0 * LinearAlgebra.I)
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λ_ref = sort(eigvals(K, M))[1:3]
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λ_mf, _ = lowest_eigenpairs(K, M; nev = 3, tol = 1e-10,
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maxiter = 100)
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relerr = maximum(abs.(λ_mf - λ_ref) ./ abs.(λ_ref))
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@test relerr < 1e-7
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println(" n=$n random SPD λ-relerr=$(round(relerr; sigdigits=3))")
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end
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end
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@testset "Float32 assembled matrices use dense convenience overload" begin
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n = 12
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K = Matrix{Float32}(SymTridiagonal(2.0f0 * ones(Float32, n),
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-ones(Float32, n - 1)))
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M = Matrix{Float32}(LinearAlgebra.I(n) * 1.0f0)
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λ_ref = sort(eigvals(Float64.(K), Float64.(M)))[1:3]
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λ_mf, _ = lowest_eigenpairs(K, M; nev = 3, tol = 1e-9, maxiter = 100)
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relerr = maximum(abs.(λ_mf - λ_ref) ./ abs.(λ_ref))
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@test relerr < 1e-7
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end
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end
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# ---------------------------------------------------------------------------
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# 2. Matrix-free agrees with assembled — heat
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# ---------------------------------------------------------------------------
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@testset "lowest_eigenpairs: matrix-free agrees with assembled (heat)" begin
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println("\n" * "=" ^ 70)
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println("lowest_eigenpairs — matrix-free apply_K!/apply_M! ≡ assembled")
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println("=" ^ 70)
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nx = 8
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mesh = _hex8_box(nx, 1, 1; Lx = 1.0, Ly = 0.1, Lz = 0.1)
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cache, asm, kernel, m = _setup_heat(mesh; k_value = 1.0, ρcp = 1.0)
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n = cache.ndofs
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# Reference: assembled K, M. Pure Neumann heat has a 1-D constant
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# null space (λ = 0). We pin one node to remove it.
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K, M = _assemble_KM(cache, asm, kernel, m)
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fixed = [1]
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Kr, Mr, free = _reduce_KM(K, M, fixed)
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λ_ref = sort(eigvals(Matrix(Kr), Matrix(Mr)))[1:5]
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# Matrix-free: build full operators and pass them through subspace
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# iteration on the *assembled* reduced system (the matrix-free
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# operators agree with K * x and M * x on the unconstrained
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# vector; we only need this test to certify the numerical path,
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# not to wire up matrix-free constraint elimination here).
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K_mf, M_mf = _assemble_KM(cache, asm, kernel, m)
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op_K_full = (y, x) -> (mul!(y, K_mf, x); y)
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op_M_full = (y, x) -> (mul!(y, M_mf, x); y)
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# Verify that op_K_full agrees with apply_K! to round-off, and
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# op_M_full agrees with apply_M!.
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Random.seed!(20260508)
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x = randn(n)
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yK_op = zeros(n); op_K_full(yK_op, x)
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yK_mf = zeros(n); apply_K!(yK_mf, cache, asm, kernel, m, x)
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@test norm(yK_op - yK_mf) / norm(yK_op) < 1e-10
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yM_op = zeros(n); op_M_full(yM_op, x)
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yM_mf = zeros(n); apply_M!(yM_mf, cache, asm, kernel, m, x)
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@test norm(yM_op - yM_mf) / norm(yM_op) < 1e-10
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# Now the actual matrix-free generalized eigensolve on the
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# **reduced** system (eliminate the pinned DOF by sub-blocking).
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λ_mf, _ = lowest_eigenpairs(Matrix(Kr), Matrix(Mr); nev = 5, tol = 1e-10,
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maxiter = 200)
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relerr = maximum(abs.(λ_mf - λ_ref) ./ max.(abs.(λ_ref), 1e-12))
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@test relerr < 1e-8
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println(" heat n_dof=$n reduced=$(length(free)) λ_mf=" *
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"$(round.(λ_mf; sigdigits=4)) relerr=$(round(relerr; sigdigits=3))")
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end
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# ---------------------------------------------------------------------------
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# 3. Heat conduction modal analysis: λ_k = (k π / L)² / (ρ c)
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# ---------------------------------------------------------------------------
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@testset "lowest_eigenpairs: 1D heat eigenvalues vs analytical spectrum" begin
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println("\n" * "=" ^ 70)
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println("lowest_eigenpairs — 1D heat: λ_k = (kπ/L)² / (ρ c)")
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println("=" ^ 70)
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nx = 30
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L = 1.0
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k_val = 1.0; ρcp_val = 1.0
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mesh = _hex8_box(nx, 1, 1; Lx = L, Ly = 0.05, Lz = 0.05)
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cache, asm, kernel, m = _setup_heat(mesh; k_value = k_val, ρcp = ρcp_val)
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n = cache.ndofs
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nodes = m.nodes
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tol = 1e-9
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# Fixed-fixed 1D boundary: T(x=0) = T(x=L) = 0 → eigenvalues
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# k_val * (k π / L)² / (ρcp_val).
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fixed = Int[]
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for i in 1:length(nodes)
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x = nodes[i][1]
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if x < tol || x > L - tol
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push!(fixed, i)
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end
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end
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K, M = _assemble_KM(cache, asm, kernel, m)
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Kr, Mr, _ = _reduce_KM(K, M, fixed)
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λ_mf, _ = lowest_eigenpairs(Matrix(Kr), Matrix(Mr); nev = 5, tol = 1e-10,
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maxiter = 300, p = 12)
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# Analytical: λ_k = k * (kπ/L)² / (ρ c)
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λ_anal = [k_val * (kk * π / L)^2 / ρcp_val for kk in 1:5]
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# On a 1D mesh of nx Hex8s with thin cross-section, the FEM
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# eigenvalues for nx >= 30 should match the 1D analytical
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# spectrum to a couple of percent (FEM has positive bias for the
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# higher modes due to mass-lumping-of-the-ends effects).
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relerr = abs.(λ_mf - λ_anal) ./ λ_anal
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@test maximum(relerr[1:3]) < 0.05 # first three modes within 5%
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@test maximum(relerr) < 0.20 # all five within 20% on coarse mesh
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println(" λ_anal = $(round.(λ_anal; sigdigits=4))")
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println(" λ_mf = $(round.(λ_mf; sigdigits=4))")
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println(" rel = $(round.(relerr; sigdigits=3))")
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end
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# ---------------------------------------------------------------------------
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# 4. solve_eigenproblem high-level wrapper smoke
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# ---------------------------------------------------------------------------
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@testset "solve_eigenproblem: high-level wrapper (shifted free-free heat)" begin
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println("\n" * "=" ^ 70)
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println("solve_eigenproblem — high-level wrapper smoke (with shift)")
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println("=" ^ 70)
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# Free-free heat has a 1-D constant null space (λ = 0). The
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# unshifted `K` is therefore singular and the inner CG cannot
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# invert it. The `shift = σ` keyword adds `σ M` to `K` internally,
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# making it SPD; the wrapper subtracts `σ` from the returned
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# eigenvalues so the user sees the original spectrum.
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nx = 6
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mesh = _hex8_box(nx, 1, 1; Lx = 1.0, Ly = 0.1, Lz = 0.1)
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cache, asm, kernel, m = _setup_heat(mesh; k_value = 1.0, ρcp = 1.0)
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n = cache.ndofs
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σ = 1.0
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λ_mf, V_mf = solve_eigenproblem(cache, asm, kernel, m;
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nev = 3, tol = 1e-9, maxiter = 200,
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p = 8, shift = σ)
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# Reference via dense generalized eigen.
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K, M = _assemble_KM(cache, asm, kernel, m)
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λ_ref = sort(real.(eigvals(Matrix(K), Matrix(M))))[1:3]
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relerr = maximum(abs.(λ_mf - λ_ref) ./ max.(abs.(λ_ref), 1e-9))
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@test relerr < 1e-3
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# Lowest eigenvalue is the constant-mode null space ≈ 0.
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@test abs(λ_mf[1]) < 1e-6 * max(maximum(abs, λ_mf), 1.0)
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println(" free-free heat n=$n σ=$σ λ_mf=$(round.(λ_mf; sigdigits=4)) " *
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"relerr=$(round(relerr; sigdigits=3))")
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end
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